Sound Analysis
79
bin
resynthesis
where the third member of the equality is obtained by defining r = m-n, and m
is a variable accounting for the temporal dislocation of the window. Therefore,
the STFT turns out to be a function of two variables, one can be thought of as
frequency, the other is essentially a time shift.
The DTFT is a periodic function of a continuous variable, and it can be
inverted by means of an integral computed over a period
w(m - n)y(n) =
1
2
-
Y
m
()e
jn
d .
(3)
By a proper alignment of the window (m = n) we can compute, if w(0) = 0
y(n) =
1
2w(0)
-
Y
n
()e
jn
d .
(4)
The STFT in its formulation (2) can be seen as convolution
Y
m
() = (w y
e
)(m) ,
(5)
where y
e
(n) = y(n)e
-jn
is the demodulated signal. If w is set to the impulse
response of the ideal lowpass filter, and if we set =
k
, we get a channel of the
filterbank of fig. 2. In general, w(·) will be the impulse response of a non-ideal
lowpass filter, but the filterbank view will keep its validity.
In practice, we need to compute the STFT on a finite set of N points. In
what follows we assume that the window is R N samples long, so that we
can use the DFT on N points, thus obtaining a sampling of the frequency axis
between 0 and 2 in multiples of 2/N .
The k-th point in the transform domain (said the k-th bin of the DFT) is
given by
Y
m
(k) =
N -1
n=0
w(m - n)y(n)e
-j
2kn
N
(6)
and, by means of an inverse DFT
w(m - n)y(n) =
1
N
N -1
k=0
Y
m
(k)e
j
2kn
N
.
(7)
By a proper alignment of the window (m = n), and assuming that w(0) = 0
we get
y(n) =
1
N w(0)
N -1
k=0
Y
n
(k)e
j2kn
N
.
(8)
More generally, we can reconstruct (resynthesis) the time-domain signal by
means of
y(n) =
1
N w(m - n)
N -1
k=0
Y
m
(k)e
j2kn
N
,
(9)
where w(m - n) = 0, which is true, given an integer n
0
, for a non-trivial window
defined for
m + n
0
n m + n
0
+ R - 1 .
(10)