Discrete Time System

Discrete time Fourier Series1.

Determination of Fourier series representation of a periodic signal2.

Fourier series coefficients3.

Resolving the input in terms of elementary functioni.

Techniques of analysis of discrete time systemsa.

Resolution of discrete time signal into impulsesi.

Impulse response of systemii.

Output of LTI system and convolution sumiii.

Discrete time impulse as elementary functionb.

Causality of LTI systemi.

Stability of LTI systemii.

Impulse response of systemc.

Finite impulse responsei.

Infinite impulse responseii.

Types of impulse responsed.

Techniques of analysis of discrete time system4.

Recursive and non recursive systemsa.

Discrete time system represented by constant coefficient difference

equations

b.

Forced and Free response of discrete time Systemc.

System function and FIR and IIR Filtersd.

FIR Systemsi.

Structure for realization of Discrete time systemse.

Discrete time System described by Difference Equation5.

INDEX24 April 201112:48

Fourier Series Page 1

Discrete Time Fourier Series

Discrete time periodic signal

a discrete time periodic signal is periodic with period N is

x[n]=x[N+n]

N is the smallest positive integer for which the equation holds .

Wo=(2*pi/N) is the fundamental frequency

Discrete time complex exponential sequence

Consider a discrete time complex exponential sequence

This sequence is periodic with period N

The signal has fundamental frequency of 2*pi/N

The discrete time complex exponential sequence have distinct

values over a range of N successful values of n from 0 - (N-1)

Harmonically related set of discrete time complex exponentialConsider a set of harmonically related discrete time complex exponentials

All signals have fundamental frequencies that are multiples

of 2*pi/N

Thus all the signals are harmonically related

DIGITAL SIGNAL PROCESSING23 April 201121:43


Thus all the signals are harmonically related

Discrete time complex exponentials which differ in

frequency by 2pi are identical

exp(jwn)=exp(j[w+2pi]n)

In general

If k changes by integral multiples of N

Then identical sequence is generated

Linear combination of discrete time complex exponential

Consider a linear combination of harmonically related

discrete time complex exponential sequence

The summation is carried over a range of N successive values

of k from 0-(N-1)

This equation indicates than a arbitrary periodic discrete

time signal x[n] can be represented by linear combination of

harmonically related complex exponential


harmonically related complex exponential

This representation if called discrete time Fourier series

representation of signal x[n]

For now we assume any discrete time signal can be expressed

as a linear combination of harmonically related complex

exponential sequence. We need to explore if this holds true

Questions

Can any discrete time signal be expressed

Also since the summation will have a finite number of terms

as k ranges from 0- (N-1). The summation will converge

absolutely


A periodic discrete time sequence can be

expressed as a arbitrary sum of discrete

time complex exponential sequence.

23 April 201122:07


Suppose we are given a signal x[n] that is periodic with fundamental period N

We need to find a way to determine if the Fourier series representation of the signal

x[n] exists•

Determine the Fourier series coefficients•

Thus we need to find the solution to a set of linear equations .

The Fourier series representation of signal provides us with N linear equations with

N unknown coefficients as k ranges a values over n successive integers

If these set of equations are linearly independent we can solve the equations to obtain

The coefficients in terms of given values of signal x[n]

One ways is to solve the equations to obtain the Fourier series coefficients

We also can obtain a closed form expression to obtain the coefficients

For now we will assume all the N equations are linearly independent

And matrix comprising of coefficients of N linearly independent equations has a inverse

x=AW

A=W^-1x

A=(1/n)W*x

Thus we can use the inverse operation to find out the fourier series coefficients of a

periodic discrete time sequence

We will demonstrate the linear independence and existence of inverse of the matrix by

using example and linear algebra concepts

Determination of Fourier series representation of a

periodic signal24 April 201112:49


Below are the expressions for analysis and synthesis equation for Fourier Series representation of

the periodic discrete time signal x[n]

The fourier series coefficients are also called the spectral coefficients

of x[n]

The coefficients specify the decomposition of a periodic discrete time

signal into sum of N harmonically related complex exponentials

Values of k range from 0 to N-1

The Fourier series coefficients are also a discrete periodic sequence which

are periodic with period N

Since there are only N discrete complex exponentials that are periodic with

period N

Discrete time Fourier Series is a finite series with N terms.

Thus if we define N consecutive values of k over which define the Fourier

Series ,we will obtain exactly N Fourier Series coefficients

Fourier series coefficients and periodicity of Fourier

Series coefficients24 April 201113:19


We can regard the sum over arbitrary sum of N successive values of k

Thus we can think Ak as a sequence defined for all values of k but only N

successive elements of the sequence will be used in the Fourier Series

representation of the sequence

Furthermore the Sequence of Fourier Series coefficients is periodic with

period N as the discrete time complex exponential Is periodic in k with

period N


There are two basic methods for analyzing the

behavior or response of a discrete time system

to a given input signal.

One method is based on direct solution of then

input-output equation for the system

For LTI system general form of input-output

relationship is

The input output relationship is called difference

equation and represents one way to characterize

behavior of discrete time LTI system.

Second method of analyzing is given a input signal

to decompose of resolve the inputs into sum of

elementary signals

The elementary signal are selected so that

response of system to elementary signals can be

easily determined

Using the linearity property response of a system

to entire input is computed as superposition of

response of system due to elementary signals

Techniques of analysis of discrete time system24 April 201116:19


Thus we look to represent the input signal as

arbitrary sum of impulse function

•

We calculate the response of system to unit

impulse function

•

We obtain the total response of the system by

superposition of response of system due to

decomposed signal components

•

One class of elementary signal is unit impulse

functions

We decompose in the input periodic signals into •

Harmonically related Complex exponentials

We find the response of system to complex

exponential signal.

•

We compute the total output of LTI system as

superposition of outputs due to individual

frequency components

•

If we restrict ourselves to a subclass of input

signal,we only consider signals that are periodic in

nature,we consider Discrete time complex

exponential sequence as elementary signal


frequency components

Resolving a discrete time signal in terms of

impulse function

Any discrete time signal can be expressed as

arbitrary sum of time shifted impulse functions

x[n]=x1[n] + x2[n] + ……

Were

Thus x(n) is represented as weighted sum of time

shifted impulse function

Let h[n] be the output of the LTI system to unit

impulse function

If x[n-m] is the input to LTI system

The output of system will also be shifted in time

by m sample ie y[n-m]

Thus if x[n] is the input to the LTI system

The output y[n] is given by


The above expression Is called convolution sum

Of sequences x[n] and h[n]

Thus the output of LTI system can be expressed

as convolution sum of the input signal and impulse

response of LTI system

IMPULSE RESPONSE OF SYSTEM

A LTI system can be characterized by its impulse

response

If we know the output of the system to unit

impulse function,we can determine the output of

the LTI system to any arbitrary input signal

Causality of LTI system

A LTI system is causal if and only if its impulse

response if causal

Causality is required in any real time processing system. Fourier Series Page 12

Causality is required in any real time processing system.

In general if input to a LTI system is causal

Output of LTI system is also causal

STABILITY OF LTI SYSTEM

A LTI system is said to be BIBO stable if a

stable input results in a stable output

Given a bounded input to LTI system we

investigate the conditions for stability of

LTI system


Thus a LTI system is BIBO stable if its

impulse response is absolutely summable

This condition is necessary and sufficient

condition to ensure stability of LTI system

If the impulse response goes to zero as n

approaches infinity

Output of LTI system also approaches zero

as n approaches infinity

Excitation at the input to system which if of

finite duration produces a output that is

transient in nature .

It amplitude decays with time and eventually

approaches steady values

And system if said to be a stable system

System with finite duration and infinite

duration impulse response

FIR Impulse response•

IIR Impulse response•

Class of LTI system can be divided into two

types

Consider a Causal FIR system


Consider a Causal FIR system

h(n)=0 n>M

Output of LTI system at any time can be

expressed as weighted linear combination of

input signal samples

System simply weights by values of impulse

response h(n) the most recent M samples

And sums the resulting products

The systems acts like a window that only

views only most recent M samples of input

signal in forming the output

It neglects prior input samples

Thus FIR system is said to have a finite

memory of length M

IIR system has infinite duration impulse

response

System output is a weighted linear

combination of input signal samples

Since sum involves all the present and past

inputs ,the system has infinite memory

Thus in case of IIR filter input signal is the

window function and window length is length

of the input signal Fourier Series Page 15

of the input signal


LTI systems are characterized by their impulse responses

Knowing Impulse response of the system we can calculate the output of system

to any arbitrary input signal

Output is determined by means of convolution sum

LTI system is characterized by input output relationship

Convolution sum also provide a means of realization of system

In case of FIR system, system is implemented directly by means of adder,

multiplier and finite number of memory locations.

IIR systems cannot be implemented directly

Since it requires infinite number of memory locations.

We need to find a way to realize IIR systems

Within general class of IIR systems we have a class of systems that can be

described by difference equation

Convolution sum expresses the output of the system explicitly and only in terms

of input system .

y[n]=h[0]x[n]+h[1]x[n-1]+h[2]x[n-2]+……..

Since system is a causal system output at any time only depends on preset and

past values of input signal

y[0]=h[0]x[0]+h[1]x[-1]+.. = h[0]x[0]

y[1]=h[0]x[1]+h[1]x[0]

y[2]=h[0]x[2]+h[1]x[1]+h[2]x[0]

Eg if h(n)=[1 1 1 1 ………]

y[0]=x[0]

y[1]=x[1]+x[0]=x[1]+y[0]

y[2]=x[2]+x[1]+x[0]=x[2]+y[1]

Discrete time System described by Difference Equation24 April 201118:03


y[2]=x[2]+x[1]+x[0]=x[2]+y[1]

y(n)-y(n-1)=x(n)

H(z)=1/1-z^-1

And h(n)=u(n)

This is example of IIR recursive system

If a system is a recursive system output depends on current input value as well

as previous output values

A system is said to be non recursive if it only depends on present and past input

values

Causal FIR filters represented by convolution sum represent a non recursive

system

The basic difference between and recursive and non recursive system is that

recursive system has a feedback loop,which feeds back the output to the input

of the system

Feedback loop basically consists of delay elements

Also the output of recursive system must be computed in order

y[0],y[1] …...y[N-1]

Thus we require N iterations to compute output y[N-1] in a recursive system

Where as in a non recursive system we may be able to compute y[N-1] directly

using the convolution sum expression

Linear Time Invariant System Characterized by Constant coefficient Difference

equations

Systems represented by constant coefficient difference equations are subclass

of recursive and non recursive systems

Consider a simple recursive system described by the equation

y(n)=ay(n-1)+x(n)

Let us apply the causal input to the system and determine the output

y[0]=ay[-1]+x[0]

For recursive system or system described by constant coefficient difference

equation we need to assume existence of initial condition


Thus to calculate the present value of the

output we require all the previous values of

the input signal

First term contains the initial term y(-1)

And represents the output of system due to

initial condition in the system before the

current input signal was applied

If the system is initially relaxed then y(-1)=0

Recursive system is said to be relaxed if it

starts with zero initial condition

If the System is initially relaxed the

response of system is called called zero

state response or forced response and the

system is said to be in zero state

Forced response of the system is given by

This equation describes the convolution sum of input signal and its impulse response


This equation describes the convolution sum of input signal and its impulse response

Zero state implies the LTI system is causal

Hence upper limit on convolution sum is n

Thus system described is a IIR system

If is a causal system

It is a first order system

System is initially relaxed

If we assume the system is not initially relaxed the output is represented by sum of two

terms,

One term is the zero state or forced response of the system

And another terms is called zero input response or natural response of the system

Thus system produces an output without being exited

The Zero input response is due to memory of the system

Thus in general the output of any LTI system described by constant coefficient difference

equations can be categorized as sum of free and forced response

The Forced response of the system is due to the present and past inputs of the system

It is the output of system which is initially relaxed

The Free response of the system is due to initial conditions of system

The free response if obtained by setting the input signal to zero making the output

independent of input

The free response of the system depends only on the nature of system and initial conditions

Zero input response is a characteristics of the system itself and thus is known as natural or

free response of the system

Zero state or forced response depends on nature of system and the input signal

The total response of the system can be expressed as


In general a LTI system can be represented by following constant coefficient difference equation

N is called the order difference equation

Initial condition summarized all that we need to know about the past history of response of the system

to compute the present and future outputs of the system

A Discrete time system characterized by constant coefficient difference equations Is linear and time

invariant

Recursive system indicate a system with feedback



A non recursive system with FIR of length M

Has M zeros in the Z plane and M poles at the origin

A recursive system ,IIR system has M zeros in the Z planes

And M poles which can lie anywhere in the Z plane .

If IIR system has to be causal,the order of numerator polynomial must be less than the denominator

polynomial,

The system has more poles that zeros In the Z plane except origin

It will have N-M zeros at the origin

And since we need to have a stable system all the poles lie within the unit circle in the Z plane

System function and FIR and IIR Filters24 April 201119:50


We have described Discrete time system using linear constant coefficient difference equations

We need to realize or implemented these discrete time system in hardware of software

FIR Discrete time system

FIR discrete time system is described by the difference equation

FIR system can also be represented by system function

Unit impulse response of FIR system is identical to coefficient Bk of

Length of FIR filter or unit impulse response is M

Direct Form 1•

Cascade Form•

Frequency sampling form•

Lattice realization•

Linear phase form•

Methods of implementing FIR discrete time systems

Direct Form Realization

Direct Form 1 structure can be realized directly from non recursive difference equation

By means of convolution sum of signals x(n) and h(n)

y(n)=h(0)x(n)+h(1)x(n-1)+h(2)x(n-2)+………+h(M-1)x(n-M-1)

As Stated earlier

System simply weights by values of impulse response h(n) the most recent M samples

And sums the resulting products

The systems acts like a window that only views only most recent M samples of input signal in

forming the output


Structure for realization of Discrete time systems24 April 201119:51



Thus FIR system is said to have a finite memory of length M

Thus it requires M-1 memory locations to store previous inputs

It requires M multiplications and M-1 additions to carry out calculation for each output sample.

Impulse response of FIR system is as long as maximum delayed input term in the

difference equation or K+1 where K is number of delay elements

Maximum possible gain for FIR filter is given by sum of inputs scaled by coefficients

in its difference equations

Y(z)=H(Z)X(Z)

FIR filter is said to have order equivalent to number of delay elements .

Cascade Form

discrete time LTI system can also be represented using System function

Screen clipping taken: 24-04-2011 22:14

We factor the system function into second order FIR system so that


Where K=[(M+1)/2] give the number of second order filter sections

The filter parameter bo may be equally distributed among K filter sections or it may be assigned

to single filter section

Zeros of H(z) are grouped in pairs to obtain second order filter section.

Complex Zeros of H(z) are grouped in complex conjugate pairs to produce second order FIR

system so that second order filter coefficients are real values

Real roots or Zeros of H(z) can be grouped in any arbitrary manner

Second order filter section are implemented in direct form

Entire system is realized as cascade connection of second order system

A digital filter is a LTI system

Consider a second order filter function


Consider a second order filter function

y(n)=Aox(n)+A1x(n-1)+A2x(n-2)

H(Z)=A+A1Z^-1 + A2 Z-2

H(Z)=AoZ2 +A1Z+A2/Z2

H(z)=[h(0)Z2+h(1)Z+h(2) ] /Z2

Thus we have zeros in the Z place and Two

poles at the origin

Many FIR filters have a zero on the unit circle

Consider a biquadratic section consisting of complex conjugate zeros on which lie on the unit circle

Thus we find that no multiplication operation is required for first and the last terms

Implementing a higher order filter with many zeros on the unit circle as a cascade of biquadratic section

requires fewer total multiplications than the direct form realization

We begin by computing the coefficient of x(n) •

The coefficient bo can be distributed over K filter sections .We divide entire equation by bo and then

proceed to find the roots•

We calculate root of the polynomial in z of system function•

Find number of second order filter sections that will be required•

Order roots in terms of complex conjugate pairs •

Consider pair of roots to obtain second order section•

Find coefficients of second order filter sections•

To realize a cascade form structure from direct form structure

Frequency Sampling Structure


•A digital filter structure is said to be canonic if the number of delays in the block

diagram representation is equal to the order of the transfer function

•Otherwise, it is a noncanonic truct

A direct form realization of an FIR filter can be readily developed from the convolution

sum description

Structures in which the multiplier coefficients are precisely the coefficients of the

transfer function are called direct form structures

Digital Filter24 April 201122:09


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Discrete Time System