ee 350 continuous-time linear systems recitation...

Recitation 3.School of Electrical Engineering and Computer Science

Jeffrey Schiano 2015-2017 All rights reserved.

EE 350

Continuous-Time Linear Systems

Recitation 3

1



Problem 1• Consider the following circuit that contains an ideal diode

1. Sketch the output y(t) as a function of the input f(t)2. Is the system zero-state linear or nonlinear?3. Is the system instantaneous (memoryless) or dynamic?

2

R R

f(t)

2V

y(t)



Problem 1 Solution

3



Problem 1 Solution

4



Problem 1 Solution

5



Problem 2• Consider the circuit below with input f(t) and output y(t)

1. Is the system linear or nonlinear? 2. Is the system time-invariant or time-varying?3. Is the system instantaneous (memoryless) or dynamic?4. Is the system causal or noncausal?

6

Rf(t)

y(t)C

v ( )C



Problem 2 Solution

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Problem 2 Solution

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Problem 2 Solution

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Problem 2 Solution

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Problem 2 Solution

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Problem 2 Solution

12



Problem 3• The relationship between the input f(t) and zero-state

response y(t) of two systems are given below. Determine if each system is

1. Zero-state linear or nonlinear2. Time-invariant or time-varying3. Causal or noncausal4. Instantaneous (memoryless) or dynamic

13

| | 2(a) ( ) ( 1)

(b) ( ) ( )e

ty t e f t

y t f t d



Problem 3 Solution

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Problem 3 Solution

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Problem 3 Solution

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Problem 3 Solution

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Problem 3 Solution

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Problem 3 Solution

19



Problem 4• Consider a linear time-invariant (LTI) system with input f(t)

and zero-state response y(t)

1. Can we use f1(t) = 2f(t/2) to find y1(t) in terms of y(t)?2. Sketch the zero-state response y1(t) to the input f1(t)3. Sketch the input f2(t) yielding the zero-state response y2(t)

20

f(t)

t

1

10

y(t)

t

1

1

LTISystem

f(t)y(t)

0

1f (t)

t2

20

2y (t)

t

3

11



Problem 4 Solution

21



Problem 4 Solution

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Problem 4 Solution

23



Complex Number Review• A complex number is a number that can be expressed in

the form a + j b, where a and b are real numbers and jsatisfies the equation j2 = 1

• Notation

• Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part

24

Complex Number:

Real part of Re , Imaginary part of Im

z a jb

z z a z z b



Rectangular and Polar Form• A complex number z can be represented either in

rectangular or polar form

25

z

Re

Im

a

b

Rectangular Form z a jb

z

Re

Im

Polar Form r z

r

2 2

1

cos( ) | |

sin( ) = z = Tan

a r r z a bbb ra



Polar Form and Euler’s Identity

26

Rectangular Form:

cos( ) sin( )

r cos( ) sin( )

Polar Form: j

z a jb

z r jr

z j

z re

Euler's Identity: cos( ) sin( )je j



Operations on Complex Numbers• Use rectangular form for addition and subtraction

• Use polar form for multiplication and division

• The complex conjugate of the complex number z = a + jb is defined as z* = a – jb

27

1 2

1 2

1 2

If and , then z ( ) ( ) z ( ) ( )

z a jb z c jdz a c j b dz a c j b d

1 2

1 2 1 2

11 2

2

1 1 2 2

( )1 2 1 2 1 2

( )1 1 1

2 2 2

If e and e , then z e e

e e

j j

j j j

jj

j

z r z r

z r r r r e

z r r ez r r



Problem 5

• Given z = a + jb = r ej show that1. z* = r e-j

2. z + z* = 2a = 2Re{z}3. z z* = 2jb = j2Im{z}4. z z* = a2 + b2 = r2

28



Problem 5 Solution

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Entering Complex Numbersin MATLAB

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• The complex number z = 2 + j3 may be entered into MATLAB as

>> z = 2 + 3*i>> z = 2 + 3i

>> z = 2 + 3*j>> z = 2 + 3j

>> z = 2 + 3*1i



Complex Functions in MATLAB

31

Operation Functionmagnitude (r) of z abs(z)angle (q) of z in radians angle(z)construct z from a and b complex(a,b)conjugate of z conj(z)imaginary part of z imag(z)real part of z real(z)



Problem 6• Use MATLAB to determine

• Explain the difference between the results in parts 3 and 4

32

1 21. The rectangular form of 1 2

2 2 32. The polar form of

1 3

3. Execute the MATLAB command >>exp(pi/2*i)

4. Execute the MATLAB command >>exp(pi/2i)

jj

j j

j j



Problem 6 Solution

33



EE 350

Continuous-Time Linear Systems

Recitation 3

1



Problem 1• Consider the following circuit that contains an ideal diode

1. Sketch the output y(t) as a function of the input f(t)2. Is the system zero-state linear or nonlinear?3. Is the system instantaneous (memoryless) or dynamic?

2



Problem 1 Solution

3



Problem 1 Solution

4



Problem 1 Solution

5



Problem 2• Consider the circuit below with input f(t) and output y(t)

1. Is the system linear or nonlinear? 2. Is the system time-invariant or time-varying?3. Is the system instantaneous (memoryless) or dynamic?4. Is the system causal or noncausal?

6



Problem 2 Solution

7



Problem 2 Solution

8



Problem 2 Solution

9



Problem 2 Solution

10



Problem 2 Solution

11



Problem 2 Solution

12



Problem 3• The relationship between the input f(t) and zero-state

response y(t) of two systems are given below. Determine if each system is

1. Zero-state linear or nonlinear2. Time-invariant or time-varying3. Causal or noncausal4. Instantaneous (memoryless) or dynamic

13



Problem 3 Solution

14



Problem 3 Solution

15



Problem 3 Solution

16



Problem 3 Solution

17



Problem 3 Solution

18



Problem 3 Solution

19



Problem 4• Consider a linear time-invariant (LTI) system with input f(t)

and zero-state response y(t)

1. Can we use f1(t) = 2f(t/2) to find y1(t) in terms of y(t)?2. Sketch the zero-state response y1(t) to the input f1(t)3. Sketch the input f2(t) yielding the zero-state response y2(t)

20



Problem 4 Solution

21



Problem 4 Solution

22



Problem 4 Solution

23



Complex Number Review• A complex number is a number that can be expressed in

the form a + j b, where a and b are real numbers and jsatisfies the equation j2 = 1

• Notation

• Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part

24



Rectangular and Polar Form• A complex number z can be represented either in

rectangular or polar form

25



Polar Form and Euler’s Identity

26



Operations on Complex Numbers• Use rectangular form for addition and subtraction

• Use polar form for multiplication and division

• The complex conjugate of the complex number z = a + jb is defined as z* = a – jb

27



Problem 5

• Given z = a + jb = r ej show that1. z* = r e-j

2. z + z* = 2a = 2Re{z}3. z z* = 2jb = j2Im{z}4. z z* = a2 + b2 = r2

28



Problem 5 Solution

29



Entering Complex Numbersin MATLAB

30

• The complex number z = 2 + j3 may be entered into MATLAB as



Complex Functions in MATLAB

31

Operation Functionmagnitude (r) of z abs(z)angle (q) of z in radians angle(z)construct z from a and b complex(a,b)conjugate of z conj(z)imaginary part of z imag(z)real part of z real(z)



Problem 6• Use MATLAB to determine

• Explain the difference between the results in parts 3 and 4

32



Problem 6 Solution

33

ee 350 continuous-time linear systems recitation...

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