EE 353 Signal and system Problem Set 3

EE 353 Signal and system Problem Set 3.

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Problem 21:

(20 points)

1. Simplify the following expressions:

(a) (4 points) [

δ

(

t

2)

δ

(

t

+ 3) +

u

(

t

2)

δ

(

t

3)

u

(

t

+ 3)]

e

jωt

(b) (4 points)

u

(

t

+ 1)

δ

(1

t

)

e

3

t

3+sin

(

π

3

)

+

e

(3

)

t

δ

(

t

1)

2. Evaluate the following integrals:

(a) (4 points)

−∞

g

(

τ

)

δ

(

t

τ

)

(b) (4 points)

−∞

δ

(2

τ

3)

δ

(

t

+ 1

τ

)

(c) (4 points)

t

0

e

τ

δ

(

τ

1)

Problem 22:

(30 points) We use the symbol

to represent the convolution operation on two signals. Math

ematically

the convolution of two signals,

x

(

t

) and

y

(

t

) is given as:

x

(

t

)

y

(

t

)

−∞

x

(

τ

)

y

(

t

τ

)

1. (16 points) A system with impulse response

h

(

t

) has input

f

(

t

) and zero state response

y

ZSR

(

t

). The system’s

input-output relation is:

y

ZSR

(

t

) =

f

(

t

)

h

(

t

)

(a) (8 points) Using the definition of the convolution given a

bove, prove that the system is linear.

(b) (8 points) Using the definition of the convolution given a

bove, prove that the system is time invariant.

2. (4 points) Show that

f

(

t

)

δ

(

t

T

) =

f

(

t

T

)

.

3. (6 points) If

f

(

t

)

g

(

t

) =

c

(

t

), show the derivative property of convolution

̇

f

(

t

)

g

(

t

) =

f

(

t

)

̇

g

(

t

) = ̇

c

(

t

)

.

4. (4 points) Derive the identity

du

(

t

)

dt

=

δ

(

t

)

,

where

u

(

t

) is the unit-step function. In order to obtain this identity

, you need to show that the functionals

g

(

t

) =

du/dt

and

δ

(

t

) have the same effect on an arbitrary function

f

(

t

), that is

−∞

f

(

t

)

g

(

t

T

)

dt

=

f

(

T

)

where

T

is a real-valued constant parameter.

Problem 23:

(30 points)

Systems can be represented by both an ODE and an impulse respo

nse function. Given either representation, you

can find the zero-state response for a given input. For exampl

e, consider the RC circuit in Figure 1.

f

(

t

)

y

(

t

)

R

2

R

1

C

Figure 1: RC circuit with input voltage

f

(

t

) and output voltage

y

(

t

).

1. (5 points) Derive the ODE representation of the system and

show that it can be expressed as

̇

y

+

1

τ

y

=

K

τ

f.

Express the time constant

τ

and parameter

K

in terms of

R

1

,

R

2

, and

C

. What is the physical significance of

the parameter

K

?

2. (5 points) Solve the ODE in part 1 to determine the zero-sta

te unit-step response.

3. (10 points) Determine the impulse response function

h

(

t

) of the circuit by:

(a) (5 points) Take the derivative of the zero-state unit-st

ep response.

(b) (5 points) Using the method described in Section 2.3 of La

thi.

Note that your results should be identical.

4. (10 points) Determine the zero-state unit-step response

using the convolution integral. Check your answer

against the result obtained in part 2. They should be identic

al.

Problem 24:

(20 points)

Using the graphical convolution method discussed in sectio

n 2.4-2 of the text and lecture, find and sketch

y

(

t

) =

f

(

t

)

h

(

t

) for the following signals.

1. (10 points)

f

(

t

) =

u

(

t

+ 1)

u

(

t

2)

h

(

t

) =

e

t/

5

u

(

t

)

2. (10 points)

f

(

t

) = (

t

+ 2) (

u

(

t

+ 2)

u

(

t

)) + (

t

+ 2) (

u

(

t

)

u

(

t

2))

h

(

t

) =

e

t

u

(

t

)

Problem 25:

(30 points) A future lecture demonstrates that any real-val

ued periodic signal

f

(

t

) with fundamental

period

T

o

may be expresses as a superposition of an infinite number of si

nusoids,

f

(

t

) =

a

o

+

n

=1

a

n

cos(

n ω

o

t

) +

n

=1

b

n

sin(

n ω

o

t

)

,

where

a

0

, a

1

, a

2

, . . . , b

1

, b

2

. . .

are real-valued constant coefficients given by

a

o

=

1

T

o

T

o

f

(

t

)

dt

a

n

=

2

T

o

T

o

f

(

t

) cos

o

tdt

b

n

=

2

T

o

T

o

f

(

t

) sin

o

tdt,

and

ω

o

= 2

π/T

o

. As an example, the coefficients for the periodic waveform sho

wn in Figure 3.10(a) of Lathi are:

a

o

=

1

3

a

n

= (

1)

n

(

2

)

2

b

n

= 0

ω

o

= 2

π/T

o

=

π

. As it not possible to numerically determine

f

(

t

) for an infinite number of terms, consider an

approximation that utilizes the first N terms of the summatio

n,

f

N

(

t

) =

a

o

+

N

n

=1

a

n

cos(

n ω

o

t

) +

N

n

=1

b

n

sin(

n ω

o

t

)

,

If

N <

, then

f

(

t

) =

f

N

(

t

) +

e

(

t

)

,

where

e

(

t

) is the approximation error.

1. (12 points) Write a MATLAB primary function find

fN that determines

f

N

(

t

) given an integer value of

N

and

a time vector. The syntax for the calling the function must be

fN = find

fN(t,N);

Implement the computation of

f

N

using a For-Loop.

Here is a basic example of a Matlab primary function using a fo

r loop:

function y = f_N(N)

y = 0;

for(i = 1:N)

y = y+i;

end

return

In Matlab, one way to create the function above is to save the f

unction as an m-file with the same name as the

function; in this case we would save it as

f_N.m

. This function can then be called either from the command

line or from other scripts. For additional information on pr

imary functions see:

http://matlab.izmiran.ru/help/techdoc/matlab_prog/…

h_func8.html

2. (18 points) Write a Matlab script m-file that:

(a) (9 points) Computes

f

(

t

) over the interval

1

t

1 using the equation

f

(

t

) =

t

2

.

The time vector must consist of 10,001 points equally spaced

between -1 and 1. Plot

f

(

t

) using a dot-dash

black line.

(b) (3 points) Call the function find

fN, using the time vector generated in part (a) and

N

= 1. Using subplots,

plot

f

(

t

), using a dashed red line, and

f

1

(

t

), using a solid blue line, in the upper subplot. Plot the erro

r

e

(

t

), in the lower subplot using a dash-dot black line.

(c) (6 points)Repeat (b) using

N

= 10 and

N

= 100. Notice that, as

N

increases, the approximation error

is reduced. It is possible to reduce the approximation error

to any acceptable level by including enough

terms in the approximation.

To earn full credit for Problem 25:

Include your three figures, your script m-file and your functi

on file with your homework solutions.

Include a comment containing your name, your section number

and the due date at the top of both the script

m-file and function file.

Appropriately label each graph; no credit is given for MATLA

B plots whose axes are unlabeled! An example

is provided in Figure 2.

Use the MATLAB command

gtext

to place your name and section name within the figure.

Time

-1

-0.5

0

0.5

1

Amplitude

-0.5

0

0.5

1

The function f(t) compared to its approximation f

1

(t).

f(t)

f

1

(t)

Time

-1

-0.5

0

0.5

1

Amplitude

-0.1

0

0.1

0.2

0.3

The approximation error, e(t).

Figure 2: Example of the plot comparing

f

(

t

) to

f

1

(

t

)

Problem 26:

(25 points)

Using the relationship

f

(

t

)

δ

(

t

T

) =

f

(

t

T

)

(1)

derived in lecture, evaluation of the convolution integral

y

(

t

) =

f

(

t

)

h

(

t

) =

−∞

f

(

τ

)

h

(

t

τ

)

is simple when either

f

(

t

) or

h

(

t

) is a sum of weighted impulses. This problem extends this res

ult to the case where

either the derivative of either

f

(

t

) or

h

(

t

) yields a sum of weighted impulses.

1. (10 points) As as an example of the utility of equation (1),

suppose that

y

(

t

) =

f

(

t

)

h

(

t

) where

f

(

t

) = 2 (

δ

(

t

+ 3) +

δ

(

t

3))

h

(

t

) =

1

1 +

t

2

.

Determine

y

(

t

), and sketch

f

(

t

),

h

(

t

), and

y

(

t

) on a single plot.

2. (15 points) Now suppose we apply the input

f

(

t

) =

u

(

t

+ 2)

u

(

t

1)

to a LTI system that has the impulse response function

h

(

t

) =

u

(

t

2)

.

Neither

f

(

t

) or

h

(

t

) is expressed directly as a sum of weighted impulses.

(a) (3 points) Find an expression for

dh/dt

in terms of an impulse and sketch

dh/dt

.

(b) (3 points) Let

g

(

t

) denote the response of a system with impulse response

dh/dt

to the input

f

, that is

g

(

t

) =

f

(

t

)

dh

dt

.

Calculate and sketch

g

(

t

).

(c) (3 points) Use the derivative property of covolution to s

how the zero-state response of the system with

impulse response

h

(

t

) to the input

f

(

t

) can be expressed as

y

(

t

) =

t

−∞

g

(

τ

)

dτ.

(d) (6 points) Using the last two results, calculate and sket

ch

y

(

t

).

Problem 27:

(20 points)

When the impulse response function

h

(

t

) is a causal signal, then the system is causal. Conversely, i

f the system

impulse response is noncausal then the system is noncausal.

To illustrate this important concept, consider two LTI

systems that are represented by the impulse response functi

ons

System 1:

h

1

(

t

) =

e

t

u

(

t

)

System 2:

h

2

(

t

) =

e

t

u

(

t

)

.

1. (6 points) Sketch

h

1

(

t

) and

h

2

(

t

), and specify whether or not each impulse response function

is a causal or

noncausal signal.

2. (14 points) Let the input to each system be

f

(

t

) =

e

t/

2

u

(

t

)

(a) (5 points) Use convolution to determine the zero state re

sponse

y

1

(

t

), for the system,

h

1

(

t

).

(b) (5 points) Use convolution to determine the zero state re

sponse

y

2

(

t

), for the system,

h

2

(

t

).

(c) (4 points) Compare the sketch of both

y

1

(

t

) and

y

2

(

t

) to the input

f

(

t

). Determine if the systems

h

1

(

t

)

and

h

2

(

t

) are causal or noncausal. Explain your answers in a clearly w

ritten sentence.

Problem 28:

(20 points)

1. (10 points) Consider two linear time-invariant systems w

hose impulse responses

h

(

t

) are specified as

h

1

(

t

) =

u

(

t

)

h

2

(

t

) =

2

t

(

t

2

+1)

2

Classify each system, corresponding to the impulse functio

ns considered above, as either BIBO stable or not

BIBO stable. In order to receive credit, justify your answer

.

2. (10 points) The system shown in Figure 3 is composed of four

LTI systems whose impulse response functions

are

h

1

(

t

),

h

2

(

t

),

h

3

(

t

), and

h

4

(

t

). The input to the overall system is

f

(

t

), and the output is

y

(

t

). Using the

distributive, commutative, and associative properties of

convolution, represent the composite system by a single

block with an impulse response function

h

(

t

) so that

y

(

t

) =

f

(

t

)

h

(

t

)

,

and express

h

(

t

) in terms of

h

1

(

t

),

h

2

(

t

),

h

3

(

t

), and

h

4

(

t

).

f

(

t

)

h

1

(

t

)

h

2

(

t

)

h

3

(

t

)

h

4

(

t

)

Σ

Σ

y

(

t

)

Figure 3: Block diagram for a system containing subsystems.

EE 353 Signal and system Problem Set 3

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