def many_psds(k=2,fs=1.0, b0=1.0, N=1024):
""" compute average of many PSDs """
psd=[]
for j in range(k):
print j
x = noise.white(N=2*4096,b0=b0,fs=fs)
f, tmp = noise.numpy_psd(x,fs)
if j==0:
psd = tmp
else:
psd = psd + tmp
return f, psd/k
def many_psds(k=2, fs=1.0, b0=1.0, N=1024):
""" compute average of many PSDs """
psd = []
for j in range(k):
print(j)
x = noise.white(num_points=2 * 4096, b0=b0, fs=fs)
f, tmp = noise.numpy_psd(x, fs)
if j == 0:
psd = tmp
else:
psd = psd + tmp
return f, psd / k
fs=100
h0=2e-16
N=16*4096
v0 = 1e6 # nominal oscillator frequency
y = noise.white(N=N,b0=h0,fs=fs) # fractional frequency
x = allantools.frequency2phase(y,fs) # phase in seconds
fi = [2*math.pi*v0*xx for xx in x] # phase in radians
t = np.linspace(0, (1.0/fs)*N, len(y))
plt.figure()
plt.plot(t,y)
plt.xlabel('Time / s')
plt.ylabel('Fractional frequency')
f_y, psd_y = noise.numpy_psd(y,fs)
f_fi, psd_fi = noise.numpy_psd(fi,fs)
f_x, psd_x = noise.numpy_psd(x,fs)
fxx, Pxx_den = noise.scipy_psd(y, fs)
f_fi2, psd_fi2 = noise.scipy_psd(fi, fs)
f_x2, psd_x2 = noise.scipy_psd(x, fs)
plt.figure()
plt.semilogy(f_y,psd_y,label='numpy.fft()')
plt.semilogy(fxx,Pxx_den,label='scipy.signal.welch()')
plt.semilogy(f_y,[h0]*len(f_y),label='h_0 = %.3g' % h0 )
plt.legend(framealpha=0.5)
plt.title('PSD of fractional frequency')
plt.xlabel('Frequeny / Hz')
plt.ylabel('one-sided PSD / S_y(f)')
plt.ylabel('Count')
plt.title('Frequency histogram')
plt.subplot(2, 2, 4)
phist, pbins = np.histogram([1e9 * yy for yy in x], bins=100, density=True)
plt.plot(pbins[1:], phist, 'ro')
plt.xlabel('Phase / ns')
plt.ylabel('Count')
plt.title('Phase histogram')
rmsphase = 1e12 * np.std(x)
print("RMS phase = ", rmsphase, " ps")
plt.text(0, 0.01, 'RMS = %f ps' % rmsphase)
# plt.show()
f_y, psd_y = noise.numpy_psd(y, fs)
f_fi, psd_fi = noise.numpy_psd(fi, fs)
f_x, psd_x = noise.numpy_psd(x, fs)
fxx, Pxx_den = noise.scipy_psd(y, fs)
f_fi2, psd_fi2 = noise.scipy_psd(fi, fs)
f_x2, psd_x2 = noise.scipy_psd(x, fs)
print("PSD calc done.")
# S_y
plt.figure()
plt.loglog(f_y, psd_y, label='numpy.fft()')
plt.loglog(fxx, Pxx_den, label='scipy.signal.welch()')
plt.loglog(f_y, [h2 * ff * ff for ff in f_y], label='h2 * f^2 ')
plt.legend(framealpha=0.5)
plt.title('PSD of fractional frequency')
# time-series figure
plt.figure()
plt.plot(t,y,label='y')
plt.plot(t,x[1:],label='x')
plt.legend()
plt.xlabel('Time / s')
plt.ylabel('Fractional frequency')
# note: calculating the PSD of an 1/f^4 signal using fft seems to be difficult
# the welch method returns a correct 1/f^4 shaped PSD, but fft often does not
# things that may help
# - using a longer time-series (increase N above)
# - detrend using signal.detrend()
# - calculate PSD for a short window of data, and average over windows (this is done in the Welch method)
# - read the Welch code to see what is going on
f_y, psd_y = noise.numpy_psd(y, fs)
f_fi, psd_fi = noise.numpy_psd( signal.detrend( fi[:len(fi)/20] ),fs)
f_x, psd_x = noise.numpy_psd(x[:len(x)/20], fs)
fxx, Pxx_den = noise.scipy_psd(y, fs)
f_fi2, psd_fi2 = noise.scipy_psd(fi, fs)
f_x2, psd_x2 = noise.scipy_psd(x, fs)
# Fractional frequency PSD
plt.figure()
plt.loglog(f_y,psd_y,label='numpy.fft()')
plt.loglog(fxx,Pxx_den,label='scipy.signal.welch()')
plt.loglog(f_y[1:],[h2/(ff*ff) for ff in f_y[1:]],label='h_2/f^2 with h_2 = %.3g' % h2 )
plt.legend(framealpha=0.5)
plt.title('PSD of fractional frequency')
fig, ax1 = plt.subplots()
ax1.plot(t, y, label='y')
ax2 = ax1.twinx()
ax2.plot(t, x[1:], label='x')
plt.legend()
plt.xlabel('Time / s')
plt.ylabel('Fractional frequency')
# note: calculating the PSD of an 1/f^4 signal using fft seems to be difficult
# the welch method returns a correct 1/f^4 shaped PSD, but fft often does not
# things that may help
# - using a longer time-series (increase N above)
# - detrend using signal.detrend()
# - calculate PSD for a short window of data, and average over windows (this is done in the Welch method)
# - read the Welch code to see what is going on
f_y, psd_y = noise.numpy_psd(y, fs)
f_fi, psd_fi = noise.numpy_psd( signal.detrend( fi[:len(fi)/20] ),fs)
f_x, psd_x = noise.numpy_psd(x[:len(x)/20], fs)
fxx, Pxx_den = noise.scipy_psd(y, fs)
f_fi2, psd_fi2 = noise.scipy_psd(fi, fs)
f_x2, psd_x2 = noise.scipy_psd(x, fs)
# Fractional frequency PSD
plt.figure()
plt.loglog(f_y,psd_y,label='numpy.fft()')
plt.loglog(fxx,Pxx_den,label='scipy.signal.welch()')
plt.loglog(f_y[1:],[h2/(ff*ff) for ff in f_y[1:]],label='h_2/f^2 with h_2 = %.3g' % h2 )
plt.legend(framealpha=0.5)
plt.title('PSD of fractional frequency')