Systems, Sampling and Quantization
5
aliasing
foldover
frequency modulation
digital frequencies
Discrete-Time Fourier
Transform
DTFT
· Convolution with an ideal sinc function.
The sinc function is ideal because its temporal extension is infinite on both sides,
thus implying that the reconstruction process can not be implemented exactly.
However, it is possible to give a practical realization of the reconstruction filter
by an impulse response that approximates the sinc function.
Whenever the condition (8) is violated, the periodic replicas of the spec-
trum have components that overlap with the base band. This phenomenon is
called aliasing or foldover and is avoided by forcing the continuous-time original
signal to be bandlimited to the Nyquist frequency. In other words, a filter in
the continuous-time domain cuts off the frequency components exceeding the
Nyquist frequency. If aliasing is allowed, the reconstruction filter can not give a
perfect copy of the original signal.
Usually, the word aliasing has a negative connotation because the aliasing
phenomenon can make audible some spectral components which are normally
out of the frequency range of hearing. However, some sound synthesis techniques,
such as frequency modulation, exploit aliasing to produce additional spectral
lines by folding onto the base band spectral components that are outside the
Nyquist bandwidth. In this case where the connotation is positive, the term
foldover is preferred.
1.3
Discrete-Time Spectral Representations
We have seen how the sampling operation essentially changes the nature of the
signal domain, which switches from a continuous to a discrete set of points. We
have also seen how this operation is transposed in the frequency domain as a
periodic replication. It is now time to clarify the meaning of the variables which
are commonly associated to the word "frequency" for signals defined in both the
continuous and the discrete-time domain. The various symbols are collected in
table 1.1, where the limits imposed by the Nyquist frequency are also indicated.
With the term "digital frequencies" we indicate the frequencies of discrete-time
signals.
Nyquist Domain
Symbol
Unit
[-F
s
/2
. . .
0
. . .
F
s
/2]
f
[Hz] = [cycles/s]
[-1/2
. . .
0
. . .
1/2]
f /F
s
[cycles/sample]
digital
[-
. . .
0
. . .
]
= 2f /F
s
[radians/sample]
frequencies
[-F
s
. . .
0
. . .
F
s
]
= 2f
[radians/s]
Table 1.1: Frequency variables
Appendix A.8.3 shows how it is possible to define a Fourier transform for
functions of a discrete variable. Here we can re-express such definition, as
a function of frequency, for discrete-variable functions obtained by sampling
continuous-time signals with sampling interval T . This transform is called the
Discrete-Time Fourier Transform (DTFT) and is expressed by
Y (f ) =
+
n=-
y(nT )e
-j2
f
Fs
n
.
(10)