mlab functions: psd, csd, cohere, corrcoef
I needed some spectral analysis functions, and finding none available, wrote my own. I use matlab a lot, so I wrote them to be matlab compatible. If you all think these look OK, I'm happy to submit them for inclusion into MLab.

"""
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
The functions are designed to work for real and complex valued Numeric arrays.
One of the major differences between this code and matlab's is that I use functions for 'detrend' and 'window', and matlab uses vectors. This can be easily changed, but I think the functional approach is a bit more elegant.
Please send comments, questions and bugs to:
Author: John D. Hunter jdhunter@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
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 = NFFTnoverlap 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
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 = NFFTnoverlap 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.
"""
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))))[0,1] try: return r.real except AttributeError: return r

I wrote a little test code comparing the output of matlab's equivalent functions. Basically, I compute the psd or cohere in matlab and python and do the rms difference on the resultant vectors
RMS cohere python/matlab difference 0.000854587104587 RMS psd python/matlab difference 0.00210783306638
I am not sure where these differences are arising, but they are quite small. I'm going to keep trying to track them down.
For corrcoef, the answers are the same past 8 significant digits.
Hope this helps! John Hunter
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John Hunter