Signals and System Homework #9 - Solutions

Problems:

1) For each of the systems be low

band pass filter

stop band filter

all pass filter

2) Text 9.45

(a) ROC for X(s) is Re{s}<2 because x(t)=0, t>0.

ROC could be Re{s}<-2, -2<Re{s}<-1, -1<Re{s}, empty set

ROC_Y = ROCX ∩ ROC_H ∴ROC_H is Re{s} >-1.

(b) So

(c) x(t) is an eigenfunction of the LTI system,

so from page 183, Eq 3.5,

because x(t)= e^3tu(t)+e^3tu(-t), the transform will be 1/(s-3) + -1/(s-3) with different

ROC’s which isn’t functional.

1) Text 9.61 (Note you will do problem 10.62 next week)

or with a change of variables

so

change variables

or with another change of variables: h(t) = x(-t)

-or- look at it graphically

Autocorrelation:

Convolution:

(b) L{x(t)} = X(s)

L{x(-t)} is not in Table 9.1. But, using Equation 9.3

Do a change of variables

so

2) Text 10.23 a, b

so

Long division results in

(see how many times 1 goes into 1, 1 goes into –z ^-1…)

so x[n] = 0,0,…,1, -1, 1/4, -1/4, 1/16, -1/16…} with the first non- zero term located at

n=0.

Also note (optional) that

So using the Geometric series,

same function as above, but different ROC .



long division to get positive exp onent series (see how many times –1/4z ^-2 goes into –z ^-1)