6
D. Rocchesso: Sound Processing
window
Uncertainty Principle
frequency resolution
We have already seen that the function Y (f ) is periodic
with period F
s
.
Therefore, it is easy to realize that the DTFT can be inverted by an integral
calculated on a single period:
y(nT ) =
1
F
s
F
s
/2
-F
s
/2
Y (f )e
j2f nT
df .
(11)
In practice, in order to compute the Fourier transform with numeric means
we must consider a finite number of points in (10). In other words, we have to
consider a window of N samples and compute the discrete Fourier transform on
that signal portion:
^
Y (f ) =
N -1
n=0
^
y(n)e
-j2
f
Fs
n
.
(12)
In (12) we have taken a window of N samples (i.e., N T seconds) of the signal,
starting from instant 0, thus forming an N -point vector. The result is still a
function of continuos variable: the larger the window, the closer is the function
to Y (f ). Therefore, the "windowing" operation introduces some loss of precision
in frequency analysis. On the other hand, it allows to localize the analysis in the
time domain. There is a tradeoff between the time domain and the frequency
domain, governed by the Uncertainty Principle which states that the product
of the window length by the frequency resolution f is constant:
f N = 1 .
(13)
Example 2. This example should clarify the spectral effects induced by
sampling and windowing. Consider the causal complex exponential function in
continuous time
y(t) =
e
s
0
t
t 0
0
t < 0
,
(14)
where s
0
is the complex number s
0
= a + jb. To visualize such complex signal
we can consider its real part
(y(t)) = (e
at
e
jbt
) = e
at
cos (bt) ,
(15)
and obtain fig. 3.a from it.
The Laplace transform of function (14) has been calculated in appendix A.8.1.
It can be reduced to the Fourier transform by the substitution s = j:
Y () =
1
j - s
0
.
(16)
The magnitude of the complex function (16) is drawn in solid line in fig. 3.
The sampled signal is also Fourier-transformable in closed form, by reducing
the Z transform obtained in appendix A.8.3 by the substitution z = e
j
. The
Y () =
1
1 - e
s
0
/F
s
e
-j
,
(17)
4
Indeed, the expression (10) can be read as the Fourier series expansion of the periodic
signal Y (f ) with coefficients y(nT ) and components which are "sinusoidal" in frequency and
are multiples of the fundamental 1/F
s
.
5
If we compare this formula with (57) of the appendix A, we see that here the variable
s
0
in the exponent is divided by F
s
. Indeed, the discrete-variable functions of appendix A.8.3
correspond to signals sampled with unit sampling rate.