ee 350 continuous-time linear systems recitation...

Recitation 2.School of Electrical Engineering and Computer Science

Jeffrey Schiano 2015-2017 All rights reserved.

EE 350

Continuous-Time Linear Systems

Recitation 2

1



Recitation 2 Topics

• MATLAB Programming– Vector Manipulation– Built-in Housekeeping Functions

• Solved Problems– Classification of Signals– Basic Signal Operations

2



Vector Manipulation• Elements are accessed by specifying an index ranging from

1 to the number of elements in the vector

• Display elements 2 through 3 of vector

• Delete elements 2 through 3

3

>> x = [2, 4, 6, 8];

>> x(2:3)ans =

4 6

>> x = [2, 4, 6, 8];

>> x(2:3) = []ans =

2 8



Vector Manipulation • Display the last two elements of the vector x

• Display every other element starting with the first element of x

4

>> x = [2, 4, 6, 8];

>> x(end-1 : end)ans =

6 8

>> x = [2, 4, 6, 8];

>> x(1 : 2 : end) ans =

2 6



Vector Functions in MATLAB

5

Operation Functionlength of vector x; returns the number of elements length(x)

size of vector x; returns the number of rows and columns size(x)transpose of x x’largest element in x max(x)smallest element in x min(x)sum of elements in vector x sum(x)indices of nonzero elements find(x)indices of elements in a specified range find(x>3)first n indices corresponding to the nonzero elements in x find(x, n)Last n indices corresponding to the nonzero elements in x find(x, n, ‘last’)



Problem 1• For the vector x = -4 : 2 : 10, use MATLAB to determine

1. length(x)2. size(x)3. x’4. size(x’)5. min(x)6. max(x)7. sum(x)8. find(x)9. find(x < 0)10. find(x,3)11. find(x,3,’last’)

6



Housekeeping Functions

7

Operation Functionlists variables currently in the workspace wholists variables currently in the workspace with their size whosclear all variables from the workspace clearclear command window, cursor moves to the top clcclear the current figure window clfclose the current figure window closeclose all the figure windows close all



Problem 2

• The response of system to a unit-step input is often characterized by the rise-time, which is the time for the output to rise from 10% to 90% of its final value

• Write an m-file that determines the rise-time of

8

( ) 1 0ty t e t



Problem 2• The m-file must

– Clear the workspace and command window, and close open figures

– Generate a time vector t of 1000 points uniformly spaced between 0 and 10

– Generate a vector y whose elements represent the values of the response at the time instants in t

– Numerically determine the rise-time using find

– Plot y versus t, and in the legend, specify the rise-time use the strcat and num2str functions

9



Problem 2

10



Problem 2

11



Problem 3• Consider the signal

1. Sketch f(t)2. Classify f(t) as a causal or noncausal signal3. Is f(t) either an even or odd function?4. Is f(t) periodic or aperiodic?5. Is f(t) an energy signal? If so, determine the energy

metric Ef

6. What is Pf, the power of the signal f(t)?

12

( ) ( )tf t e u t



Problem 3 Solution

13



Problem 3 Solution

14



Problem 3 Solution

15




1. Sketch f(t)2. Classify f(t) as a causal or noncausal signal3. Is f(t) either an even or odd function?4. Is f(t) periodic or aperiodic? If periodic, what is the

fundamental period?5. Is f(t) an energy signal? If so, determine the energy metric Ef

6. Is f(t) a power signal? If so, determine the power metric Pf

16

( ) 5 ( 10 2) u(t 10 2)n

n

f t u t n n



Problem 4 Solution

17



Problem 4 Solution

18



Problem 4 Solution

19



Problem 5• Express the signal f(t) as the sum of even and odd functions

1. Verify, for an arbitrary f(t), that the functions fe(t) and fo(t) are even and odd functions, respectively

2. Determine expressions for fe(t) and fo(t) for the given f(t)3. Sketch f(t), fe(t) and fo(t)

20

-( ) ( ) ( ) ( )

where

( ) ( ) ( ) ( )( ) and ( )2 2

te o

e o

f t e u t f t f t

f t f t f t f tf t f t



Problem 5 Solution

21



Problem 5 Solution

22



Problem 5 Solution

23



Problem 6• Consider the signal shown below

1. Find an expression for f(t)2. Sketch the following signals: f(t-1), 2f(t+1), f(t/3), f(-t), f(1-t)

24

( )f t

t101

1



Problem 6 Solution

25



Problem 6 Solution

26



Problem 6 Solution

27



EE 350

Continuous-Time Linear Systems

Recitation 2

1



Recitation 2 Topics

• MATLAB Programming– Vector Manipulation– Built-in Housekeeping Functions

• Solved Problems– Classification of Signals– Basic Signal Operations

2



Vector Manipulation• Elements are accessed by specifying an index ranging from

1 to the number of elements in the vector

• Display elements 2 through 3 of vector

• Delete elements 2 through 3

3



Vector Manipulation • Display the last two elements of the vector x

• Display every other element starting with the first element of x

4



Vector Functions in MATLAB

5

Operation Functionlength of vector x; returns the number of elements length(x)

size of vector x; returns the number of rows and columns size(x)transpose of x x’largest element in x max(x)smallest element in x min(x)sum of elements in vector x sum(x)indices of nonzero elements find(x)indices of elements in a specified range find(x>3)first n indices corresponding to the nonzero elements in x find(x, n)Last n indices corresponding to the nonzero elements in x find(x, n, ‘last’)



Problem 1• For the vector x = -4 : 2 : 10, use MATLAB to determine

1. length(x)2. size(x)3. x’4. size(x’)5. min(x)6. max(x)7. sum(x)8. find(x)9. find(x < 0)10. find(x,3)11. find(x,3,’last’)

6



Housekeeping Functions

7

Operation Functionlists variables currently in the workspace wholists variables currently in the workspace with their size whosclear all variables from the workspace clearclear command window, cursor moves to the top clcclear the current figure window clfclose the current figure window closeclose all the figure windows close all



Problem 2

• The response of system to a unit-step input is often characterized by the rise-time, which is the time for the output to rise from 10% to 90% of its final value

• Write an m-file that determines the rise-time of

8



Problem 2• The m-file must

– Clear the workspace and command window, and close open figures

– Generate a time vector t of 1000 points uniformly spaced between 0 and 10

– Generate a vector y whose elements represent the values of the response at the time instants in t

– Numerically determine the rise-time using find

– Plot y versus t, and in the legend, specify the rise-time use the strcat and num2str functions

9



Problem 2

10



Problem 2

11




1. Sketch f(t)2. Classify f(t) as a causal or noncausal signal3. Is f(t) either an even or odd function?4. Is f(t) periodic or aperiodic?5. Is f(t) an energy signal? If so, determine the energy

metric Ef

6. What is Pf, the power of the signal f(t)?

12



Problem 3 Solution

13



Problem 3 Solution

14



Problem 3 Solution

15




1. Sketch f(t)2. Classify f(t) as a causal or noncausal signal3. Is f(t) either an even or odd function?4. Is f(t) periodic or aperiodic? If periodic, what is the

fundamental period?5. Is f(t) an energy signal? If so, determine the energy metric Ef

6. Is f(t) a power signal? If so, determine the power metric Pf

16



Problem 4 Solution

17



Problem 4 Solution

18



Problem 4 Solution

19



Problem 5• Express the signal f(t) as the sum of even and odd functions

1. Verify, for an arbitrary f(t), that the functions fe(t) and fo(t) are even and odd functions, respectively

2. Determine expressions for fe(t) and fo(t) for the given f(t)3. Sketch f(t), fe(t) and fo(t)

20



Problem 5 Solution

21



Problem 5 Solution

22



Problem 5 Solution

23



Problem 6• Consider the signal shown below

1. Find an expression for f(t)2. Sketch the following signals: f(t-1), 2f(t+1), f(t/3), f(-t), f(1-t)

24



Problem 6 Solution

25



Problem 6 Solution

26



Problem 6 Solution

27

ee 350 continuous-time linear systems recitation...

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