Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 67

Note that the signal x ( t ) is in general not bandlimited unless f ( t ) also hap-
pens to be bandlimited. This problem shows that even a non-bandlimited
signal can sometimes be reconstructed from its uniform samples if we know
the model which generates the signal.

2.5. Show how the 3-bit Gray code in Fig. 2.17 can be extended to 4-bit PAM.
Hint . You only have to make sure that the 16 codewords are distinct and
that adjacent codewords differ exactly in one bit location.

2.6. Consider the system shown in the figure, where H ( jω ) is a linear time-
invariant system.

cos ω t
c

cos ω t
c

x ( t )

x

x

H ( j ω)

y ( t )

+

x

x

H ( j ω )

sin ω t
c

sin ω t

c

Figure P2.6 .

1. Show that the modulator cos ω c t is a linear time varying device, that
is, the operation y 1 ( t )= x ( t )cos ω c t is linear time varying. Similarly
the modulator sin ω c t is time varying.
2. Show, however, that the overall system from x ( t )to y ( t ) in the figure
is linear and time invariant (LTI). What is its transfer function and
what is the impulse response? Thus the system is an example of an
LTI system, with time varying building blocks in it.

2.7. Consider the system shown in Fig. P2.7, where H ( jω ) is a linear time
invariant system. This is a modified version of the system in Problem 2.6.
Is this an LTI system?