Introduction to Signal Theory

8/6/2019 Introduction to Signal Theory

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Introduction to Signals & Systems

Signals & Systems

Examples of practicalCommunication andControl systems

Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 1

Notions

SignalSystem

Noise


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Scope

Study of mathematical concepts and

techni ues, useful for anal sis of


communication andcontrolsystems


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Introduction to Si nal Theor



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Signal ConceptsWhat is a Signal?What is a Signal?What is a Signal?What is a Signal?

Variation of aphysical quantity,containing someinformation


Signal DescriptionSignal DescriptionSignal DescriptionSignal Description

Mathematical Model:Single-valued function

ConceptsConceptsConceptsConcepts

DomainDimension

Energy/ PowerCross Energy/PowerNorm


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Signal ClassificationDimensionDimensionDimensionDimension

Single/ multi-dimensional

DomainDomainDomainDomain

Time/ frequency / spatial

ValueValueValueValue

Real/ Complex

Extent in magnitudeExtent in magnitudeExtent in magnitudeExtent in magnitude

Bounded/ unbounded

PredictabilityPredictabilityPredictabilityPredictability

Deterministic/ Random

Relative to time originRelative to time originRelative to time originRelative to time origin

Causal/ non-causal


Extent in timeExtent in timeExtent in timeExtent in time

Finite/ eternal/ semi- infinite

Finiteness of Energy/ PowerFiniteness of Energy/ PowerFiniteness of Energy/ PowerFiniteness of Energy/ Power

Energy/ power

Continuity/ quantizationContinuity/ quantizationContinuity/ quantizationContinuity/ quantization

Continuous/ discrete

Symmetry about originSymmetry about originSymmetry about originSymmetry about origin

Even/ odd

PeriodicityPeriodicityPeriodicityPeriodicity

Aperiodic/ periodic


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Some Properties of Periodic SignalsA periodic function is also periodic with integer multiples of

the fundamental period.

Sum of two periodic signals has a fundamental period = LCMof the two periods

Product of two periodic signals has a period = LCM of the


two perio s not necessari y un amenta perio .

Sum or product of periodic signals would not be periodic ifratio of the periods is irrational.

A periodic signal with period T is said to have half-wave orrotational symmetry if f(tT/2)= f(t)

All periodic signals bounded in amplitude are power signals.


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Some Notes of Even & Odd Signalseven + even = evenodd + odd = oddodd + even = neither odd nor even

An even function expressed as a sum cannot have any odd components; and, an oddfunction can not have any even components.

Any arbitrary signal can be split into an even

An odd function alwayspasses through the origin.Integral of an odd functionequals zero.

Constant is an evenfunction.

Multiplication by an odd


( ) ( ) ( )

( ) ( )( )

2( ) ( )

( )2

even odd

even

odd

f t f t f t

f t f t f t

f t f t f t

= +

+ =

=

even x even = even

odd x odd = eveneven x odd = odd

part and an odd part in a unique manner:function alters evenness tooddness and vice versa,while multiplication by aneven function does not.


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Some Elementary/ Standard Signalsexponential sinusoidal

step u(t) impulseramp r(t)


rectangular signum sgn(x)

sinc(t)

sin( )

=

t

t


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Elementary Signals contd..Unit Impulse:

(t) = 0 for t 0

(t) dt = 1

Unit Step:

u(t) = 0 for t 0

1 for t > 0

Unit Ramp:

r(t) = 0 for t 0

t for t > 0

( )0as a Some functions which approach (t) in the limit


1 11. Rectangular pulse for 2. Triangular pulse 1 for2 2

1 13. Exponential pulse 4. Double exponential pulse

2

5. Sinc pulse

tt

aa

ta at ta a a

e ea a

<


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Elementary Signals contd..Ex. Classify u(t) and exp(-2t)u(t) as energy/ power signals

Ex. Find even and odd components of exponential and step functions

Ex. Show that derivative of u(t) equals (t); also verify that running integralof(t) equals u(t).

Ex. Find the area under sinc(t) by integration


Some notesSignal value at a discontinuity (such as in a step) for practical purposes maybe equated to the left limit, right limit or the mid-value.

Integral value of a continuous function equals area under its graph (positivearea - negative area)

Random waveforms produced by sources of finite average power, are powersignals.


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Mathematical Operations on Signals

Operations ona signal

Operations ontwo signals

Amplitude Shift

Convolution


me

Amplitude Scaling

Time Scaling

Correlation


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Transformation of Independent Variable

A function once defined, is invariant under transformation of independentvariable. That is, the function value for any value of argument remains thesame even after transformation.

f(t-1) is f(t) shifted right by one, and f(t+1) is f(t) shifted left by one.

f(at) is compression of f(t) for a>0, expansion for a


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Reversal about

t = 0.5

Transformation of Independent Variable:

Example

0

1 1

1

1 2

t

t t

f(t)

f(2t-1) f(-2t+1)


0

0

0

1

0.5 1.5 - 0.5 0.5

0- 0.5 0.5 - 0.5-1.5

1

t t

f(2t+1)

f(-2t-1)Reversal about

t = - 0.5

Reversal about t= 0


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Signal Operations contd..

Ex: For the signals x(t) and h(t) sketched below, find and sketcha) x(t)h(t+1) b)x(2- 0.5t) c) x(t-1) h(1-t) d) x(t)h(-t)

Ex.: For the f(t) given below sketch f(2t-1), f(2t+1), f(-2t+1) and f(-2t-1).Also verify.

0 1 2

1

t

f(t)


32-1 1

1

2

210-1-2

1

-1

t t

x(t) h(t)

Verify your answers by substituting suitable values for t.

0

Ex: Express f(t), x(t) and h(t) given above, as linear combinations of other

basic signals with suitable transformations.


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Convolution of Signals:

Example of Graphical Method

Procedure

Change to

dummy variable

Time-reverse

either signal

y(t) = x(t) h(t) = x( )h(t- )d


Shift

Multiply

Find area

Repeat

1 3

product function

for t

Documents

Introduction to Signal Theory