Lecture 03 - Discrete-time Systems

7/27/2019 Lecture 03 - Discrete-time Systems

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Discrete-time Systems

Lecture 3


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Discrete Systems

Mathematically described as T[.] that take

sequence x(n) and transform to anothersequence y[n]

)]([)( nxTny =


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Linear System

A system is linear if and only if

)]([)]([)]()([ 22112211 nxLanxLanxanxaL +=+

)(),(,,2121

nxnxaa


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Linear System

Output of linear system to an arbitrary input x(n)

L[(n-k)] = h(n,k) is the response of linear system at time n toa unit sample at time k

Called impulse response

This is a time-varying impulse response h(n,k) which is notvery convenient

=

=

=

==nn

knLkxknkxLnxLny )]([)()()()]([)(

=

=n

knhkxny )()()(


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Linear time-invariant (LTI) System

A linear system in which an input-output pair x(n) and y(n) is

invarient to a shift kin time

The L[.] and shifting operators are reversable

)()]([)]([)( knyknxLnxLny ==

L[.] Shift by k

Shift by k L[.]

x(n) y(n)

x(n) y(n-k)

y(n-k)

x(n-k)


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Linear time-invariant (LTI) System

Time-invariant function: h(n-k)

Output form:

Impulse response of an LTI system is given by h(n)

the operation is called linear convolution sum *

=

==n

knhkxnxLTIny )()()]([)(

h(n)x(n) y(n) = x(n) * h(n)

===

n

knhkxnhnxny )()()(*)()(


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Stability

Important to consider stability to avoid building harmful

systems or avoid burnout or saturation in system operation

A system is boundary-input boundary-output (BIBO) stable if

every bounded input produces a bounded output

An LTI system is BIBO if and only if its impulse response is

absolutely sumable

yxnynx ,,)()(


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Causality

Necessary to make sure the system can be built

A system is causal if the output at index n0 depends only on

the input up to and including index n0 (output does not depend

on future value)

An LTI system is causal if and only if

0,0)(


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Example of Convolution

let the rectangle pulse x(n)=u(n)-u(n-10) be an input to an LTI systemwith impulse response

Determine the output y(n)

Almost geometric series sum except that the term u(n-k) takes differentvalues depending on n and k

)()9.0()( nunh n=

==

==9

0

9

0

)()9.0()9.0()()9.0)(1()(k

k

k

nkn knuknuny


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When n


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Given the following two sequence

Determine the convolution y(n) = x(n)*h(n)

33],2,4,1,0,7,11,3[)( = nnx 41],1,2,5,0,3,2[)( = nnh

]2,8,3,22,18,41,5,51,6,47,31,6[)( =ny

011)5(3)1()( +=k

khkx

2037 ++

)1(6 == y

27111)2()( +=k

khkx

0)1()5(0 ++2234 ++

)2(41 y==


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MATLAB Implementation

>> x=[ 3, 11, 7, 0, - 1, 4, 2] ;

>> h=[ 2, 3, 0, - 5, 2, 1] ;

>> y = conv( x, h)

y =

6 31 47 6 - 51 - 5 41 18 - 22 - 3 8 2

>>

conv function computes convolution between two finiteduration sequences, assuming the two sequences begin atn=0


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MATLAB Implementation

f unct i on [ y, ny] = conv_m( x, nx, h, nh)

nyb = nx( 1) +nh( 1) ;nye = nx( l engt h( nx) ) + nh( l engt h( nh) ) ;ny = [ nyb: nye] ;

y = conv( x, h) ;

Given a finite duration x(n) and h(n),

The beginning and ending points of y(n) are

>> [ y, ny] =conv_m( x, nx, h, nh)

y =

6 31 47 6 - 51 - 5 41 18 - 22 - 3 8 2

ny =- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7

});({ xexb nnnnx });({ hehb nnnnh and

hbxbyb nnn += hexeye nnn +=and


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Sequence Correlation Revisited

The cross correlation can be put in the form

With autocorrelation in the form

)(*)()( lxlylryx =

)(*)()( lxlxlrxx =


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example of the crosscorrelation sequence

Let this be a prototype sequence

Let y be its noise-corrupted-and-shifted version

Where w(n) is a Gaussian sequence with mean 0 and variance 1

Compute crosscorrelation between y(n) and x(n)

]2,4,1,0,7,11,3[)( =

nx

)()2()( nwnxny +=


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Example of the crosscorrelation sequence>> x = [ 3 11 7 0 - 1 4 2] ;>> nx = [ - 3: 3] ;>> [ y, ny] = si gshi f t ( x, nx, 2) ;>> w=r and( 1, l ength( y) ) ; nw=ny;>> [ y, ny] =si gadd( y, ny, w, nw) ;>> [ x, nx] =si gf ol d( x, nx) ;

>> [ r xy, nr xy] =conv_m( y, ny, x, nx) ;>> subpl ot ( 1, 1, 1) , subpl ot ( 2, 1, 1) ; st em( nr xy, r xy) ;>> axi s( [ - 5, 10, - 50, 250] ) ; xl abel ( ' l ag var i abl e l ' ) ;>> yl abel ( ' r xy' ) ;


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Example of the crosscorrelation sequence>> x = [ 3 11 7 0 - 1 4 2] ;>> nx = [ - 3: 3] ;>> [ y, ny] = si gshi f t ( x, nx, 2) ;>> [ r xy, nr xy] =xcor r ( x, y) ;>> subpl ot ( 1, 1, 1) , subpl ot ( 2, 1, 1) ; st em( nr xy, r xy) ;>> axi s( [ - 5, 10, - 50, 250] ) ; xl abel ( ' l ag var i abl e l ' ) ;

>> yl abel ( ' r xy' ) ;


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crosscorrelation and autocorrelation>> x = [ 3 11 7 0 - 1 4 2] ;>> nx = [ - 3: 3] ;>> [ y, ny] = si gshi f t ( x, nx, 2) ;>> [ r xy, nr xy] =xcor r ( x, y) ;>> st em( nr xy, r xy) ; axi s( [ - 5, 10, - 50, 250] ) ; xl abel ( ' l ag var i abl e l ' ) ; yl abel ( ' r xy' ) ;>> [ r xx, nr xx] =xcor r ( x) ;

>> st em( nr xx, r xx) ; axi s( [ - 5, 10, - 50, 250] ) ; xl abel ( ' l ag var i abl e l ' ) ; yl abel ( ' r xx' ) ;


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Different Equations

An LTI discrete system can also be described by a linear

constant coefficient different equation of the form

If aN 0, then the equation is of order N

Another form is

nmnxbknyaN

k

M

m

mk = =

=0 0

),()(

== =N

k

k

M

m

m knyamnxbny00

)()()(


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Different Equations

The solution of this equation can be obtained in the form

The homogenous part of the solution, yH(n) is given by

Where zk, k=1..N are N roots (called natural frequencies) ofthe characteristic equation

)()()( nynyny PH +=

=

=N

k

n

kkH zcny0

)(

=

=N

k

k

kza0

0


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Different Equations

The system is stable if the roots zk satisfy the condition

Then the causal system is stable

Where zk, k=1..N are N roots (called natural frequencies) of

the particular part yP(n) is determined from

Nkzk ,...,1,1 => b=[ 1] ; a=[ 1, - 1, 0. 9] ; n=[ - 20: 120] ;>> h=i mpz( b, a, n) ;>> xl abel ( ' n' ) ; yl abel ( ' h( n) ' ) ;


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Example

Given a difference equation

Calculate and plot the unit step response s(n) at n = -20,,120nnxnynyny

=+ );()2(9.0)1()(

>> b=[ 1] ; a=[ 1, - 1, 0. 9] ; n=[ - 20: 120] ;

>> x=st epseq( 0, - 20, 120) ; s= f i l t er ( b, a, x) ;>> st em( n, s) ;>> xl abel ( ' n' ) ; yl abel ( ' s(n) ' ) ;


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Example

Given a difference equation

Is the system specified by h(n) stable?

We can use the plot of the impulse response to observe that h(n) ispractically zero for n > 120. We can check if

Therefore, the system is stable

nnxnynyny =+

);()2(9.0)1()(

>> sum( abs( h) )ans =

14. 8785

>> z = r oot s( a) ; magz = abs( z)magz =0. 94870. 9487

The sum is less than infinityThe roots are less than one


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Example (cont.)

By definition h(n) is the output of an LTI system when the

input is (n)

Substituting x(n) for (n) and y(n) for h(n), the difference eq. is

)()1(9.0)( nxnyny =

>> a=[ 1, - 1, 0. 9] ; b=1;>> x=i mpseq( 0, - 20, 120) ; n=[ - 20: 120] ;>> h=f i l t er ( b, a, x) ;>> st em( n, h) ;>> axi s( [ - 20, 120, - 1. 1, 1. 1] )

>> t i t l e( ' I mpul se Response' ) ;>>xl abel ( ' n' ) ; yl abel ( ' h( n) ' ) ;


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Zero-Input and Zero State Response Difference eq. is solved forward in time from n=0

Initial condition on x(n) and y(n) are necessary to determine the output for

n >=0. The initial condition is given by

Subject to the initial conditions:

A solution can be obtained in the form

yZI(n) is called zero-input solution: due to the initial conditions alone(assuming they exist)

yZS(n) is called zero state solution: due to input x(n) alone(assuming the initial conditions are zero)

0;)()()(00

= ==

nknyamnxbnyN

k

k

M

m

m

}1);({ nNny }1);({ nMnx

)()()( nynyny ZSZI +=


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Digital Filters

Filter means LTI system designed for a

specific job of frequency selection or

discrimination


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FIR (Finite-duration Impulse Response) Filters

the unit impulse response of the LTI system is of finite duration

h(n) = 0 for n < n1 and n > n2

Causal FIR filter:

h(0) = b0, h(1) = b1, h(2) = b2 h(M) = bM. Other h(n) are 0

Also called nonrecursive or moving average (MA) filter.

Represented as impulse response {h(n)} or as difference eq. coefficient

{bm} and {a0 = 1}

We can use either conv(x,h) or filter(b,1,x)

Output of conv(x,h) has longer length than both x(n) and h(n)

Output of filter(b,1,x) has same length as x(n)

=

=M

m

m mnxbny0

)()(


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IIR (Infinite-duration Impulse Response) Filters

the impulse response of the LTI system is of infinite duration

The part of difference eq:

output y(n) is recursively computed from previously computed values

Also called autoregressive (AR) filter.

This is also decribes IIR filter

called autoregressive moving average (ARMA) filter

Described by difference eq. coefficients {bm} and {ak}, implemented byfilter(b,a,x)

)()(0

nxknyaN

k

k ==

nmnxbknya

N

k

M

mmk

= = =0 0 ),()(

Lecture 03 - Discrete-time Systems

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