digital signal processing lab manual ece students
DESCRIPTION
DSP lab using MATLAB software for all ECE studentsTRANSCRIPT
FLOWCHART:
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START
ENTER THE SIGNAL PARAMETERS (AMPLITUDE,
TIME AND FREQUENCY)
GENERATE THE WAVEFORM BY USING THE
APPROPRIATE LIBRARY FUNCTION
PLOT THE WAVEFORMS
STOP
AIM:
Write a program in MATLAB to generate the following waveforms
(Discrete – Time signal and Continuous – Time
signal)
1. Unit Impulse
sequence,
2. Unit step
sequence,
3. Unit Ramp
sequence,
4. Sinusoidal
sequence,
5. Exponential
sequence,
6. Random
sequence,
1. Pulse signal,
2. Unit step
signal
3. Ramp signal
4. Sinusoidal
signal,
5. Exponential
signal,
6. R
an
do
m
si
gn
al
APPARATUS REQUIRED:
Pentium 4 Processor, MATLAB software
THEORY:
Real signals can be quite complicated. The study of signals therefore starts with the analysis of basic and fundamental signals. For linear systems, a complicated signal and its behaviour can be studied
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EX. NO :1DATE:09-12-13EX. NO :1DATE:09-12-13 WAVEFORM GENERATIONWAVEFORM GENERATION
by superposition of basic signals. Common basic signals are:
Discrete – Time signals:
Unit impulse sequence. x n nn
( ) ( ),
1 0
0
for
, otherwise
Unit step sequence. x n u nn
( ) ( ),
1 0
0
for
, otherwise
Unit ramp sequence. x n r nn n
( ) ( ),
for
, otherwise
0
0
Sinusoidal sequence. x n A n( ) sin( ) .
Exponential sequence. x(n) = A an, where A and a are constant.
Continuous – time signals:
Unit impulse signal.
Unit step signal. Unit ramp signal.
Sinusoidal signal. .
Exponential signal. , where A
and a are constant.
LIBRARY FUNCTIONS:
clc:CLC Clear command window.CLC clears the command window and homes
the cursor.
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x t r tt t
( ) ( ),
for , otherwise
00
x t A
( ) sin(
) t
x t at
( ) = A e at
x t tt
( ) ( ),
1 00
for , otherwise
x u tt
(t ) ( ),
1 0
0
for
, otherwise
clear all:CLEAR Clear variables and functions from
memory. CLEAR removes all variables from the workspace.CLEAR
VARIABLES does the same thing.
close all:CLOSE Close figure.CLOSE, by itself, closes
the current figure window.CLOSE ALL closes all the open figure
windows.
exp:EXP Exponential.EXP(X) is the exponential of the elements of
X, e to the X.
input:INPUT Prompt for user input.R = INPUT('How many apples') gives the user the prompt in the text string and then waits for input from the keyboard. The input can be any MATLAB expression, which is evaluated,using the variables in the current workspace, and the result returned in R. If the user presses the return key without entering anything, INPUT returns an empty matrix.
linspace:LINSPACE Linearly spaced vector.LINSPACE(X1, X2) generates a row vector of 100 linearly equally spaced points between X1 and X2.
rand:The rand function generates arrays of random
numbers whose elements are uniformly distributed in the interval (0,1).
ones:
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ONES(N) is an N-by-N matrix of ones.ONES(M,N) or ONES([M,N]) is an M-by-N
matrix of ones.
zeros:ZEROS(N) is an N-by-N matrix of Zeros.ZEROS(M,N) or ZEROS([M,N]) is an M-by-
N matrix of zeros
plot:PLOT Linear plot.PLOT(X,Y) plots vector Y versus vector X. If X or Y is a matrix, then the vector is plotted versus the rows or columns of the matrix, whichever line up.
subplot:SUBPLOT Create axes in tiled positions.H = SUBPLOT(m,n,p), or SUBPLOT(mnp), breaks the Figure window into an m-by-n matrix of small axes, selects the p-th axes for the current plot, and returns the axis handle. The axes are counted along the top row of the Figure window, then the second row, etc.
stem:STEM Discrete sequence or "stem" plot.STEM(Y) plots the data sequence Y as stems from the x axis terminated with circles for the data value.STEM(X,Y) plots the data sequence Y at the
values specified in X.
title:TITLE Graph title.TITLE('text') adds text at the top of the current
axis.
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xlabel:XLABEL X-axis label.XLABEL('text') adds text beside the X-axis on
the current axis.
ylabel:YLABEL Y-axis label.YLABEL('text') adds text beside the Y-axis on
the current axis.
ALGORITHM/PROCEDURE:
1. Start the program
2. Get the inputs for signal generation
3. Use the appropriate library function
4. Display the waveform
Source code :
%WAVE FORM GENERATION%CT SIGNAL%UNIT IMPULSEclc;clear all;close all;t1=-3:1:3;x1=[0,0,0,1,0,0,0];subplot(2,3,1);plot(t1,x1);xlabel('time');ylabel('Amplitude');title('Unit impulse signal');%UNIT STEP SIGNALt2=-5:1:25;x2=[zeros(1,5),ones(1,26)];subplot(2,3,2);plot(t2,x2);
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xlabel('time');ylabel('Amplitude');title('Unit step signal');%EXPONENTIAL SIGNALa=input('Enter the value of a:');t3=-10:1:20;x3=exp(-1*a*t3);subplot(2,3,3);plot(t3,x3);xlabel('time');ylabel('Amplitude');title('Exponential signal');%UNIT RAMP SIGNALt4=-10:1:20;x4=t4;subplot(2,3,4);plot(t4,x4);xlabel('time');ylabel('Amplitude');title('Unit ramp signal');%SINUSOIDAL SIGNALA=input('Enter the amplitude:');f=input('Enter the frequency:');t5=-10:1:20;x5=A*sin(2*pi*f*t5);subplot(2,3,5);plot(t5,x5)xlabel('time');ylabel('Amplitude');title('Sinusoidal signal');%RANDOM SIGNALt6=-10:1:20;x6=rand(1,31);subplot(2,3,6);plot(t6,x6);xlabel('time');ylabel('Amplitude');title('Random signal');
%WAVE FORM GENERATION
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%DT SIGNAL%UNIT IMPULSEclc;clear all;close all;n1=-3:1:3;x1=[0,0,0,1,0,0,0];subplot(2,3,1);stem(n1,x1);xlabel('time');ylabel('Amplitude');title('Unit impulse signal');%UNIT STEP SIGNALn2=-5:1:25;x2=[zeros(1,5),ones(1,26)];subplot(2,3,2);stem(n2,x2);xlabel('time');ylabel('Amplitude');title('Unit step signal');%EXPONENTIAL SIGNALa=input('Enter the value of a:');n3=-10:1:20;x3=power(a,n3);subplot(2,3,3);stem(n3,x3);xlabel('time');ylabel('Amplitude');title('Exponential signal');%UNIT RAMP SIGNALn4=-10:1:20;x4=n4;subplot(2,3,4);stem(n4,x4);xlabel('time');ylabel('Amplitude');title('Unit ramp signal');%SINUSOIDAL SIGNALA=input('Enter the amplitude:');f=input('Enter the frequency:');
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n5=-10:1:20;x5=A*sin(2*pi*f*n5);subplot(2,3,5);stem(n5,x5);xlabel('time');ylabel('Amplitude');title('Sinusoidal signal');%RANDOM SIGNALn6=-10:1:20;x6=rand(1,31);subplot(2,3,6);stem(n6,x6);xlabel('time');ylabel('Amplitude');title('Random signal');
CONTINUOUS TIME:
DISCRETE TIME :
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OUTPUT WAVEFORM(CONTINUOUS TIME):
OUTPUT WAVEFORM (DISCRETE TIME):
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RESULT:
The program to generate various
waveforms is written, executed and the output is
verified.
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FLOWCHART:
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START
READ THE INPUT SEQUENCE
PLOT THE WAVEFORMS
STOP
READ THE CONSANT FOR (SCALAR) AMPLITUDE
AND TIME MANIPULATION
READ THE (VECTOR) SEQUENCE FOR SIGNAL
ADDTION AND MULTIPLICATION
PERFORM OPERTAION ON THE D.T. SIGNAL
AIM:
Write a program in MATLAB to study the
basic operations on the Discrete – time signals.
(Operation on dependent variable (amplitude
manipulation) and Operation on independent variable
(time manipulation)).
APPARATUS REQUIRED:
Pentium 4 Processor, MATLAB software
THEORY:
Let x(n) be a sequence with finite length.
1. Amplitude manipulation
Amplitude scaling:y[n] =ax[n], where a is a
constant.
If a > 1, then y[n] is
amplified sequence
If a < 1, then y[n] is
attenuated sequence
If a = - 1, then y[n] is
amplitude reversal
sequence
Offset the signal: y[n] =a+x[n], where a is a
constant
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EX. NO :2DATE :21-12-13EX. NO :2DATE :21-12-13 BASIC OPERATIONS ON D.T SIGNALSBASIC OPERATIONS ON D.T SIGNALS
Two signals x1[n] and x2[n] can also be added
and multiplied: By adding the values y1[n]=
x1[n] + x2[n] at each corresponding sample
and by multiplying the values y2[n]= x1[n] X
x2[n] at each corresponding sample.
2. Time manipulation
Time scaling: y[n]=x[an],
where a is a constant.
Time shifting: y[n]=x[n - ],
where is a constant.
Time reflection (folding):y[n]=x[-n]
Arithmetic Operations
* Matrix multiplication
.* Array multiplication (element-wise)
LIBRARY FUNCTIONS:date Current date as date string.
S = date returns a string containing the date in
dd-mmm-yyyy format
tic & toc Start a stopwatch timer.
The sequence of commands
TIC, operation, TOC
prints the number of seconds required for the
operation.
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Fprintf
Write formatted data to file. The special
formats \n,\r,\t,\b,\f can be used to produce
linefeed, carriage return, tab, backspace, and
formfeed characters respectively.
Use \\ to produce a backslash character and %
% to produce the percent character.
ALGORITHM/PROCEDURE:
1. Start the program
2. Get the input for signal manipulation
3. Use the appropriate library function
4. Display the waveform
Source code :
clc;clear all;close all;%operations on the amplitude of signalx=input('Enter input sequence:');a=input('Enter amplification factor:');b=input('Enter attenuation factor:');c=input('Enter amplitude reversal factor:');y1=a*x;y2=b*x;y3=c*x;n=length(x);subplot(2,2,1);stem(0:n-1,x);
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xlabel('time');ylabel('amplitude');title('Input signal');subplot(2,2,2);stem(0:n-1,y1);xlabel('time');ylabel('Amplitude');title('Amplified signal');subplot(2,2,3);stem(0:n-1,y2);xlabel('time');ylabel('Amplitude');title('Attenuated signal');subplot(2,2,4);stem(0:n-1,y3);xlabel('time');ylabel('Amplitude');title('Amplitude reversal signal');%scalar additiond=input('Input the scalar to be added:');y4=d+x;figure(2);stem(0:n-1,y4);xlabel('time');ylabel('Amplitude');title('Scalar addition signal');
clc;clear all;close all;%Operations on the independent variable%Time shifting of the independent variablex=input('Enter the input sequence:');n0=input('Enter the +ve shift:');n1=input('Enter the -ve shift:');l=length(x);subplot(2,2,1);stem(0:l-1,x);xlabel('time');
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ylabel('Amplitude');title('Input signal');i=n0:l+n0-1;j=n1:l+n1-1;subplot(2,2,2);stem(i,x);xlabel('time');ylabel('Amplitude');title('Positive shifted signal');subplot(2,2,3);stem(j,x);xlabel('time');ylabel('Amnplitude');title('Negative shifted signal');%Time reversalsubplot(2,2,4);stem(-1*(0:l-1),x);xlabel('time');ylabel('Amplitude');title('Time reversal signal');
clc;clear all;close all;%Arithmetic operations on signals%Addition and multiplication of two signalsx1=input('Enter the sequence of first signal:');x2=input('Enter the sequence of second signal:');l1=length(x1);l2=length(x2);subplot(2,2,1);stem(0:l1-1,x1);xlabel('time');ylabel('Amplitude');title('Input sequence 1');subplot(2,2,2);stem(0:l2-1,x2);
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xlabel('time');ylabel('Amplitude');title('Input sequence 2');if l1>l2 l3=l1-l2; x2=[x2,zeros(1,l3)]; y1=x1+x2; subplot(2,2,3); stem(0:l1-1,y1);xlabel('time');ylabel('Amplitude');title('Addition of two signals');y2=x1.*x2; subplot(2,2,4); stem(0:l1-1,y2);xlabel('time');ylabel('Amplitude');title('Multiplication of two signals');endif l2>l1 l3=l2-l1; x1=[x1,zeros(1,l3)]; y1=x1+x2; subplot(2,2,3); stem(0:l2-1,y1);xlabel('time');ylabel('Amplitude');title('Addition of two signals');y2=x1.*x2; subplot(2,2,4); stem(0:l2-1,y2);xlabel('time');ylabel('Amplitude');title('Multiplication of two signals');else y1=x1+x2; subplot(2,2,3); stem(0:l1-1,y1);xlabel('time');ylabel('Amplitude');
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title('Addition of two signals');y2=x1.*x2; subplot(2,2,4); stem(0:l1-1,y2); xlabel('time');ylabel('Amplitude');title('Multiplication of two signals');end
operations on the amplitude of signal :
Time shifting of the independent variable :
Addition and multiplication of two signals :
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OUTPUT (operations on the amplitude of
signal ):
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OUTPUT(Time shifting of the independent
variable) :
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OUTPUT(Addition and multiplication of two
signals) :
RESULT:
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The program to perform various operations on
discrete time signal is written, executed and the
output is verified
FLOWCHART:
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START
READ THE INPUT SEQUENCE
PLOT THE WAVEFORMS
STOP
READ THE CONSANT FOR (SCALAR) AMPLITUDE
AND TIME MANIPULATION
READ THE (VECTOR) SEQUENCE FOR SIGNAL
ADDTION AND MULTIPLICATION
PERFORM OPERTAION ON THE D.T. SIGNAL USING
THE APPROPRIATE LIBRARY FUNCTION
AIM:
To check for linearity, Causality and stability
of various systems given bellow:
Linearity: System1 n.X(n), System2 An.X2(n)+B
System3: Log (X),sin(x),5X(n) …etc
Causality: System1 U(-n) System2 X(n-4)+U(n+5)
Stability: System1 Z / (Z2 + 0.5 Z+1)
APPARATUS REQUIRED:
Pentium 4 Processor, MATLAB software
THEORY:
LINEARITY:
The response of the system to a weighted sum
of signals is equal to the corresponding weighted
sum of the responses (outputs) of the system to each
of the individual input signals.
METHODS OF PROOF:
Individual inputs are applied and the weighted
sum of the outputs is taken. Then the weighted sum
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EX. NO : 3DATE: 06-1-14
EX. NO : 3DATE: 06-1-14
PROPERTIES OF DISCRETE TIME SYSTEMPROPERTIES OF DISCRETE TIME SYSTEM
of signals input is applied to the system and the two
outputs are checked to see if they are equal.
CAUSALITY:
A system is said to be causal, if the output of
the system at any time n(y(n)) depends only on the
present and past inputs and past outputs [x(n),x(n-1)
…….y(n-1),…..]
But does not depend on future inputs [x
(n+1),x(n+2),…..]
y(n) = F[ x(n),x(n-1),x(n-2)….] F[ ] –
Arbitrary function.
METHODS OF PROOF:
1. If the difference equation is given, the
arguments of the output y (n) are compared
with the arguments (time instant) of the input
signals. In the case of only present and past
inputs, the system is causal. If future inputs are
present then the system is non-causal.
2. If the impulse response is given, then it is
checked whether all the values of h (n) for
negative values of n are zero. (i.e.) if h(n)=0,
for <0. If this is satisfied, then the system is
causal.
STABILITY:
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An arbitrary relaxed system is said to be
bounded input – bounded output (BIBO) stable, if
and only if every bounded input produces a bounded
output.
METHODS OF PROOF:
1. If the impulse response is given, then the
summation of responses for n ranging from -
to + is taken and if the sum is finite, the
system is said to be BIBO stable.
2. It the transfer function of the system is given,
the poles of the transfer function is plotted. If
all the poles lie within the unit circle, the
system is stable.
A single order pole on the boundary of
unit circle makes the systems marginally
stable. If there are multiple order poles on the
boundary of unit circle, the system is
unstable.If any pole is lying outside the unit
circle, the system is unstable.
LIBRARY FUNCTION:
.^ Array power.
Z = X.^Y denotes element-by-element
powers. X and Y must have the same
dimensions unless one is a scalar. A scalar
can operate into anything.
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C = POWER(A,B) is called for the syntax
'A .^ B' when A or B is an object.
residuez Z-transform partial-fraction
expansion.
[R,P,K] = residuez(B,A) finds the residues,
poles and direct terms of the partial-fraction
expansion of B(z)/A(z),
zplane Z-plane zero-pole plot.
zplane(Z,P) plots the zeros Z and poles P (in
column vectors) with the unit circle for
reference. Each zero is represented with a 'o'
and each pole with a 'x' on the plot.
tf Creation of transfer functions or conversion
to transfer function.
SYS = tf(NUM,DEN,TS) creates a discrete-
time transfer function with sample time TS
(set TS=1 to get it in z). Z = tf('z',TS) specifies
H(z) = z with sample time TS.
syms Short-cut for constructing symbolic
objects.
subs Symbolic substitution. subs(S,NEW)
replaces the free symbolic variable in S with
NEW.
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ALGORITHM/PROCEDURE:
1. Click on the MATLAB icon on the desktop (or
go to start – all programs and click
on MATLAB) to get into the Command
Window
2. Type ‘edit’ in the MATLAB prompt ‘>>’ that
appears in the Command window.
3. Write the program in the ‘Edit’ window and
save it in ‘M-file’
4. Run the program
5. Enter the input in the command window
6. The result is displayed in the Command
window and the graphical output is displayed
in the Figure Window
Source code :1
clc;clear all;close all; %Properties of DT Systems(Linearity)%y(n)=[x(n)]^2+B; x1=input('Enter first input sequence:');n=length(x1);x2=input('Enter second input sequence:');a=input('Enter scaling constant(a):');b=input('Enter scaling constant(b):');B=input('Enter scaling constant(B):');
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y1=power(x1,2)+B;y2=power(x2,2)+B;rhs=a*y1+b*y2;x3=a*x1+b*x2;lhs=power(x3,2)+B; subplot(2,2,1);stem(0:n-1,x1);xlabel('Time');ylabel('Amplitude');title('First input sequence');subplot(2,2,2);stem(0:n-1,x2);xlabel('Time');ylabel('Amplitude');title('Second input sequence');subplot(2,2,3);stem(0:n-1,lhs);xlabel('Time');ylabel('Amplitude');title('LHS');subplot(2,2,4);stem(0:n-1,rhs);xlabel('Time');ylabel('Amplitude');title('RHS'); if(lhs==rhs) display('system is linear');else display('system is non-linear');end; Source code :2
clc;clear all;close all; %Properties of DT Systems(Linearity)%y(n)=x(n);
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x1=input('Enter first input sequence:');x2=input('Enter second input sequence:');a=input('Enter scaling constant(a):');b=input('Enter scaling constant(b):'); subplot(2,2,1);stem(x1);xlabel('time');ylabel('Amplitude');title('First signal');subplot(2,2,2);stem(x2);xlabel('time');ylabel('Amplitude');title('Second signal'); y1=x1;y2=x2;rhs=a*y1+b*y2;x3=a*x1+b*x2;lhs=x3; if(lhs==rhs) display('system is linear');else display('system is non-linear');end;subplot(2,2,3);stem(lhs);xlabel('time');ylabel('Amplitude');title('L.H.S');subplot(2,2,4);stem(rhs);xlabel('time');ylabel('Amplitude');title('R.H.S');
Source code :3
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clc;clear all;close all; %Properties of DT Systems(Time Invariance)%y(n)=x(n); x1=input('Enter input sequence x1:');n0=input('Enter shift:');x2=[zeros(1,n0),x1];y1=x1;y2=x2;y3=[zeros(1,n0),y1]; if(y2==y3) display('system is time invariant');else display('system is time variant');end;subplot(2,2,1);stem(x1);xlabel('time');ylabel('Amplitude');title('Input signal');subplot(2,2,2);stem(x2);xlabel('time');ylabel('Amplitude');title('Signal after shift');subplot(2,2,3);stem(y2);xlabel('time');ylabel('Amplitude');title('L.H.S');subplot(2,2,4);stem(y3);xlabel('time');ylabel('Amplitude');title('R.H.S');
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Source code :4
clc;clear all;close all; %Properties of DT Systems(Time Invariance)%y(n)=n*[x(n)]; x1=input('Enter input sequence x1:');n1=length(x1);for n=1:n1 y1(n1)=n.*x1(n);end;n0=input('Enter shift:');x2=[zeros(1,n0),x1];for n2=1:n1+n0 y2(n2)=n2.*x2(n2);end;y3=[zeros(1,n0),y1]; if(y2==y3) display('system is time invariant');else display('system is time variant');end;subplot(2,2,1);stem(x1);xlabel('time');ylabel('Amplitude');title('Input signal');subplot(2,2,2);stem(x2);xlabel('time');ylabel('Amplitude');title('Signal after shift');subplot(2,2,3);stem(y2);xlabel('time');ylabel('Amplitude');
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title('L.H.S');subplot(2,2,4);stem(y3);xlabel('time');ylabel('Amplitude');title('R.H.S');
Source code :5
clc;clear all;close all; %Properties of DT Systems(Causality)%y(n)=x(-n); x1=input('Enter input sequence x1:');n1=input('Enter lower limit n1:');n2=input('Enter lower limit n2:');flag=0;for n=n1:n2 arg=-n; if arg>n; flag=1;end;end; if(flag==1) display('system is causal');else display('system is non-causal');end;
Source code :6
disp('stability');nr=input('input the numerator coefficients:');dr=input('input the denominator coefficients:');z=tf(nr,dr,1);[r,p,k]=residuez(nr,dr);figure
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zplane(nr,dr);if abs(p)<1 disp('the system is stable');else disp('the system is unstable');end;Non-Linear System :1
Output :
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Linear System :2
Output :
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Time invariant system :3
Output :
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Time variant system : 4
Output :
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Non-Causal system :5
Unstable system :6
Output :
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RESULT:
The properties of Discrete – Time system is
verified using MATLAB
FLOWCHART:
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EX. NO : 4DATE: 13-1-14
EX. NO : 4DATE: 13-1-14
SAMPLING RATE CONVERSIONSAMPLING RATE CONVERSION
START
ENTER THE SIGNAL PARAMETERS (AMPLITUDE,
TIME AND FREQUENCY)
FIND THE SPECTRUM OF
ALL THE SIGNALS
PLOT THE WAVEFORMS
STOP
PERFORM THE SAMPLING RATE CONVERSION
ON THE INPUT BY USING UPSAMPLE,
DOWNSAMPLE AND RESAMPLE
PERFORM INTERPOLATION AND DECIMATION
ON THE INPUT
AIM:
Write a MATLAB Script to perform sampling
rate conversion for any given arbitrary sequence
(D.T) or signal (C.T) by interpolation, decimation,
upsampling, downsampling and resampling (i.e.
fractional value)
.
APPARATUS REQUIRED:
PC, MATLAB software
THEORY:
SAMPLING PROCESS:
It is a process by which a continuous time
signal is converted into discrete time signal. X[n] is
the discrete time signal obtained by taking samples
of the analog signal x(t) every T seconds, where T is
the sampling period.
X[n] = x (t) x p (t)
Where p(t) is impulse train; T – period of
the train
SAMPLING THEOREM:
It states that the band limited signal x(t)
having no frequency components above Fmax Hz is
specified by the samples that are taken at a uniform
rate greater than 2 Fmax Hz (Nyquist rate), or the
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frequency equal to twice the highest frequency of
x(t).
Fs ≥ 2 Fmax
SAMPLING RATE CONVERSION:
Sampling rate conversion is employed to
generate a new sequence with a sampling rate higher
or lower than that of a given sequence. If x[n] is a
sequence with a sampling rate of F Hz and it is used
to generate another sequence y[n] with desired
sampling rate F’ Hz, then the sampling rate alteration
is given by,
F’/F = R
If R > 1, the process is called interpolation and
results in a sequence with higher sampling rate. If R<
1, the process is called decimation and results in a
sequence with lower sampling rate.
DOWNSAMPLE AND DECIMATION:
Down sampling operation by an integer factor
M (M>1) on a sequence x[n] consists of keeping
every Mth sample of x[n] and removing M-1 in
between samples, generating an output sequence y[n]
according to the relation
y [n] = x[nM]
y [n] – sampling rate is 1/M that of x[n]
If we reduce the sampling rate, the resulting signal
will be an aliased version of x[n]. To avoid aliasing,
the bandwidth of x[n] must be reduced to Fmax =Fx/2 π
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or max = π /M. The input sequence is passed through
LPF or an antialiasing filter before down sampling.
x [n] -
y[n]
UPSAMPLE AND INTERPOLATION:
Upsampling by an integer factor L (L > 1) on a
sequence x[n] will insert (L–1) equidistant samples
between an output sequence y[n] according to the
relation
x[n/L], n = 0, 1, 2 ….
y[n] = 0, otherwise
The sampling rate of y[n] is L times that of x[n].
The unwanted images in the spectra of sampled
signal must be removed by a LPF called anti-
imaging filter. The input sequence is passed through
an anti-imaging filter after up sampling.
x[n]
y[n]
SAMPLING RATE CONVERSION BY A
RATIONAL FACTOR I/O:
We achieve this conversion, by first
performing interpolation by the factor I and then
decimating the output of interpolator by the factor D,
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ANTIALIASINGFILTER H (Z)
M
L ANTI IMAGING FILTER H (Z)
interpolation has to be performed before decimation
to obtain the new rational sampling rate.
x[n]
y[n]
LIBRARY FUNCTIONS:
resample: Changes sampling rate by
any rational factor.
y = resample (x,p,q) resamples the sequence in
vector x at p/q times the original sampling rate, using
a polyphase filter implementation. p and q must be
positive integers. The length of y is equal to ceil
(length(x)*p/q).
interp: Increases sampling rate by an
integer factor (interpolation)
y = interp (x,r) increases the sampling rate of x by a
factor of r. The interpolated vector y is r times longer
than the original input x. ‘interp’ performs low pass
interpolation by inserting zeros into the original
sequence and then applying a special low pass filter.
upsample: Increases the sampling rate
of the input signal
y = upsample(x,n) increases the sampling rate of x
by inserting n-1 zeros between samples. The
upsampled y has length(x)*n samples
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UPSAMPLER ANTI
IMAGING FILTER
ANTI ALIASING
FILTER
DOWN SAMPLER
decimate: Decreases the sampling rate
for a sequence (decimation).
y = decimate (x, r) reduces the sample rate of x by a
factor r. The decimated vector y is r times shorter in
length than the input vector x. By default, decimate
employs an eighth-order low pass Chebyshev Type I
filter. It filters the input sequence in both the forward
and reverse directions to remove all phase distortion,
effectively doubling the filter order.
downsample: Decreases the sampling
rate of the input signal
y = downsample(x,n) decreases the sampling rate of
x by keeping every nth sample starting with the first
sample. The downsampled y has length(x)/n samples
ALGORITHM/PROCEDURE:
1. Generate a sinusoidal waveform
2. Using the appropriate library function for
interpolation ,decimation ,upsampling ,
downsampling and resampling, perform
sampling rate conversion for the sinusoidal
waveform
3. Find the spectrum of all the signals and
compare them in frequency domain.
4. Display the resultant waveforms
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Source code :
clc;clear all;close all;%continuous sinusoidal signala=input('Enter the amplitude:');f=input('Enter the Timeperiod:');t=-10:1:20;x=a*sin(2*pi*f*t);subplot(2,3,1);plot(t,x);xlabel('time');ylabel('Amplitude');title('Sinusoidal signal');%decimating the signald=input('Enter the value by which the signal is to be decimated:');y1=decimate(x,d);subplot(2,3,2);stem(y1);xlabel('time');ylabel('Amplitude');title('Decimated signal');%interpolating the signali=input('Enter the value by which the signal is to be interpolated:');y2=interp(x,i);subplot(2,3,3);stem(y2);xlabel('time');ylabel('Amplitude');title('Interpolated signal');%resampling the signaly3=resample(x,3,2);subplot(2,3,4);stem(y3);xlabel('time');
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ylabel('Amplitude');title('Resampled signal');%downsampling the signaly4=downsample(x,2);subplot(2,3,5);stem(y4);xlabel('time');ylabel('Amplitude');title('Downsampled signal');%upsampling the signaly5=upsample(x,3);subplot(2,3,6);stem(y5);xlabel('time');ylabel('Amplitude');title('Upsampled signal');
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Output :
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RESULT:
The program written using library functions
and the sampling rate conversion process is studied.
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FLOWCHART:
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START
ENTER THE INPUT SEQUENCE x[n] & SYSTEM
RESPONSE h[n]
PERFORM LINEAR AND CIRCULAR
CONVOLUTION IN TIME DOMAIN
PLOT THE WAVEFORMS AND ERROR
STOP
AIM:
Write a MATLAB Script to perform discrete
convolution (Linear and Circular) for the given two
sequences and also prove by manual calculation.
APPARATUS REQUIRED:
PC, MATLAB software
THEORY:
LINEAR CONVOLUTION:
The response y[n] of a LTI system for any
arbitrary input x[n] is given by convolution of
impulse response h[n] of the system and the arbitrary
input x[n].
y[n] = x[n]*h[n] = or
If the input x[n] has N1 samples and impulse
response h[n] has N2 samples then the output
sequence y[n] will be a finite duration sequence
consisting of (N1 + N2 - 1) samples. The convolution
results in a non periodic sequence called Aperiodic
convolution.
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EX. NO : 5DATE : 20-1-14
EX. NO : 5DATE : 20-1-14
DISCRETE CONVOLUTIONDISCRETE CONVOLUTION
STEPS IN LINEAR CONVOLUTION:
The process of computing convolution between x[k]
and h[k] involves four steps.
1. Folding : Fold h[k] about k=0 to obtain h[-
k]
2. Shifting : Shift h[-k] by ‘n0’to right if ‘n0’ is
positive and shift h[-k] by ‘n0’ to the left if
‘n0’ is negative. Obtain h[n0-k]
3. Multiplication : Multiply x[k] by h[n0-k] to
obtain the product sequence
yn0 [k] = x[k] h [n0 –k]
4. Summation : Find the sum of all the values
of the product sequence to obtain values of
output at n = n0
Repeat steps 2 to 4 for all possible time
shifts ‘n0’ in the range - <n<
C IRCULAR CONVOLUTION The convolution of two periodic sequences
with period N is called circular convolution of two
signals x1[n] and x2[n] denoted by
y[n] = x1[n] * x2[n] = [(n-k) mod N] x2
(k) or x2 [(n-k) mod N]
where x1[(n-k) mod N] is the reflected and circularly
translated version of x1[n].
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x1[n] * x2[n] = IDFTN { DFTN (x1[n] ) . DFTN
(x2[n])}
It can be performed only if both the sequences
consist of equal number of samples. If the sequences
are different in length then convert the smaller size
sequence to that of larger size by appending zeros
METHODS FOR CIRCULAR CONVOLUTION :
Matrix Multiplication Method and Concentric Circle Method
LIBRARY FUNCTION:
conv: Convolution and polynomial
multiplication.
C = conv (A, B) convolves vectors A and B.
The resulting vector C’s length is given by
length(A)+length(B)-1. If Aand B are vectors
of polynomial coefficients, convolving them is
equivalent to multiplying the two polynomials
in frequency domain.
length: Length of vector.
length(X) returns the length of vector X. It is
equivalent to size(X) for non empty arrays and 0 for
empty ones.
fft: Discrete Fourier transform.
fft(x) is the Discrete Fourier transform
(DFT) of vector x. For matrices, the ‘fft’
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operation is applied to each column. For N-D
arrays, the ‘fft’ operation operates on the first
non-single dimension. fft(x,N) is the N-point
FFT, padded with zeros if x has less than N
points and truncated if it has more.
ifft: Inverse Discrete Fourier
transform.
ifft(X) is the Inverse Discrete Fourier
transform of X.
ifft(X,N) is the N-point Inverse Discrete
Fourier transform of X.
ALGORITHM/PROCEDURE:
LINEAR CONVOLUTION:
1. Enter the sequences (Input x[n] and the
Impulse response h[n])
2. Perform the linear convolution between x[k]
and h[k] and obtain y[n].
3. Find the FFT of x[n] & h[n].Obtain X and H
4. Multiply X and H to obtain Y
5. Find the IFFT of Y to obtain y’[n]
6. Compute error in time domain e=y[n]-y’[n]
7. Plot the Results
CIRCULAR CONVOLUTION
1. Enter the sequences (input x[n] and the
impulse response h[n])
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2. Make the length of the sequences equal by
padding zeros to the smaller length sequence.
3. Perform the circular convolution between x[k]
and h[k]and obtain y[n].
4. Find the FFT of x[n] & h[n].Obtain X and H
5. Multiply X and H to obtain Y
6. Find the IFFT of Y to obtain y’[n]
7. Compute error in time domain e=y[n]-y’[n]
8. Plot the Results
SOURCE CODE : 1
clc;clear all;close all;%Program to perform Linear Convolution x1=input('Enter the first sequence to be convoluted:');subplot(3,1,1);stem(x1);xlabel('Time');ylabel('Amplitude');title('First sequence'); x2=input('Enter the second sequence to be convoluted:');subplot(3,1,2);stem(x2);xlabel('Time');ylabel('Amplitude');title('Second sequence'); f=conv(x1,x2);disp('The Linear convoluted sequence is');
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disp(f);subplot(3,1,3);stem(f);xlabel('Time');ylabel('Amplitude');title('Linear Convoluted sequence');
Command window :
OUTPUT :
SOURCE CODE :2
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clc;clear all;close all;%Program to perform Circular Convolution x1=input('Enter the first sequence to be convoluted:');subplot(3,1,1);l1=length(x1);stem(x1);xlabel('Time');ylabel('Amplitude');title('First sequence'); x2=input('Enter the second sequence to be convoluted:');subplot(3,1,2);l2=length(x2);stem(x2);xlabel('Time');ylabel('Amplitude');title('Second sequence'); if l1>l2 l3=l1-l2; x2=[x2,zeros(1,l3)];elseif l2>l1 l3=l2-l1; x1=[x1,zeros(1,l3)];end
f=cconv(x1,x2);
disp('The Circular convoluted sequence is');disp(f);subplot(3,1,3);stem(f);xlabel('Time');ylabel('Amplitude');
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title('Circular Convoluted sequence');Command window :
OUTPUT :
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SOURCE CODE :3
clc;clear all;close all;%Program to perform Linear Convolution using Circular Convolution x1=input('Enter the first sequence to be convoluted:');subplot(3,1,1);l1=length(x1);stem(x1);xlabel('Time');ylabel('Amplitude');title('First sequence'); x2=input('Enter the second sequence to be convoluted:');subplot(3,1,2);l2=length(x2);stem(x2);xlabel('Time');ylabel('Amplitude');title('Second sequence'); if l1>l2 l3=l1-l2; x2=[x2,zeros(1,l3)];elseif l2>l1 l3=l2-l1; x1=[x1,zeros(1,l3)];endn=l1+l2-1;f=cconv(x1,x2,n);disp('The convoluted sequence is');disp(f);subplot(3,1,3);stem(f);
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xlabel('Time');ylabel('Amplitude');title('Convoluted sequence');
Command window :
OUTPUT:
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SOURCE CODE :4clc;clear all;close all;%Program to perform Linear Convolution x=input('Enter the first input sequence:');l1=length(x);subplot(3,2,1);stem(x);xlabel('Time index n---->');ylabel('Amplitude');title('input sequence'); h=input('Enter the system response sequence:');l2=length(h);subplot(3,2,2);stem(h);xlabel('Time index n---->');ylabel('Amplitude');title('system response sequence'); if l1>l2 l3=l1-l2; h=[h,zeros(1,l3)];elseif l2>l1 l3=l2-l1; x=[x,zeros(1,l3)];end y=conv(x,h);disp('The time domain convoluted sequence is:');disp(y);subplot(3,2,3);stem(y);xlabel('Time index n---->');
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ylabel('Amplitude');title('convoluted output sequence'); X=fft(x,length(y));H=fft(h,length(y));Y=X.*H;disp('The frequency domain multiplied sequence is:');disp(Y);subplot(3,2,4);stem(Y);xlabel('Time index n---->');ylabel('Amplitude');title('frequency domain multiplied response'); y1=ifft(Y,length(Y));disp('The inverse fourier transformed sequence is:');disp(y1);subplot(3,2,5);stem(y1);xlabel('Time index n---->');ylabel('Amplitude');title('output after inverse fourier transform'); e=y-y1;disp('The Error sequence:')disp(abs(e));subplot(3,2,6);stem(abs(e));xlabel('Time index n---->');ylabel('Amplitude');title('error sequence');
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Command window :
OUTPUT :
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SOURCE CODE :5
clc;clear all;close all;%PROGRAM FOR CIRCULAR CONVOLUTION USING DISCRETE CONVOLUTION EXPRESSION x=input('Enter the first sequence:');n1=length(x);h=input('Enter the second sequence:');n2=length(h);n=0:1:n1-1;subplot(3,1,1);stem(n,x);xlabel('Time');ylabel('Amplitude');title('First sequence Response x(n)');n=0:1:n2-1;subplot(3,1,2);stem(n,h);
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xlabel('Time');ylabel('Amplitude');title('Second sequence Response h(n)');n=n1+n2-1;if n1>n2 n3=n1-n2; h=[h,zeros(1,n3)];elseif n2>n1 n3=n2-n1; x=[x,zeros(1,n3)];endx=[x,zeros(1,n-n1)];h=[h,zeros(1,n-n2)];for n=1:n y(n)=0; for i=1:n j=n-i+1; if(j<=0) j=n+j; end y(n)=y(n)+x(i)*h(j); endenddisp('Circular Convolution of x&h is');disp(y);subplot(3,1,3);stem(y);xlabel('Time');ylabel('Amplitude');title('Circular Convoluted sequence Response');
Command window :
OUTPUT :
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RESULT:
The linear and circular convolutions are
performed by using MATLAB script and the
program results are verified by manual calculation.
FLOWCHART:
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START
ENTER THE INPUT SEQUENCE
PERFORM THE DFT USING IN-BUILT FFT
AND USING DIRECT FORMULA ON THE
GIVEN INPUT SEQUENCE
PLOT THE WAVEFORMS AND ERROR
STOP
AIM:
Write a MATLAB program to perform the
Discrete Fourier Transform for the given sequences.
.
APPARATUS REQUIRED:
PC, MATLAB software
THEORY:
DISCRETE FOURIER TRANSFORM
Fourier analysis is extremely useful for data
analysis, as it breaks down a signal into constituent
sinusoids of different frequencies. For sampled
vector data Fourier analysis is performed using the
Discrete Fourier Transform (DFT).
The Discrete Fourier transform computes the
values of the Z-transform for evenly spaced points
around the circle for a given sequence.
If the sequence to be represented is of finite
duration i.e. it has only a finite number of non-zero
values, the transform used is Discrete Fourier
transform.
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EX. NO : 6DATE : 27-1-14
EX. NO : 6DATE : 27-1-14
DISCRETE FOURIER TRANSFORM DISCRETE FOURIER TRANSFORM
It finds its application in Digital Signal
processing including Linear filtering, Correlation
analysis and Spectrum analysis.
Consider a complex series x [n] with N samples of
the form Where x is a
complex number Further,
assume that the series outside the range 0, N-1 is
extended N-periodic, that is, xk = xk+N for all k. The
FT of this series is denoted as X (k) and has N
samples. The forward transform is defined as
The inverse transform is defined as
Although the functions here are described as
complex series, setting the imaginary part to 0 can
represent real valued series. In general, the transform
into the frequency domain will be a complex valued
function, that is, with magnitude and phase.
LIBRARY FUNCTIONS:
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exp: Exponential Function.
exp (X) is the exponential of the elements of X, e to the power X. For complex Z=X+i*Y, exp (Z) = exp(X)*(COS(Y) +i*SIN(Y)).
disp: Display array.
disp (X) is called for the object X when the semicolon is not used to terminate a statement.
max: Maximum elements of an array
C = max (A, B) returns an array of the same size as A and B with the largest elements taken from A or B.
fft: Discrete Fourier transform.
fft(x) is the discrete Fourier transform (DFT)
of vector x. For the matrices, the FFT
operation is applied to each column. For N-
Dimensional arrays, the FFT operation
operates on the first non-singleton
dimension.
ALGORITHM/PROCEDURE:
1. Click on the MATLAB icon on the desktop (or
go to Start - All Programs and click on
MATLAB) to get into the Command Window
2. Type ‘edit’ in the MATLAB prompt ‘>>’ that
appears in the Command window.
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3. Write the program in the ‘Edit’ window and
save it as ‘m-file’
4. Run the program
5. Enter the input in the command window
6. The result is displayed in the Command
window and the graphical output is displayed
in the Figure Window
Source Code :
clc;clear all;close all;%Get the sequence from user disp('The sequence from the user:');xn=input('Enter the input sequence x(n):'); % To find the length of the sequenceN=length(xn); %To initilise an array of same size as that of input sequenceXk=zeros(1,N);iXk=zeros(1,N); %code block to find the DFT of the sequencefor k=0:N-1 for n=0:N-1 Xk(k+1)=Xk(k+1)+(xn(n+1)*exp((-i)*2*pi*k*n/N)); endend %code block to plot the input sequencet=0:N-1;subplot(3,2,1);
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stem(t,xn);ylabel ('Amplitude');xlabel ('Time Index');title ('Input Sequence'); %code block to plot the X(k)disp('The discrete fourier transform of x(n):');disp(Xk);t=0:N-1;subplot(3,2,2);stem(t,Xk);ylabel ('Amplitude');xlabel ('Time Index');title ('X(k)'); % To find the magnitudes of individual DFT pointsmagnitude=abs(Xk); %code block to plot the magnitude responsedisp('The magnitude response of X(k):');disp(magnitude);t=0:N-1;subplot(3,2,3);stem(t,magnitude);ylabel ('Amplitude');xlabel ('K');title ('Magnitude Response'); %To find the phases of individual DFT pointsphase=angle(Xk); %code block to plot the phase responsedisp('The phase response of X(k):');disp(phase);t=0:N-1;subplot(3,2,4);stem(t,phase);ylabel ('Phase');xlabel ('K');
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title ('Phase Response'); % Code block to find the IDFT of the sequencefor n=0:N-1 for k=0:N-1 iXk(n+1)=iXk(n+1)+(Xk(k+1)*exp(i*2*pi*k*n/N)); endend iXk=iXk./N; %code block to plot the output sequencet=0:N-1;subplot(3,2,5);stem(t,xn);ylabel ('Amplitude');xlabel ('Time Index');title ('IDFT sequence'); %code block to plot the FFT of input sequence using inbuilt functionx2=fft(xn);subplot(3,2,6);stem(t,x2);ylabel ('Amplitude');xlabel ('Time Index');title ('FFT of input sequence');
Command Window :
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OUTPUT :
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RESULT:
The program for DFT calculation was
performed with library functions and without library
functions. The results were verified by manual
calculation.
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FLOWCHART:
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START
ENTER THE INPUT SEQUENCE IN TIME
DOMAIN OR FREQUENCY DOMAIN
PERFORM DIT/DIF-FFT FOR TIME SAMPLES
OR PERFORM IDIT/IDIT-FFT FOR
FREQUENCY SAMPLES
PLOT THE WAVEFORMS
STOP
AIM:
Write a MATLAB Script to compute Discrete
Fourier Transform and Inverse Discrete Fourier
Transform of the given sequence using FFT
algorithms (DIT-FFT & DIF-FFT)
.
APPARATUS REQUIRED:
PC, MATLAB software
THEORY:
DFT is a powerful tool for performing
frequency analysis of discrete time signal and it is
described as a frequency domain representation of a
DT sequence.
The DFT of a finite duration sequence x[n] is
given by
X (k) = k=0,
1….N-1
which may conveniently be written in the form
X (k) = k=0,
1….N-1
where WN=e-j2/N which is known as
Twiddle or Phase factor.
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EX. NO:7DATE :3-2-14
EX. NO:7DATE :3-2-14
FAST FOURIER TRANSFORM ALGORITHMS FAST FOURIER TRANSFORM ALGORITHMS
COMPUTATION OF DFT:
To compute DFT, it requires N2
multiplication and (N-1) N complex addition. Direct
computation of DFT is basically inefficient precisely
because it does not exploit the symmetry and
periodicity properties of phase factor WN.
FAST FOURIER TRANSFORM (FFT):
FFT is a method of having computationally
efficient algorithms for the execution of DFT, under
the approach of Divide and Conquer. The number of
computations can be reduced in N point DFT for
complex multiplications to N/2log2N and for
complex addition to N/2log2N.
Types of FFT are,
(i) Decimation In Time (DIT)
(ii)Decimation In Frequency (DIF)
IDFT USING FFT ALGORITHMS:
The inverse DFT of an N point sequence
X (k), where k=0,1,2…N-1 is defined as ,
x [n] =
where, wN=e-j2/N.
Taking conjugate and multiplying by N, we get,
N x*[n] =
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The right hand side of the equation is the DFT of
the sequence X*(k). Now x[n] can be found by
taking the complex conjugate of the DFT and
dividing by N to give,
x [n]=
RADIX-2 DECIMATION IN TIME FFT:
The idea is to successively split the N-
point time domain sequence into smaller sub
sequence. Initially the N-point sequence is split
into xe[n] and xo[n], which have the even and odd
indexed samples of x[n] respectively. The N/2
point DFT’s of these two sequences are evaluated
and combined to give N-point DFT. Similarly N/2
point sequences are represented as a combination
of two N/4 point DFT’s. This process is
continued, until we are left with 2 point DFT.
RADIX-2 DECIMATION IN FREQUENCY
FFT:
The output sequence X(k) is divided into
smaller sequence.. Initially x[n] is divided into
two sequences x1[n], x2[n] consisting of the first
and second N/2 samples of x[n] respectively.
Then we find the N/2 point sequences f[n] and
g[n] as
f[n]= x1[n]+x2[n],
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g[n]=( x1[n]-x2[n] )wNk
The N/2 point DFT of the 2 sequences gives even
and odd numbered output samples. The above
procedure can be used to express each N/2 point
DFT as a combination of two N/4 point DFTs. This
process is continued until we are left with 2 point
DFT.
LIBRARY FUNCTION:
fft: Discrete Fourier transform.
fft(x) is the discrete Fourier transform (DFT) of vector x. For matrices, the FFT operation is applied to each column. For N-Dimensional arrays, the FFT operation operates on the first non-singleton dimension.
ditfft: Decimat ion in t ime (DIT)ff t
ditfft(x) is the discrete Fourier transform (DFT) of vector x in time domain decimation
diffft: Decimat ion in f requency
(DIF)f f t
diffft(x) is the discrete Fourier transform (DFT) of vector x in Frequency domain decimation
ALGORITHM/PROCEDURE:
1. Input the given sequence x[n]
2. Compute the Discrete Fourier Transform
using FFT library function (ditfft or diffft) and
obtain X[k]
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3. Compute the Inverse Discrete Fourier
Transform using FFT library function (ditfft
or diffft) and obtain X[n] by following steps
a. Take conjugate of X [k] and obtain
X[k]*
b. Compute the Discrete Fourier
Transform using FFT library function
(ditfft or diffft) for X[k]* and obtain
N.x[n]*
c. Once again take conjugate for N.x[n]*
and divide by N to obtain x[n]
4. Display the results.
SOURCE CODE:(DITFFT)
clc;clear all;close all; N=input('Enter the number of elements:');for i=1:N re(i)= input('Enter the real part of the element:'); im(i)= input('Enter the imaginary part of the element:');end%% Call Dit_fft function[re1,im1]= ditfft(re,im,N);disp(re1);disp(im1); figure(1);subplot(2,2,1);stem(re1);xlabel('Time period');
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ylabel('Amplitude');title('Real part of the output');subplot(2,2,2);stem(im1);xlabel('Time period');ylabel('Amplitude');title('Imaginary part of the output'); %%dit_ifft N=input('Enter the number of elements:');for i=1:N re(i)= input('Enter the real part of the element:'); im(i)= input('Enter the imaginary part of the element:');endfor i=1:N re(i)=re(i); im(i)=-im(i);end%% call dit_ifft function [re1,im1]=ditifft(re,im,N);for i=1:N re1(i)=re1(i)/N; im1(i)=-im1(i)/N;enddisp(re1);disp(im1);%figure(2)subplot(2,2,3);stem(re1);xlabel('Time period');ylabel('Amplitude');title('Real part of the output'); subplot(2,2,4);stem(im1);xlabel('Time period');ylabel('Amplitude');
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title('Imaginary part of the output');
Function Table:(DITFFT)
function [ re, im ] = ditfft( re, im, N)%UNTITLED5 Summary of this function goes here% Detailed explanation goes hereN1=N-1;N2=N/2;j=N2+1;M=log2(N); % Bit reversal sorting for i=2:N-2 if i<j tr=re(j); ti=im(j); re(j)=re(i); im(j)=im(i); re(i)=tr; im(i)=ti; end k=N2; while k<=j j=j-k; k=k/2;endj=j+k;j=round(j);endfor l=1:M le=2.^l; le2=le/2; ur=1; ui=0; sr=cos(pi/le2); si=sin(pi/le2); for j=2:(le2+1) jm=j-1;
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for i=jm:le:N ip=i+le2; tr=re(ip)*ur-im(ip)*ui; ti=re(ip)*ui-im(ip)*ur; re(ip)=re(i)-tr; im(ip)=im(i)-ti; re(i)=re(i)+tr; im(i)=im(i)+ti; end tr=ur; ur=tr*sr-ui*si; ui=tr*si+ui*sr; endend
Function Table:(DITIFFT)
function [ re,im] = ditifft(re,im,N)%UNTITLED2 Summary of this function goes here% Detailed explanation goes hereN1=N-1;N2=N/2;j=N2+1;M=log2(N); %Bit reversal sorting for i=2:N-2 if i<j tr=re(j); ti=im(j); re(j)=re(i); im(j)=im(i); re(i)=tr; im(i)=ti; end k=N2; while k<=j j=j-k; k=k/2;
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end j=j+k; j=round(j);endfor l=1:M le=2.^l; le2=le/2; ur=1; ui=0; sr=cos(pi/le2); si=-sin(pi/le2); for j=2:(le2+1) jm=j-1; for i=jm:le:N ip=i+le2; tr=re(ip)*ur-im(ip)*ui; ti=re(ip)*ui+im(ip)*ur; re(ip)=re(i)-tr; im(ip)=im(i)-ti; re(i)=re(i)+tr; im(i)=im(i)+ti; end tr=ur; ur=tr*sr-ui*si; ui=tr*si+ui*sr; endendend
Command Window:
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Output:
Source code :(DIFFFT)
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%% DIF_FFT clc;clear all;close all;%% N=input('Enter the number of points in DIF DFT:'); for i=1:N re(i)=input('Enter the real part of the element:'); im(i)=input('Enter the imaginary part of the element:');end %% % Call DIf_FFT Function[re1, im1]=diffft(re,im,N);display(re1);display(im1);figure(1);subplot(2,2,1);stem(re1);xlabel('Time');ylabel('Amplitude');title('Real part of the output'); subplot(2,2,2);stem(im1);xlabel('Time');ylabel('Amplitude');title('Imaginary part of the output'); %% DIF IFFTN=input('Enter the number of points in DIF IFFT:');for i=1:N re(i)=input('Enter the real part of the element:'); im(i)=input('Enter the imaginary part of the element:');end
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for i=1:N re(i)=re(i); im(i)=-im(i);end%% Call dif_ifft function[re1, im1]=ditifft(re,im,N);for i=1:N re1(i)=re1(i)/N; im1(i)=-im1(i)/N;enddisplay(re1)display(im1);% figure(2);subplot(2,2,3);stem(re1);xlabel('Time');ylabel('Amplitude');title('Real part of the output'); subplot(2,2,4);stem(im1);xlabel('Time');ylabel('Amplitude');title('Imaginary part of the output');
Function Table:(DIFFFT)
function [re, im ] = diffft(re, im, N)%UNTITLED5 Summary of this function goes here% Detailed explanation goes hereN1=N-1;N2=N/2;M=log2(N); %% % for l=M:-1:1; le=2.^l; le2=le/2; ur=1;
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ui=0; sr=cos(pi/le2); si=-sin(pi/le2); for j=2:(le2+1) jm=j-1; for i=jm:le:N ip=i+le2; tr=re(ip); ti=im(ip); re(ip)=re(i)-re(ip); im(ip)=im(i)-im(ip); re(i)=re(i)+tr; im(i)=im(i)+ti; tr=re(ip); re(ip)=re(ip)*ur-im(ip)*ui; im(ip)=im(ip)*ur+tr*ui; end tr=ur; ur=tr*sr-ui*si; ui=tr*si+ui*sr; endendj=N2+1;for i=2:N-2 if i<j tr=re(j); ti=im(j); re(j)=re(i); im(j)=im(i); re(i)=tr; im(i)=ti; end k=N2; while k<=j j=j-k; k=k/2;endj=j+k;end
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Function Table:(DIFIFFT)
function [ re, im ] = dififft( re, im, N)%UNTITLED5 Summary of this function goes here% Detailed explanation goes hereN1=N-1;N2=N/2;M=log2(N); %% % for l=M:-1:1; le=2.^l; le2=le/2; ur=1; ui=0; sr=cos(pi/le2); si=-sin(pi/le2); for j=2:(le2+1) jm=j-1; for i=jm:le:N ip=i+le2; tr=re(ip); ti=im(ip); re(ip)=re(i)-re(ip); im(ip)=re(i)-re(ip); re(i)=re(i)+tr; im(i)=im(i)+ti tr=re(ip); re(ip)=re(ip)*ur-im(ip)*ui; im(ip)=im(ip)*ur+tr*ui; end tr=ur; ur=tr*sr-ui*si; ui=tr*si+ui*sr; endendj=N2+1;for i=2:N-2
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if i<j tr=re(j); ti=im(j); re(j)=re(i); im(j)=im(i); re(i)=tr; im(i)=ti; end k=N2; while k<=j j=j-k; k=k/2;endj=j+k;end
Command Window:
Enter the number of points in DIF DFT:4Enter the real part of the element:1Enter the imaginary part of the element:0Enter the real part of the element:1Enter the imaginary part of the element:0Enter the real part of the element:1Enter the imaginary part of the element:0Enter the real part of the element:1Ent
er
the
im
agi
nar
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y
par
t of
the
ele
me
nt:
0
re1
=
4 0 0 0
im1 =
0 0 0 0
Enter the number of points in DIF IFFT:4Enter the real part of the element:4Enter the imaginary part of the element:0Enter the real part of the element:0Enter the imaginary part of the element:0Enter the real part of the element:0Enter the imaginary part of the element:0Enter the real part of the element:0
Enter the imaginary part of the element:0
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re1
=
1 1 1 1
im1 =
0 0 0 0
Output:
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RESULT:
The DFT and IDFT of the given sequence are
computed using FFT algorithm both DITFFT and
DIFFFT.
FLOWCHART:
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START
ENTER THE FILTER SPECIFICATIONS (ORDER OF
THE FILTER, CUT-OFF FREQUENCY)
DESIGN THE FILTER
PLOT THE WAVEFORMS
STOP
AIM:
Write a MATLAB Script to design a low pass
FIR filter using Window Method for the given
specifications
.
APPARATUS REQUIRED:
PC, MATLAB software
THEORY:
A digital filter is a discrete time LTI system. It
is classified based on the length of the impulse
response as
IIR filters:
Where h [n] has infinite number of samples
and is recursive type.
FIR filters:
They are non-recursive type and h [n] has
finite number of samples.
The transfer function is of the form:
This implies that it has (N-1) zeros located
anywhere in the z-plane and (N-1) poles at Z = h.
THE FIR FILTER CAN BE DESIGNED BY:
99 UR11EC098
EX. NO:8DATE :10-2-14
EX. NO:8DATE :10-2-14
DESIGN OF FIR FILTERSDESIGN OF FIR FILTERS
Fourier series method
Frequency sampling method
Window method
Most of the FIR design methods are
interactive procedures and hence require more
memory and execution time. Also implementation of
narrow transition band filter is costly. But there are
certain reasons to go for FIR.
TYPES OF WINDOWS:1. Rectangular
2. Triangular
3. Hamming
4. Hanning
5. Blackman
6. Kaiser
LIBRARY FUNCTIONS:
fir1 FIR filter design using the Window
method.
B = fir1(N,Wn) designs an Nth order low pass FIR digital filter and returns the filter coefficients of vector B of length (N+1). The cut-off frequency Wn must be between 0 < Wn < 1.0, with 1.0 corresponding to half the sample rate. The filter B is real and has linear phase. The normalized gain of the filter at Wn is -6 dB.B = fir1(N,Wn,'high') designs an Nth order high pass filter. You can also use B = fir1(N,Wn,'low') to design a low pass filter.
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If Wn is a two-element vector, Wn = [W1 W2], fir1 returns an order N band pass filter with pass band W1 < W < W2.You can also specify B = fir1(N,Wn,'bandpass'). If Wn = [W1 W2], B = fir1(N,Wn,'stop') will design a band-stop filter. If Wn is a multi-element vector, Wn = [W1 W2 W3 W4 W5 ... WN], fir1 returns a N-order multi-band filter with bands 0 < W < W1, W1 < W < W2, ..., WN < W < 1.B = fir1(N,Wn,'DC-1') makes the first band a pass band.B = fir1(N,Wn,'DC-0') makes the first band a stop band.B = fir1(N,Wn,WIN) designs an N-th order FIR filter using the vector WIN of (N+1) length to window the impulse response. If empty or omitted, fir1 uses a Hamming window of length N+1. For a complete list of available windows, see the Help for the WINDOW function. KAISER and CHEBWIN can be specified with an optional trailing argument. For example, B = fir1(N,Wn,kaiser(N+1,4)) uses a Kaiser window with beta=4. B = fir1(N,Wn,'high',chebwin(N+1,R)) uses a Chebyshev window with R decibels of relative sidelobe attenuation.For filters with a gain other than zero at Fs/2, e.g., high pass and band stop filters, N must be even. Otherwise, N will be incremented by one. In this case, the window length should be specified as N+2. By default, the filter is scaled so the center of the first pass band has magnitude exactly one after windowing. Use a trailing 'noscale' argument to prevent this scaling, e.g.
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B = fir1(N,Wn,'noscale'), B =
fir1(N,Wn,'high','noscale'),
B = fir1(N,Wn,wind,'noscale'). You can also
specify the scaling explicitly, e.g.
fir1(N,Wn,'scale'), etc.
We can specify windows from the Signal
Processing Toolbox, such as boxcar,
hamming, hanning, bartlett, blackman, kaiser
or chebwin
w = hamming(n) returns an n-point symmetric
Hamming window in the column vector w. n
should be a positive integer.
w = hanning(n) returns an n-point symmetric
Hann window in the column vector w. n must
be a positive integer.
w=triang(n) returns an n-point triangular
window in the column vector w. The
triangular window is very similar to a Bartlett
window. The Bartlett window always ends
with zeros at samples 1 and n, while the
triangular window is nonzero at those points.
For n odd, the center (n-2) points of triang(n-
2) are equivalent to bartlett(n).
w = rectwin(n) returns a rectangular window
of length n in the column vector w. This
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function is provided for completeness. A
rectangular window is equivalent to no
window at all.
ALGORITHM/PROCEDURE:
1. Click on the MATLAB icon on the desktop
(or go to Start – All Programs and
click on MATLAB) to get into the Command
Window.
2. Type ‘edit’ in the MATLAB prompt ‘>>’ that
appears in the Command window.
3. Write the program in the ‘Edit’ window and
save it in ‘M-file’.
4. Run the program.
5. Enter the input in the Command Window.
6. The result is displayed in the Command
window and the graphical output is displayed
in the Figure Window.
Source code :1
%windows technique of Rectangular window using low pass filterclc;clear all;close all;N=input('Size of window:');wc=input('Cut off frequency:');h=fir1(N-1,wc/pi,boxcar(N));tf(h,1,1,'variable','z^-1');freqz(h);
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xlabel('Frequency');ylabel('Magnitude');title('FIR Filter');
Source code :2
%windows technique of Triangular(Bartlet Window) using High pass filterclc;clear all;close all;N=input('Size of window:');wc=input('Cut off frequency:');h=fir1(N-1,wc/pi,'high',triang(N));tf(h,1,1,'variable','z^-1');freqz(h);xlabel('Frequency');ylabel('Magnitude');title('FIR Filter');
Source code :3
%windows technique of Hamming using Band pass filterclc;clear all;close all;N=input('Size of window:');wc1=input('Lower Cut off frequency:');wc2=input('Upper Cut off frequency:');wc=[wc1 wc2];h=fir1(N-1,wc/pi,'bandpass',hamming(N));tf(h,1,1,'variable','z^-1');freqz(h);xlabel('Frequency');ylabel('Magnitude');title('FIR Filter');
Source code :4
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%windows technique of Hanning using Band stop filterclc;clear all;close all;N=input('Size of window:');wc1=input('Lower Cut off frequency:');wc2=input('Upper Cut off frequency:');wc=[wc1 wc2];h=fir1(N-1,wc/pi,'stop',hanning(N));tf(h,1,1,'variable','z^-1');freqz(h);xlabel('Frequency');
ylabel('Magnitude');title('FIR Filter');
Source code :5
%windows technique of Blackman window using low pass filterclc;clear all;close all;N=input('Size of window:');wc=input('Cut off frequency:');h=fir1(N-1,wc/pi,blackman(N));tf(h,1,1,'variable','z^-1');freqz(h);xlabel('Frequency');ylabel('Magnitude');title('FIR Filter');
Command Window :1
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Output:1
Command Window :2
OUTPUT: 2
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Command Window :3
Output:3
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Command Window :4
Output:4
Command Window :5
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Output:
RESULT:
The given low pass filter was designed using
Window method and manually verified filter co-
efficient of the filters.
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FLOWCHART:
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START
ENTER THE FILTER SPECIFICATIONS (PASS, STOP BAND GAINS AND
EDGE FREQUENCIES)
DESIGN THE ANALOG BUTTERWORTH
AND CHEBYSHEV FILTER
PLOT THE WAVEFORMS
STOP
AIM:
Write a MATLAB Script to design Analog
Butterworth filters for the given specifications.
APPARATUS REQUIRED:
PC, MATLAB software
THEORY:
BUTTERWORTH FILTER:
Low pass Analog Butterworth filters are all
pole filters characterised by magnitude frequency
response by
2 =
where N is the order of the filter and is the
cut-off frequency.
As N , the low pass filter is said to be
normalized. All types of filters namely-Low pass,
High pass, Band pass and Band elimination filters
can be derived from the Normalized Low pass filter.
STEPS IN DESIGNING ANALOG FILTER:
111 UR11EC098
EX. NO : 9DATE: 17-02-14
EX. NO : 9DATE: 17-02-14
9.DESIGN OF BUTTERWORTH FILTERS 9.DESIGN OF BUTTERWORTH FILTERS
1. Transform the given specification to a
Normalized Low pass specification
2. Find the order of the filter N and cut-off
frequency c
3. Find the transfer function H(s) of normalized
LPF.
4. Use the applicable analog-to-analog
transformation to get the transfer function of
the required filter.
LIBRARY FUNCTIONS: butter: Butterworth digital and analog
filter design.
[B, A] = butter (N,Wn) designs an Nth order Low pass digital Butterworth filter and returns the filter coefficient vectors B (numerator) and A (denominator) in length N+1. The coefficients are listed in descending powers of z. The cut-off frequency Wn must be in the range 0.0 < Wn < 1.0, with 1.0 corresponding to half the sample rate.
butter (N,Wn,'s'),butter (N,Wn,'low','s'),butter (N,Wn,'high','s'),butter (N,Wn,'pass','s')and butter (N,Wn,'stop','s')design analog Butterworth filters. In this case, Wn is in [rad/s] and it can be greater than 1.0.
buttord: Butterworth filter order
selection.
[N, Wn] = buttord (Wp, Ws, Rp, Rs) returns the order N of the lowest order digital
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Butterworth filter that loses no more than Rp dB in the pass band and has at least Rs dB of attenuation in the stop band. Wp and Ws are the pass band and stop band edge frequencies, normalized from 0 to 1 (where 1 corresponds to pi radians/sample). For example,
Lowpass: Wp = .1, Ws = .2
Highpass: Wp = .2, Ws = .1
Bandpass: Wp = [.2 .7], Ws = [.1 .8]
Bandstop: Wp = [.1 .8], Ws = [.2 .7]
buttord: also returns Wn, the Butterworth natural frequency (or) the "3 dB frequency" to be used with BUTTER to achieve the specifications.
[N, Wn] = buttord (Wp, Ws, Rp, Rs, 's') does the computation for an analog filter, in which case Wp and Ws are in radians/second. When Rp is chosen as 3 dB, the Wn in BUTTER is equal to Wp in BUTTORD.
angle : Phase angle.
Theta=angle (H) returns the phase angles, in radians, of a matrix with complex elements.
freqs : Laplace-transform (s-domain)
frequency response.
H = freqs(B,A,W) returns the complex frequency response vector H of the filter B/A:
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B(s) b (1)s nb-1 + b(2)s nb-2
+ ... + b(nb) H(s) = ---- = -------------------------------------------------- A(s) a(1)s na-1 + a(2)s na-2
+ ... + a(na)
given the numerator and denominator
coefficients in vectors B and A. The frequency
response is evaluated at the points specified in
vector W (in rad/s). The magnitude and phase
can be graphed by calling freqs(B,A,W) with
no output arguments.
tf: Transfer function
SYS = tf(NUM,DEN) creates a continuous-
time transfer function SYS with
numerator(s) NUM and denominator(s) DEN.
The output SYS is a tf object.
ALGORITHM/PROCEDURE:
1. Click on the MATLAB icon on the desktop (or
go to Start – All programs and click on
MATLAB) to get into the Command Window.
2. Type ‘edit’ in the MATLAB prompt ‘>>’ that
appears in the Command window.
114 UR11EC098
3. Write the program in the ‘Edit’ window and
save it in ‘M-file’.
4. Run the program.
5. Enter the input in the command window.
6. The result is displayed in the Command
window and the graphical output is displayed
in the Figure Window.
Butterworth Filters
SOURCE CODE:1
clc;clear all;close all;%% Butterworth low pass Filter% Filter Specifications k1=input('Enter the passband gain in db:');k2=input('Enter the stopband gain in db:');w1=input('Enter the passband edge frequency in rad/Sec:');w2=input('Enter the stopband edge frequency in rad/Sec:'); %Find the order and Cutofrf frequency using the given specification of%filter[n,Wc]=buttord(w1,w2,k1,k2,'s');disp('The order is:');disp(n);disp('The cutoff frequency is:');disp(Wc); % Low pass filtering[b,a]=butter(n,Wc,'low','s'); %Plotting magnitude & phase response
115 UR11EC098
f=linspace(1,512,1000);h=freqs(b,a,f); m=20*log(abs(h));subplot(2,1,1);semilogx(f,m);xlabel('Frequency');ylabel('Magnitude');title('Magnitude response of Butterworth LPF'); % Phase responsep=angle(h);subplot(2,1,2);semilogx(f,p);xlabel('Frequency');ylabel('Phase');title('Phase response of Butterworth LPF');
SOURCE CODE:2
clc;clear all;close all;%% Butterworth high pass Filter% Filter Specifications k1=input('Enter the passband gain in db:');k2=input('Enter the stopband gain in db:');w1=input('Enter the passband edge frequency in rad/Sec:');w2=input('Enter the stopband edge frequency in rad/Sec:'); %Find the order and Cutofrf frequency using the given specification of%filter[n,Wc]=buttord(w1,w2,k1,k2,'s');disp('The order is:');disp(n);
116 UR11EC098
disp('The cutoff frequency is:');disp(Wc); % Low pass filtering[b,a]=butter(n,Wc,'high','s'); %Plotting magnitude & phase response f=linspace(1,512,1000);h=freqs(b,a,f); m=20*log(abs(h));subplot(2,1,1);semilogx(f,m);xlabel('Frequency');ylabel('Magnitude');title('Magnitude response of Butterworth HPF'); % Phase responsep=angle(h);subplot(2,1,2);semilogx(f,p);xlabel('Frequency');ylabel('Phase');title('Phase response of Butterworth HPF');
SOURCE CODE:3
clc;clear all;close all;%% Bandpass Filter SpecificationsWp=input('Enter the pass band edge frequency : ');Ws=input('Enter the stop band edge frequency : ');Rp=input('Enter the Pass band ripple: ');Rs=input('Enter the stop band gain: '); %To find order of the filter[N]=buttord(Wp,Ws,Rp,Rs,'s')
117 UR11EC098
%To find cut off frequencyWc=[Wp Ws]; %Band pass Filtering[b,a]=butter(N,Wc,'bandpass','s'); %plotting magnitude and phase responsefigure(1);freqs(b,a);
SOURCE CODE:4
clc;
clear all;close all;%% Bandstop Filter SpecificationsWp=input('Enter the pass band edge frequency : ');Ws=input('Enter the stop band edge frequency : ');Rp=input('Enter the Pass band ripple: ');Rs=input('Enter the stop band gain: '); %To find order of the filter[N]=buttord(Wp,Ws,Rp,Rs,'s') %To find cut off frequencyWc=[Wp Ws]; %Band stop Filtering[b,a]=butter(N,Wc,'stop','s'); %plotting magnitude and phase responsefigure(1);freqs(b,a);
Command Windows :
Using Low pass filter
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Using High pass filter
Using Band pass filter
Using Band stop filter
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OUTPUTS:
Using Low pass filter
Using High pass filter
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Using Band pass filter
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Using Band stop filter
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RESULT:
Analog Butterworth Filter is designed for the
given specifications, and manually verified the order,
cut off frequency and filter co-efficient of the filter.
FLOWCHART:
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START
ENTER THE FILTER SPECIFICATIONS (PASS, STOP BAND GAINS AND
EDGE FREQUENCIES)
DESIGN THE ANALOG BUTTERWORTH
AND CHEBYSHEV FILTER
PLOT THE WAVEFORMS
STOP
124 UR11EC098
AIM:
Write a MATLAB Script to design Analog
Chebyshev filter for the given specifications.
APPARATUS REQUIRED:
PC, MATLAB software
THEORY:
Chebyshev Filters :
There are two types of Chebyshev filters.
Type I are all-pole filters that exhibit equi-ripple
behaviour in pass band and monotonic characteristics
in stop band.
Type II are having both poles and zeros and exhibit
monotonic behavior in pass band and equi-ripple
behavior in stop band. The zero lies on the imaginary
axis.
The magnitude-squared function is given as
is the ripple parameter in pass band
CN(x) is the Nth order Chebyshev polynomial defined
as
125 UR11EC098
EX. NO : 10DATE: 17-02-14
EX. NO : 10DATE: 17-02-14
10.DESIGN OF CHEBYSHEV FILTERS 10.DESIGN OF CHEBYSHEV FILTERS
CN(x) =
STEPS IN DESIGNING ANALOG FILTER:
1.Transform the given specification to a
Normalized Low pass specification
2. Find the order of the filter N and cut-off
frequency c
3. Find the transfer function H(s) of normalized
LPF.
4. Use the applicable analog-to-analog
transformation to get the transfer function of
the required filter.
LIBRARY FUNCTIONS: cheb1ord: Chebyshev Type I filter
order selection.
[N, Wn] = cheb1ord (Wp, Ws, Rp, Rs) returns the order N of the lowest order digital Chebyshev Type I filter that loses no more than Rp dB in the pass band and has at least Rs dB of attenuation in the stop band. Wp and Ws are the pass band and stop band edge frequencies, normalized from 0 to 1 (where 1 corresponds to pi radians/sample). For example,
Lowpass: Wp = .1, Ws = .2
126 UR11EC098
Highpass: Wp = .2, Ws = .1
Bandpass: Wp = [.2 .7], Ws = [.1 .8]
Bandstop: Wp = [.1 .8], Ws = [.2 .7]
cheb1ord also returns Wn, the Chebyshev natural frequency to use with cheby1 to achieve the specifications.
[N, Wn] = cheb1ord (Wp, Ws, Rp, Rs, 's') does the computation for an analog filter, in which case Wp and Ws are in radians/second.
cheby1 Chebyshev Type I digital and analog filter design.
[B,A] = cheby1 (N,R,Wn) designs an Nth order Low pass digital Chebyshev filter with R decibels of peak-to-peak ripple in the pass band. cheby1 returns the filter coefficient vectors B (numerator) and A (denominator) of length N+1. The cut-off frequency Wn must be in the range 0.0 < Wn < 1.0, with 1.0 corresponding to half the sample rate. Use R=0.5 as a starting point, if you are unsure about choosing R.
cheby1 (N,R,Wn,'s'), cheby1 (N,R,Wn,'low','s'), cheby1 (N,R,Wn,'high','s'), cheby1 (N,R,Wn,'pass','s') and cheby1 (N,R,Wn,'stop','s') design analog Chebyshev Type I filters. In this case, Wn is in [rad/s] and it can be greater than 1.0.
ALGORITHM/PROCEDURE:
1. Click on the MATLAB icon on the desktop (or
go to Start – All programs and click on
127 UR11EC098
MATLAB) to get into the Command Window.
2. Type ‘edit’ in the MATLAB prompt ‘>>’ that
appears in the Command window.
3. Write the program in the ‘Edit’ window and
save it in ‘M-file’.
4. Run the program.
5. Enter the input in the command window.
6. The result is displayed in the Command
window and the graphical output is displayed
in the Figure Window.
CHEBYSHEV FILTERS :
Source code:1clc;clear all;close all;%% Chebyshev low pass Filter% Filter Specifications k1=input('Enter the passband ripple in db:');k2=input('Enter the stopband attenuation in db:');w1=input('Enter the passband edge frequency in rad/Sec:');w2=input('Enter the stopband edge frequency in rad/Sec:'); %Find the order and Cutofrf frequency using the given specification of%filter[n,Wc]=cheb1ord(w1,w2,k1,k2,'s');disp('The order is:');disp(n);disp('The cutoff frequency is:');disp(Wc);
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% Low pass filtering[b,a]=cheby1(n,k1,w1,'low','s');figure(1);freqs(b,a); Source code:2 clc;clear all;close all;%% Chebyshev High pass Filter% Filter Specifications k1=input('Enter the passband ripple in db:');k2=input('Enter the stopband attenuation in db:');w1=input('Enter the passband edge frequency in rad/Sec:');w2=input('Enter the stopband edge frequency in rad/Sec:'); %Find the order and Cutofrf frequency using the given specification of%filter[n,Wc]=cheb1ord(w1,w2,k1,k2,'s');disp('The order is:');disp(n);disp('The cutoff frequency is:');disp(Wc); % High pass filtering[b,a]=cheby1(n,k1,w1,'high','s');figure(1);freqs(b,a);
Source code: 3clc;clear all;close all;%% Bandpass Filter SpecificationsWp=input('Enter the pass band edge frequency : ');Ws=input('Enter the stop band edge frequency : ');Rp=input('Enter the Pass band ripple: ');
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Rs=input('Enter the stop band gain: '); %To find order of the filter[N]=cheb1ord(Wp,Ws,Rp,Rs,'s') %To find cut off frequencyWc=[Wp Ws]; %Band pass Filtering[b,a]=cheby1(N,Rp,Wc,'bandpass','s'); %plotting magnitude and phase responsefigure(1);freqs(b,a);
Source code: 4clc;clear all;close all;%% Bandstop Filter SpecificationsWp=input('Enter the pass band edge frequency : ');Ws=input('Enter the stop band edge frequency : ');Rp=input('Enter the Pass band ripple: ');Rs=input('Enter the stop band gain: '); %To find order of the filter[N]=cheb1ord(Wp,Ws,Rp,Rs,'s') %To find cut off frequencyWc=[Wp Ws]; %Bandstop Filtering[b,a]=cheby1(N,Rp,Wc,'stop','s'); %plotting magnitude and phase responsefigure(1);freqs(b,a);
Command Window :1(LPF)
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Output:
Command Window :2(HPF)
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Output:
Command Window :3(BPF)
Output:
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Command Window :4(BSF)
Output:
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RESULT:
Analog Chebyshev Filter is designed for the
given specifications, and manually verified the order,
cut off frequency and filter co-efficient of the filter.
FLOWCHART:
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START
ENTER THE FILTER SPECIFICATIONS (PASS, STOP BAND GAINS AND
EDGE FREQUENCIES)
DESIGN THE ANALOG BUTTERWORTH AND
CHEBYSHEV LOW PASS FILTER
PLOT THE WAVEFORMS
STOP
CONVERT THE LOW PASS FILTERS IN TO
DIGITAL FILTERS BY USING THE IMPULSE
INVARIANT AND BILINEAR
TRANSFORMATIONS
AIM:
Write a MATLAB Script to design
Butterworth and Chebyshev low pass filters using
Bilinear Transformation
Impulse Invariant Transformation
.
APPARATUS REQUIRED:
PC, MATLAB software
THEORY:
A digital filter is a linear time invariant
discrete time system. The digital filters are classified
into two, based on their lengths of impulse response
1. Finite Impulse response (FIR)
They are of non-recursive type and h [n]
has finite number of samples
2. Infinite Impulse response (IIR)
h[n] has finite number of samples. They
are of recursive type. Hence, their
transfer function is of the form
0
)()(n
nznhzH
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EX. NO : 11DATE: 03-03-14
EX. NO : 11DATE: 03-03-14
11.DESIGN OF IIR FILTERS11.DESIGN OF IIR FILTERS
The digital filters are designed from analog
filters. The two widely used methods for digitizing
the analog filters include
1. Bilinear transformation
2. Impulse Invariant transformation
The bilinear transformation maps the s-plane into the z-plane by
H(Z) =
This transformation maps the jΩ axis (from Ω = -∞
to +∞) repeatedly around the unit circle (exp ( jw),
from w=-π to π ) by
2tan
2 T
BILINEAR TRANSFORMATION:
DESIGN STEPS:
1. From the given specifications, Find pre-
warped analog frequencies using
2tan
2 T
2. Using the analog frequencies, find H(s)
of the analog filter
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3. Substitute in the H(s) of
Step:2
IMPULSE INVARIANT TRANSFORMATION:
DESIGN STEPS:
1. Find the analog frequency using = T/
2. Find the transfer function of analog filter
Ha(s)
3. Express the analog filter transfer function
as a sum of single pole filters
4. Compute H(Z) of digital filter using the
formula
LIBRARY FUNCTIONS:
Impinvar: Impulse Invariant method for
analog-to-digital filter conversion [bz,az] =
impinvar(b,a,fs) creates a digital filter with
numerator and denominator coefficients bz
and az, respectively, whose impulse response
is equal to the impulse response of the analog
filter with coefficients b and a, scaled by 1/fs.
If you leave out the argument fs (or) specify fs
as an empty vector [ ], it takes the default
value of 1 Hz.
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Bilinear: Bilinear transformation method for
analog-to-digital filter conversion. The bilinear
transformation is a mathematical mapping of
variables. In digital filtering, it is a standard
method of mapping the s or analog plane into
the z or digital plane. It transforms analog
filters, designed using classical filter design
ALGORITHM/PROCEDURE:
1. Calculate the attenuation in dB for the given
design parameters
2. Design the analog counterpart
3. Using Impulse Invariant /Bilinear
transformation design the digital filter
4. Display the transfer function. Plot the
magnitude response and phase response
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SOURCE CODE:
/ Butterworth Lowpass Impulse invariant
method
clc;clear all;close all;warning off; % Design of IIR Filters%% Filter Specifications% Input Wp,Ws,Sp,Ss,T% T=1,bothe ripple gains should be b/w .1 to .3disp(' Butterworth Lowpass filter using Impulse invariant method '); T=input('Enter the Sampling Frequency in rad/sec: ');Sp=input('Enter the Pass-band Ripple Gain: ');Wp=input('Enter the Pass-band Edge Frequency in rad/sec: ');Ss=input('Enter the Stop-band Ripple Gain: ');Ws=input('Enter the Stop-band Edge Frequency in rad/sec: '); % Calculation of ohmp,ohms,Ap,AsAp=abs(20*log10(1-Sp));As=abs(20*log10(Ss));
ohmp=Wp/T; ohms=Ws/T;% Butterworth Filter % Calculation of order and cutoff freq. for the above filter specs.[n,Wc]=buttord(ohmp,ohms,Ap,As,'s'); % Low Pass Filtering[b,a]=butter(n,Wc,'low','s');[bz,az] = impinvar(b,a,T);tf(bz,az,T);
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O=linspace(-pi,pi,50);% O is the freq. axisH=freqz(bz,az,O); % Magnitude ResponseHm=20*log10(abs(H));subplot(2,1,1);semilogx(O,Hm);xlabel('Frequency');ylabel('Magnitude');title('Magnitude Response of IIR Filter using Impulse Invariant Method'); % Phase ResponseHa=angle(H);subplot(2,1,2);semilogx(O,Ha);xlabel('Frequency');ylabel('Phase');title('Phase Response of IIR Filter using Impulse Invariant Method');
/ Butterworth Lowpass Bilinear Transformation Method
clc;clear all;close all;warning off; % Design of IIR Filters%% Filter Specifications% Input Wp,Ws,Sp,Ss,T% T=1,both the ripple gains should have band width( .1 to .3)disp(' Butterworth Lowpass filter using Bilnear transformation method '); T=input('Enter the Sampling Frequency in rad/sec: ');Sp=input('Enter the Pass-band Ripple Gain: ');
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Wp=input('Enter the Pass-band Edge Frequency in rad/sec: ');Ss=input('Enter the Stop-band Ripple Gain: ');Ws=input('Enter the Stop-band Edge Frequency in rad/sec: '); % Calculation of ohmp,ohms,Ap,AsAp=abs(20*log10(1-Sp));As=abs(20*log10(Ss)); ohmp=Wp/T; ohms=Ws/T;% Butterworth Filter % Calculation of order and cutoff freq. for the above filter specs.[n,Wc]=buttord(ohmp,ohms,Ap,As,'s'); % Low Pass Filtering[b,a]=butter(n,Wc,'low','s');[bz,az] = bilinear(b,a,1/T);tf(bz,az,T);O=linspace(-pi,pi,50);% O is the freq. axisH=freqz(bz,az,O); % Magnitude ResponseHm=20*log10(abs(H));subplot(2,1,1);semilogx(O,Hm);xlabel('Frequency');ylabel('Magnitude');title('Magnitude Response of IIR Filter using Bilinear Transformation Method'); % Phase ResponseHa=angle(H);subplot(2,1,2);semilogx(O,Ha);xlabel('Frequency');ylabel('Phase');
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title('Phase Response of IIR Filter using Bilinear Transformation Method');
/ Chebyshev Lowpass Impulse invariant method
clc;clear all;close all;warning off; % Design of IIR Filters%% Filter Specifications% Input Wp,Ws,Sp,Ss,T% T=1,bothe ripple gains should be b/w .1 to .3disp(' Chebyshev Lowpass filter using Impulse invariant method '); T=input('Enter the Sampling Frequency in rad/sec: ');Sp=input('Enter the Pass-band Ripple Gain: ');Wp=input('Enter the Pass-band Edge Frequency in rad/sec: ');Ss=input('Enter the Stop-band Ripple Gain: ');Ws=input('Enter the Stop-band Edge Frequency in rad/sec: '); % Calculation of ohmp,ohms,Ap,AsAp=abs(20*log10(1-Sp));As=abs(20*log10(Ss)); ohmp=Wp/T; ohms=Ws/T;% Chebyshev Filter % Calculation of order and cutoff freq. for the above filter specs.[n,Wc2]=cheb1ord(ohmp,ohms,Ap,As,'s') % Low Pass Filtering[b,a]=cheby1(n,Ap,Wc2,'low','s');[bz,az] = impinvar(b,a,T);tf(bz,az,T);
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O=linspace(-pi,pi,50);% O is the freq. axisH=freqz(bz,az,O); % Magnitude ResponseHm=20*log10(abs(H));subplot(2,1,1);semilogx(O,Hm);xlabel('Frequency');ylabel('Magnitude');title('Magnitude Response of IIR Filter using Impulse Invariant Method'); % Phase ResponseHa=angle(H);subplot(2,1,2);semilogx(O,Ha);xlabel('Frequency');ylabel('Phase');title('Phase Response of IIR Filter using Impulse Invariant Method');
/Chebyshev Lowpass Bilinear Transformation Method
clc;clear all;close all;warning off; % Design of IIR Filters%% Filter Specifications% Input Wp,Ws,Sp,Ss,T% T=1,bothe ripple gains should be b/w .1 to .3disp(' Chebyshev Lowpass filter using Bilnear transformation method '); T=input('Enter the Sampling Frequency in rad/sec: ');Sp=input('Enter the Pass-band Ripple Gain: ');
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Wp=input('Enter the Pass-band Edge Frequency in rad/sec: ');Ss=input('Enter the Stop-band Ripple Gain: ');Ws=input('Enter the Stop-band Edge Frequency in rad/sec: '); % Calculation of ohmp,ohms,Ap,AsAp=abs(20*log10(1-Sp));As=abs(20*log10(Ss)); ohmp=Wp/T; ohms=Ws/T;% Chebyshev Filter % Calculation of order and cutoff freq. for the above filter specs.[n,Wc2]=cheb1ord(ohmp,ohms,Ap,As,'s') % Low Pass Filtering[b,a]=cheby1(n,Ap,Wc2,'low','s');[bz,az] = bilinear(b,a,1/T);tf(bz,az,T);O=linspace(-pi,pi,50);% O is the freq. axisH=freqz(bz,az,O); % Magnitude ResponseHm=20*log10(abs(H));subplot(2,1,1);semilogx(O,Hm);xlabel('Frequency');ylabel('Magnitude');title('Magnitude Response of IIR Filter using Bilinear Transformation Method'); % Phase ResponseHa=angle(H);subplot(2,1,2);semilogx(O,Ha);xlabel('Frequency');ylabel('Phase');
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title('Phase Response of IIR Filter using Bilinear Transformation Method');
/ Butterworth Lowpass Impulse invariant method
Command window:
Output:
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/ Butterworth Lowpass Bilinear Transformation Method
Command Window :
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Output:
/ Chebyshev Lowpass Impulse invariant
method
Command Window:
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Output:
/Chebyshev Lowpass Bilinear
Transformation Method
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Command Window:
Output:
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RESULT:
Butterworth and Chebyshev Lowpass filters
were designed using Bilinear and Impulse Invariant
transformations, and manually verified the order, cut
off frequency and filter co-efficient of the filters.
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FLOWCHART:
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START
ENTER THE SYSTEM’S NUMERATOR AND
DENOMINATOR COEFFICIENTS
GENERATE THE WAVEFORM OF TIME
DOMAIN RESPONSE BY USING THE
APPROPRIATE LIBRARY FUNCTION
PLOT THE WAVEFORMS
STOP
AIM:
Write a MATLAB Script to find the time
domain response (impulse response and step
response) for the given FIR and IIR systems (filters).
.
APPARATUS REQUIRED:
PC, MATLAB software
THEORY:
IMPULSE RESPONSE:
δ[n]
y1[n]=T[δ[n]]=h[n]
If the input to the system is a unit impulse
(ie) x[n] = [n], then the output of the system, known
as the impulse response, is denoted by h [n] where,
h[n]=T[[n]]
STEP RESPONSE:
u[n]
y2[n]=T[u[n]]=s[n]
153 UR11EC098
EX. NO:12(A)DATE:10-03-14
EX. NO:12(A)DATE:10-03-14 TIME DOMAIN RESPONSE OF LTI SYSTEMSTIME DOMAIN RESPONSE OF LTI SYSTEMS
T
T
If the input to the system is a unit step (ie)
x[n] = u[n], then the output of the system, known as
step response, is denoted by s[n] where,
s[n]=T[u[n]]
The relation between the impulse response and
step response is given by
s[n] = u[n]*h[n]
* is the Convolution operator
LIBRARY FUNCTIONS: filter: Filters data with an infinite
impulse response (IIR) or finite impulse
response (FIR) filter .The filter function filters
a data sequence using a digital filter which
works for both real and complex inputs. The
filter is a direct form II transposed
implementation of the standard difference
equation.
y = filter (b, a, X) filters the data in
vector X with the filter described by numerator
coefficient vector b and denominator
coefficient vector a. If a(1) is not equal to 1,
filter normalizes the filter coefficients by a(1).
If a(1) equals 0, filter returns an error.
ALGORITHM/PROCEDURE:
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1. Click on the MATLAB icon on the desktop (or
go to Start – All Programs and click on
MATLAB) to get into the Command Window
2. Type ‘edit’ in the MATLAB prompt ‘>>’ that
appears in the Command window.
3. Write the program in the ‘Edit’ window and
save it in ‘M-file’
4. Run the program
5. Enter the input in the command window
6. The result is displayed in the Command
window and the graphical output is displayed
in the Figure Window
SOURCE CODE:
PROGRAM 1
clc;clear all;close all;%% Input the samples% Time domain response of FIR filterN=16; %Input samplesk=0:N-1;x=(k==0);b0=1; b1= -1; b2=-2;B=[b0,b1,b2]; %Numerator coeff.A=1; %Denominator coeff.%Filteringy=filter(B, A ,x);%Plot the graphsubplot(2,2,1), stem(k,x,'r');xlabel('Time');ylabel('Unit Impulse');
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title('Impulse input');subplot(2,2,2), stem(k,y,'r');xlabel('Frequency');ylabel('Magnitude');title('Impulse Response FIR Filter'); % Time domain Response of IIR FilterN1=10; %input samplesk1=0:N1-1;x1=(k1==0);B1=1;a=0.8;A1=[1,-a];y1=filter(B1, A1 ,x1);%plot the graphsubplot(2,2,3), stem(k1,x1,'r');xlabel('Time');ylabel('Unit Impulse');title('Impulse input');subplot(2,2,4), stem(k1,y1,'r');xlabel('Frequency');ylabel('Magnitude');title('Impulse Response IIR Filter');
PROGRAM 2clc;clear all;close all;%% Input the samples% Time domain Response of FIR filterN=16; %Input samplesk=0:N-1;x(1:N)=1;b0=1; b1= -1; b2=-2;B=[b0,b1,b2]; %Numerator coeff.A=1; %Denominator coeff.%Filteringy=filter(B, A ,x);%plot the graphsubplot(2,2,1);
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stem(k,x,'r');xlabel('Time');ylabel('Unit step signal');title('Unit step input signal');subplot(2,2,2);stem(k,y,'r');xlabel('Frequency');ylabel('Magnitude');title('Step Response FIR filter'); % Time domain response of IIR filterN1=10; %Input samplesk1=0:N1-1;x1(1:N1)=1;B1=1;a=0.8;A1=[1,-a];y1=filter(B1, A1 ,x1);%plot the graphsubplot(2,2,3);stem(k1,x1,'r');xlabel('Time');ylabel('Unit step input');title('Unit step input signal');subplot(2,2,4);stem(k1,y1,'r');xlabel('Frequency');ylabel('Magnitude');title('Step Response IIR Filter');
Output for Unit impulse input:
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Output for Unit step input:
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RESULT:
The Time domain responses of the given
systems were found using MATLAB and manually
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verified.
FLOWCHART:
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START
ENTER THE SYSTEM’S NUMERATOR AND
DENOMINATOR COEFFICIENTS
GENERATE THE WAVEFORM OF
FREQUENCY
PLOT THE WAVEFORMS
STOP
AIM:
Write a MATLAB Script to find the frequency
domain response (magnitude response and phase
response) for the given FIR and IIR systems (filters).
.
APPARATUS REQUIRED:
PC, MATLAB software
THEORY:
IMPULSE RESPONSE:
x[n]
y[n]=T[x[n]]
If the input to the system is a unit impulse
(ie) x[n]=[n], then the output of the system is
known as impulse response denoted by h[n] where,
h[n]=T[[n]]
We know that any arbitrary sequence x[n]
can be represented as a weighted sum of discrete
161 UR11EC098
EX. NO : 12(B)DATE:10-03-14
EX. NO : 12(B)DATE:10-03-14 FREQUENCY DOMAIN RESPONSE OF LTI SYSTEMSFREQUENCY DOMAIN RESPONSE OF LTI SYSTEMS
T
T
impulses. Now the system response is given by,
y[n]=T[x[n]]= T[ ]
where x(k) denotes the kth sample. The
response of DTLTI system to sequence x(k) [n-k]
will be x(k)h[n-k].i.e. T[x(k) [n-k]] = x(k) T[[n-k]]
= x(k) h[n-k].So response y[n] of DTLTI system to
x[n] is
y [n]=
This is known as convolution sum and can be
represented as
y[n] = x[n]*h[n]
* is the Convolution operator
FREQUENCY RESPONSE:
y[n] can also be written as
y[n]=
If x[n] is a complex exponential of the form x[n]
=ejn where n varies from - to
Then y[n] =
=
=
y[n]=H(ej) ejn
where H(ej)=
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H(ej) is called the frequency response of DTLTI
system. It is a complex function
H(ej) =Hr(ej )+j Him(ej)
H(ej)= ej()
is the magnitude response. is the phase
response.
LIBRARY FUNCTIONS:
exp: Exponential.exp (X) is the exponential of the elements of X, e to the power X. For complex Z=X+i*Y, exp (Z) = exp (X)*(COS(Y) +i*SIN(Y)).
disp: Display array.
DISP (X) is called for the object X when the semicolon is not used to terminate a statement. Freqz: Compute the frequency response
of discrete-time filters,
[h,w] = freqz (hd) returns the frequency response vector h and the corresponding frequency vector w for the discrete-time filter hd. When hd is a vector of discrete-time filters, freqz returns the matrix h. Each column of h corresponds to one filter in the vector hd.
Angle: Phase angle
P = angle (Z) returns the phase angle, in radians, for each element of complex array Z.
log10:
The log10 function operates element-by-element on arrays. Its domain includes
163 UR11EC098
complex numbers, which may lead to unexpected results if used unintentionally
ALGORITHM/PROCEDURE:
1. Click on the MATLAB icon on the desktop (or
go to Start – All programs and click on
MATLAB) to get into the Command Window
2. Type ‘edit’ in the MATLAB prompt ‘>>’ that
appears in the Command window.
3. Write the program in the ‘Edit’ window and
save it in ‘M-file’.
4. Run the program.
5. Enter the input in the command window.
6. The result is displayed in the Command
window and the graphical output is displayed
in the Figure Window.
SOURCE CODE:
clc;clear all;close all; %Frequency Responsenum=input('Enter num:');denum=input('Enter denum:'); n=linspace(0,pi,1000); h=freqz(num,denum,n); mag=20*log(abs(h));subplot(2,2,1),semilogx(n,mag);
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xlabel('Frequency index'),ylabel('Magnitude'),title('Magnitude Response'); pha=angle(h);subplot(2,2,2),semilogx(n,pha);xlabel('Frequency index'),ylabel('Phase'),title('Phase Response'); z=tf(num,denum,1)
COMMAND WINDOW:
OUTPUT :
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RESULT:
The frequency response (magnitude and phase
response) of the given system was found using
MATLAB.
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