78
D. Rocchesso: Sound Processing
analysis window
f
|W|
1
0
F /N
F
s
s
Figure 1: Frequency response of a prototype lowpass filter
filter in order to isolate a specific slice of the frequency spectrum. This procedure
is reported in fig. 2.
0
e,
y
N-1
0
e,
y
N-1
m
N-1
Y ( ) = (w * )(m)
e,
y
0
Y ( )
m
0
e,
y
= (w * )(m)
W( )
W( )
-j
e
N-1
n
-j
n
e
y(n)
Figure 2: Decomposition of a signal into a set of non-overlapping frequency slices.
0
, . . . ,
N -1
are the central frequencies of the bands of the analysis channels.
4.1.2
The DFT View
The scheme of fig. 2 can be obtained by Fourier transformation of a "windowed"
sequence. We recall from section 1.3 that the DTFT of an infinite sequence is
Y () =
+
n=-
y(n)e
-jn
.
(1)
If the DTFT is computed on a portion of y(), weighted by an analysis
window w(m - n), we get a frame of the STFT:
Y
m
() =
+
n=-
w(m - n)y(n)e
-jn
= e
-jm
+
r=-
w(r)y(m - r)e
-jr
,
(2)
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