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] = divide(fx[:numFreqs], norm(windowVals)**2)
# 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)
freqs = Fs/NFFT*arange(numFreqs)
Pxx.shape = len(freqs),
return Pxx, freqs
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] = divide(fx[:numFreqs], norm(windowVals)**2)
# 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)
freqs = Fs/NFFT*arange(numFreqs)
Pxx.shape = len(freqs),
return Pxx, freqs
def fftsurr(x):
"""
Compute an FFT phase randomized surrogate of x
"""
z = fft(x)
a = 2.*pi*1j
phase = a*rand(len(x))
z = z*exp(phase)
return inverse_fft(z).real
def fftsurr(x, detrend=detrend_none, window=window_none):
"""
Compute an FFT phase randomized surrogate of x
"""
x = window(detrend(x))
z = fft(x)
a = 2.*pi*1j
phase = a*rand(len(x))
z = z*exp(phase)
return inverse_fft(z).real
def fftsurr(x, detrend=detrend_none, window=window_none):
"""
Compute an FFT phase randomized surrogate of x
"""
x = window(detrend(x))
z = fft(x)
a = 2. * pi * 1j
phase = a * rand(len(x))
z = z * exp(phase)
return inverse_fft(z).real
def specgram(x,
NFFT=256,
Fs=2,
detrend=detrend_none,
window=window_hanning,
noverlap=128):
"""
Compute a spectrogram of data in x. Data are split into NFFT
length segements and the PSD of each section is computed. The
windowing function window is applied to each segment, and the
amount of overlap of each segment is specified with noverlap
See pdf for more info.
The returned times are the midpoints of the intervals over which
the ffts are calculated
"""
x = asarray(x)
assert (NFFT > noverlap)
if log(NFFT) / log(2) != int(log(NFFT) / log(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 typecode(x) == Complex: numFreqs = NFFT
else: numFreqs = NFFT // 2 + 1
windowVals = window(ones((NFFT, ), typecode(x)))
step = NFFT - noverlap
ind = arange(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
# Scale the spectrum by the norm of the window to compensate for
# windowing loss; see Bendat & Piersol Sec 11.5.2
Pxx[:, i] = divide(fx[:numFreqs], norm(windowVals)**2)
t = 1 / Fs * (ind + NFFT / 2)
freqs = Fs / NFFT * arange(numFreqs)
return Pxx, freqs, t
def specgram(x, NFFT=256, Fs=2, detrend=detrend_none,
window=window_hanning, noverlap=128):
"""
Compute a spectrogram of data in x. Data are split into NFFT
length segements and the PSD of each section is computed. The
windowing function window is applied to each segment, and the
amount of overlap of each segment is specified with noverlap
See pdf for more info.
The returned times are the midpoints of the intervals over which
the ffts are calculated
"""
assert(NFFT>noverlap)
if log(NFFT)/log(2) != int(log(NFFT)/log(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 = arange(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
# Scale the spectrum by the norm of the window to compensate for
# windowing loss; see Bendat & Piersol Sec 11.5.2
Pxx[:,i] = divide(fx[:numFreqs], norm(windowVals)**2)
t = 1/Fs*(ind+NFFT/2)
freqs = Fs/NFFT*arange(numFreqs)
return Pxx, freqs, t
def levypdf(x, gamma, alpha):
"Returm the levy pdf evaluated at x for params gamma, alpha"
N = len(x)
if N % 2 != 0:
raise ValueError, 'x must be an event length array; try\n' + \
'x = linspace(minx, maxx, N), where N is even'
dx = x[1] - x[0]
f = 1 / (N * dx) * arange(-N / 2, N / 2, Float)
ind = concatenate([arange(N / 2, N, Int), arange(N / 2, Int)])
df = f[1] - f[0]
cfl = exp(-gamma * absolute(2 * pi * f)**alpha)
px = fft(take(cfl, ind) * df).astype(Float)
return take(px, ind)
def levypdf(x, gamma, alpha):
"Returm the levy pdf evaluated at x for params gamma, alpha"
N = len(x)
if N%2 != 0:
raise ValueError, 'x must be an event length array; try\n' + \
'x = linspace(minx, maxx, N), where N is even'
dx = x[1]-x[0]
f = 1/(N*dx)*arange(-N/2, N/2, Float)
ind = concatenate([arange(N/2, N, Int),
arange(N/2,Int)])
df = f[1]-f[0]
cfl = exp(-gamma*absolute(2*pi*f)**alpha)
px = fft(take(cfl,ind)*df).astype(Float)
return take(px, ind)
def cohere_pairs( X, ij, NFFT=256, Fs=2, detrend=detrend_none,
window=window_hanning, noverlap=0,
preferSpeedOverMemory=True,
progressCallback=donothing_callback,
returnPxx=False):
"""
Cxy, Phase, freqs = cohere_pairs( X, ij, ...)
Compute the coherence for all pairs in ij. X is a
numSamples,numCols Numeric array. ij is a list of tuples (i,j).
Each tuple is a pair of indexes into the columns of X for which
you want to compute coherence. For example, if X has 64 columns,
and you want to compute all nonredundant pairs, define ij as
ij = []
for i in range(64):
for j in range(i+1,64):
ij.append( (i,j) )
The other function arguments, except for 'preferSpeedOverMemory'
(see below), are explained in the help string of 'psd'.
Return value is a tuple (Cxy, Phase, freqs).
Cxy -- a dictionary of (i,j) tuples -> coherence vector for that
pair. Ie, Cxy[(i,j) = cohere(X[:,i], X[:,j]). Number of
dictionary keys is len(ij)
Phase -- a dictionary of phases of the cross spectral density at
each frequency for each pair. keys are (i,j).
freqs -- a vector of frequencies, equal in length to either the
coherence or phase vectors for any i,j key. Eg, to make a coherence
Bode plot:
subplot(211)
plot( freqs, Cxy[(12,19)])
subplot(212)
plot( freqs, Phase[(12,19)])
For a large number of pairs, cohere_pairs can be much more
efficient than just calling cohere for each pair, because it
caches most of the intensive computations. If N is the number of
pairs, this function is O(N) for most of the heavy lifting,
whereas calling cohere for each pair is O(N^2). However, because
of the caching, it is also more memory intensive, making 2
additional complex arrays with approximately the same number of
elements as X.
The parameter 'preferSpeedOverMemory', if false, limits the
caching by only making one, rather than two, complex cache arrays.
This is useful if memory becomes critical. Even when
preferSpeedOverMemory is false, cohere_pairs will still give
significant performace gains over calling cohere for each pair,
and will use subtantially less memory than if
preferSpeedOverMemory is true. In my tests with a 43000,64 array
over all nonredundant pairs, preferSpeedOverMemory=1 delivered a
33% performace boost on a 1.7GHZ Athlon with 512MB RAM compared
with preferSpeedOverMemory=0. But both solutions were more than
10x faster than naievly crunching all possible pairs through
cohere.
See test/cohere_pairs_test.py in the src tree for an example
script that shows that this cohere_pairs and cohere give the same
results for a given pair.
"""
numRows, numCols = X.shape
# zero pad if X is too short
if numRows < NFFT:
tmp = X
X = zeros( (NFFT, numCols), X.typecode())
X[:numRows,:] = tmp
del tmp
numRows, numCols = X.shape
# get all the columns of X that we are interested in by checking
# the ij tuples
seen = {}
for i,j in ij:
seen[i]=1; seen[j] = 1
allColumns = seen.keys()
Ncols = len(allColumns)
del seen
# for real X, ignore the negative frequencies
if X.typecode()==Complex: numFreqs = NFFT
else: numFreqs = NFFT//2+1
# cache the FFT of every windowed, detrended NFFT length segement
# of every channel. If preferSpeedOverMemory, cache the conjugate
# as well
windowVals = window(ones((NFFT,), X.typecode()))
ind = range(0, numRows-NFFT+1, NFFT-noverlap)
numSlices = len(ind)
FFTSlices = {}
FFTConjSlices = {}
Pxx = {}
slices = range(numSlices)
normVal = norm(windowVals)**2
for iCol in allColumns:
progressCallback(i/Ncols, 'Cacheing FFTs')
Slices = zeros( (numSlices,numFreqs), Complex)
for iSlice in slices:
thisSlice = X[ind[iSlice]:ind[iSlice]+NFFT, iCol]
thisSlice = windowVals*detrend(thisSlice)
Slices[iSlice,:] = fft(thisSlice)[:numFreqs]
FFTSlices[iCol] = Slices
if preferSpeedOverMemory:
FFTConjSlices[iCol] = conjugate(Slices)
Pxx[iCol] = divide(mean(absolute(Slices)**2), normVal)
del Slices, ind, windowVals
# compute the coherences and phases for all pairs using the
# cached FFTs
Cxy = {}
Phase = {}
count = 0
N = len(ij)
for i,j in ij:
count +=1
if count%10==0:
progressCallback(count/N, 'Computing coherences')
if preferSpeedOverMemory:
Pxy = FFTSlices[i] * FFTConjSlices[j]
else:
Pxy = FFTSlices[i] * conjugate(FFTSlices[j])
if numSlices>1: Pxy = mean(Pxy)
Pxy = divide(Pxy, normVal)
Cxy[(i,j)] = divide(absolute(Pxy)**2, Pxx[i]*Pxx[j])
Phase[(i,j)] = arctan2(Pxy.imag, Pxy.real)
freqs = Fs/NFFT*arange(numFreqs)
if returnPxx:
return Cxy, Phase, freqs, Pxx
else:
return Cxy, Phase, 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] = conjugate(fx[:numFreqs])*fy[: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(numFreqs)
Pxy.shape = len(freqs),
return Pxy, freqs
def cohere_pairs(X,
ij,
NFFT=256,
Fs=2,
detrend=detrend_none,
window=window_hanning,
noverlap=0,
preferSpeedOverMemory=True,
progressCallback=donothing_callback,
returnPxx=False):
"""
Cxy, Phase, freqs = cohere_pairs( X, ij, ...)
Compute the coherence for all pairs in ij. X is a
numSamples,numCols Numeric array. ij is a list of tuples (i,j).
Each tuple is a pair of indexes into the columns of X for which
you want to compute coherence. For example, if X has 64 columns,
and you want to compute all nonredundant pairs, define ij as
ij = []
for i in range(64):
for j in range(i+1,64):
ij.append( (i,j) )
The other function arguments, except for 'preferSpeedOverMemory'
(see below), are explained in the help string of 'psd'.
Return value is a tuple (Cxy, Phase, freqs).
Cxy -- a dictionary of (i,j) tuples -> coherence vector for that
pair. Ie, Cxy[(i,j) = cohere(X[:,i], X[:,j]). Number of
dictionary keys is len(ij)
Phase -- a dictionary of phases of the cross spectral density at
each frequency for each pair. keys are (i,j).
freqs -- a vector of frequencies, equal in length to either the
coherence or phase vectors for any i,j key. Eg, to make a coherence
Bode plot:
subplot(211)
plot( freqs, Cxy[(12,19)])
subplot(212)
plot( freqs, Phase[(12,19)])
For a large number of pairs, cohere_pairs can be much more
efficient than just calling cohere for each pair, because it
caches most of the intensive computations. If N is the number of
pairs, this function is O(N) for most of the heavy lifting,
whereas calling cohere for each pair is O(N^2). However, because
of the caching, it is also more memory intensive, making 2
additional complex arrays with approximately the same number of
elements as X.
The parameter 'preferSpeedOverMemory', if false, limits the
caching by only making one, rather than two, complex cache arrays.
This is useful if memory becomes critical. Even when
preferSpeedOverMemory is false, cohere_pairs will still give
significant performace gains over calling cohere for each pair,
and will use subtantially less memory than if
preferSpeedOverMemory is true. In my tests with a 43000,64 array
over all nonredundant pairs, preferSpeedOverMemory=1 delivered a
33% performace boost on a 1.7GHZ Athlon with 512MB RAM compared
with preferSpeedOverMemory=0. But both solutions were more than
10x faster than naievly crunching all possible pairs through
cohere.
See test/cohere_pairs_test.py in the src tree for an example
script that shows that this cohere_pairs and cohere give the same
results for a given pair.
"""
numRows, numCols = X.shape
# zero pad if X is too short
if numRows < NFFT:
tmp = X
X = zeros((NFFT, numCols), typecode(X))
X[:numRows, :] = tmp
del tmp
numRows, numCols = X.shape
# get all the columns of X that we are interested in by checking
# the ij tuples
seen = {}
for i, j in ij:
seen[i] = 1
seen[j] = 1
allColumns = seen.keys()
Ncols = len(allColumns)
del seen
# for real X, ignore the negative frequencies
if typecode(X) == Complex: numFreqs = NFFT
else: numFreqs = NFFT // 2 + 1
# cache the FFT of every windowed, detrended NFFT length segement
# of every channel. If preferSpeedOverMemory, cache the conjugate
# as well
windowVals = window(ones((NFFT, ), typecode(X)))
ind = range(0, numRows - NFFT + 1, NFFT - noverlap)
numSlices = len(ind)
FFTSlices = {}
FFTConjSlices = {}
Pxx = {}
slices = range(numSlices)
normVal = norm(windowVals)**2
for iCol in allColumns:
progressCallback(i / Ncols, 'Cacheing FFTs')
Slices = zeros((numSlices, numFreqs), Complex)
for iSlice in slices:
thisSlice = X[ind[iSlice]:ind[iSlice] + NFFT, iCol]
thisSlice = windowVals * detrend(thisSlice)
Slices[iSlice, :] = fft(thisSlice)[:numFreqs]
FFTSlices[iCol] = Slices
if preferSpeedOverMemory:
FFTConjSlices[iCol] = conjugate(Slices)
Pxx[iCol] = divide(mean(absolute(Slices)**2), normVal)
del Slices, ind, windowVals
# compute the coherences and phases for all pairs using the
# cached FFTs
Cxy = {}
Phase = {}
count = 0
N = len(ij)
for i, j in ij:
count += 1
if count % 10 == 0:
progressCallback(count / N, 'Computing coherences')
if preferSpeedOverMemory:
Pxy = FFTSlices[i] * FFTConjSlices[j]
else:
Pxy = FFTSlices[i] * conjugate(FFTSlices[j])
if numSlices > 1: Pxy = mean(Pxy)
Pxy = divide(Pxy, normVal)
Cxy[(i, j)] = divide(absolute(Pxy)**2, Pxx[i] * Pxx[j])
Phase[(i, j)] = arctan2(Pxy.imag, Pxy.real)
freqs = Fs / NFFT * arange(numFreqs)
if returnPxx:
return Cxy, Phase, freqs, Pxx
else:
return Cxy, Phase, 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 typecode(x) == Complex: numFreqs = NFFT
else: numFreqs = NFFT // 2 + 1
windowVals = window(ones((NFFT, ), typecode(x)))
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] = conjugate(fx[:numFreqs]) * fy[: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(numFreqs)
Pxy.shape = len(freqs),
return Pxy, freqs
