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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
Fourier Series Page 2
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
Fourier Series Page 3
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
Fourier Series Page 4
A periodic discrete time sequence can be
expressed as a arbitrary sum of discrete
time complex exponential sequence.
23 April 201122:07
Fourier Series Page 5
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
Fourier Series Page 6
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
Fourier Series Page 7
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
Fourier Series Page 8
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
Fourier Series Page 9
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
Fourier Series Page 10
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
Fourier Series Page 11
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
Fourier Series Page 13
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
Fourier Series Page 14
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
Fourier Series Page 16
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
Fourier Series Page 17
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
Fourier Series Page 18
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
Fourier Series Page 19
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
Fourier Series Page 20
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
Fourier Series Page 21
Fourier Series Page 22
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
Fourier Series Page 23
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
It neglects prior input samples
Structure for realization of Discrete time systems24 April 201119:51
Fourier Series Page 24
It neglects prior input samples
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
Fourier Series Page 25
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
Fourier Series Page 26
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
Fourier Series Page 27
•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
Fourier Series Page 28