Sound Analysis
81
resynthesis
filterbank summation
phase vocoder
decimation
hop size
side lobes
The real part of the function sinc
R
() is plotted in figure 4 for different values
of the window length R.
0
5
10
15
-3
-2
-1
0
1
2
3
sinc_R
radian frequency
R = 4
R = 8
R = 16
Figure 4: sinc
R
() for different values of window length R.
In figure 4, it can be noticed that 2/R is the zero closest to dc. Therefore, we
can say that if we use the rectangular window as a prototype of filter represented
in figure (2), the equivalent bandwidth is 2/R. If we neglect aliasing for a
moment, we realize that we can decimate each channel Y
m
(
k
) by a factor R
without loosing any information.
A superficial look at the expression (12) seems to indicate that the shifted
replicas of sinc
R
produce aliasing in the base band -
2
R
,
2
R
. Indeed, if we sum
R shifted replicas we verify that the aliasing components cancel out. Therefore,
with this window, it is possible to decimate the output channels by a factor
equal to the window length. Furthermore, if we choose N = R, we can perform
one FFT per frame and advance the window by N samples at each step.
According to (7), the reconstruction (resynthesis) of the analyzed signal can
be obtained by filterbank summation, as depicted in figure 5. The reconstruction
can be interpreted as a bank of oscillators driven by the analysis data. The two
stages represented in figures 2 and 5, taken as a whole, are often called the phase
vocoder.
Between the analysis stage of figure 2 and the synthesis stage of figure 5, a
decimation stage can be inserted. Namely, with the rectangular window we can
reduce the intermediate sampling rate down to F
s
/R. Of course, in order to do
the filter bank summation of figure 5, an interpolation stage will be needed to
take the sampling rate back to F
s
.
For the rectangular window, the window is shifted in time by R samples
after each DFT computation. This temporal shift is technically called hop size.
In the case of the rectangular window, hop sizes smaller than R do not add any
information to the analysis.
Commonly-used windows
In practice, signal analysis is seldom performed using rectangular windows, be-
cause its frequency response has side lobes that are significantly high thus po-