Signals and Systems - Electrical Engineering

Problems 409

6.6 What Have We Accomplished? What Is Next?

In this chapter we have illustrated the application of the Laplace and the Fourier analysis to the
theories of control, communications, and filtering. As you can see, the Laplace transform is very
appropriate for control problems where transients as well as steady-state responses are of interest.
On the other hand, in communications and filtering there is more interest in steady-state responses
and frequency characterizations, which are more appropriately treated using the Fourier transform.
It is important to realize that stability can only be characterized in the Laplace domain, and that it
is necessary when considering steady-state responses. The control examples show the importance of
the transfer function and transient and steady-state computations. Block diagrams help to visualize
the interconnection of the different systems. Different types of modulation systems are illustrated
in the communication examples. Finally, this chapter provides an introduction to the design of
analog filters. In all the examples, the application of MATLAB was illustrated.

Although the material in this chapter does not have sufficient depth, reserved for texts in control,
communications, and filtering, it serves to connect the theory of continuous-time signals and systems
with applications. In the next part of the book, we will consider how to process signals using comput-
ers and how to apply the resulting theory again in control, communications, and signal processing

Problems............................................................................................

PROBLEMS

6.1. Cascade implementation and loading

The transfer function of a filterH(s)= 1 /(s+ 1 )^2 is to be implemented by cascading two first-order filters

Hi(s)= 1 /(s+ 1 ),i=1, 2.

(a) ImplementHi(s)as a series RC circuit with inputvi(t)and outputvi+ 1 (t),i=1, 2. Cascade two of

these circuits and find the overall transfer functionV 3 (s)/V 1 (s). Carefully draw the circuit.

(b) Use a voltage follower to connect the two circuits when cascaded and find the overall transfer function

V 3 (s)/V 1 (s). Carefully draw the circuit.

(c) Use the voltage follower circuit to implement a new transfer function

G(s)=

1

(s+ 1000 )(s+ 1 )

Carefully draw your circuit.

6.2. Cascading LTI and LTV systems

The receiver of an AM system consists of a band-pass filter, a demodulator, and a low-pass filter. The

received signal is

r(t)=m(t)cos( 40000 πt)+q(t)

where m(t) is a desired voice signal with bandwidth BW=5 KHz that modulates the carrier

cos(40, 000πt)andq(t)is the rest of the signals available at the receiver. The low-pass filter is ideal with

magnitude 1 and bandwidthBW. Assume the band-pass filter is also ideal and that the demodulator is

cos(ct).

(a) What is the value ofcin the demodulator?

(b) Suppose we input the received signal into the band-pass filter cascaded with the demodulator and

the low-pass filter. Determine the magnitude response of the band-pass filter that allows us to recover

m(t). Draw the overall system and indicate which of the components are LTI and which are LTV.