COSC 3451: Signals and Systems

Course Instructor: Amir Asif Teaching Assistant: TBA

Contact Information: Instructor: Teaching Assistant:Office: CSB 3028 Durwas, Nehaasif@cse.yorku.ca cs231070@cse.yorku.ca(416) 736-2100 X70128

URL: http://www.cs.yorku.ca/course/3451

Text: A. V. Oppenheim and A. S. Willsky with S. H. Nawab, Signals andSystems, NY: Prentice Hall, 1997.

Class Schedule: TR 13:00 − 14:30 (CB 115)

Assessment: Assignment / Quiz: 20% Projects: 15%Mid-term: 25%Final: 40%

Office Hours: Instructor: CSB 3028, TR 12:00 – 13:00 TA: TBA

2

Course Objectives

Introduce CT and DT signals and systems in terms of physical phenomena

Review elementary signals and see how these can be represented in domains other than time

Understand different transforms (Fourier, Laplace, Z) used to analyze signals

Complete analysis of a linear time invariant (LTI) systems in time and other domains

Design filters (systems) to process a signal for different applications

Present state-of-art technologies used in communication systems

Use computers for digital signal processing

Demystify terminology !!!!!!

3

What is a Signal?

Signal is a waveform that contains information

System is a model for physical phenomena that generates, processes, or receive signals.

Speech Waveformx(t): Continuous Time

4

Examples of Signals

ImageI[m,n]: 2D DT signal

(a) 1D CT signal x(t)(b) 1D DT signal x[n]

Activity 1: For each of the representations: (a) z[m,n,k] (b) I(x,y,z,t), establish if the signal is CT or DT. Specify the independent and dependent variables. Also, think of an example from the real world that will have the same mathematical representation.

5

Power vs. Energy Signals (1)

Energy:

Power:

Activity 2:a. Consider the sinusoidal signal x(t) = cos(0.5πt). Choosing T = 4, determine the average power of x(t).b. Consider the signal x(t) = 5 sin(2πt) for the interval −1 <= t <= 1 and is 0 elsewhere. Calculate the energy

of x(t).c. Calculate the energy of the signal x[n] = (0.8)n for n >= 0 and is 0 elsewhere.

⎪⎪⎩

⎪⎪⎨

⎧

=

∑

∫∞

∞−

∞

∞−∞

Signals DTfor |][|

Signals CTfor |)(|

2

2

nx

dttxE

⎪⎪⎩

⎪⎪⎨

⎧

+∞→

∞→=

∑

∫

−

−∞

Signals DTfor |][|12

1lim

Signals CTfor |)(|21lim

2

2

N

N

T

T

nxNN

dttxTT

P

6

Power vs. Energy Signals (2)

1. Energy Signals: have finite total energy for the entire duration of the signal. As a consequence, total power in an energy signal is 0.

2. Power Signals: have non-zero power over the entire duration of the signal. As a consequence, the total energy in a power signal is infinite.

Activity 3:Classify the signals defined in Activity 2 as a power or an energy signal.

7

Linear Transformations

There are two types of linear transformations that we will consider

1. Time Shifting:

2. Scaling:

)()( ottxty −=

)()( atxty =

8

Linear Transformations (2)

Scaling for DT Signals:

Note that y[n] = x[n/2] is not completely defined

][][ onnxny −=

9

Linear Transformations (3)

Precedence Rule:

1. Establishes the order of shifting and scaling in relationships involving both shifting and scaling operations.

2. Time-shifting takes precedence over time-scaling. In the above representation, the time shift b is performed first on x(t), resulting in an intermediate signal v(t) defined by

v(t) = x(t + b)

followed by time-scaling by a factor of a, that is,

y(t) = v(at) = x(at + b).

3. Example: Sketch y(t) = x(2t + 3) for x(t) given in (a)

)()( batxty +=

10

Linear Transformations (4)

For the DT signal x[n] illustrated below:

Activity 4: Draw the following:(a) x[−n]; (b) x[2n]; (c) x[n + 3 ]; (d) x[2n + 3]

11

Periodic Signals

1. Periodic signals: A periodic signal x(t) is a function of time that satisfies the condition x(t) = x(t + T) for all t

where T is a positive constant number and is referred to as the fundamental period of the signal.

Fundamental frequency (f) is the inverse of the period of the signal. It is measured in Hertz (Hz =1/s).2. Nonperiodic (Aperiodic) signals: are those that do not repeat themselves.

Activity 5: For the sinusoidal signals (a) x[n] = sin (5πn) (b) y[n] = sin(πn/3), determine the fundamental period N of the DT signals.

12

Even vs. Odd Signals (1)

1. Even Signal: A CT signal x(t) is said to be an even signal if it satisfies the conditionx(−t) = x(t) for all t.

2. Odd Signal: The CT signal x(t) is said to be an odd signal if it satisfies the condition x(−t) = −x(t) for all t.

3. Even signals are symmetric about the vertical axis or time origin.4. Odd signals are antisymmetric (or asymmetric) about the time origin.

13

Even vs. Odd Signals (2)

5. Signals that satisfy neither the even property nor the odd property can be divided into even and odd components based on the following equations:

Even component of x(t) = 1/2 [ x(t) + x(−t) ]Odd component of x(t) = 1/2 [ x(t) − x(−t) ]

Activity 6: For the signal

do the following:(a) sketch the signal(b) evaluate the odd part of the signal(c) evaluate the even part of the signal.

⎩⎨⎧ ≤≤−−= elsewhere0

20for |1|1)( tttx

14

CT Exponential Signals (1)

CT exponential signals are of the form

where C and a can both be complex numbers.

atCetx =)(

.

complexcomplexGeneral Complex Exponential

imaginaryrealPeriodic Complex Exponential

realrealReal Exponential Signals

aCCharacteristic

15

CT Exponential Signals: Real (2).

Real CT exponential signals:

where C and a are both positive numbers.

atCetx =)(

16

CT Exponential Signals: Periodic Complex (3).

1. Periodic complex exponential signals:

where C is a positive number but (a = j ωo) is an imaginary number.

2. Difficult to draw. Magnitude and phase are drawn separately.Activity 7: Show that the periodic complex signal x(t) = C exp(jωot) has a fundamental period given by:

atCetx =)(

οωπ= 2T

17

CT Exponential Signals: Sinusoidal (4).

1. Sinusoidal Signals:

where C is real and a is complex.2. Types of Sinusoidal Signals:

}{IM)(or}{RE)( atat CetxCetx ==

⎩⎨⎧

φ+ωφ+ω=

ο

ο waveSine)sin(

waveCosine)cos()( tCtCtx

18

CT Exponential Signals: Sinusoidal (5).

Activity 8: Sketch the following sinusoidal signal

What is the value of the magnitude, fundamental frequency, fundamental phase, and power for the sinusoidal signal.

Activity 9: Show that the power of a sinusoidal signal

is given by A2 / 2.

)2/10sin(6)( π+π= ttx

)2sin()( tfAtx οπ=

19

CT Exponential Signals: Complex (6).

1. General Complex Exponential signals:

where C and a are both complex numbers.2. By substituting C = |C|ejθ and a = (r + jωo), the magnitude and phase of x(t) can be expressed as

atCetx =)(

)()(:Phase|||)(|:Magnitude

φ+ω==

οttxeCtx rt

p

20

CT Exponential Signals (7).

Activity 10: Derive and plot the magnitude and phase of the composite signal

tjtj eetx 35.2)( +=

21

DT Exponential Signals (1).

nCnx α=][

DT exponential signals are of the form

where C and α can both be complex numbers.

Complex ComplexGeneral Complex Exponential

Imaginary (α = jω)RealSinusoidal Signals

RealRealReal Exponential Signals

αCCharacteristic

22

DT Exponential Signals: Real (2).

nCnx β=][

1. For real exponential signals, C and α are both real

2. Depending upon the value of a, a different waveform is produced.α > 1 gives a rising exponentialα = 1 gives a constant line0 < α < 1 gives a decaying exponential−1 < α < 0 gives an alternating sign decaying exponentialα < −1 gives an alternating sign rising exponential

23

DT Exponential Signals: Sinusoidal (3).

1. Sinusoidal Signals:

where C is real and α is complex.2. Types of Sinusoidal Signals:

3. Not all DT sinusoidal signals are periodic

}{IM)(or}{RE][ nn CtxCnx α=α=

⎩⎨⎧

φ+ωφ+ω=

ο

ο waveSine)sin(

waveCosine)cos(][ nCnCnx

24

DT Exponential Signals: Sinusoidal (4).

4. Condition for periodicity: A DT sinusoidal signal

is periodic if (ωo / 2π) is a rational number with the period given by

Activity 11: Consider the signal

x[n] =cos(πn/3) sin(πn/6).

Determine if the signal is periodic. If yes, calculate the period N.

⎩⎨⎧

φ+ωφ+ω=

ο

ο waveSine)sin(

waveCosine)cos(][ nCnCnx

οωπ= mN 2

25

Unit Step Function

DT domain:

CT domain

Activity: For the discrete time signal

Describe x[n] as a function of two step functions. Ans: U[n + 5] − U[n − 10]

⎩⎨⎧

<≥= 00

01][ nnnU

⎩⎨⎧

<≥= 00

01)( tttU

⎩⎨⎧ ≤≤−= elsewhere0

105for 1][ nnx

26

Unit Sample (Impulse) Function

DT domain:

CT domain: Impulse function is defined as

⎩⎨⎧

≠==δ 00

01][ nnn

∫

∫

∞∞− οο

οοο

∞∞−

=−δ

−δ=−δ

δ=δ

=δ

)()()(.4)()()()(.3

)(||1)(.2

1)(.1

txdttttxtttxtttx

taatdtt

27

Gate Function

DT domain:

CT domain

⎩⎨⎧ ≥=Π

elsewhere02/1][ Nnn

⎩⎨⎧ <=Π

elsewhere01)( Ttt

28

Ramp Function

DT domain:

CT domain

⎩⎨⎧ ≥= elsewhere0

0][ nnnr

⎩⎨⎧ ≥= elsewhere0

0)( tttr