# spectral density, periodograms, Numeric

John Hunter jdhunter at ace.bsd.uchicago.edu
Thu Jan 30 23:26:48 CET 2003

```>>>>> "John" == John Hunter <jdhunter at ace.bsd.uchicago.edu> writes:

John> Is anyone aware of cross and power spectral density
John> functions for Numeric arrays using an averaged periodogram
John> method.

Well, I never had any luck finding these so I ended up writing my
own.  Hope these help out the next fella who ends up here

Requires python2.2 and Numeric

"""
Spectral analysis functions for Numerical python written for
compatability with matlab commands with the same names.

psd - Power spectral density uing Welch's average periodogram
csd - Cross spectral density uing Welch's average periodogram
cohere - Coherence (normalized cross spectral density)
corrcoef - The matrix of correlation coefficients

All functions should work for real or complex valued Numeric arrays.

The primary difference between the arguments for these functions and
those of matlab is that 'window' and 'detrend' are both functions,
whereas in matlab they are vectors.

Author: John D. Hunter <jdhunter at ace.bsd.uchicago.edu>

"""

from __future__ import division
from MLab import mean, hanning, cov
from Numeric import zeros, ones, diagonal, transpose, matrixmultiply, \
resize, sqrt, divide, array, Float, Complex, concatenate, \
convolve, dot, conjugate, absolute, arange, reshape
from FFT import fft

def norm(x):
return sqrt(dot(x,x))

def window_hanning(x):
return hanning(len(x))*x

def window_none(x):
return x

def detrend_mean(x):
return x - mean(x)

def detrend_none(x):
return x

def detrend_linear(x):
"""Remove the best fit line from x"""
# I'm going to regress x on xx=range(len(x)) and return
# x - (b*xx+a)
xx = arange(len(x), typecode=x.typecode())
X = transpose(array([xx]+[x]))
C = cov(X)
b = C[0,1]/C[0,0]
a = mean(x) - b*mean(xx)
return x-(b*xx+a)

def psd(x, NFFT=256, Fs=2, detrend=detrend_none,
window=window_hanning, noverlap=0):
"""
The power spectral density by Welches average periodogram method.
The vector x is divided into NFFT length segments.  Each segment
is detrended by function detrend and windowed by function window.
noperlap gives the length of the overlap between segments.  The
absolute(fft(segment))**2 of each segment are averaged to compute Pxx,
with a scaling to correct for power loss due to windowing.  Fs is
the sampling frequency.

-- NFFT must be a power of 2
-- detrend and window are functions, unlike in matlab where they are
vectors.
-- if length x < NFFT, it will be zero padded to NFFT

Returns the tuple Pxx, freqs

Refs:
Bendat & Piersol -- Random Data: Analysis and Measurement
Procedures, John Wiley & Sons (1986)

"""

if NFFT % 2:
raise ValueError, 'NFFT must be a power of 2'

# zero pad x up to NFFT if it is shorter than NFFT
if len(x)<NFFT:
n = len(x)
x = resize(x, (NFFT,))
x[n:] = 0

# for real x, ignore the negative frequencies
if x.typecode()==Complex: numFreqs = NFFT
else: numFreqs = NFFT//2+1

windowVals = window(ones((NFFT,),x.typecode()))
step = NFFT-noverlap
ind = range(0,len(x)-NFFT+1,step)
n = len(ind)
Pxx = zeros((numFreqs,n), Float)

# do the ffts of the slices
for i in range(n):
thisX = x[ind[i]:ind[i]+NFFT]
thisX = windowVals*detrend(thisX)
fx = absolute(fft(thisX))**2
Pxx[:,i] = fx[:numFreqs]

# Scale the spectrum by the norm of the window to compensate for
# windowing loss; see Bendat & Piersol Sec 11.5.2
if n>1: Pxx = mean(Pxx,1)
Pxx = divide(Pxx, norm(windowVals)**2)
freqs = Fs/NFFT*arange(0,numFreqs)
return Pxx, freqs

def csd(x, y, NFFT=256, Fs=2, detrend=detrend_none,
window=window_hanning, noverlap=0):
"""
The cross spectral density Pxy by Welches average periodogram
method.  The vectors x and y are divided into NFFT length
segments.  Each segment is detrended by function detrend and
windowed by function window.  noverlap gives the length of the
overlap between segments.  The product of the direct FFTs of x and
y are averaged over each segment to compute Pxy, with a scaling to
correct for power loss due to windowing.  Fs is the sampling
frequency.

NFFT must be a power of 2

Returns the tuple Pxy, freqs

Refs:
Bendat & Piersol -- Random Data: Analysis and Measurement
Procedures, John Wiley & Sons (1986)

"""

if NFFT % 2:
raise ValueError, 'NFFT must be a power of 2'

# zero pad x and y up to NFFT if they are shorter than NFFT
if len(x)<NFFT:
n = len(x)
x = resize(x, (NFFT,))
x[n:] = 0
if len(y)<NFFT:
n = len(y)
y = resize(y, (NFFT,))
y[n:] = 0

# for real x, ignore the negative frequencies
if x.typecode()==Complex: numFreqs = NFFT
else: numFreqs = NFFT//2+1

windowVals = window(ones((NFFT,),x.typecode()))
step = NFFT-noverlap
ind = range(0,len(x)-NFFT+1,step)
n = len(ind)
Pxy = zeros((numFreqs,n), Complex)

# do the ffts of the slices
for i in range(n):
thisX = x[ind[i]:ind[i]+NFFT]
thisX = windowVals*detrend(thisX)
thisY = y[ind[i]:ind[i]+NFFT]
thisY = windowVals*detrend(thisY)
fx = fft(thisX)
fy = fft(thisY)
Pxy[:,i] = fy[:numFreqs]*conjugate(fx[:numFreqs])

# Scale the spectrum by the norm of the window to compensate for
# windowing loss; see Bendat & Piersol Sec 11.5.2
if n>1: Pxy = mean(Pxy,1)
Pxy = divide(Pxy, norm(windowVals)**2)
freqs = Fs/NFFT*arange(0,numFreqs)
return Pxy, freqs

def cohere(x, y, NFFT=256, Fs=2, detrend=detrend_none,
window=window_hanning, noverlap=0):
"""
cohere the coherence between x and y.  Coherence is the normalized
cross spectral density

Cxy = |Pxy|^2/(Pxx*Pyy)

The return value is (Cxy, f), where f are the frequencies of the
coherence vector.  See the docs for psd and csd for information
about the function arguments NFFT, detrend, windowm noverlap, as
well as the methods used to compute Pxy, Pxx and Pyy.

Returns the tuple Cxy, freqs

"""

Pxx,f = psd(x, NFFT=NFFT, Fs=Fs, detrend=detrend,
window=window, noverlap=noverlap)
Pyy,f = psd(y, NFFT=NFFT, Fs=Fs, detrend=detrend,
window=window, noverlap=noverlap)
Pxy,f = csd(x, y, NFFT=NFFT, Fs=Fs, detrend=detrend,
window=window, noverlap=noverlap)

Cxy = divide(absolute(Pxy)**2, Pxx*Pyy)
return Cxy, f

def corrcoef(*args):
"""

corrcoef(X) where X is a matrix returns a matrix of correlation
coefficients for each row of X.

corrcoef(x,y) where x and y are vectors returns the matrix or
correlation coefficients for x and y.

Numeric arrays can be real or complex

The correlation matrix is defined from the covariance matrix C as

r(i,j) = C[i,j] / (C[i,i]*C[j,j])
"""

if len(args)==2:
X = transpose(array([args[0]]+[args[1]]))
elif len(args==1):
X = args[0]
else:
raise RuntimeError, 'Only expecting 1 or 2 arguments'

C = cov(X)
d = resize(diagonal(C), (2,1))
r = divide(C,sqrt(matrixmultiply(d,transpose(d))))
try: return r.real
except AttributeError: return r

```