EE 350 Continuous-Time Linear Systems Recitation 7

Recitation 7.School of Electrical Engineering and Computer Science

Jeffrey Schiano 2015-2017 All rights reserved.

EE 350

Continuous-Time Linear Systems

Recitation 7

1



Recitation 7 Topics

• Solved Problems – Convolution– EE 210 Review: Sinusoidal Steady-State Analysis– Relationship between the zero-state response and the

sinusoidal steady-state response

• MATLAB Exercises– Approximation of the convolution integral

2



Problem 1

• A LTI system with impulse response h(t) is driven by the input f(t)

1. Determine the zero-state response y(t) using graphical convolution

2. Verify that the width property holds for part 1

3

( ) ( ) ( 2)

( ) 2u(t) 2u(t 2)

h t u t u t

f t



Problem 1 Solution

4



Problem 1 Solution

5



Problem 1 Solution

6



Problem 1 Solution

7



Problem 1 Solution

8



Numerical Approximation of the Convolution Integral

• The zero-state response y(t) of a casual LTI system with impulse response h(t) to the causal input f(t) is

• As discussed in lecture, y(t) represents the area under the integrand, and can be approximated by the sum of area under rectangles of width T and height f(kT) h(t-kT) using the Riemann sum

9

0( ) ( ) ( )

ty t f h t d

0

(t) (kT) (t )n

k

y f h kT T



MATLAB conv Function• From the Riemann sum, the zero-state response at t = nT is

• The built-in MATLAB function conv computes the summation

• The syntax of the function conv is

where s1 is a vector of length N1, s2 is a vector of length N2, and the length of the returned vector is N1+N2-1

10

0

(nT) (kT) (nT )n

k

y T f h kT

c = conv(s1, s2)



Operation of the conv Function• For example, given the vectors

the function y = conv(f,h) returns the vector

where

11

[ (0), ( ), (2 )]

[ (0),h(T),h(2T)]

f f f T f T

h h

[ (0), ( ) y(2T), y(3 ), (4 )]y y y T T y T

0

0

1

0

(0) ( ) (0 - ) (0) (0),

( ) ( ) ( - ) (0) ( ) ( ) (0),

k

k

y f kT h kT f h

y T f kT h T kT f h T f T h



Problem 2• As in Problem 1, consider a LTI system with

impulse response h(t) is driven by the input f(t)

• Write an m-file that1. Uses the conv function to approximate y(t) from

t = 0 to t = 4 using T = 0.001 2. Plots the approximate zero-state response y(t)

• Does the numerical result approximate the result obtained in Problem 1?

12

( ) ( ) ( 2)

( ) 2u(t) 2u(t 2)

h t u t u t

f t



Problem 2 m-file

13



Problem 2 Numerical Results

14

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5EE 350 Recitation 7 Problem 2

y(t)

t



Problem 3


• Determine the zero-state response y(t) using graphical convolution

15

| | 2( )

h( ) u(t 2) u(t)

tf t e

t



Problem 3 Solution

16



Problem 3 Solution

17



Problem 3 Solution

18



Problem 3 Solution

19



• Consider a LTI system with impulse response

• Determine if the system is BIBO stable

20

5( ) 2 2 ( )th t e u t

Problem 4



Problem 4 Solution

21



Problem 5• It is known that a certain physical system with input f(t)

and output y(t) is represented by an ODE of the form

• In order to determine the unknown parameters α and β, the an engineer applies a unit-step input and observes the zero-state response. Using this response, the engineer determines that the impulse response of the system is

• Determine the value of the parameters α and β

22

( ) ( )y y t f t

4( ) 3 ( )th t e u t



Problem 5 Solution

23



Problem 5 Solution

24



Problem 5 Solution

25



Problem 6• The sinusoidal signal

may be represented by the phasor

• The phasor is a complex-valued constant

• Using Euler’s identity, show that

26

( ) cosf t A t

jF Ae

( ) Re j tf t Fe



Problem 6 Solution

27



Problem 7

• Determine the phasor representation of the following signals

28

(a) ( ) 3sin 10 30

(b) ( ) 2cos 90

f t t

f t t



Problem 7 Solution

29



Problem 8• Consider the RC network with input f(t) and output y(t)

1. Determine the zero-state response for the input

2. Determine the zero-state response for times much larger than the network RC time constant

3. Derive the frequency response function, and use it to determine the sinusoidal steady-state response

4. Compare the results from parts 2 and 330

f(t)R

y(t)C

( ) cos of t A t



Problem 8 Solution

31



Problem 8 Solution

32



Problem 8 Solution

33



Problem 8 Solution

34



Problem 8 Solution

35



EE 350

Continuous-Time Linear Systems

Recitation 7

1



Recitation 7 Topics

• Solved Problems – Convolution– EE 210 Review: Sinusoidal Steady-State Analysis– Relationship between the zero-state response and the

sinusoidal steady-state response

• MATLAB Exercises– Approximation of the convolution integral

2



Problem 1


1. Determine the zero-state response y(t) using graphical convolution

2. Verify that the width property holds for part 1

3



Problem 1 Solution

4



Problem 1 Solution

5



Problem 1 Solution

6



Problem 1 Solution

7



Problem 1 Solution

8



Numerical Approximation of the Convolution Integral

• The zero-state response y(t) of a casual LTI system with impulse response h(t) to the causal input f(t) is

• As discussed in lecture, y(t) represents the area under the integrand, and can be approximated by the sum of area under rectangles of width T and height f(kT) h(t-kT) using the Riemann sum

9



MATLAB conv Function• From the Riemann sum, the zero-state response at t = nT is

• The built-in MATLAB function conv computes the summation

• The syntax of the function conv is

where s1 is a vector of length N1, s2 is a vector of length N2, and the length of the returned vector is N1+N2-1

10



Operation of the conv Function• For example, given the vectors

the function y = conv(f,h) returns the vector

where

11



Problem 2• As in Problem 1, consider a LTI system with

impulse response h(t) is driven by the input f(t)

• Write an m-file that1. Uses the conv function to approximate y(t) from

t = 0 to t = 4 using T = 0.001 2. Plots the approximate zero-state response y(t)

• Does the numerical result approximate the result obtained in Problem 1?

12



Problem 2 m-file

13



Problem 2 Numerical Results

14



Problem 3


• Determine the zero-state response y(t) using graphical convolution

15



Problem 3 Solution

16



Problem 3 Solution

17



Problem 3 Solution

18



Problem 3 Solution

19



• Consider a LTI system with impulse response

• Determine if the system is BIBO stable

20

Problem 4



Problem 4 Solution

21



Problem 5• It is known that a certain physical system with input f(t)

and output y(t) is represented by an ODE of the form

• In order to determine the unknown parameters α and β, the an engineer applies a unit-step input and observes the zero-state response. Using this response, the engineer determines that the impulse response of the system is

• Determine the value of the parameters α and β

22



Problem 5 Solution

23



Problem 5 Solution

24



Problem 5 Solution

25



Problem 6• The sinusoidal signal

may be represented by the phasor

• The phasor is a complex-valued constant

• Using Euler’s identity, show that

26



Problem 6 Solution

27



Problem 7

• Determine the phasor representation of the following signals

28