
Single channel signal 53
where h is the window. If the window has a finite energy then the STFT can be
reversed:
x(t)=
1
E
h
Z
∞
−∞
Z
∞
−∞
F
x
(u, f;h)h(t −u)e
i2πtf
dud f (2.82)
where E
h
=
R
∞
−∞
|h(t)|
2
dt. Thus we can view the STFT as a tool that decomposes the
signal into waveforms, time-frequency atoms, of the form:
h
t, f
(u)=h(u −t)e
i2π fu
(2.83)
This concept is illustrated in Figure 2.15. Each atom is obtained by translation of a
single window h and its modulation with frequency f .
STFT can be used to obtain the distribution of energy in the time-frequency space.
If the window function has a unit energy (i.e., E
h
= 1) then a squared modulus of
STFT, a spectrogram, is ...