80
D. Rocchesso: Sound Processing
analysis window
rectangular window
N=8
R=5
0
n
n
w(n) = w(-n)
w(3 - n)
y(n)
n
n
0
0
0
m = 3
Y (0)
Y (7)
3
3
8
8
8
8
7
DFT
IDFT
time-centered window
reconstruction of 5 samples
of y(n)
0
1
1/w(2)
1/w(-2)
y(5)
y(1)
w(3 - n) y(n)
5
Figure 3: Analysis and resynthesis of a frame of STFT.
Example
Figure 3 illustrates the operations involved in analysis and resynthesis of a frame
of STFT (R = 5, N = 8).
4.1.3
Windowing
The rectangular window
The simplest analysis window is the rectangular window
w
R
(n) =
1 n = 0, . . . , R - 1
0 elsewhere
,
(11)
Considered a filter having 11 as its impulse response, the frequency response is
found by Fourier-transformation of w
R
(n):
W
R
() =
+
n=-
w
R
(n)e
-jn
=
R-1
n=0
e
-jn
=
1 - e
-jR
1 - e
-j
=
= sinc
R
() = e
-j
R-1
2
sin
R
2
sin
2
.
(12)