Solution 7 August 2015

ECE301 Signals and Systems: Final Exam

Cover Sheet

Test Duration: 120 minutes

Coverage: Chap. 1, 2, 3, 4, 5, 7

One 8.5" x 11" crib sheet is allowed.

Calculators, textbooks, notes are not allowed.

Fourier Transform Tables are attached at the end.

All work should be done on the provided sheets.

You must show work or explain answer for each problem to receive full credit.

Problem No. Topics Points

1 Linear and Time-Invariant system 20

2 Fourier Series Representation 25

3 Discrete-Time Fourier Transform 30

4 Sampling Theory 25

Prob. 1. [20 pts] Consider the cascade interconnection of three causal LTI systems as below:

The impulse response is

and the impulse response of the overall system is shown below:

(The values are 0 when and )

(a) Find the impulse resposne of .

(Answer)

where

Now since

we have the impulse response of the overall system,

Pluggin in the values of h[n] on the given plot,

Since each system is causal, the overall system is also causal meaning that we have

Solving for the seven equations,

(b) Find the response of the overall system to the input

(Answer)

When the input is

the output is

Since we have

where indicates that it is the value when n=0.

Your answer can be either in a sequence form, a plot, or an equation.

Prob. 2. [25 pts] Let

be a periodic signal with fundamental period and Fourier coefficients .

(a) Determine the value of .

(Answer)

(b) Determine the Fourier series representation of

.

(Answer)

Let

, then

with period T=2.

Then, let the coefficients of .

where are defined above.

(c) Use the result of part (b) and the differentiation property of the continuous-time Fourier series to

determine the Fourier series (FS) coefficients of .

(Hint: If has FS coefficients , then

has FS coefficients

.)

(Answer)

Since

has FS coefficients ,

Solving the equation,

Prob. 3. [30 pts] The signal below is defined for all parts of this problem.

For each sub-part, answer the three questions (i),(ii), and (iii).

(a) For

,

(i) Compute the energy

. Show work below.

CTFT of x(t), , is a rectangle.

From Parseval's Theorem,

(ii) What is the Nyquist sampling rate (in angular frequency, w) for x ?

(iii) Define . Plot the magnitude of the DTFT of , where

. Plot over

and properly label the values on the plot.

is mapped to

in DT frequency.

(b) For

(i) Compute the energy

. Show work below.

So the energy is same as

(ii) What is the Nyquist sampling rate (in angular frequency, w) for y ?

(iii) Define . Plot the magnitude of the DTFT of , where

. Plot over

and properly label the values on the plot.

Similarly as in (a), is mapped to

in DT frequency.

(c) For

(i) Compute the energy

. Show work below.

Using the differentiation property of CTFT,

(ii) What is the Nyquist sampling rate (in angular frequency, w) for y ?

(iii) Define . Plot the magnitude of the DTFT of , where

. Plot over

and properly label the values on the plot.

is mapped to

in DT frequency.

(d) For

(i) Compute the energy

. Show work below.

From Parseval's Theorem,

(ii) What is the Nyquist sampling rate (in angular frequency, w) for y ?

(iii) Define . Plot the magnitude of the DTFT of , where

. Plot over

and properly label the values on the plot.

is mapped to

in DT frequency.

(e) For

(i) Compute the energy

. Show work below.

From Parseval's Theorem,

(ii) What is the Nyquist sampling rate (in angular frequency, w) for y ?

(iii) Define . Plot the magnitude of the DTFT of , where

. Plot over

and properly label the values on the plot.

Since , aliasing exists.

is mapped to

and is mapped to

.

Prob. 4. [25 pts]

The signal

is sampled every

to form . Then the

discrete-time (DT) signal is input to the DT system which has impulse response as below:

Note that is a constant real number.

(Answer)

(a) Check if there is an aliasing when sampling the signal.

Plot to find the maximum frequency

Since , there is an aliasing.

(b) Plot where is the DTFT of .

There are overlaps between triangles so can be plotted as below:

Using the relationship where

,

(For example, is mapped to

)

(c) Plot where Y is the DTFT of and y[n] is the output of x[n] through the DT system.

Since

Then since .

(d) Determine

Using Parseval's Theorem,