def problem4():
# read in tada.wav
rate, tada = wavfile.read('tada.wav')
# upon inspection, we find that tada.wav is a stereo audio file.
# we create stereo white noise that lasts 10 seconds
L_white = sp.int16(sp.random.randint(-32767,32767,rate*10))
R_white = sp.int16(sp.random.randint(-32767,32767,rate*10))
white = sp.zeros((len(L_white),2))
white[:,0] = L_white
white[:,1] = R_white
# pad tada signal with zeros
padded_tada = sp.zeros_like(white)
padded_tada[:len(tada)] = tada
ptada = padded_tada
# fourier transforms
ftada = sp.fft(ptada,axis=0)
fwhite = sp.fft(white,axis=0)
# inverse transform of convolution
out = sp.ifft((ftada*fwhite),axis=0)
# prepping output and writing file
out = sp.real(out)
scaled = sp.int16(out / sp.absolute(out).max() * 32767)
wavfile.write('my_tada_conv.wav',rate,scaled)
def get_envelope(R,dim=1):
"""
Returns the complex version of the input signal R.
@param R: The input data matrix.
@param dim: The dimension along which the envelope is to be taken. default: dim=1
"""
if dim==0:
R=R.T
if len(R.shape)==1:
freqs=scipy.fft(R)
length=len(R)/2
freqs[length:]=0
freqs[1:length]=2*freqs[1:length]
## freqs[1:length]=freqs[1:length]
env=scipy.ifft(freqs)
else:
freqs=scipy.fft(R)
length=R.shape[dim]/2
#Something is fishy here:
freqs[:,length:]=0
freqs[:,1:length]=2*freqs[0,1:length]
## freqs[:,1:length]=freqs[0,1:length]
env=scipy.ifft(freqs)
if dim==0:
return env.T
return env
def test_fft_function():
# Many NumPy symbols are imported into the scipy namespace, including
# numpy.fft.fft as scipy.fft, conflicting with this module (gh-10253)
np.random.seed(1234)
# Callable before scipy.fft is imported
import scipy
x = np.random.randn(10) + 1j * np.random.randn(10)
with pytest.deprecated_call(match=r'1\.5\.0'):
X = scipy.fft(x)
with pytest.deprecated_call(match=r'2\.0\.0'):
y = scipy.ifft(X)
assert_allclose(y, x)
# Callable after scipy.fft is imported
import scipy.fft
assert_allclose(X, scipy.fft.fft(x))
with pytest.deprecated_call(match=r'1\.5\.0'):
X = scipy.fft(x)
assert_allclose(X, scipy.fft.fft(x))
with pytest.deprecated_call(match=r'2\.0\.0'):
y = scipy.ifft(X)
assert_allclose(y, x)
# Callable when imported using from
from scipy import fft
with pytest.deprecated_call(match=r'1\.5\.0'):
X = fft(x)
with pytest.deprecated_call(match=r'2\.0\.0'):
y = scipy.ifft(X)
assert_allclose(y, x)
def forward_propagation():
"""
Returns:
y: received signal (shape = [Nsamp_d, 2], separate real and imaginary part)
x: symbol vector (shape = [Nsym], complex)
P: launch power (in W)
"""
np.random.seed() # new seed is necessary for multiprocessor
P = P_W_r[np.random.randint(P_W_r.shape[0])] # get random launch power
# [SOURCE] random points from the signal constellation
if modulation == "QAM":
x = const[np.random.randint(const.shape[0], size=[1, Nsym])]
elif modulation == "Gaussian":
x = (np.random.normal(0, 1, size=[1, Nsym]) +
1j * np.random.normal(0, 1, size=[1, Nsym])) / np.sqrt(2)
else:
raise ValueError("wrong modulation format: " + modulation)
# [MODULATION] upsample + pulse shaping
x_up = np.zeros([1, Nsamp_a], dtype=np.complex64)
x_up[:, ::OS_a] = x * np.sqrt(OS_a)
u = sp.ifft(sp.fft(x_up) * ps_filter_tx_freq) * np.sqrt(P)
# [CHANNEL] simulate forward propagation
for NN in range(Nsp): # enter a span
for MM in range(fw.model_steps): # enter a segment
u = sp.ifft(fw.get_cd_filter_freq(MM) * sp.fft(u))
u = u * np.exp(1j * fw.nl_param[MM] * np.abs(u)**2)
#u = u*np.exp(1j*(8/9)*fw.nl_param[MM]*(np.abs(u[0,:])**2+np.abs(u[1,:])**2))
# add noise, NOTE: amplifier gain (u = u*np.exp(alpha_lin*Lsp/2.0)) is absorbed in nl_param
u = u + np.sqrt(sigma2 / 2 / Nsp) * (np.random.randn(1, Nsamp_a) +
1j * np.random.randn(1, Nsamp_a))
# [RECEIVER] low-pass filter + downsample
u = sp.ifft(sp.fft(u) * lp_filter_freq)
y = u[0, ::OS_a // OS_d]
y = np.stack([np.real(y), np.imag(y)], axis=1)
return y, x[0, :], P
def get_envelope(R, dim=1):
"""
Returns the complex version of the input signal R.
@param R: The input data matrix.
@param dim: The dimension along which the envelope is to be taken. default: dim=1
"""
if dim == 0:
R = R.T
if len(R.shape) == 1:
freqs = scipy.fft(R)
length = len(R) / 2
freqs[length:] = 0
freqs[1:length] = 2 * freqs[1:length]
## freqs[1:length]=freqs[1:length]
env = scipy.ifft(freqs)
else:
freqs = scipy.fft(R)
length = R.shape[dim] / 2
#Something is fishy here:
freqs[:, length:] = 0
freqs[:, 1:length] = 2 * freqs[0, 1:length]
## freqs[:,1:length]=freqs[0,1:length]
env = scipy.ifft(freqs)
if dim == 0:
return env.T
return env
def GaussianPulseDR(epsilon, Bx, c0, zm, zs, LOG='no'):
''' Analytical Propgation of a gaussion pulse near a rigid boundary.
epsilon : amplitude of the gaussian
Bx : width of the gaussian
c0 : celerity of wave
zm : location of the microphone
zs : location of the source
LOG : if LOG=='log', display complementary informations
'''
ext = 10
Nfreq = 2**12
Npadd = 2**14
B = np.sqrt(Bx**2/np.log(2))
k0 = np.sqrt(2)/B
fmin = 0.00001
fmax = k0*c0/(2*np.pi)
r1 = abs(zm-zs)
r2 = abs(zm+zs)
# Axes
k = np.linspace(2*np.pi*fmin/c0, ext*k0, Nfreq)
f = np.linspace(fmin, ext*fmax, Nfreq)
df = f[2] - f[1]
wtime = np.linspace(0, 1/df, Npadd)
# Puissance de la source % omega
Sw = 1j*k*np.pi*epsilon*B**2*np.exp(-k**2*B**2/4.)/c0
# Pression dans le domaine freq.
Pdw = -1j*Sw*hankel1(0, k*r1)/4.
Prw = -1j*Sw*hankel1(0, k*r2)/4.
Ai = fmax*Nfreq/(2*np.pi)
Pdt = Ai*ifft(Pdw, Npadd)[::-1]
Prt = Ai*ifft(Prw, Npadd)[::-1]
if LOG == 'log':
print('Max. frequency : ' + repr(fmax))
pl.figure('Source Strenght')
pl.subplot(311)
pl.plot(f, abs(Sw), 'k')
pl.ylabel('Strength $S_w$')
pl.subplot(312)
pl.plot(f, abs(Pdw), 'k', label='Direct')
pl.plot(f, abs(Prw), color='0.5', label='Reflected')
pl.legend()
pl.ylabel(r'Pressure $\tilde{p}(r,w)$')
pl.subplot(313)
pl.plot(wtime, Pdt.real/Pdt.real.max(), color='0.8', label='Direct')
pl.plot(wtime, Prt.real/Prt.real.max(), color='0.5', label='Reflected')
pl.plot(wtime, Pdt.real/Pdt.real.max() + Prt.real/Prt.real.max(), 'k--', linewidth=4, label='Sum')
pl.legend()
pl.ylabel(r'Pressure $\tilde{p}(r,t)$')
return Pdt+Prt, Pdt, Prt, f, wtime
def start():
global signal
global heartrate
global spo2
easypulse = EasyPulse()
beats = easypulse.readPulse()
heartrate = easypulse.computeHeartrate(beats)
spo2 = spo2()
#print ("Number of beats: " + str(len(beats)))
#print( "Heartrate: " + str(heartrate))
signal=[]
for i in range(1,100):
reading = easypulse.readadc(2, 11, 10, 9, 8)
signal.append(reading*3.3/1023)
fft=scipy.fft(signal)
bp=fft[:]
for i in range(len(bp)):
if i>=10:bp[i]=0
ibp=scipy.ifft(bp)
def prob4(filename='saw.wav', new_rate = 11025, outfile='prob4.wav'):
"""Down-samples a given .wav file to a new rate and saves the resulting
signal as another .wav file.
Parameters
----------
filename : string, optional
The name of the .wav sound file to be down-sampled.
Defaults to 'saw.wav'.
new_rate : integer, optional
The down-sampled rate. Defaults to 11025.
outfile : string, optional
The name of the new file. Defaults to prob4.wav.
Returns
-------
None
"""
old_rate, in_sig = wavfile.read(filename)
fin = fftw.fft(sp.float32(in_sig))
# Use if scipy_fftpack is unavailable
# fin = sp.fft(sp.float32(in_sig))
nsiz = sp.floor(in_sig.size * new_rate / old_rate)
nsizh = sp.floor(nsiz / 2)
fout = sp.zeros(nsiz) + 0j
fout[0:nsizh] = fin[0:nsizh]
fout[nsiz-nsizh+1:] = sp.conj(sp.flipud(fout[1:nsizh]))
out = sp.real(sp.ifft(fout))
out = sp.int16(out/sp.absolute(out).max() * 32767)
plot_signal(filename)
wavfile.write('prob4.wav',new_rate,out)
print ""; plot_signal('prob4.wav')
def sampling_reverb(signal_array, impulse_response):
sigL = len(signal_array)
irL = len(impulse_response)
### インパルス応答の長さ調整
new_irL = int((2**nextpow2(irL)) * 2) #2フレーム分
frameL = new_irL / 2
#zeros:0で初期化した配列を作成（shape, dtype）
new_IR = zeros(new_irL, dtype=float64)
#[:]はスライス．今回の場合は一番最初からirLまでの長さの部分にimpulse＿responseを代入
new_IR[:irL] = impulse_response
###入力信号を適度な長さにする．
frame_num = int(ceil((sigL + frameL) / float(frameL)))
new_sigL = frameL * frame_num
new_sig = zeros(new_sigL, dtype=float64)
new_sig[frameL:frameL + sigL] = signal_array
###インパルス応答の畳み込み
ret = zeros(new_sigL - frame_num, dtype=float64)
ir_fft = fft(new_IR) #インパルス応答のFFT
for ii in xrange(frame_num - 1):
s_ind = frameL * ii
e_ind = frameL * (ii + 2)
sig_fft = fft(new_sig[s_ind:e_ind]) #信号のFFT
###畳み込み
ret[s_ind:s_ind + frameL] = ifft(sig_fft * ir_fft)[frameL:].real
print(new_irL, sigL, new_sigL)
print len(sig_fft)
print len(ir_fft)
return ret[:sigL]
def IMRpeakAmp(m1, m2, spin1z, spin2z, d):
"""
IMRpeakAmp finds the peak amplitude of the waveform for a given source parameters and the source distance.
usage: IMRpeakAmp(m1,m2,spin1z,spin2z,distance)
e.g.
spawaveApp.IMRpeakAmp(30,40,0.45,0.5,100)
"""
chi = spawaveform.computechi(m1, m2, spin1z, spin2z)
imrfFinal = spawaveform.imrffinal(m1, m2, chi, 'fcut')
fLower = 10.0
order = 7
dur = 2**numpy.ceil(
numpy.log2(spawaveform.chirptime(m1, m2, order, fLower)))
sr = 2**numpy.ceil(numpy.log2(imrfFinal * 2))
deltaF = 1.0 / dur
deltaT = 1.0 / sr
s = numpy.empty(sr * dur, 'complex128')
spawaveform.imrwaveform(m1, m2, deltaF, fLower, s, spin1z, spin2z)
s = scipy.ifft(s)
#s = numpy.abs(s)
s = numpy.real(s)
max = numpy.max(s) / d
return max
def propagate(self):
r"""
Given the wavefunction values :math:`\Psi` at time :math:`t`, calculate new
values at time :math:`t + \tau`. We perform exactly one timestep :math:`\tau` here.
"""
# How many states we have
nst = self.Psi.get_number_components()
# Read values out of current WaveFunction state
vals = self.Psi.get_values()
# Do the propagation
tmp = [ zeros(vals[0].shape, dtype=complexfloating) for item in vals ]
for row in xrange(0, nst):
for col in xrange(0, nst):
tmp[row] = tmp[row] + self.VE[row*nst+col] * vals[col]
tmp = tuple([ fft(item) for item in tmp ])
tmp = tuple([ self.TE * item for item in tmp ])
tmp = tuple([ ifft(item) for item in tmp ])
values = [ zeros(tmp[0].shape, dtype=complexfloating) for item in tmp ]
for row in xrange(0, nst):
for col in xrange(0, nst):
values[row] = values[row] + self.VE[row*nst+col] * tmp[col]
# Write values back to WaveFunction object
self.Psi.set_values(values)
def removenoise(self, data):
nototalframes = np.floor(
len(data) / (self.framelength)) # the total no of frames in sound
dataclean = np.array([]) # storage for final sound
for i in range(int(nototalframes)):
start = i * (self.framelength)
dft = scipy.fft(
data[start:start +
self.framelength]) # no windowing .. required?
dmag = np.abs(dft)
dang = np.angle(dft)
dmag = dmag - (
SUBTRACTION_FACTOR * self.noiseprofile
) # subtract the noise spectrum by 1.5 for greater effect
#over subtraction above 2kh
dmag[self.subfeqlevelmin:self.subfeqlevelmax] = dmag[
self.subfeqlevelmin:self.subfeqlevelmax] - (
OVERSUBTRACTION *
self.noiseprofile[self.subfeqlevelmin:self.subfeqlevelmax])
dmag[dmag < 0] = 0 # because magnitude spectrum cannot be -ve
new = dmag * np.exp(
dang *
1j) # integrate new magnitude and old phase mag*e^(j*phase)
idft = scipy.ifft(new) # time domain
dataclean = np.append(dataclean, np.real(idft))
return dataclean
def istft(spectrogram, w_size, step):
if spectrogram.shape[0] != w_size:
print ("Mismatch w_size and spectrogram")
eps = np.finfo(float).eps
window = np.hanning(w_size)
# spectrogram.shape = w_size , bins
spectr_len = spectrogram.shape[1]
reconst_len = w_size + (spectr_len - 1) * step
reconst_x = np.zeros(reconst_len, dtype=float)
windowsum = np.zeros(reconst_len, dtype=float)
windowsq = window * window
# Overlap add
for i in range(0, spectr_len):
s = i * step
e = i * step + w_size
r = ifft(spectrogram[:, i]).real
# r = abs(ifft(spectrogram[:, i]))
reconst_x[s:e] += r * window
windowsum[s:e] += windowsq
# Normalize by window
# for i in range(0,reconst_len)
# if windowsum[i] > eps
# reconst_x[i] /= windowsum[i]
pos = (windowsum != 0)
reconst_x[pos] /= windowsum[pos]
return reconst_x.astype("int16")
def down_sample(filename, new_rate, outputfile=None):
"""
Create a down-sampled copy of the provided .wav file. Unless overridden, the output
file will be of the form "down_<orginalname>.wav"
Parameters
----------
filename : string
input .wav file
new_rate : int
sample rate of output file
outputfile : string
name of output file
"""
if outputfile is None:
outputfile = "down_" + filename
old_rate, in_sig = wavfile.read(filename)
in_sig = sp.float32(in_sig)
fin = sp.fft(in_sig)
nsiz = sp.floor(in_sig.size * new_rate / old_rate)
nsizh = sp.floor(nsiz / 2)
fout = sp.zeros(nsiz)
fout = fout + 0j
fout[0:nsizh] = fin[0:nsizh]
fout[nsiz - nsizh + 1 :] = sp.conj(sp.flipud(fout[1:nsizh]))
out = sp.ifft(fout)
out = sp.real(out) # Take the real component of the signal
out = sp.int16(out / sp.absolute(out).max() * 32767)
wavfile.write(outputfile, new_rate, out)
def ffacorr(a):
"""Returns the autocorrelation of a. Expects raw data"""
z=np.zeros(2*len(a))
z[:len(a)]=a
fft=sc.fft(z)
out=sc.ifft(fft*sc.conj(fft))
return (out[:len(out)/2])
def process_data(start, sum_data):
input_data = read('junk.wav')
audio_in = input_data[1]
samples = len(audio_in)
intvl = start / seg
k = 0
var_thres = 2.2
data_out = []
#print "intvl = ",intvl,start,seg
for i in xrange(intvl):
buffer_out = []
for j in xrange(seg):
buffer_out.append(audio_in[k])
k = k + 1
cbuffer_out = fft(buffer_out)
for j in xrange(seg):
if (abs(cbuffer_out[j]) < var_thres * sum_data[j]):
cbuffer_out[j] = 0.02 * cbuffer_out[j]
buf_out = ifft(cbuffer_out)
for j in xrange(seg):
data_out.append(buf_out[j].real)
sar = numpy.array(data_out, dtype=numpy.int16)
write("junk_out.wav", 44100, sar)
cmd4 = 'lame junk_out.wav junk_out.mp3 >enc.log 2>&1 '
os.system(cmd4)
def cwt_freq(data, wavelet, widths, dt, axis):
# compute in frequency
# next highest power of two for padding
N = data.shape[axis]
pN = int(2 ** np.ceil(np.log2(N)))
# N.B. padding in fft adds zeros to the *end* of the array,
# not equally either end.
fft_data = scipy.fft(data, n=pN, axis=axis)
# frequencies
w_k = np.fft.fftfreq(pN, d=dt) * 2 * np.pi
# sample wavelet and normalise
norm = (2 * np.pi * widths / dt) ** .5
wavelet_data = norm[:, None] * wavelet(w_k, widths[:, None])
# Convert negative axis. Add one to account for
# inclusion of widths axis above.
axis = (axis % data.ndim) + 1
# perform the convolution in frequency space
slices = [slice(None)] + [None for _ in data.shape]
slices[axis] = slice(None)
out = scipy.ifft(fft_data[None] * wavelet_data.conj()[slices],
n=pN, axis=axis)
# remove zero padding
slices = [slice(None) for _ in out.shape]
slices[axis] = slice(None, N)
if data.ndim == 1:
return out[slices].squeeze()
else:
return out[slices]
def delayedsignalF(x, t0_pts):
#==============================================================
"""
Delay a signal with a non integer value
(computation in frequency domain)
Synopsis:
y=delayedsignalF(x,t0_pts)
Inputs: x vector of length N
t0_pts is a REAL delay
expressed wrt the sampling time Ts=1:
t0_pts = 1 corresponds to one time dot
t0_pts may be positive, negative, non integer
t0_pts>0: shift to the right
t0_pts<0: shift to the left
Rk: the length of FFT is 2^(nextpow2(N)+1
"""
#
# M. Charbit, Jan. 2010
#==============================================================
N = len(x)
p = ceil(log2(N)) + 1
Lfft = int(2.0**p)
Lffts2 = Lfft / 2
fftx = fft(x, Lfft)
ind = concatenate((range(Lffts2 + 1), range(Lffts2 + 1 - Lfft, 0)), axis=0)
fftdelay = exp(-2j * pi * t0_pts * ind / Lfft)
fftdelay[Lffts2] = real(fftdelay[Lffts2])
ifftdelay = ifft(fftx * fftdelay)
y = ifftdelay[range(N)]
if isreal(any(x)):
y = real(y)
return y
def istft(X, size):
x = np.zeros(size)
framesamp = X.shape[1]
hopsamp = framesamp / 2
for n, i in enumerate(range(0, len(x) - framesamp, hopsamp)):
x[i:i + framesamp] += scipy.real(scipy.ifft(X[n]))
return x
def prob4(filename='saw.wav', new_rate=11025, outfile='prob4.wav'):
"""Down-samples a given .wav file to a new rate and saves the resulting
signal as another .wav file.
Parameters
----------
filename : string, optional
The name of the .wav sound file to be down-sampled.
Defaults to 'saw.wav'.
new_rate : integer, optional
The down-sampled rate. Defaults to 11025.
outfile : string, optional
The name of the new file. Defaults to prob4.wav.
Returns
-------
None
"""
old_rate, in_sig = wavfile.read(filename)
fin = fftw.fft(sp.float32(in_sig))
# Use if scipy_fftpack is unavailable
# fin = sp.fft(sp.float32(in_sig))
nsiz = sp.floor(in_sig.size * new_rate / old_rate)
nsizh = sp.floor(nsiz / 2)
fout = sp.zeros(nsiz) + 0j
fout[0:nsizh] = fin[0:nsizh]
fout[nsiz - nsizh + 1:] = sp.conj(sp.flipud(fout[1:nsizh]))
out = sp.real(sp.ifft(fout))
out = sp.int16(out / sp.absolute(out).max() * 32767)
plot_signal(filename)
wavfile.write('prob4.wav', new_rate, out)
print ""
plot_signal('prob4.wav')
def ffacorr(a): """Returns the autocorrelation of a. Expects raw data""" z=np.zeros(2*len(a)) z[:len(a)]=a fft=sc.fft(z) out=sc.ifft(fft*sc.conj(fft)) return (out[:len(out)/2])
def HT(y): """ HT(y) Computes the Hilbert transform of the array y using the frequency-domain approach, as in http://library2.usask.ca/theses/available/etd-09272006-135609/unrestricted/XianglingWang.pdf, pages 36-37 For a good transform, the following requirements must be satisfied: 1. The numbers of zeros and local extrema in y must differ by at most one. 2. y must be symmetric about zero (i.e. the mean value of the envelopes defined the by the local maxima and minima of y must be zero). """ from scipy import fft, ifft from numpy import zeros_like #Take the Fourier transform of the function ffty = fft(y) #Write as the FFT of the Hilbert transform Y = zeros_like(ffty) N = len(ffty) for i in range(-N/2+1,N/2+1): Y[i] = -1j*sgn(i)*ffty[i] #Take the inverse Fourier transform HT = ifft(Y) return HT
def mmse_stsa(infile, outfile, noise_sum):
signal, params = read_signal(infile, WINSIZE)
nf = len(signal)/(WINSIZE/2) - 1
sig_out=sp.zeros(len(signal),sp.float32)
G = sp.ones(WINSIZE)
prevGamma = G
alpha = 0.98
window = sp.hanning(WINSIZE)
gamma15=spc.gamma(1.5)
lambdaD = noise_sum / 5.0
percentage = 0
for no in xrange(nf):
p = int(math.floor(1. * no / nf * 100))
if (p > percentage):
percentage = p
print "{}%".format(p),
y = get_frame(signal, WINSIZE, no)
Y = sp.fft(y*window)
Yr = sp.absolute(Y)
Yp = sp.angle(Y)
gamma = Yr**2/lambdaD
xi = alpha * G**2 * prevGamma + (1-alpha)*sp.maximum(gamma-1, 0)
prevGamma = gamma
nu = gamma * xi / (1+xi)
G = (gamma15 * sp.sqrt(nu) / gamma ) * sp.exp(-nu/2) * ((1+nu)*spc.i0(nu/2)+nu*spc.i1(nu/2))
idx = sp.isnan(G) + sp.isinf(G)
G[idx] = xi[idx] / (xi[idx] + 1)
Yr = G * Yr
Y = Yr * sp.exp(Yp*1j)
y_o = sp.real(sp.ifft(Y))
add_signal(sig_out, y_o, WINSIZE, no)
write_signal(outfile, params, sig_out)
def ApplyFiltersWall(self, data, lowFreq, highFreq, SampleFreq=1250000.0):
'''Applies FFT software filters.
In -> Data, lowFreq, highFreq
Out-> filteredSignal
Warning: Appling a blank wall filter is the same as
convolving with sinc function!!
THIS PRODUCES RINGING!!
'''
#Perform FFT
fftsigs=numpy.fft.fft(data)
fftfreqs = numpy.fft.fftfreq(len(data),1.0/SampleFreq)
#Apply blank wall filter by cutting off the unwanted frequencies
if lowFreq is not None:
lowfreq_pos = [i for i,x in enumerate(fftfreqs) if lowFreq-10<=x<=lowFreq+10]
for i in range(lowfreq_pos[0]):
fftsigs[i]=0
#for i in range(len(fftsigs)):
# if abs(fftfreqs[i])<=lowFreq:
# fftsigs[i]=0
if highFreq is not None:
for i in range(len(fftsigs)):
if fftfreqs[i]>=highFreq:
fftsigs[i]=0
#Regenerate Signal
return scipy.ifft(fftsigs, n=len(fftsigs)/2)
def istft(X, win, step):
""" ISTFT (Short Term Fourie Transform)
逆一次元短時間フーリエ変換，stft関数の逆変換を行う
Args:
X (ndarray): 入力信号(M x N 行列)
win (int): 窓関数
step (int): シフト幅
Returns:
ndarray: 逆変換後の信号(l)
出力信号の長さ: l = [(M - 1) * step + N]
"""
M, N = X.shape
assert len(win) == N, "FFT length and window length are different."
l = int((M - 1) * step + N)
x = zeros(l, dtype=float64)
wsum = zeros(l, dtype=float64)
for m in range(M):
start = int(step * m)
### 滑らかな接続
x[start:start + N] = x[start:start + N] + ifft(X[m, :]).real * win
wsum[start:start + N] += win
pos = wsum != 0
x_pre = x.copy()
### 窓分のスケール合わせ
x[pos] /= wsum[pos]
return x
def timescale(data, scaling=1):
"""Scales the playback_duration of input_filename, while keeping pitch constant."""
length = len(data)
phi = scipy.zeros(N)
out = scipy.zeros(N, dtype=complex)
sigout = scipy.zeros(length / scaling + N)
amplitude = max(data)
window = scipy.hanning(N)
for index in scipy.arange(0, length - (N + H), H * scaling):
spec1 = scipy.fft(window * data[index:index + N])
spec2 = scipy.fft(window * data[index + H:index + N + H])
phi += scipy.angle(spec2 / spec1)
phi %= 2 * scipy.pi
out.real, out.imag = scipy.cos(phi), scipy.sin(phi)
out_index = int(index / scaling)
sigout[out_index:out_index +
N] += (window * scipy.ifft(scipy.absolute(spec2) * out)).real
sigout *= amplitude / max(sigout)
return scipy.array(sigout, dtype='int16')
def istft(X, fs, T, hop):
x = scipy.zeros(T * fs)
framesamp = X.shape[1]
hopsamp = int(hop * fs)
for n, i in enumerate(range(0, len(x)-framesamp, hopsamp)):
x[i:i+framesamp] += scipy.real(scipy.ifft(X[n]))
return x
def bandpass_ifft(X, Low_cutoff, High_cutoff, F_sample, M=None):
"""Bandpass filtering on a real signal using inverse FFT
Inputs
=======
X: 1-D numpy array of floats, the real time domain signal (time series) to be filtered
Low_cutoff: float, frequency components below this frequency will not pass the filter (physical frequency in unit of Hz)
High_cutoff: float, frequency components above this frequency will not pass the filter (physical frequency in unit of Hz)
F_sample: float, the sampling frequency of the signal (physical frequency in unit of Hz)
Notes
n=====
1. The input signal must be real, not imaginary nor complex
2. The Filtered_signal will have only half of original amplitude. Use abs() to restore.
3. In Numpy/Scipy, the frequencies goes from 0 to F_sample/2 and then from negative F_sample to 0.
"""
import numpy, scipy
if M == None: # if the number of points for FFT is not specified
M = X.size # let M be the length of the time series
#Spectrum = scipy.fft(X, n=M)
Spectrum = X
[Low_cutoff, High_cutoff, F_sample] = map(float, [Low_cutoff, High_cutoff, F_sample])
#Convert cutoff frequencies into points on spectrum
#[Low_point, High_point] = map(lambda F: F/F_sample * M /2, [Low_cutoff, High_cutoff])# the division by 2 is because the spectrum is symmetric
[Low_point, High_point] = map(lambda F: F/F_sample * M /1, [Low_cutoff, High_cutoff])# the division by 2 is because the spectrum is symmetric
Filtered_spectrum = [Spectrum[i] if i >= Low_point and i <= High_point else 0.0 for i in xrange(M)] # Filtering
Filtered_signal = scipy.ifft(Filtered_spectrum, n=M) # Construct filtered signal
return Spectrum, Filtered_spectrum, Filtered_signal, Low_point, High_point
def get_noise_freq_domain_1NSD( P , df , inittime , parityN , seed ) :
"""
returns the noise time-sereis given the power spectral density
INPUT:
P --- power spectral density. numpy array
df --- frequency resolution
inittime --- initial time of the noise time-series
parityN --- is the length of the time-series 'Odd' or 'Even'
seed --- seed for the noise
OUPUT:
t --- time [s]
n --- noise time-series
"""
Nf = P.shape[0]
if parityN == 'Odd' :
N = 2 * Nf + 1
elif parityN == 'Even' :
N = 2 * ( Nf + 1 )
else :
raise InputError , "parityN must be either 'Odd' or 'Even'!"
stime = 1 / ( N*df )
t = inittime + stime * np.arange( N )
np.random.seed( seed )
z = ( np.random.standard_normal( P.shape ) + 1j * np.random.standard_normal( P.shape ) ) / np.sqrt( 2 )
ntilde_fplus = np.sqrt( N / ( 2*stime ) * P ) * z
if N % 2 == 0 :
ntilde = np.array( [0] + list( ntilde_fplus ) + [0] + list( np.flipud(np.conj(ntilde_fplus)) ) )
else :
ntilde = np.array( [0] + list( ntilde_fplus ) + list( np.flipud(np.conj(ntilde_fplus)) ) )
n = np.real( sp.ifft( ntilde ) )
return t , n
def istft(X, fs, T, hop):
x = scipy.zeros(T * fs)
framesamp = X.shape[1]
hopsamp = int(hop * fs)
for n, i in enumerate(range(0, len(x) - framesamp, hopsamp)):
x[i:i + framesamp] += scipy.real(scipy.ifft(X[n]))
return x
def filter_wav(input_file, output_file = "filtered.wav"):
""" filters input_file and save resulting data to output_file """
raw = invert(readwave(input_file))
fftr = numpy.fft.fft(raw)
fft = fftr[:]
fftx= numpy.fft.fftfreq(len(raw), d=(1.0/(rate)))
lowfreq = 58
highfreq = 62
lowspectrum = [(lowfreq+(item*60)) for item in range(5)]
highspectrum = [(highfreq+(item*60)) for item in range(5)]
fft=bandStop(fft,fftx,0,20)
fft=bandStop(fft,fftx,-20,0)
for i in range(5):
fft = bandStop(fft,fftx,lowspectrum[i],highspectrum[i])
fft=bandStop(fft,fftx,300,max(fftx))
fix = scipy.ifft(fft)
smoothed = Hanning(fix)
gain = 1.0/max(numpy.absolute(smoothed))
nsamples = len(smoothed)
asamples = numpy.zeros(nsamples,dtype=numpy.int16)
numpy.multiply(smoothed,32767.0 * gain,asamples)
wavfile.write(output_file,rate,asamples)
def delayedsignalF(x,t0_pts):
#==============================================================
"""
Delay a signal with a non integer value
(computation in frequency domain)
Synopsis:
y=delayedsignalF(x,t0_pts)
Inputs: x vector of length N
t0_pts is a REAL delay
expressed wrt the sampling time Ts=1:
t0_pts = 1 corresponds to one time dot
t0_pts may be positive, negative, non integer
t0_pts>0: shift to the right
t0_pts<0: shift to the left
Rk: the length of FFT is 2^(nextpow2(N)+1
"""
#
# M. Charbit, Jan. 2010
#==============================================================
N = len(x)
p = ceil(log2(N))+1;
Lfft = int(2.0**p);
Lffts2 = Lfft/2;
fftx = fft(x, Lfft);
ind = concatenate((range(Lffts2+1),
range(Lffts2+1-Lfft,0)),axis=0)
fftdelay = exp(-2j*pi*t0_pts*ind/Lfft);
fftdelay[Lffts2] = real(fftdelay[Lffts2]);
ifftdelay = ifft(fftx*fftdelay);
y = ifftdelay[range(N)];
if isreal(any(x)):
y=real(y)
return y
def cwt_freq(data, wavelet, widths, dt, axis):
# compute in frequency
# next highest power of two for padding
N = data.shape[axis]
pN = int(2**np.ceil(np.log2(N)))
# N.B. padding in fft adds zeros to the *end* of the array,
# not equally either end.
fft_data = scipy.fft(data, n=pN, axis=axis)
# frequencies
w_k = np.fft.fftfreq(pN, d=dt) * 2 * np.pi
# sample wavelet and normalise
norm = (2 * np.pi * widths / dt)**.5
wavelet_data = norm[:, None] * wavelet(w_k, widths[:, None])
# Convert negative axis. Add one to account for
# inclusion of widths axis above.
axis = (axis % data.ndim) + 1
# perform the convolution in frequency space
slices = [slice(None)] + [None for _ in data.shape]
slices[axis] = slice(None)
out = scipy.ifft(fft_data[None] * wavelet_data.conj()[slices],
n=pN,
axis=axis)
# remove zero padding
slices = [slice(None) for _ in out.shape]
slices[axis] = slice(None, N)
if data.ndim == 1:
return out[slices].squeeze()
else:
return out[slices]
def violent_multi_band_pass(signal, mask):
""" A filter that lets you define a bunch of bands:
everything outside of these frequency ranges gets mercylessly filtered"""
assert(len(signal) == len(mask))
f = fft(signal)
filtered = [mask[i] * f[i] for i in range(len(signal))]
return scipy.ifft(filtered).real.tolist()
def denoisedSignals(inputData):
#IMPORTANT !! needs to be computationnally optimized by using the operations shown in the exercises
normalizedOutput = np.zeros(inputData.shape)
numberSamples = (np.array(inputData[:, 0, 0])).size
numberElectrodes = (np.array(inputData[0, :, 0])).size
for i in range(0, numberSamples):
for j in range(0, numberElectrodes):
signal = np.array(inputData[i, j, :])
data = np.array(signal)
fft = scipy.fft(data) #signal denoising
bp = fft[:]
for p in range(len(bp)):
if p >= 10:
bp[p] = 0
ibp = scipy.ifft(bp)
#ibp = (ibp-ibp[0])/max(max(ibp), abs(min(ibp))) #signal normalization with initial offset suprresion
ibp = (ibp - np.mean(ibp)) / np.std(
ibp) #signal normalization with initial offset suprresion
normalizedOutput[i, j, :] = ibp.real
return normalizedOutput
def istft(X, chunk_size, hop, w=None):
"""
Naively inverts the short time fourier transform using an overlap and add
method. The overlap is defined by hop
Args:
X: STFT windows to invert, overlap and add.
chunk_size: size of analysis window.
hop: hop distance between analysis windows
w: windowing function to apply. Must be of length chunk_size
Returns:
ISTFT of X using an overlap and add method. Windowing used to smooth.
Raises:
ValueError if window w is not of size chunk_size
"""
if not w:
w = sp.hanning(chunk_size)
else:
if len(w) != chunk_size:
raise ValueError(
"window w is not of the correct length {0}.".format(
chunk_size))
x = sp.zeros(len(X) * (hop))
i_p = 0
for n, i in enumerate(range(0, len(x) - chunk_size, hop)):
x[i:i + chunk_size] += w * sp.real(sp.ifft(X[n]))
return x
def istft(X, chunk_size, hop, w=None):
"""
Naively inverts the short time fourier transform using an overlap and add
method. The overlap is defined by hop
Args:
X: STFT windows to invert, overlap and add.
chunk_size: size of analysis window.
hop: hop distance between analysis windows
w: windowing function to apply. Must be of length chunk_size
Returns:
ISTFT of X using an overlap and add method. Windowing used to smooth.
Raises:
ValueError if window w is not of size chunk_size
"""
if not w:
w = sp.hanning(chunk_size)
else:
if len(w) != chunk_size:
raise ValueError("window w is not of the correct length {0}.".format(chunk_size))
x = sp.zeros(len(X) * (hop))
i_p = 0
for n, i in enumerate(range(0, len(x)-chunk_size, hop)):
x[i:i+chunk_size] += w*sp.real(sp.ifft(X[n]))
return x
def time_mod(arr, factor, win_size=1024, hop=None):
if not hop:
hop = float(win_size) * 0.25
window = np.hanning(win_size)
stft = [
fft(window * arr[i:i + win_size])
for i in range(0,
len(arr) - win_size, win_size)
]
if len(stft) < 2:
raise RuntimeError("Win size too large")
stft = [(stft[i + 1], stft[i]) for i in range(len(stft) - 1)]
frames = len(stft)
omegas = [2 * np.pi * np.arange(len(x[0])) / len(x[0]) for x in stft]
deltas = [np.angle(x[0]) - np.angle(x[1]) for x in stft]
unwrapped = [deltas[i] - hop * omegas[i] for i in range(frames)]
rewrapped = [np.mod(x + np.pi, 2 * np.pi) - np.pi for x in unwrapped]
freqs = [omegas[i] + rewrapped[i] / hop for i in range(frames)]
phase_acc = list(
accumulate(freqs, func=lambda acc, x: acc + x * factor * hop))
cartesians = [
np.abs(stft[i][0]) * np.exp(phase_acc[i] * 1j) for i in range(frames)
]
corrected = [window * np.real(ifft(x)) for x in cartesians]
x = np.zeros(int(len(corrected) * hop * factor))
print(len(x))
print(len(corrected * win_size))
for i, j in enumerate(range(0, len(x) - win_size, int(factor * hop))):
x[j:j + win_size] += corrected[i]
return x
def BarkerLag(rawbeams,
numtype=sp.complex128,
pulse=GenBarker(13),
lagtype=None):
"""This will process barker code data by filtering it with a barker code pulse and
then sum up the pulses.
Inputs
rawbeams - A complex numpy array size NpxNs where Np is the number of pulses and
Ns is the number of samples.
numtype - The type of numbers being used for processing.
pulse - The barkercode pulse.
Outputs
outdata- A Nrx1 size numpy array that holds the processed data. Nr is the number
of range gates """
# It will be assumed the data will be pulses vs rangne
rawbeams = rawbeams.transpose()
(Nr, Np) = rawbeams.shape
pulsepow = sp.power(sp.absolute(pulse), 2.0).sum()
# Make matched filter
filt = sp.fft(pulse[::-1] / sp.sqrt(pulsepow), n=Nr)
filtmat = sp.repeat(filt[:, sp.newaxis], Np, axis=1)
rawfreq = sp.fft(rawbeams, axis=0)
outdata = sp.ifft(filtmat * rawfreq, axis=0)
outdata = outdata * outdata.conj()
outdata = sp.sum(outdata, axis=-1)
#increase the number of axes
return outdata[len(pulse) - 1:, sp.newaxis]
def test_fft_function():
# Many NumPy symbols are imported into the scipy namespace, including
# numpy.fft.fft as scipy.fft, conflicting with this module (gh-10253)
import scipy
scipy.random.seed(1234)
# Callable before scipy.fft is imported
x = scipy.randn(10) + 1j * scipy.randn(10)
y = scipy.ifft(scipy.fft(x))
assert_allclose(y, x)
import scipy.fft
# Callable after scipy.fft is imported
z = scipy.ifft(scipy.fft(x))
assert_allclose(z, x)
assert_allclose(scipy.fft(x), scipy.fft.fft(x))
def wiener_deconvolution(signal, kernel, snr):
"lambd is the SNR"
from scipy import fft,ifft
kernel = np.hstack((kernel, np.zeros(len(signal) - len(kernel)))) # zero pad the kernel to same length
H = fft(kernel)
deconvolved = np.real(ifft(fft(signal)*np.conj(H)/(H*np.conj(H) + snr**2)))
return deconvolved
def pCodePhaseSearch(sample, LOFreq, PRNSpectrumConjugate):
""" Search in a given LO freq space all the code phases
at once
"""
I, Q, _ = generateIQ(sample.size, LOFreq)
# mix is down with the LO
sampledMixedI = sample * I
sampledMixedQ = sample * Q
# munge them into a single array of complex numbers for the fft
combinedMixed = sampledMixedI + 1j*sampledMixedQ
# do the fft
signalSpectrum = scipy.fft(combinedMixed)
# circulator correlation in da frequency space
correlatedSpectrum = signalSpectrum * PRNSpectrumConjugate
# and back to time domain
timeDomainReconstructed = np.abs(scipy.ifft(correlatedSpectrum))**2
return timeDomainReconstructed
def IMRpeakAmp(m1,m2,spin1z,spin2z,d):
"""
IMRpeakAmp finds the peak amplitude of the waveform for a given source parameters and the source distance.
usage: IMRpeakAmp(m1,m2,spin1z,spin2z,distance)
e.g.
spawaveApp.IMRpeakAmp(30,40,0.45,0.5,100)
"""
chi = spawaveform.computechi(m1, m2, spin1z, spin2z)
imrfFinal = spawaveform.imrffinal(m1, m2, chi, 'fcut')
fLower = 10.0
order = 7
dur = 2**numpy.ceil(numpy.log2(spawaveform.chirptime(m1,m2,order,fLower)))
sr = 2**numpy.ceil(numpy.log2(imrfFinal*2))
deltaF = 1.0 / dur
deltaT = 1.0 / sr
s = numpy.empty(sr * dur, 'complex128')
spawaveform.imrwaveform(m1, m2, deltaF, fLower, s, spin1z, spin2z)
s = scipy.ifft(s)
#s = numpy.abs(s)
s = numpy.real(s)
max = numpy.max(s)/d
return max
def gft1d(signal, windowType):
N = len(signal)
# Compute the windows signal
if (windowType == 'gaussian'):
windowFct = gaussian
else:
raise NotImplemented
#windowFct = &box
win = windows(N, windowFct)
# Do the initial FFT of the signal
signal = fft(signal, N)
# Apply the windows
signal = signal * win
# For each of the GFT frequency bands
fstart = 0
for fend in diadicPartitions(N):
# frequency band that we're working with : signal[fstart*2:(fstart+fend)]
# inverse FFT to transform to S-space
signal[fstart:fend] = ifft(signal[fstart:fend], fend - fstart)
fstart = fend
return roll(signal, N / 2)
def process(signal, low, high):
spectrum = get_spectrum(signal)
peak = is_there_a_dart(spectrum, low, high)
filtered = [0] * BUFFER_SIZE
for i in range(index_from_freq(peak / FACTOR),index_from_freq(peak * FACTOR)):
filtered[i] = spectrum[i]
return scipy.ifft(filtered).real.tolist()
def prob5():
rate, sig = wavfile.read('tada.wav')
sig = sp.float32(sig)
noise = sp.float32(sp.random.randint(-32767, 32767, sig.shape))
out = sp.ifft(sp.fft(sig) * sp.fft(noise))
out = sp.real(out)
out = sp.int16(out / sp.absolute(out).max() * 32767)
wavfile.write('white-conv.wav', rate, out)
def cepstrum(self): ms1=self.fs/1000.0; # maximum speech Fx at 1000Hz ms20=self.fs/50.0; # minimum speech Fx at 50Hz xfft=scipy.fft(self.frame); x_hat=scipy.ifft(numpy.log(numpy.abs(xfft)+EPS)); x_hat = numpy.real(x_hat[ms1:ms20]); q=scipy.r_[ms1:ms20]/float(self.fs); return (q,x_hat)
def prob5():
rate, sig = wavfile.read('tada.wav')
sig = sp.float32(sig)
noise = sp.float32(sp.random.randint(-32767,32767,sig.shape))
out = sp.ifft(sp.fft(sig)*sp.fft(noise))
out = sp.real(out)
out = sp.int16(out/sp.absolute(out).max() * 32767)
wavfile.write('white-conv.wav',rate,out)
def low_pass_fft(curve, low_freqs):
"""Filters the curve by setting to zero the high frequencies"""
center = len(curve) / 2
a = scp.fft(curve)
for i in range(0, center - low_freqs):
a[center + i] = 0
a[center - i] = 0
return scp.array(scp.ifft(a))
def fft_lowpass_filter(signal, min_period=2):
"""Applies a simple as a log FFT lowpass filter to a signal.
INPUT:
- ``signal`` -- a list of data representing the sampled signal;
- ``min_period`` -- a threshold for the lowpass filter (the minimal period
of the oscillations which should be left intact) expressed in a number of
samples per one full oscillation.
EXAMPLES:
If you have a sampling frequency equal to 1 Hz (one sample per second)
and want to filter out all the frequencies higher than 0.5 Hz (two samples
per one oscillation period) you should call this function like this::
sage: fft_lowpass_filter(signal, min_period=2)
If you, for example, have a sampling frequency of 1 kHz (1000 samples per
second) and wish to filter out all frequencies hihger than 50 Hz, you should
use the value of ``min_period`` = <sampling frequency> / <cutoff frequency> = 1000 Hz / 50 Hz = 20::
sage: fft_lowpass_filter(signal, min_period=20)
"""
import scipy
signal_fft = scipy.fft(signal)
spectrum_array_length = len(signal_fft)
i = 0 # This is used instead of ' for i in range()'
while i < spectrum_array_length: # because the 'signal' array can be rather long
if i >= spectrum_array_length / min_period:
signal_fft[i] = 0
i += 1
signal_filtered = scipy.ifft(signal_fft)
fft_filtered_signal = []
filtered_signal_length = len(signal_filtered)
i = 0
while i < filtered_signal_length:
fft_filtered_signal.append(float(signal_filtered[i]))
i += 1
norm = max(signal) / max(fft_filtered_signal) # In case we need
# to renormalize
fft_filtered_signal_length = len(
fft_filtered_signal) # the filtered signal
i = 0 # in order to obtain
while i < fft_filtered_signal_length: # the same amplitude
fft_filtered_signal[i] *= norm # as the inintial
i += 1 # signal had.
return fft_filtered_signal
def prob1():
rate,data = wavfile.read('Noisysignal2.wav')
fsig = sp.fft(data,axis = 0)
f = sp.absolute(fsig)
plt.plot(f[0:f.shape[0]/2])
for j in xrange(14020,50001):
fsig[j]=0
fsig[-j]=0
newsig=sp.ifft(fsig)
f = sp.absolute(fsig)
plt.figure()
plt.plot(f[0:f.shape[0]/2])
plt.show()
plt.close()
newsig = sp.ifft(fsig).astype(float)
scaled = sp.int16(newsig/sp.absolute(newsig).max() * 32767)
wavfile.write('cleansig2.wav',rate,scaled)
def istft(X, N):
""" Transform an array of spectra to the original signal
"""
framesz = X.shape[1]
hop = int(float(framesz)/2)
x = scipy.zeros(N + (2*framesz))
for n,i in enumerate(range(hop, len(x)-(framesz+hop), hop)):
x[i:i+framesz] += scipy.real(scipy.ifft(X[n]))
return x[framesz:-framesz]
def idct(v,axis=-1):
n = len(v.shape)
N = v.shape[axis]
even = (N%2 == 0)
slices = [None]*4
for k in range(4):
slices[k] = []
for j in range(n):
slices[k].append(slice(None))
k = arange(N)
if even:
ak = scipy.r_[1.0,[2]*(N-1)]*exp(1j*pi*k/(2*N))
newshape = ones(n)
newshape[axis] = N
ak.shape = newshape
xhat = real(scipy.ifft(v*ak,axis=axis))
x = 0.0*v
slices[0][axis] = slice(None,None,2)
slices[1][axis] = slice(None,N/2)
slices[2][axis] = slice(N,None,-2)
slices[3][axis] = slice(N/2,None)
for k in range(4):
slices[k] = tuple(slices[k])
x[slices[0]] = xhat[slices[1]]
x[slices[2]] = xhat[slices[3]]
return x
else:
ak = 2*scipy.exp(1j*pi*k/(2*N))
newshape = ones(n)
newshape[axis] = N
ak.shape = newshape
newshape = list(v.shape)
newshape[axis] = 2*N
Y = zeros(newshape,scipy.Complex)
#Y[:N] = ak*v
#Y[(N+1):] = conj(Y[N:0:-1])
slices[0][axis] = slice(None,N)
slices[1][axis] = slice(None,None)
slices[2][axis] = slice(N+1,None)
slices[3][axis] = slice((N-1),0,-1)
Y[slices[0]] = ak*v
Y[slices[2]] = conj(Y[slices[3]])
x = scipy.real(scipy.ifft(Y,axis=axis))[slices[0]]
return x
def spec_interp(coef_spec,factor=1,x1=1.0):
"""Interpolate a function"""
assert isinstance(factor,int) and factor > 0,\
"factor must be a positive integer"
npts = len(coef_spec)
padded_fft = S.zeros(npts*factor,'D')
padded_fft[:npts] = coef_spec
return factor*S.ifft(padded_fft).real
def _istft(X): time_blocks = [] for block in X: num_elems = block.shape[0] / 2 real_chunk = block[0:num_elems] imag_chunk = block[num_elems:] new_block = real_chunk + 1.0j * imag_chunk time_block = scipy.ifft(new_block) time_blocks.append(time_block) return np.concatenate(time_blocks)
def istft(self, X):
"""
Apply short-time Fourier transform to X
Parameters
----------
X : list of lists or ndArrays
"""
X = np.array([scipy.real(scipy.ifft(x)) for x in X])
return X
