563489578934

96
Signals and Spectra Chap. 2
W(f)
–B B
1
–– 2
f s
f
(a) Spectrum of Unsampled Waveform
Low-pass filter
T s W s (f)
3 –f s 1
1 f
–– f 2 s ––2 f s 3
––– f –
2 s s ––2 f s
B
2f s
f
(b) Spectrum of Impulse Sampled Waveform (f s < 2B)
Figure 2–19 Undersampling and aliasing.
independent pieces of information that will describe the waveform over a T 0 interval.
N is said to be the number of dimensions required to specify the waveform, and B is the
absolute bandwidth of the waveform. [Shannon, 1949; Wozencraft and Jacobs, 1965;
Wyner and Shamai (Shitz), 1998].
The dimensionality theorem of Eq. (2–174) says simply that the information which can
be conveyed by a bandlimited waveform or a bandlimited communication system is proportional
to the product of the bandwidth of that system and the time allowed for transmission of
the information. The dimensionality theorem has profound implications in the design and performance
of all types of communication systems. For example, in radar systems, it is well
known that the time–bandwidth product of the received signal needs to be large for superior
performance.
There are two distinct ways in which the dimensionality theorem can be applied. First,
if any bandlimited waveform is given and we want to store some numbers in a table (or a computer
memory bank) that could be used to reconstruct the waveform over a T 0 -s interval, at