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.