Re: Considerations in the design of a ShortTimeFFT window?

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From: Dietrich Brunn <[email protected]>
To: [email protected]
Cc: [email protected], Edward Richards <[email protected]>
Date: Mon, 04 Mar 2024 16:51:42 +0100
Message-ID: <4203513.1IzOArtZ34@rechentux>
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Hi Ned,
I took the time to think about this a little bit: Unfortunately I am not aw=
are of useful=20
literature for choosing STFT windows. Note that the concept of a dual windo=
w for the=20
ISTFT does not seem to be widely known in the signal processing community. =
The only=20
other implementation I know, which provides a dual window is LTFAT [1]=20

Designing a STFT based differentiator is a bit involving, so let's discuss =
an FFT based one=20
first: The continuous-time signal representation of a sampled signal of fin=
ite length can be=20
expressed by a complex-valued Fourier series  E. g., for signal of duration=
 one, we can=20
write

x(t) =3D Sum[  X[l]  exp(2j=CF=80(l =CE=94 f) t ) ] with l being the summat=
ion index.=20

The coefficients X[l] can be calculated with an FFT ([2] discusses this fro=
m a different=20
angle). Differentiation with respect to time gives=20

d/dt x(t) =3D Sum[  X[l]  2j=CF=80(l =CE=94 f) exp(2j=CF=80(l =CE=94 f) t) =
] .

Note that x(t) is assumed to be periodic. Hence a discontinuity between sta=
rt and end of=20
signal produces ringing due to Gibb's phenomenon.


The key insight of the ShortTimeFFT implementation is that any signal can b=
e represented=20
by a series expansion of time- and frequency-shifted dual windows, i.e.,

x(t) =3D Sum[  S[q, p] d(t - p =CE=94t) exp(2j=CF=80(q =CE=94 f) t ) ]=20

with (p =CE=94t) representing the time shift,  (q =CE=94 f) the frequency s=
hift, d(t) the dual window and=20
S[p, q] the STFT coefficient (consult [3] for details).
Differentiation with respect to time gives=20

d/dt x(t) =3D Sum[ S[q, p] 2j=CF=80(q =CE=94 f)  d(t - p =CE=94t) exp(2j=CF=
=80(q =CE=94 f) t ) ]  +=20
                            S[q, p] exp(2j=CF=80(q =CE=94 f) t )  d/dt d(t =
=2D p =CE=94t) ]=20

Note that:

* The window dependent term is parameterized by the derivative of the  dual=
 window d/dt=20
d(t - p =CE=94t).

* If the signal is not periodic in each slice which is stenciled out by the=
 sliding window,=20
Gibb's phenomenon will strike again (this is what you probably observe in y=
our=20
simulation). Hence it is a good idea to choose a dual window, like the Hann=
 window, which=20
suppresses discontinuities at the beginning and end of the signal slice. =20

* Not only the dual window but also its derivative needs to suppress those =
discontinuities.=20

The following example shows an example of STFT based differentiation. The h=
op width is=20
chosen small enough to supress Gibb's phenomenon in the dual window derivat=
ive. Note =20
that Gibb's phenomenon  can be observed at the beginning and the end of the=
 signal.

import matplotlib.pyplot as plt

import numpy as np

from scipy.signal import ShortTimeFFT

from scipy.signal import windows

from scipy.fft import rfft, rfftfreq, irfft


# Create periodic test signal and its derivative:

n, T =3D 1000, 1/1000  # samples and sampling interval for 1 second signal

t =3D np.arange(n) * T  # time stamps
# Create single frequency signal and derivative:

f =3D rfftfreq(n, T)
k_c =3D f.searchsorted(7)
omega_c =3D 2*np.pi*f[k_c]
X =3D np.zeros(len(f))
X[k_c] =3D n/2/omega_c
x =3D irfft(X, n=3Dn)
y =3D irfft(2j*np.pi*f*X, n=3Dn)  # dx / dt

# Two ShortTimeFFT instances are needed:

kw =3D dict(hop=3D10, fs=3D1/T,fft_mode=3D'onesided', phase_shift=3DNone)
SFT =3D ShortTimeFFT.from_dual(windows.hann(32, sym=3DFalse), **kw)

# Differentiate dual window per FFT:

diff_dual_win =3D irfft(rfft(SFT.dual_win) * 2j*np.pi*rfftfreq(SFT.m_num))
dSFT =3D ShortTimeFFT.from_dual(diff_dual_win, **kw)


# Perform the filtering:

S_x =3D SFT.stft(x)
dS_x =3D 2j*np.pi*SFT.f[:, np.newaxis] * S_x
dx =3D SFT.istft(dS_x, k1=3Dn) + dSFT.istft(S_x, k1=3Dn)


# Plot windows:

fg0, axx0 =3D plt.subplots(2, 1, sharex=3D'all', tight_layout=3DTrue)
axx0[0].set(title=3D"Windows")
axx0[1].set(title=3D"Dual Windows")
axx0[0].plot(SFT.win, '.-', alpha=3D0.5, label=3D'SFT')
axx0[0].plot(dSFT.win, '.-', alpha=3D0.5, label=3D'dSFT')
axx0[1].plot(SFT.dual_win, '.-', alpha=3D0.5, label=3D'SFT')
axx0[1].plot(dSFT.dual_win, '.-', alpha=3D0.5, label=3D'dSFT')


# Plot signal:

fg1, ax1 =3D plt.subplots()
ax1.set(title=3Drf"STFT-based Differentiator for $f_c=3D{f[k_c]}\,$Hz Signa=
l",
        xlabel=3D"Time $t$", ylabel=3D"Amplitude")
ax1.plot(t, omega_c * x, alpha=3D0.5, label=3Dr"$x(t) \cdot 2\pi f_c$")
ax1.plot(t, y, '--', alpha=3D0.5, label=3D"$y =3D dx/dt$")
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