Digital communication systems and filter banks - Signal Processing and Optimization for Transceiver Systems - page 106

Problems

3.1. The decimator and expander in cascade are in general not interchangeable.
That is, the two systems in Fig. P3.1 are in general not equivalent.

y ( n ) y ( n )

x ( n ) x ( n )

1

2

M L M

L

Figure P3.1.

1. For M = L = 1, prove that the systems are not equivalent by con-

structing an input x ( n ) such that y 1 ( n ) = y 2 ( n ) .

2. For M = L , prove that the two systems are equivalent if M and L

are coprime (i.e., they do not have any common integer factors other
than unity).

3. For M = L , prove that the two systems are equivalent only if M and
L are coprime.

3.2. Consider the multirate system shown in part (a) of Fig. P3.2. Assume that
M and L are coprime integers. In view of Euclid's theorem this implies
that there exist integers N and K such that NL + MK = 1. Show then
that this system can be drawn as in part (b). With z replaced in part (a)
by z −k

for arbitrary integer k , how does the system in part (b) change?

z N z K L z M M

L

(a) (b)

Figure P3.2.

3.3. Bandpass sampling. For the signal shown in Fig. P3.3, the total bandwidth
(sum of passband widths) is less than 2 π/M and the bands are located as
shown, that is, the band edges are integer multiples of π/M. Show that
M -fold decimation causes no aliasing. This result is called the bandpass
sampling theorem.