Chapter_2_Part2_DT Signals & Systems.pdf

8/12/2019 Chapter_2_Part2_DT Signals & Systems.pdf

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Chapter 2 - Discrete-Time

Signals & Systems

Digital Signal Processing

Discrete-Time Signals.

Discrete-Time Systems.

Analysis of DT Linear Invariant Systems.

Implementation of DT Systems.

(PART II)

Discrete-Time Signals & Systems


2.3. Analysis of DT Linear Invariant Systems

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In this section we will analyze the Linear Time-Invariant

(LTI) systems.

2.3.1. System Analysis Techniques:

Two methods are presented in for analyzing the response of a

system to a given input:

Direct Solution of the Input-Output Equation (or

difference equation).

Signal Decomposition (Convolution).



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System Input output equation:

The general input output equation for any system:

where F[ ] denotes some functions of the quantities.

We can rewrite the general input output equation as

( ) ( ) ( ) ( ) ( )1

N M

k kk k L

y n a n y n k b n x n k= =

= +

( ) ( ) ( ) ( ) ( )1 0

N M

k kk k

y n a n y n k b n x n k= =

= +

where akand bkare constantparameters

If the system is casualL = 0, so the equation:

This system is linear , time-variantsystem. Thissystem called Adaptive Linear System.

Note that both a andbvary with time



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( ) ( ) ( )1 0

N M

k kk k

y n a y n k b x n k= =

= +

If a and b are constant over time, then the previous equationfurther simplifies into the general equation for causal, Linear,Time-Invariant(LTI) Systems:

Linear Time-Invariant (LTI) Systems:

Note: akand bkare independent of time (n) or simplyconstant for all time n.

Input, Output Equation is called Difference Equation.

The order for the LTI system is N.


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System Interconnections:

Cascade

Parallel

y(n) = T2{y1(n)} = T2{T1{x(n)}}

y1(n) = T1{x(n)}

Tc= T2T1T1T2

Specifically, for LTISystems: T2T1 = T1T2

y1(n) = T

1{x(n)} y

2(n) = T

2{x(n)}

y3(n) = y1(n)+y2(n) = T1{x(n)} + T2{x(n)}

y(n)= (T2+ T1)x(n)

Tp= (T2+ T1)



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Non-linear Difference Equation:

Difference Equations can also describe non linear systems:

( ) ( )y n A x n B= +

( ) ( )y n x n=

( ) ( ) ( )22 3y n x n x n=

( ) ( ) ( )( )3 logy n x n x n= +



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Signal Decomposition:

Decompose input into weighted basis functions:

Generally we choose the basis

functions and compute ck.

( ) ( )k kk

x n c x n=

Decomposition to Impulses:Decompose input sequenceto sum of unit samples.

( ) { }3,5,2,1x n =

( ) { } ( )1 3,0,0,0 3x n n= =

( ) { } ( )2 0,5,0,0 5 1x n n= = ( ) { } ( )3 0,0,2,0 2 2x n n= =

( ) { } ( )4 0,0,0,1 3x n n= =

( ) ( ) ( ) ( ) ( )1 2 3 4x n x n x n x n x n= + + +

( ) ( ) ( ) ( ) ( )3 5 1 2 2 3x n n n n n = + + +

Consider



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Decomposition to Impulses:

( ) ( ) ( )3

0k

x n x k n k=

=

( ) ( ) ( )k

x n x k n k

==

( ) ( ) ( ),k kc x k x n n k = =

We can write the followingequation for the previousexpression:

In general form

where


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Convolution Summation:

( ) ( ) ( )k

x n x k n k

= =

( ) ( ) ( )kk

y n x k h n

==

Consider the following system:

Recall input can be decomposedas follows:

Therefore:

If Tsystem is Linear:

Impulse Response:

Useless, infinite set of responses:(dependent on both n and k)

Ty(n)x(n)


)}({),( knTknh = )}({)( knTknh =

)(),( knhknh =


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Convolution Summation:

Impulse Response

Note if system is time invariant

Therefore

and

For LTI systems, the convolution sum is as follows:

( ) ( ) ( ) ( ) ( ) ( ) * ( ) ( ) * ( )k k

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

= =

= = = =



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Convolution Summation Calculation:

Graphical method:

To calculate y(n0) using the convolution summation, do

the following steps:

Folding: Fold h(k) about k = 0 to obtain h(-k).

Shifting: Shift h(-k) by n0to the right (left) if n0is

positive (negative), to obtain h(n0k)

Multiplication: Multiply x(k) by h(n0k) to obtain the

product sequence x(k)h(n0k)

Summation: Sum all the values of the product

sequence x(k)h(n0k) to obtain the value of the output

at time n = n0



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Graphical Convolution Example

Consider the following input and impulse response

{ }( ) 1,2,1, 1h n =

{ }( ) 1,2,3,1x n =

Input Sequence Impulse Response


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Graphical Convolution Example Cont.

Folded Impulse Shifted Impulse



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Graphical Convolution Example Cont.

Multiply input by folded shifted h(n):

Product Sequence x(k)h(n-k):

( ) ( ) ( )3

0

1 1 1 4 3 8k

y x k h k=

= = + + =

Sum of Product Sequence:

( )1y ( )0y



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Mathematical Convolution Example

Consider the following input and impulse responsex(n) = [5 ,2 , 4 , -1] h(n) = [5 , 2 , 4 , -1]

y[0] y[1] y[2] y[3] y[4] y[5] y[6]

25 20 44 6 12 -8 1

h(k)

x(k)

5 2 4 -1

5 25 10 20 -5

2 10 4 8 -2

4 20 8 16 -4

-1 -5 -2 -4 1

y(n) = [25, 20 , 44 , 6 , 12 , -8 , 1]

Note:The convolution of two FINITE-LENGTH sequences is that if

x(n) is of length L1and h(n) is of length L2, y(n) = x(n)*h(n) will be

of length : L = L1+ L2-1

x(n) = [4 ,2 , 3 , -1] h(n) = [2 , 3 , 4 , -1]



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Convolution Properties

( ) ( )

( ) ( ) ( )ky n x k h n k x n h n

=

= =

( ) ( ) ( ) ( )x n h n h n x n =

( ) ( ) ( ) ( )k k

x k h n k h k x n k

= = =

Convolution Symbol:

Convolution is Commutative:

Convolution is Distributive:

( ) ( ) ( )

( ) ( ) ( ) ( )1 2 1 2h n x n x n h n x n h n x n + = +

Convolution is Associative:

( ) ( ) ( ) ( ) ( ) ( )1 2 2 33 1x n x n x n x n x n x n =


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Useful Geometric Summation Formulas



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Causal Linear Time-Invariant Systems

An LTI system is Causal IFF its impulseresponse is 0 for negative values of n. ( ) 0, 0h n for n= 0:

For example if x(n) = anu(n) the particular solution will

be in the form:



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The Particular Solution:

The particular solution to a DE for different several inputs:



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Example:

Determine the total solution for n 0 of a DT system characterized

by the following difference equation:

For x(n) = u(n) assuming the initial conditions of

y(-1) = 0 and y(-2) = 0.

Solution:

For x(n) = u(n)

Particular solution:

Substitute thissolution into the DE

( ) 2nx n =



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Example: (cont.)

Homogenous solution:

( ) ny n = 025.0 )2( = nn

0)25.0( 2)2( = n

set

0)5.0)(5.0( =+ n

ny )5.0()(1 =nny )5.0()(2 = )()()( 2211 nyAnyAnyh +=

nn

h AAny )5.0()5.0()( 21 +=

then

substitute

The homogenous solution

5.01=

5.02 =


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Example: (cont.)

Total Solution:

The total solution is:

at n = 0 and n = 1

The solution is:



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Difference Equations:

Zero-Input & Zero-State Response:

An alternate approach to determining the totalsolution of DE.

)()()( nynyny zszi +=yzi(n) : zero-input response

yzs(n) : zero-state response

yzi(n)is obtained by solving DE by setting the input x(n) = 0.

yzs(n)is obtained by solving DE by applying the specified input

with all initial conditions set to zero(0).



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Determining Impulse Response:

Given a LTI system in the form of a

difference equation,

For LTI Systems:

For input:

( )n ( )h nLTI

zero state

( ) ( ) ( )1

MN

k k

k Lk

y n a y n k b x n k

==

= +

( ) ( )x n n=



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Case 1 Non Recursive Systems

For unit impulse input, output:

For non recursive system:Expanding above:

Impulse Response:

Impulse Response:

Length = M+L+1 (FIR)

( ) ( )M

kk L

h n b n k =

=

( ) ( )y n h n=

( ) ( ) ( ) ( ) ( ) ( )1 0 1... 1 0 1 ...L Mh n b n L b n b b n b n M = + + + + + + + +

( ) ( ) ( ) ( ) ( )0 1 1;... 0 ; 1 ; 1 ;..M Lh M b h b h b h b h L b = = = = =

( ) { }1 0 1,... , , ,...,L Mh n b b b b b =

Impulse response is same as coefficients in difference equation

( ) ,kh k b for L k M = " "


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Case 2 Recursive Systems

Procedure for finding Impulse Response given recursivedifference equation:

Find Homogeneous Solution:

( ) 1 1 2 2 ...n n n

h N Ny n C C C = + + +

( ) ( )x n n= 1 2, ,..., NC C Clet and compute



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2.4. implementation of DT Systems

Structure for the Realization of LTI Systems:

Here, the LCCDE structure for the realizationof systems isdescribed, additional structures for these system will introduce inlater chapters.

Consider the first-order system:

Direct Form I Structure



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The previous system can be viewed as 2 LTI systems in cascade,the first is a non-recursivesystem described by the equation:

Whereas the second is a recursivesystem described by theequation:



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If we interchangethe order of the recursive and non-recursivesystems, we obtain an alternative structurefor the realization ofthe system described previously:


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Direct Form II Structure

Minimizing the 2 delayin the structure form Ito 1 delayinstructure form II.

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Chapter_2_Part2_DT Signals & Systems.pdf