{ECE 120: Signals

and Spectra

Summer A.Y. 2014-2015

Exam 70 %

Quizzes 15

HW/SW 10

Attendance 5

Total 100 %

Grading system

Passing Rate: 60%

1. Signals and Systems

2. Continuous-Time Signals and Systems

LTI Systems

Frequency Analysis

Application

3. Discrete-Time Signals and Systems

COURSE OUTLINE

Signal and its Classification

Operations on Signals

Basic Continuous Signals

Basic Discrete Signals

System and Classification of Systems

Signals and Systems

a function representing a physical quantity or variable

contains/conveys information

represented as a function of an independent variable (e.g. t)

Ex:

Speech is represented by acoustic pressure as a function of time

Signal

1. Continuous-Time vs. Discrete Time

Continuous-time independent variable is continuous

signal itself need not be continuous

Classification of Signals

t : continuous-time variable

Discrete-time independent variable takes on only a discrete set of values

defined at discrete instant of time

can be obtained by sampling continuous-time signals

can be defined in two ways:

n : discrete-time variable

2. list the values of the sequence1. specify a rule for calculating the nth value of the sequence

2. Analog and Digital Signals:

Analog

a continuous-time signal that can take on any value in the continuous interval (a, b), where a may be - and b may be +

Digital

a discrete-time signal that can take on only a finite number of distinct values

discrete in time and quantized in amplitude

3. Real and Complex Signals:

Real

value is a real number

Complex

value is a complex number.

general complex signal is a function of the form

x(t) = x1(t) + jx2(t)

5. Deterministic vs Random

- according to the predictability of their behavior

Deterministic

values are completely specified for any given time

can be modeled or described by a known function of time t

Random

takes random values at any given time and must be characterized statistically

6. Causal vs Anti-Causal vs Non-causal

Causal

zero for all negative time

Anti-Causal

zero for all positive value of time

Non-Causal

non zero values in both positive and negative time

7. Even vs Odd

- according to the symmetry with respect to the time origin

Even

x(t) = x(-t)

x[n] = x[-n]

Odd

x(t) = - x(-t)

x[n] = - x[-n]

Graph is unchanged after 180rotation

7. Even vs Odd

Any signal () is representable as a sum of an even component ()and an odd component ()

= + ()

where

=1

2[ + ]

=1

2[ ]

Example:

Determine the even and odd components of the signal = 2 + 2.

7. Even vs Odd

Example: Determine the even and odd components of the signal = 2 + 2.

For the even component:

=1

2 +

=1

22 + 2 + 2 + 2

=1

22 + 2 + 2 + + 2 =

1

222 + 4 ]

= 2 + 2

For the odd component:

=1

2

=1

22 + 2 2 + 2

=1

22 + 2 2 + + 2 =

1

2[2]

=

Eulers Identity

= +

Proof: Let = +

So

= + = + =

=

=

=

=

*Review

From = + and =

+ = 2

=+

2

and

= 2

=

2

8. Periodic vs Aperiodic

- according to whether the signals exhibit repetitive behavior or not

Periodic

CT

defined for all possible values of t, - < t < +, and

there is a positive real value T, the period of x(t), such that

x(t + mT) = x(t), for any integer m.

DT

x[n + T] = x[n] or x[n + mT] = x[n]

Aperiodic

Classify:

= 7

Continuous or piecewise continuous?

Even or odd?

Deterministic or random?

Causal, anti-causal, non-causal?

Periodic or not?

Example

1. =

Exercise

Classify:

=

Continuous or piecewise continuous?

Even or odd?

Deterministic or random?

Causal, anti-causal, non-causal?

Periodic or not?

Example

1. =

Exercise

1. Amplitude Scaling

Ax(t) , Ax[n]

multiply the entire signal by a constant value A

operations

2. Time Reversal

x(-t) , x[-n]

x(-t) signal is obtained by reflecting original signal x(t) about t = 0 (n = 0 for discrete)

3. Time Shifting

x(t-t0) , x[n-n0]

x(t-t0) time shifted version of x(t)

t0 > 1 : shifted right (time delayed)

t0 < 1 : shifted left (time advanced)

4. Time Scaling

compresses (speeds up) or dilates (slows down) a signal by multiplying the time variable by some quantity

x(at) , x[an]

|a| > 1 : compress

0 < |a| < 1 : dilate

5. Addition

6. Multiplication

Example

1. : Determine:

a. x(t + 3)b. x(t/2 2)c. x(1 2t)d. 4x(t)

1. Unit Step Function

a.k.a Heaviside unit function

defined as

2. Ramp Function

defined as

Basic ct Signals

3. Unit Impulse Function

a.k.a Dirac delta function

zero everywhere except at zero

defined as

and which is also constrained to satisfy the identity

properties:

4. Complex Exponential

= 0

By Eulers Theorem

= 0 = cos 0 + 0 (periodic)

General Complex Exponential Signals

Let s = + j0 be a complex number

can be rewritten as

= = (+0) = cos 0 + 0

5. Sinusoidal Signals

= cos 0 +

where:

A = amplitude

0 = radian frequency

= phase angle (radians)

Periodic at

0 =2

0(fundamental period)

cos 0 + = Re{ 0+ }

Im 0+ = sin 0 +

1. Unit Step Sequence

defined as

shifted unit step sequence u[n - k]

Basic Dt Signals

*defined at n = 0 (unlike that of CT)

2. Unit Ramp Sequence

value of unit ramp sequence increases linearly with the sample number n

Denoted by [] and is defined by

Ex: unit ramp sequence with four samples:

[] = {0,1,2,3,4}

3. Unit Impulse Sequence

unit impulse (or unit sample) sequence

[n] is defined as

shifted unit impulse sequence [n k]

properties

4. Complex Exponential

[] = 0

With Eulers Formula,

Periodicity:

In order for 0 to be periodic with period N ( > 0 ) ,

N and m should have no factors in common

A very important distinction between CT and DT complex exponentials:

CT: 0 - all distinct for distinct values 0DT: 0 - identical signals for values of 0

separated by 2

Discrete-time sinusoid with = /3 (what is the period if periodic?)

Discrete-time sinusoid with = 1 (what is the period if periodic?)

Discrete-time sinusoid with = 1/6 (what is the period if periodic?)

General Complex Exponential Sequences

defined as =

where C and are general complex numbers

4 special cases of

5. Sinusoidal Signals

[] = cos 0 + cos 0 + = Re{ 0+ }

Im{ 0+ } = sin 0 +

1. Classify = ( + 2) as Continuous or piecewise continuous?

Even or odd?

Deterministic or random?

Causal, anti-causal, non-causal?

Periodic or not?

2. Express the signal in the figure in terms of step functions.

Quiz

3. Given x(t):

Determine:

a. x(2t + 2)

b. x(2 t/3)

c. x(1 t)

4. Determine the even and odd components of 0 +

4.

5. Determine whether or not each of the following signals is periodic. If a signal is periodic, determine its fundamental period:

a. = 2

b. [] = (1)

3)

-4 -3 -2 -1 0 1 2 3 4-2

-1

0

1

2

3

4

t

heaviside(t - 1) -...+ (heaviside(t - 2) - heaviside(t - 3)) (t - 3)

Answer

-4 -3 -2 -1 0 1 2 3 4-2

-1

0

1

2

3

4

t

heaviside(t + 1) +...- (2 t + 3) (heaviside(t + 1) - heaviside(t + 3/2))

0

0

3a

3b

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-2

-1

0

1

2

3

4

t

heaviside(3 - t) -...+ (t/3 - 3) (heaviside(6 - t) - heaviside(9 - t))

0

0

3c

-4 -3 -2 -1 0 1 2 3 4-2

-1

0

1

2

3

4

t

heaviside(-t) -...- (t + 2) (heaviside(- t - 1) - heaviside(- t - 2))

0

0

http://iitg.vlab.co.in/?sub=59&brch=166&sim=618&cnt=1

http://en.wikipedia.org/wiki/Dirac_delta_function#Definitions

http://hitoshi.berkeley.edu/221a/delta.pdf

Schaums Outlines: Signals and Systems by Hwei P. Hsu

Signals and Systems by Oppenheim

References