MAE143A Signals & Systems - Homework 2, Winter 2013 due by the end of class Tuesday January 22, 2012.

This homework should be handed in at the class not at my office!

This weeks homework is meant to be a little lighter than normal to accommodate the MLKJ Day holiday and correspond-ing hiatus on problem session and office hours.

Question 1 Composite signals

Use the basic functions, step function 1(t) and ramp function r(t), together with scaling (multiplication by a constant)and time-shifting (t t a) to provide expressions for the waveforms in Figure 1 (a), (b) and (c). Accordingly, computetheir Laplace transforms using, say, the Table 3.2 on Page 199 of Chaparro or the equivalent from elsewhere. Please donot compute the integrals from first principles.

2

1

3

2 5 8

(a)

2 3

-2

62 10

-3

6 10

(c)(b)

Figure 1: Question 1

Tuesday, 15 January 2013

Question 2 Energy and power

Prove each of the following properties of signals, where E() and P () are the energy and power of a signal and a > 0 isa positive constant.

(a) E[x(t+ a)] = E[x(t)] and P [x(t+ a)] = P [x(t)]. That is, time-shifts do not affect energy or power.

(b) E[ax(t)] = a2E[x(t)] and P [ax(t)] = a2P [x(t)]. That is, scaling by a scales the energy and power by a2.

(c) E[x(at)] = 1aE[x(t)] and P [x(at)] = P [x(t)]. That is, time-scaling by1a scales the energy by

1a but does not affect

the power.

Question 3 Sinusoidal test for LTI (Chaparro 2.8)

A fundamental property of linear time-invariant (LTI) systems is that whenever the input of the system is a sinusoid of acertain frequency, the output will also be a sinusoid of the same frequency but with an amplitude and phase determined bythe system. For the following systems, let the input x(t) = cos(t), < t < , and find the output y(t) to determineif the system is LTI.

(a) y(t) = |x(t)|2,(b) y(t) = 0.5[x(t) x(t 1)],(c) y(t) = x(t)1(t), [Note that Chaparro uses u(t) to represent the step function, 1(t). Too bad for him!]

(d) y(t) = 12 tt1 x() d.

Question 4 Testing time-invariance of systems (Chaparro 2.9)

Consider the following systems and find the response, y1(t) and y2(t), to each of the two inputs x1(t) = 1(t) andx2(t) = 1(t 1). Determine from the corresponding outputs whether the system is time-invariant or not.

(a) y(t) = x(t) cos(pit),

(b) y(t) = x(t)[1(t) 1(t 2)],(c) y(t) = 0.5[x(t) x(t 1)].

Plot/sketch y1(t) and y2(t) in each case.