3 DSP - System Analysis in the Shift (Time) Domain

8/11/2019 3 DSP - System Analysis in the Shift (Time) Domain

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Digital Signal Processing

C. Discrete-Time Systems

Athanassios C. Iossifides

October 2012


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Athanassios Iossifides DIGITAL SIGNAL PROCESSING C2

C.1 Discrete-time systems representation

C.2 System classification

C.3 Analysis of LSI systems

C.4 LSI systems and difference equationsC.5 System realization



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C.1 Discrete-time systems

representation



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Definition

Discrete-time system is a device or algorithm that acts on discrete-timesignal, that is called input, following a predefined rule and produces a new

discrete-time signal called outputor response.

The operand T represents the action of the system on the input.

( ) ( )x n y nT

Disrete-time

Systemx(n) y(n)

Tx(n) y(n)


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Example Proakis 2.2.1(d)

Moving average filter1

( ) [ ( 1) ( ) ( 1)]3

y n x n x n x n | |, 3 3( )0,

n nx n

( ) ...,0,3,2,1,0,1,2,3,0,...}x n ( 2) (1 / 3)[ ( 3) ( 2) ( 1)] 2

( 1) (1 / 3)[ ( 2) ( 1) (0)] 1(0) (1 / 3)[ ( 1) (0) (1)] 2 / 3

(1) (1 / 3)[ (0) (1) (2)] 1

(2) (1 / 3)[ (1) (2) (3)] 2

y x x x

y x x x y x x x

y x x x

y x x x


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Example Proakis 2.2.1(e)

Accumulator( ) ( ) ( ) ( 1) ( 2) ...

n

k

y n x k x n x n x n

| |, 3 3( )

0,

n nx n

( ) ...,0,3,2,1,0,1,2,3,0,...}x n ( 2) ( 2) ( 3) ... 5

( 1) ( 1) ( 2) ( 3) ... 6

(0) (0) ( 1) ( 2) ( 3) ... 6

y x x

y x x x

y x x x x

1

( ) ( ) ( ) ( 1) ( )

n

k

y n x k x n y n x n


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Block diagrams

The basic structural elements for the representation of a discrete-timesystem with a block diagram are:

The accumulator:

Constant multiplier:

Signal multiplier:




Block diagrams

The basic structural elements for the representation of a discrete-timesystem with a block diagram are:

Delay element:

Advance element:




Block Diagrams example

Draw the block diagram of a system with inputx(n) and output given by

We write the equations as

1( ) ( 1) ( ) ( 1)

2y n y n x n x n

1( ) ( 1) ( ) ( 1)

2y n y n x n x n

( 1)y n

( 1)x n

0.5 ( )x n


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Static/dynamic systems

A discrete-time system is called staticor memorylesswhen its response atany time instant depends solely on the corresponding sample of the input

at the same time instant.

Examples:

A discrete-time system that is not static is called dynamic system.

Dynamic systems have memory which can be either finite or infinite.

Examples:

( ) 3 ( )y n x n

2( ) 3 ( ) ( 1)y n x n nx n

0( ) ( )

n

ky n x n k

0( ) ( )N

ky n x n k

0

( ) ( )k

y n x n k

2( ) 2 ( ) ( )y n x n nx n




Shift (time) invariant/variant systems

A discrete-time system is said to be shift invariant (SI) or time invariant(TI)when the internal characteristics of the system remain unalterd with

shift or time, that is, when

A system that is not shift invariant is called shift variant or time variant

In order to identify if a system is shift invariant or not, we evaluate the

output of the system

and compare with the output

By setting nnk( )y n k

[ ( )]x n kT

( ) ( ) ( ) ( ), , ( )x n y n x n k y n k k x n T T




Shift invariance identification example Proakis 2.2.4

Calculation of

Calculation of

Calculation of

Calculation of

Calculation of

Calculation of

( ) : ( ) ( ) ( 1)y n k y n k x n k x n k

[ ( )] : [ ( )] ( ) ( 1)x n k x n k x n k x n k T T

( ) ( ) ( ) ( 1)x n y n x n x n T

( ) ( ) ( )x n y n nx n T

[ ( )] : [ ( )] ( )x n k x n k nx n k T T

( ) : ( ) ( ) ( )y n k y n k n k x n k

SI

SV

0( ) ( ) ( )cos( )x n y n x n n T

0[ ( )] : [ ( )] ( )cos( )x n k x n k x n k n T T

0( ) : ( ) ( )cos[ ( )]y n k y n k x n k n k SV




Linear / nonlinear systems

A discrete-time systems is linear when it satisfies the properties ofhomogeneity and superposition, that is

Homogeneity:

Superposition:

[ ( )] [ ( )], constantax n a x n aT T

1 2 1 2[ ( ) ( )] [ ( )] [ ( )]x n x n x n x n T T T

T

T

T

1 1 2 2 1 1 2 2 1 2[ ( ) ( )] [ ( )] [ ( )], , constantsa x n a x n a x n a x n a a T T T




Linearity identification example Proakis 2.2.5

1 1 2 2 1 1 2 2[ ( ) ( )] ( ) ( )a x n a x n a nx n a nx n T( ) ( )y n nx n

1 1 2 2 1 1 2 2[ ( )] [ ( )] ( ) ( )a x n a x n a nx n a nx n T Tlinear

2

2 2 2 2 21 1 2 2 1 1 1 1 1 1 2 2 1 2 1 2

2 2 2 21 1 2 2 1 1 2 2

( ) ( )

[ ( ) ( )] [ ( ) ( )] ( ) ( ) 2 ( ) ( )nonlinear

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

y n x n

a x n a x n a x n a x n a x n a x n a a x n x n

a x n a x n a x n a x n

T

T T

1 1 2 2 1 1 2 2 1 1 2 2

1 1 2 2 1 1 1 2 2 2

( ) ( ) , , constants

[ ( ) ( )] [ ( ) ( )] ( ) ( ) nonlinear[ ( )] [ ( )] ( ) ( )

y n Ax n B A B

a x n a x n A a x n a x n B Aa x n Aa x n Ba x n a x n a Ax n a B a Ax n a B

TT T




Causal/non causal systems

A discrete-time systems is called causal when its current output (at anytime instant) depends solely on the current and the past samples of the

input but not on the future samples of the input.

In general, a causal system is described as a function of the form

A system for which the output depends on future samples of the input is

non causal.

Real time systems are causal systems. However, when the output signal is

recorded and its processing takes places off-line (not in real time), thenthe system can be non causal.

( ) [ ( 1), ( 2),..., ( ), ( ), ( 1), ( 2),...]y n f y n y n y n N x n x n x n




Stable/unstable systems

A discrete-time system is called stableif and only if every finite inputsignal creates a finite output signal.

This type of stability is called (ounded nputounded utput)

stability.

If for any (and only one) input signal, the output signal takes infinite

values, then the system is unstable.

( ) ( )x yx n M y n M



C.3 Analysis of LSI discrete-time

systems

C. Discrete-time systems


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C.3 Analysis of LSI discrete-time systems

LSI system response

Let h(n) be the response (output) of a LSI system when the input signal is(n), that is

This response is called impulse response of the system. If the impulse

response of a system is known, then the response of the system to anyinput signal can be calculated.

Every signalx(n) can be expressed as a linear combination of properly

delayed and weighted impulse sequences, that is

Taking into account the properties of an LSI system, its response to any

signalx(n) may be calculated as follows.

T[ ( )] ( ) n h n T(n) h(n)

( ) ( ) ( )

k

x n x k n k

T( ) ( ) n h n


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LSI system response

The response y(n) of an LSI system to the inputx(n) can be calculated by

the sum

This sum is called convolution sum or simply convolution of the input

signal with the impulse response.

( ) ( )

( ) ( ), Shift invariance (SI)

( ) ( ) ( ) ( ), ( ) constant, linearity

( ) ( ) ( ) ( ), linearity

( ) ( ) ( )

k k

k

n h n

n k h n k

x k n k x k h n k x k

x k n k x k h n k

x n x k h n k

T

T

T

T

T

( ) ( ) ( ) ( ) ( )k

y n x k h n k x n h n


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Convolution properties, interconnection of systems

The convolution of two signals is defined as

The following properties can be proved:

Commutative:

Associative:

(cascade interconnection of systems)

( ) ( ) ( ) ( )k

x n h n x k h n k

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

1 2 1 2[ ( ) ( )] ( ) ( ) [ ( ) ( )]x n h n h n x n h n h n

( )x n ( )h n ( )y n ( )x n( )h n ( )y n

( )x n 1( )h n 2( )h n ( )y n ( )x n 2( )h n ( )y n1( )h n

( )x n 1( )h n 2( )h n ( )y n ( )x n ( )y n1 2( ) ( )h n h n


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Convolution properties, interconnection of systems

Distributive:

(parallel interconnection of systems)

Identity element and shift property:

1 2 1 2( ) [ ( ) ( )] ( ) ( ) ( ) ( )]x n h n h n x n h n x n h n

( )x n

1( )h n

2( )h n

( )y n

( )x n ( )y n

1 2( ) ( )h n h n

( ) ( ) ( )

( ) ( ) ( )

x n n x n

x n n k x n k


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Calculation of convolution

Convolution of two discrete-time signals is defined as

Calculation of convolution can be applied in four basic ways:

Direct calculation of the sum with mathematical operations

The graphical method

The method of sliding rule

Transformations (Fourier, Z)

( ) ( ) ( ) ( )k

x n h n x k h n k


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Direct convolution calculation example Hayes 1.4.1

Calculate the convolution of the following signals

For the calculation of the sum, nis considered to be constant.

u(k) is zero when k< 0.

u(n-k) is zero when n-k< 0 n< k

When n< 0, u(n-k) is zero for every positive k and has no common

nonzero values with u(k).

When n 0, u(n-k) is zero for k n, so common nonzero values withu(k) exist from 0 up to n.

( ) ( ), 1 ( ) ( )nx n a u n a h n u n

( ) ( ) ( ) ( ) ( ) ( ) ( )kk k

y n x n h n x k h n k a u k u n k

1

0

1( ) ( )

1

n nk

k

ay n a u n

a


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Graphical calculation of convolution

Graphical calculation consists of the following steps:

Graphical representation of the sequences with respect to k

Folding of one of the two sequences, e.g.

Shifting of the folding sequence step by step. When n< 0, the

sequence is shifted towards the left and when n> 0, the sequence is

shifted towards the right.

Sample by sample multiplication of the sequencesx(k) and h(nk). Sum of the product values for all kso that to calculate the convolution

for position (time instant) n.

Repetition of the procedure for every n.

( ) ( )h k h k

( ) ( ) ( ) ( )k

x n h n x k h n k


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Graphical calculation of convolutionexample Proakis 2.3.3

Calculate the convolution of the discrete-time signals . ( ) ( ), 1 ( ) ( )nh n a u n a x n u n


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0(0) 1y a ( 1) 0y (1) 1y a

( ) ( ), 1 ( ) ( )n

h n a u n a x n u n


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2 3(3) 1y a a a 2(2) 1y a a 2 3 4(4) 1y a a a a

( ) ( ), 1 ( ) ( )n

h n a u n a x n u n


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2(2) 1y a a 0(0) 1y a

2 3 4(4) 1y a a a a

1

0

1( ) ( )

1

n nk

k

ay n a u n

a


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Remarks on convolution

Let two finite-duration sequencesx1(n) andx2(n), with lengths (durations)L1and L2, respectively. Their convolution

Has length

Comparing the formulas of crosscorrelation and convolution

we find

1 2( ) ( )x n x n

1 2 1L L

( ) ( ) ( )xyk

r n x k y k n

( ) ( ) ( ) ( )

k

x n y n x k y n k

( ) ( ) ( )xyr n x y y n

3 DSP - System Analysis in the Shift (Time) Domain

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