elce301 lecture5(ltisystems time2)

7/28/2019 ELCE301 Lecture5(LTIsystems Time2)

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Signals and SystemsELCE 301

Time-domain Analysis of LTI Systems (p.2)

113


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Analysis of LTI Systems using inputsLimitations of ODE-based analysis of LTI systems

1. Complex systems need complex differential equations (analyticsolution difficult or impossible).

2. For LTI systems, the solutions of homogeneous equations (which

are easier to find) have limited practicality.

Solutions ofhomogeneous equations define natural (zero-input)

responses of LTI systems.

3. Particular solutions of nonhomogeneous equations must be

individually found for each input signal x(t).

Solutions ofnonhomogeneous equations define forced (zero-

state) responses of LTI systems.

114


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Analysis of LTI Systems using inputsSampling properties of signals

117

0

dttt

TdtTtt

][][][][][][]0[][][ 0000 nnnnnnnnnnnnn

For discrete signals (using Kronecker):

TdttTt

For continuous signals (using Dirac ):


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Assume we know a response h[n] of a discrete LTI system Hto [n]input, i.e.

The output signal h[n] is called impulse responseof an LTIsystem.

Then, we want to estimate the systems response y[n] to any

input signal x[n], using a sequence of mathematical

manipulations.

Analysis of LTI Systems using inputsLTI system outputs for inputs (discrete variant)

118

][][ nnh H


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119

][][][][

k

knkxnxny HH

][][ nxny H This is what we want to find.

Sampling property offunction.

][][

k

knkx H Linearity of the system.

][][

k

knkx H Linearity of the system.

][][

k

knhkx Time-invariance of the system.


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120

][][][

k

knhkxny

CONCLUSION: Response of a discrete LTI system to any input signal

x[n] is a linear combination of delayed or advanced copies of its

impulse response h[n]. The coefficients of this linear combination are

the corresponding samples ofx[n].

[ ] [ ] [ ] ... [ 1] [ 1] [0] [ ] [1] [ 1] ...k

y n x k h n k x h n x h n x h n


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121

Response of a discrete LTI system to a signal x[n] is

a linear combination of delayed or advanced copies

ofthe systems impulse response h[n].


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Assume we know a response h(t) of a CT LTI system H to (t)input, i.e.

The output signal h(t) is called impulse responseof an LTIsystem.

Then, we want to estimate the systems response y(t) to any

input signal x(t), using a similar sequence of mathematical

manipulations.

Analysis of LTI Systems using inputsLTI system outputs for inputs (continuous variant)

122

)()( tth H


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123

dtxtxty )()()()( HH

)()( txty H This is what we want to find.

Sampling property offunction.

)()(

dtxH Linearity of the system.

dtx

)()( H Linearity of the system.

)()( dthx

Time-invariance of the system.


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124

dthxty

)()()(

CONCLUSION: Response of a continuous-time LTI system to any

input signal x(t) is an integral (summation) of delayed or advanced

copies of its impulse response. The coefficients of summation are the

corresponding values of the input signal x(t).


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Analysis of LTI Systems using inputsConvolution

125

The presented results can be formally expressed using the conceptofconvolution, i.e. by convolving the input signal with the impulse

response.

Convolution of discrete functions:

Convolution of continuous functions:

][][][][][][][][][ nxnhknxkhknhkxnhnxnykk

)()()()()()()()()( txthdtxhdthxthtxty


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Analysis of LTI Systems using inputsConvolution

126

Commutative Property:

x(t)*y(t)=y(t)*x(t)

x[n]*y[n]=y[n]*x[n]

Distributive Property: x(t)*(y1(t) + y2(t))=x(t)*y1(t) + x(t)*y2(t)

x[n]*(y1[n] + y2[n])=x[n]*y1[n] + x[n]*y2[n]

Associative Property:x(t)*(y1(t)*y2(t))=(x(t)*y1(t))*y2(t)

x[n]*(y1[n]*y2[n])=(x[n]*y1[n])*y2[n]


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Analysis of LTI Systems using inputsMain steps in analysis of LTI systems

127

1. Check if the system is LTI.

2. If yes, connect (t) signalto the input andcalculate/estimate the impulse response h(t) (e.g. by

solving a differential equation, using measuring tools, etc.).

3. The impulse response h(t)provides complete information

about an LTI system. With it, you can predict the output

signal y(t) for any input signal x(t)by computing the

convolution, i.e. y(t) = x(t)*h(t).


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Analysis of LTI Systems using inputsA note about causal systems

128

What is wrong with this system? Output exists before input!

Impulse response ofa causal systemMUST BE zero for negative times.


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Analysis of LTI Systems using inputsA note about causal systems

129

Impulse response ofa causal systemMUST BE zero for negative times.

0for0)( tth

CONCLUSION: For causal LTI systems any integrationregarding the impulse response should be always done over a

limited domain. For example, the output y(t) for any input signal

x(t) would be computed by the following convolution

0

( ) ( ) ( ) ( ) ( ) ( ) ( )

t

y t x t h t x h t d x t h d


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130

Analysis of LTI Systems using inputsExample (the same as before)

)(1

)(1

)('or)(1

)(1)(

tRC

thRC

thtRC

thRCdt

tdh

Complete solution: )()()( 11 ththCth p

Already known from the related homogeneous ODE.

RC

t

eth

)(1


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Analysis of LTI Systems using inputsExample

Particular solution of the nonhomogeneous equation:

)(1

)(1

)(

)()()(

1

1 tueRC

deRC

edtth

tbthth RC

tt

RCRC

t

p

)(1

)()()( 111 tueRC

eCththCth RCt

RCt

p

Because the system must be causal, then 00)0( 11

CeCth RCt

)(1

)( tueRC

th RCt


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Analysis of LTI Systems using inputsExample

Now, we can calculate the response of the RC circuit to any

input. For example, assume that x(t)= u(t).

0 0

( ) ( ) ( ) ( ) ( )

1 1 1( ) ( ) ( ) ( )

0 0

( ) ( 1) (1 ) ( )

1 0

t

t t tt t tRC RC RC RC

t t t

RC RC RC t

RC

y t u t h t u h t d

u e u t d u t e d u t e e d RC RC RC

t

u t e e e u t

e t

The same result as before!

OBSERVATION: Convolutions are computationally complex and

tedious operations. Simplified/approximate methods are welcome.


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Analysis of LTI Systems using inputsGraphical evaluation of convolution integrals

Four basic steps to evaluate c(t)= a(t)*b(t):

1. Plot diagrams of the functions a() and b().2. Create a mirror reflection of the function b, i.e. b() => b(-).3. Slide the diagram of function b(-) along the horizontal axis,

i.e. b(-) => b(t-).4. Estimate the integral over the intersection area of diagrams

a() and b(t-).


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134

Analysis of LTI Systems using inputsGraphical evaluation of convolution integrals (example)

Find the convolution y(t) = h(t)*x(t), where


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135

Analysis of LTI Systems using inputsGraphical evaluation of convolution integrals - Example

Steps 1 and 2 (note that is used instead of)

Steps 3 and 4 (several cases shown)


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136


Steps 3 and 4 (cont.)


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137


Steps 3 and 4 (cont.)

Final result


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138

Analysis of LTI Systems using inputsEvaluation of convolution integralsDelta method

The graphical evaluation is tedious.Example:

00

tt

x(t) h(t)

-2

1

2

1

2

1 2 13

-1

2

-1-2

x(2- )

1 32

h()x(- )


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139

Analysis of LTI Systems using inputs

t

x(t-)

h()

Evaluation of convolution integralsDelta method

The graphical evaluation is tedious.


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140


The graphical evaluation is tedious.Selected case of integration (for 1< t 2)

1

0

1

1 1

3

)5.05.01(2212)(t

t

t

t

dtdddty


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321-1-2

h(t)

1 2

dx/dt

141


Delta-representation of polynomial functions.

)3(2 1 t)1(1 t

)1(1 t

)(1 t

)2(21 2 t

)(2

1 2t

-2

t

t

tt

dh/dt

0 0

)2(2 1 t

x(t))3(2)1()()( 111 tttth

)2(2)1(

)(2

1)2(

2

1)(

11

22

tt

tttx


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Analysis of LTI Systems using inputsEvaluation of convolution integralsDelta methodTable for Delta-method convolution.

2

3

2/1

3

2

2

)2(2

1)( 3 tty 2( ) 2 ( 3)y t t

)5(4)4(2)3(

)2()1(2

3)1(

2

1)3(4)1()(

2

1)2(

2

1

)5(4)4(2)3()1()3(2

)2()1(21)1(

21)3(2)1()(

21)2(

21)(

223

2332233

22332

2332233

ttt

ttttttt

ttttt

tttttttty


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Analysis of LTI Systems using inputsExtras (other operations similar to convolution)Cross-correlation

The cross-correlation (symbol is used) of functionsx(t) and h(t) is defined

by the convolution of functions orx(t) and h( t)], i.e.

Cross-correlation indicates how similartwo signals are when one of them

is delayed/advanced by various amounts of time. The maximum value of

the cross-correlation corresponds to the delay when both signals are the

most similar.


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145

Analysis of LTI Systems using inputsExtras (other operations similar to convolution)Cross-correlation

Properties and computational schemes forcross-correlationcan be easily

obtained from the corresponding properties/schemes ofconvolution.

Auto-correlation

Auto-correlationis the cross-correlation of a signal with itself.

It represents similarity between the original signal and its

delayed/advanced copies. Auto-correlation is a tool for detecting repeating

patterns in the analyzed signals (including detection of periodic signals).

elce301 lecture5(ltisystems time2)

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