ee 350 continuous-time linear systems recitation...

Recitation 3.School of Electrical Engineering and Computer Science

EE 350

Continuous-Time Linear Systems

Recitation 3

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?

Problem 1 Solution

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?

v ( )C

Problem 2 Solution

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

| | 2(a) ( ) ( 1)

(b) ( ) ( )e

ty t e f t

y t f t d

Problem 3 Solution

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)

LTISystem

f(t)y(t)

1f (t)

2y (t)

Problem 4 Solution

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

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

Rectangular Form z a jb

Polar Form r z

cos( ) | |

sin( ) = z = Tan

a r r z a bbb ra

Polar Form and Euler’s Identity

Rectangular Form:

cos( ) sin( )

r cos( ) sin( )

Polar Form: j

z a jb

z r jr

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

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 1 2 2

( )1 2 1 2 1 2

( )1 1 1

If e and e , then z e e

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

Problem 5 Solution

Entering Complex Numbersin MATLAB

• 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

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

1 21. The rectangular form of 1 2

2 2 32. The polar form of

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

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

Problem 6 Solution

EE 350

Continuous-Time Linear Systems

Recitation 3

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?

Problem 1 Solution

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?

Problem 2 Solution

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

Problem 3 Solution

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)

Problem 4 Solution

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

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

rectangular or polar form

Polar Form and Euler’s Identity

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

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

Problem 5 Solution

Entering Complex Numbersin MATLAB

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

Complex Functions in MATLAB

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

Problem 6 Solution

ee 350 continuous-time linear systems recitation...

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