<div dir="ltr"><p style="margin:0px 0px 1em;padding:0px;border:0px;font-variant-numeric:inherit;font-variant-east-asian:inherit;font-stretch:inherit;line-height:inherit;font-family:Arial,"Helvetica Neue",Helvetica,sans-serif;font-size:15px;vertical-align:baseline;box-sizing:inherit;clear:both;color:rgb(36,39,41)">This example is taken from a <a href="https://dspillustrations.com/pages/posts/misc/convolution-examples-and-the-convolution-integral.html" rel="nofollow noreferrer" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(0,89,153)">tutorial</a> related to convolution integral.</p><p style="margin:0px 0px 1em;padding:0px;border:0px;font-variant-numeric:inherit;font-variant-east-asian:inherit;font-stretch:inherit;line-height:inherit;font-family:Arial,"Helvetica Neue",Helvetica,sans-serif;font-size:15px;vertical-align:baseline;box-sizing:inherit;clear:both;color:rgb(36,39,41)">I would like to export this example as animation in mp4 format. So far, the code looks like this :</p><pre class="gmail-lang-py gmail-prettyprint gmail-prettyprinted" style="margin-top:0px;margin-bottom:1em;padding:12px 8px;border:0px;font-variant-numeric:inherit;font-variant-east-asian:inherit;font-stretch:inherit;line-height:inherit;font-family:Consolas,Menlo,Monaco,"Lucida Console","Liberation Mono","DejaVu Sans Mono","Bitstream Vera Sans Mono","Courier New",monospace,sans-serif;font-size:13px;vertical-align:baseline;box-sizing:inherit;width:auto;max-height:600px;overflow:auto;background-color:rgb(239,240,241);border-radius:3px;color:rgb(57,51,24)"><code style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:Consolas,Menlo,Monaco,"Lucida Console","Liberation Mono","DejaVu Sans Mono","Bitstream Vera Sans Mono","Courier New",monospace,sans-serif;vertical-align:baseline;box-sizing:inherit;white-space:inherit"><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">import</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> scipy</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">integrate
</span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">import</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> numpy </span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">as</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> np
</span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">import</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> matplotlib</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">pyplot </span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">as</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> plt
</span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">import</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> matplotlib</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">animation </span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">as</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> animation

</span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">def</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> showConvolution</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t0</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">f1</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> f2</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">):</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># Calculate the overall convolution result using Simpson integration</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    convolution </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> np</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">zeros</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">len</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">))</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    </span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">for</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> n</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> t_ </span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">in</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> enumerate</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">):</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
        prod </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">lambda</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> tau</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">:</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> f1</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">tau</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">*</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> f2</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t_</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">-</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">tau</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
        convolution</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">[</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">n</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">]</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> scipy</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">integrate</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">simps</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">prod</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">),</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">

    </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># Create the shifted and flipped function</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    f_shift </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">lambda</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">:</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> f2</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t0</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">-</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    prod </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">lambda</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> tau</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">:</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> f1</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">tau</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">*</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> f2</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t0</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">-</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">tau</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">

    </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># Plot the curves</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">gcf</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">().</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">clear</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">()</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># il</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">

    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">subplot</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">211</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">gca</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">().</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">set_ymargin</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">0.05</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># il</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">plot</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> f1</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">),</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> label</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">r</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'$f_1(\tau)$'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">plot</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> f_shift</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">),</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> label</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">r</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'$f_2(t_0-\tau)$'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">fill</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> prod</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">),</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> color</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'r'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> alpha</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">0.5</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> edgecolor</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'black'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> hatch</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'//'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># il</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">plot</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> prod</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">),</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'r-'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> label</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">r</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'$f_1(\tau)f_2(t_0-\tau)$'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">grid</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">True</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">);</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">xlabel</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">r</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'$\tau$'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">);</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">ylabel</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">r</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'$x(\tau)$'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># il</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">legend</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">fontsize</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">10</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># il</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">text</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(-</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">4</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">0.6</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'$t_0=%.2f$'</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">%</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> t0</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> bbox</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">dict</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">fc</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'white'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">))</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># il</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">

    </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># plot the convolution curve</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">subplot</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">212</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">gca</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">().</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">set_ymargin</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">0.05</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># il</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">plot</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> convolution</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> label</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'$(f_1*f_2)(t)$'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">

    </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># recalculate the value of the convolution integral at the current time-shift t0</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    current_value </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> scipy</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">integrate</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">simps</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">prod</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">),</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">plot</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t0</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> current_value</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'ro'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">  </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># plot the point</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">grid</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">True</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">);</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">xlabel</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'$t$'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">);</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">ylabel</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'$(f_1*f_2)(t)$'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># il</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">legend</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">fontsize</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">10</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># il</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
    plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">show</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">()</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># il</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">

</span><span class="gmail-typ" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(43,145,175)">Fs</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">50</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">  </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># our sampling frequency for the plotting</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
T </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">5</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">    </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># the time range we are interested in</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
t </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> np</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">arange</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(-</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">T</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> T</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">1</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">/</span><span class="gmail-typ" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(43,145,175)">Fs</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">  </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># the time samples</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
f1 </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">lambda</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">:</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> np</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">maximum</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">0</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">1</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">-</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">abs</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">))</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
f2 </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-kwd" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(16,16,148)">lambda</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">:</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">></span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">0</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">*</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> np</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">exp</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(-</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">2</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">*</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">

t0 </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> np</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">arange</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(-</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">2.0</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">2.0</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">0.05</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">

fig </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">figure</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">figsize</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=(</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">8</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">3</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">))</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">
anim </span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> animation</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-typ" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(43,145,175)">FuncAnimation</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">fig</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> showConvolution</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> frames</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">t0</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> fargs</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=(</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">f1</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">f2</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">),</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">interval</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">80</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">

anim</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">save</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">(</span><span class="gmail-str" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">'animation.mp4'</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">,</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> fps</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">=</span><span class="gmail-lit" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(125,39,39)">30</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">)</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)"> </span><span class="gmail-com" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(133,140,147)"># fps = frames per second</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">

plt</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">.</span><span class="gmail-pln" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">show</span><span class="gmail-pun" style="margin:0px;padding:0px;border:0px;font-style:inherit;font-variant:inherit;font-weight:inherit;font-stretch:inherit;line-height:inherit;font-family:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(48,51,54)">()</span></code></pre><p style="margin:0px 0px 1em;padding:0px;border:0px;font-variant-numeric:inherit;font-variant-east-asian:inherit;font-stretch:inherit;line-height:inherit;font-family:Arial,"Helvetica Neue",Helvetica,sans-serif;font-size:15px;vertical-align:baseline;box-sizing:inherit;clear:both;color:rgb(36,39,41)">Is there a way to change t0 so that I'd like to able to save it as animation as in <a href="https://dspillustrations.com/pages/posts/misc/convolution-examples-and-the-convolution-integral.html" rel="nofollow noreferrer" style="margin:0px;padding:0px;border:0px;font-variant-numeric:inherit;font-variant-east-asian:inherit;font-stretch:inherit;line-height:inherit;vertical-align:baseline;box-sizing:inherit;color:rgb(0,89,153)">the tutorial</a> In the example t0 decreases from -2.0 to -1.95 etc the green curve is shifted right, and the area between curves, product increases. In the tutorial there is an html animation and I would like to save as mp4 file.<br></p><p style="margin:0px 0px 1em;padding:0px;border:0px;font-variant-numeric:inherit;font-variant-east-asian:inherit;font-stretch:inherit;line-height:inherit;font-family:Arial,"Helvetica Neue",Helvetica,sans-serif;font-size:15px;vertical-align:baseline;box-sizing:inherit;clear:both;color:rgb(36,39,41)">I've also added code as an attachment.</p></div>