Short-Time Fourier
Transform
STFT
filterbank
modulation
carrier
demodulation
Chapter 4
Sound Analysis
Sounds are time-varying signals in the real world and, indeed, all of their mean-
ing is related to such time variability. Therefore, it is interesting to develop
sound analysis techniques that allow to grasp at least some of the distinguished
features of time-varying sounds, in order to ease the tasks of understanding,
comparison, modification, and resynthesis.
In this chapter we present the most important sound analysis techniques.
Special attention is reserved on criteria for choosing the analysis parameters,
such as window length and type.
4.1
Short-Time Fourier Transform
The Short-Time Fourier Transform (STFT) is nothing more than Fourier anal-
ysis performed on slices of the time-domain signal. In order to slightly simplify
the formulas, we are going to present the STFT under the assumption of unitary
sample rate (F
s
= T
-1
= 1).
There are two complementary views of STFT: the filterbank view, and the
DFT-based view.
4.1.1
The Filterbank View
Assume we have a prototype ideal lowpass filter, whose frequency response is
depicted in fig. 1. Let w() and W () be the impulse response and transfer
function, respectively, of such prototype filter.
We define modulation of a signal y(n) by a carrier signal e
j
0
n
as the (com-
plex) multiplication y(n)e
j
0
n
. This translates, in the frequency domain, into a
frequency shift by =
0
(shift theorem 1.2 of chapter 1). In other words,
modulating a signal means moving its low frequency content onto an area around
the carrier frequency. On the other hand, we call demodulation of a signal y(n)
its multiplication by e
-j
0
n
, that brings the components around
0
onto a neigh-
borhood of dc.
By demodulation we can obtain a filterbank that slices the spectrum (be-
tween 0Hz and F
s
) in N equal non-overlapping portions. Namely, we can trans-
late the input signal in frequency and filter it by means of the prototype lowpass
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