bslab basic simulation lab
DESCRIPTION
wellTRANSCRIPT
BASIC SIMULATION LAB OBSERVATION
YEAR/SEMESTER
DEPARTMENT OF ECE
IND
Department of Electronics and Communication EngineeringBasic Simulation Lab
INDEX
Experiment
No.Name of the Experiment
Page No.
Signature of Lab Incharge
01 Basic operations on Matrices 2
02Generation Of Various Signals and Sequence ( periodic and aperiodic)
7
03 Operations on signals and sequence 10
04 Even and Odd parts of signal and sequence 13
05Convolution Between Signals and Sequences
15
06 17
07Auto correlation and Cross correlation between signals and sequences
19
08Computation of Unit sample, Unit step and sinusoidal response of LTI system
23
09Reconstruction of Periodic Signal by its Fourier Series
27
10Locating Zeros and Poles on S-Plane and Z-Plane
29
11 Sampling Theorem 33
12Removal of noise by Auto correlation / Cross Correlation
44
13 Impulse response of a raised cosine filter 48
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Expt No: 01 Date:
Basic operations on Matrices
Mathematical FunctionsMATLAB contains a set of built-in mathematical functions.
All of these functions are applied to arrays on an element by element basis. Thus, they return an array having the same size as the input array with elements all modified in the same way. We have already defined and used five of the functions. These are
sqrt - square rootreal - complex number real partimag - complex number imaginary partabs - complex number magnitudeangle - complex number angleIf a number is real, then abs produces the absolute value of the number. Other available mathematical functions that are of interest to us in signal and system analysis includeexp - exponential base elog - logarithm base elog 10 - logarithm base 10sin - sinecos - cosinetan - tangentasin - arcsineacos - arccosineatan - arctangentatan2 - four quadrant arctangentround - round to nearest integer[The trigonometric functions all apply to angles expressed in radians.]
Mathematical ExpressionsWe can combine arithmetic operations, 0-1 arrays generated by logical operations, and mathematical functions into mathematical expressions. Often, these expressions take the form of equations, although they may also be used in flow control statements. Thearithmetic operations follow the usual precedence rules. Many mathematical expressions require parentheses to construct the desired sequence of operations within the precedence rules.>> t=0.1; x=2^t*sqrt(t) - sin(2*t)/3x = 0.2727
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>> y=2^(t*sqrt(t)) - sin(2*t)/3y = 0.9559We can evaluate a mathematical expression for a set of independent variable values by expressing the independent variable as a one-dimensional array (vector) and using array operations.>> f=0:2:4; w=2*pi*f;>> X=(3 - j*0.1*w)./(1.5+j*0.2*w)X = 2.0000 0.1566 - 1.1002i -0.2956 - 0. 6850iOne important use of a 0-1 array for signal and system analysis is in the representation of a piecewise defined signal with a mathematical expression.>> t=-0.5:0.5:2.5;>> x=(t+1).*(t>=0&t<1)+2*(t>=1&t<=2)x = 0 1.0000 1.5000 2.0000 2.0000 2.0000 0The notation .* is required for the first mulitplication since we want element by elementmultiplication of the two like-sized, one-dimensional arrays. We can use just * for thesecond multiplication since it is a constant times an array.
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Expt No: 02 Date:
Generation Of Various Signals and Sequence( periodic and aperiodic)
AIM: To write a MATLAB program to generate the standard discrete time signals like unit impulse, unit step, unit ramp signals, discrete time signals, sinusoidal signals and exponential signals.
Program To Plot Standard Signals:
%Create a time base vector
t = [0:0.1:2];% Create a signal as a function of time
x = sin(pi*t/2);
subplot(3,3,1);
plot(t,x);
%Nonperiodic Signals
t = linspace(0,1,11);
%Step:
y = ones(1,11);
subplot(3,3,2);
stem(y);
axis([-1 6 0 2]);
% Impulse:
y = [1 zeros(1,10)];
subplot(3,3,3);
stem(y);
axis([-1 6 0 2]);
% Ramp:
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y = 2*t;
subplot(3,3,4);
plot(y);
axis([-1 6 0 2])
%Useful Matlab functions step, impulse, gen signal
%Step function:
fs = 10;
ts = [0:1/fs:5 5:1/fs:10];
x = [zeros(1,51) ones(1,51)];
subplot(3,3,5);
stairs(ts,x);
%Impulse function with width w:fs = 10;
w = 0.1;
ts = [-1:1/fs:-w 0 w:1/fs:1];
x = [zeros(1,10) 1 zeros(1,10)];
subplot(3,3,6);
plot(ts,x);
%Sinusoids
%Sinusoid parameters
%Amplitude, A
%Frequency, f
%Phase shift,
%Vertical offset, B
%The general form of a sine wave is
%y = Asin(2 ft + ) + B
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%Example: generate a sine wave given the following speculations:
%A = 5
%f = 2 Hz
%PS=π/8 radians
t = linspace(0,1,1001);
A = 5;
f = 2;
PS = pi/8;
sinewave = A*sin(2*pi*f*t + PS);
subplot(3,3,7);
plot(t, sinewave);
%Square Waves
%Square wave generation is like sine wave generation, but you specify a duty cycle, which is the percentage of the time over one period that the amplitude is high.
%Example:
% duty cycle is 75%
%frequency is 4 Hz.
t = linspace(0,1,1001);
sqw2 = square(2*pi*4*t,75);
subplot(3,3,8);
plot(t,sqw2);
axis([-0.1 1.1 -1.1 1.1]);
%Sawtooth Waves
%Sawtooth waves are like square waves except that instead of specifying a duty cycle, you specify the location of the peak of the sawtooth.
%Example:
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%peak is halfway through the period
%frequency is 3 Hz.
t = linspace(0,1,1001);
saw2 = sawtooth(2*pi*3*t,1/2);
subplot(3,3,9);
plot(t,saw2);
RESULT
Figure :Periodic and aperiodic signals
Questions:1. Define the different signals like step, ramp, impulse,
sinusoidal and exponential.2. What is the difference between analog signals and digital
signals?3. Express the ramp signal in terms of step. 4. Draw the different signal waveforms.
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Department of Electronics and Communication EngineeringBasic Simulation Lab
Expt No: 03 Date:
Operations on signals and sequence
AIM: To perform various operations on signals and sequence such as addition, multiplication, scaling, shifting and folding computation of energy and average power using MATLAB Code.
Program demonstrating Basic Signal Manipulation:
N=128;
f1=150;
f2=450;
f3=1500;
fs=8000;
n=0:N-1;
x1=sin(2*pi*(f1/fs)*n);
x2=(1/3)*sin(2*pi*(f2/fs)*n);
x3=sin(2*pi*(f3/fs)*n);
figure(1);
subplot(1,1,1);
subplot(2,3,1);
plot(n,x1);
grid;
title('Signal, x1(n)');
subplot(2,3,2);
plot(n,x2);
grid;
title('Signal, x2(n)');
subplot(2,3,3);
plot(n,x3);
grid;
title('Signal, x3(n)');
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% Signal Delay
x1d=[zeros(1,20), x1(1:N-20)];
subplot(2,3,4);
plot(n,x1d);
grid;
title('Delayed x(n), [x1(n-20)]');
% Signal Addition
xadd=x1+x2;
subplot(2,3,5);
plot(n,xadd);
grid;
title('x1(n)+x2(n)');
% Signal Multiplication
xmult=x1.*x3;
subplot(2,3,6);
plot(xmult);
grid;
title('x1*x3');
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Department of Electronics and Communication EngineeringBasic Simulation Lab
RESULTS:
Figure : Basic signal Manipulation with respect to time
Questions:1. What are the various mathematical operations on signals?2. Define the various mathematical operations like addition,
multiplication, division and average with respect to signals.3. Define scaling of a signal.4. What is the difference between frequency scaling and
amplitude scaling?
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Department of Electronics and Communication EngineeringBasic Simulation Lab
Expt No:04 Date:
Even and Odd parts of signal or sequence
AIM: To write a MATLAB code for finding the a) Even and Odd parts of signal or sequence b) Real and imaginary parts of signal.
% Finding the even and odd part of the signal x(n)=0.8^n
n=-5: 1: 5; %specify the range of n
A=0.8;
x1=A.^(n); %generate the given signal
x2=A.^(-n); %generate the folded signal
if(x2==x1)
disp('The given signal is even signal');
else if (x2==(-x1))
disp('The given signal is odd signal');
else
disp('The given signal is neither even nor odd signal');
end
end
xe=(x1+x2)/2; %compute even part
xo=(x1-x2)/2; %compute odd part
subplot(2,2,1);stem(n,x1);
xlabel('n');ylabel('x1(n)');title('signal x(n)');
subplot(2,2,2);stem(n,x2);
xlabel('n');ylabel('x2(n)');title('signal x(-n)');
subplot(2,2,3);stem(n,xe);
xlabel('n');ylabel('xe(n)');title('even part of x(n)');
subplot(2,2,4);stem(n,xo);
xlabel('n');ylabel('xo(n)');title('odd part of x(n)');
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RESULT:
Figure : Even and odd parts of given signal
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Department of Electronics and Communication EngineeringBasic Simulation Lab
B) Finding Real and imaginary parts of signal.
t = [-0.5:0.01:0.5];w=20;y = exp(i*pi*w*t/2);
a=real(y);subplot(2,2,1);plot(t,a);b=imag(y);subplot(2,2,2);plot(t,b);c=abs(y);subplot(2,2,3);plot(t,c);d=angle(y);subplot(2,2,4);plot(t,d);
Figure: Real and imaginary parts of signal
Questions:1. Define the Even and Odd Signals.2. Give the expressions for the Even and Odd signals.3. What is the real and imaginary part of the signal?
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Department of Electronics and Communication EngineeringBasic Simulation Lab
Expt No: 05a Date:Convolution Between Signals
AIM: To write a MATLAB program to perform convolution of the following two signals.
X1(t) =1; 0<t<2
x2(t) =1; 0<t<1
=-1; 1<t<2
Program:
tmin=0; tmax=4; dt=0.01;
t=tmin:dt:tmax; %set time vector for given signal
x1=1.*(t>=0&t<=2); %generate signal x1(t)
xa=1;
xb=-1;
x2=xa.*(t>=0&t<=1)+ xb.*(t>=1&t<=2);
% generate signal x2(t)
x3=conv(x1,x2); % perform convolution of xl(t) & x2(t)
n3=length(x3);
t1=0:1:n3-1; %set time vector for signal x3(t)
subplot(3,1,1);plot(t,x1);
xlabel(‘t’);ylabel(‘x1(t)’);title(‘signal x1(t)’);
subplot(3,1,2);plot(t,x2);
xlabel(‘t’);ylabel(‘x2(t)’);title(‘signal x2(t)’);
subplot(3,1,3);plot(t1,x3);
xlabel(‘t/dt’);ylabel(‘x3(t)/dt’);title(‘signal, x3(t)=x1(t)*x2(t)’);
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Department of Electronics and Communication EngineeringBasic Simulation Lab
RESULTS:
Figure :Convolution between CT signals
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Department of Electronics and Communication EngineeringBasic Simulation Lab
B)Convolution Between Sequences
AIM: To write a MATLAB program to perform convolution of the following two discrete time signals.
x1(n)=1; 1<n<10
x2(n)=1; 2<n<10
Program to perform convolution of two signals:
clear all
n=0:1:15; %specify range of n
x1=1.*(n>=1 & n<=10) ; %generate signal x1(n)
x2=1.*(n>=2 & n<=10); % generate signal x2(n)
N1=length(x1);
N2=length(x2);
x3=conv(x1,x2); % convolution of signals x1(n)and x2(n)
n1=0: 1: N1+N2-2; %specify range of n for x3(n)
subplot(3,1,1);stem(n,x1);
xlabel('n');ylabel('x1(n)');
title('signal x1(n)');
subplot(3,1,2);stem(n,x2);
xlabel('n');ylabel('x2(n)');
title('signal x2(n)');
subplot(3,1,3);stem(n1,x3);
xlabel('n');ylabel('x3(n)');
title('signal , x3(n)=x1(n)*x2(n)');
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Department of Electronics and Communication EngineeringBasic Simulation Lab
RESULT:
Figure :Convolution between DT signals
Questions:
1. Define convolution.
2. What is the difference between sequence and signal?
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Department of Electronics and Communication EngineeringBasic Simulation Lab
Expt No: 07 Date:
Auto correlation and Cross correlation between signals and sequences
AIM: To find Auto correlation and Cross correlation between signals and sequences using MATLAB Code.
Program for auto correlation:
N=1024; % Number of samples
f1=1; % Frequency of the sine wave
FS=200; % Sampling Frequency
n=0:N-1; % Sample index numbers
x=sin(2*pi*f1*n/FS); % Generate the signal, x(n)
t=[1:N]*(1/FS); % Prepare a time axis
subplot(2,1,1); % Prepare the figure
plot(t,x); % Plot x(n)
title('Sinwave of frequency 1000Hz [FS=8000Hz]');
xlabel('Time, [s]');
ylabel('Amplitude');
grid;
Rxx=xcorr(x); % Estimate its autocorrelation
subplot(2,1,2); % Prepare the figure
plot(Rxx); % Plot the autocorrelation
grid;
title('Autocorrelation function of the sinewave');
xlable('lags');
ylabel('Autocorrelation');
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RESULT:
Figure :Auto correlation function of the Sine wave
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Department of Electronics and Communication EngineeringBasic Simulation Lab
Program for cross correlation:
N=1024; % Number of samples to generate
f=1; % Frequency of the sinewave
FS=200; % Sampling frequency
n=0:N-1; % Sampling index
x=sin(2*pi*f1*n/FS); % Generate x(n)
y=x+10*randn(1,N); % Generate y(n)
subplot(3,1,1);
plot(x);
title(‘Pure Sinewave’);
grid;
subplot(3,1,2);plot(y);
title(‘y(n), Pure Sinewave + Noise’);
grid;
Rxy=xcorr(x,y); % Estimate the cross correlation
subplot(3,1,3);
plot(Rxy);
title(‘Cross correlation Rxy’);
grid;
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RESULT:
Figure : Cross correlation function of sine wave and Noise
Questions:
1. Define auto correlation.2. Define cross correlation.3. What is the difference between auto and cross
correlations?
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Department of Electronics and Communication EngineeringBasic Simulation Lab
Expt No: 08 Date:Computation of Unit sample, Unit step and
sinusoidal response of LTI system
AIM: Write a MATLAB program to compute and sketch the impulse response of a discrete time system governed by the following transfer function,
H(Z)=1/(1-0.8Z^-1+0.16 Z^-2)
Program to find 'impulse response of a discrete time System:
clear all
syms z n
H=1/(1-0.8*(z^(-1))+0.16*(z^(-2)));
disp('impulse response h(n) is');
h=iztrans(H) ; %compute impulse response
simplify(h)
N=15;
b=[0 0 1]; %numerator coefficients
a=[1 -0.8 0.16]; %denominator coefficients
[H,n]=impz(b,a,N); %compute N samples of impulse response
Stem(n,H); %sketch impulse response
xlabel('n');
ylabel('h(n)');
title('impulse response of a DT System');
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RESULT:
Figure:Impulse Response of a DT system
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Department of Electronics and Communication EngineeringBasic Simulation Lab
AIM: To write a MATLAB program to find the step response of the first and second order LTI systems governed by the following transfer functions,
H(s)=1/(s+2) and H(s)=1/(s2+2.5s+25) .
Program to find the step response of I and II order systems:
syms s complex;
H1=1/(s+2);
disp('step response of first order system is');
h1= ilaplace(H1);
simplify(h1)
H2=1/(s^2+2.5*s+25);
disp('step response of second order system is');
h2= ilaplace(H2);
simplify(h2)
s=tf('s');
H1=1/(s+2);
H2=1/(s^2+2.5*s+25);
t1=0: 0.0005 :5; %set a time vector
s1=step(H1,t1); % step response of first order system
s2=step(H2,t1); % step response of second order system
subplot(2,1,1);plot(t1,s1);
xlabel('Time in seconds');ylabel('s1(t)');
title('step response of first order system');
subplot(2,1,2);plot(t1,s2);
xlabel('Time in seconds');
ylabel('s2(t)');
title('step response of second order system');
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RESULT:
Figure : Unit Step Response for given I and II order systems
Questions:
1. Define impulse signal and step signal.
2. What is impulse response and step response?
3. What are the time response specifications?
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Department of Electronics and Communication EngineeringBasic Simulation Lab
Expt No: 09 Date:
Reconstruction of Periodic Signal by its Fourier Series
AIM: Write a MATLAB program to reconstruct the following periodic signal represented by its fourier series , by considering only 3,5 and 59 terms.
x(t)=(1/2)+ bn sinnΩ0 . where bn=2/n; Ω0=2nF; F=1
Program:
syms t real;
N=input('Enter number of signals to reconstruct');
n_har=input('enter number of harmonics in each signal as array');
t=-1: 0.002:1;
omega_o=2*pi;
for k=1:N
n=[ ];
n=[1:2:n_har(k)];
b_n=2./(pi*n);
L_n=length(n);
x=0.5+b_n*sin(omega_o*n'*t);
subplot(N,1,k);
plot(t,x);
xlabel('t');
ylabel('recons signal');
axis( [-1 1 -0.5 1.5]);
text(.55, 1.0,['no.of har.=',num2str(n_har(k))]);
end
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RESULT:
Enter number of signals to reconstruct : 3
enter number of harmonics in each signal as array[3 5 59]
.
Figure : Gibbs phenomenon
Questions:
1. What is fourier series?
2. Define periodic and non periodic signals.
3. What is the difference between FS and DFS?
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Department of Electronics and Communication EngineeringBasic Simulation Lab
Expt No: 10 Date:
Locating Zeros and Poles on S-Plane and Z-Plane
AIM: To write a MATLAB program for finding residues and poles of s-domain transfer function
Program:
clc;close all;clear all;disp('Enter the transfer function of the original system');N=input('Enter the co efficients of the numarator:');D=input('Enter the co efficients of the denominator:');disp('The transfer function of the original system:');G=tf(N,D);ltiview(G);
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RESULT:
Enter the transfer function of the original system
Enter the co efficients of the numarator: [1 2 3]
Enter the co efficients of the denominator: [2 3 3]
The transfer function of the original system:
Figure : Pole-zero map for s-domain transfer function
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Department of Electronics and Communication EngineeringBasic Simulation Lab
AIM: To write a MATLAB program for finding residues and poles of z-domain signal is given below
(Z^2+0.8Z+0.8)/(Z^2+0.49),
And sketch the pole zero plot.
Program
clear allsyms znum_coeff=[1 0.8 0.8];disp('Roots of numerator polynomial Z^2+0.8Z+0.8 are');zeros=roots(num_coeff)
den_coeff=[1 0 0.49];disp('Roots of denominator polynomial Z^2+0.49 are');poles=roots(den_coeff)H=tf('z');Ts=0.1;
a=tf([num_coeff], [den_coeff], Ts);zgrid on;pzmap(a); %pole -zero plotRoots of numerator polynomial Z^2+0.8Z+0.8 are
zeros =
-0.4000 + 0.8000i -0.4000 - 0.8000i
Roots of denominator polynomial Z^2+0.49 are
poles =
0 + 0.7000i 0 - 0.7000i
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RESULT:
Figure : Pole- Zero map for Z-domain signal
Questions:1. Define sampling theorem in both frequency and time domain.2. What is the sampling rate?3. What is the multiple sampling rate?4. What is the Nyquist Sampling Rate?
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Expt No: 12 Date:
Removal of noise by Auto correlation / Cross Correlation
AIM: Removal of noise by Auto / Cross Correlation in a given signal corrupted by noise by using MATLAB Code.
Program
% AutoCorrelation of White Noise
% Loops are used in this program for easier portability to
% C or assembly code.
%
%************************************************************
% N = Length of Time Domain Signal
N = 512;
%************************************************************
% Create an index matrix for graphing:
%************************************************************
for n = 1: 1: N;
index(n) = n;
end
%************************************************************
% Create an expanded index matrix from -N to N for graphing:
%************************************************************
index_expanded = 1:(2*N - 1);
for n = 1: 1: (N - 1);
index_expanded(n) = -(index(N - n));
end
index_expanded(N) = 0;
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for n = (N + 1): 1: (2*N - 1);
index_expanded(n) = index(n - N);
end
%************************************************************
% Generate the flat +- 0.1 white noise:
%************************************************************
w_2 = 1:N;
w_2 = 0.1*rand(1,N);
w_2 = w_2 - mean(w_2);
%************************************************************
% Compute 0.1 white noise Autocorrelation:
%************************************************************
r_xx_w_2 = 1:N;
for m = 1: 1: N;
F = 0;
for k = 1: 1: (N+1-m);
F = F + w_2(k)*w_2(k-1+m);
end
r_xx_w_2(m) = F/(N+1-m);
end
%************************************************************
% Create an expanded 0.1 white noise Autocorrelation Function
% from -N to N for graphing purposes:
%************************************************************
r_xx_w__expanded = 1:(2*N - 1);
for m = 1: 1: N;
r_xx_w_2_expanded(m) = r_xx_w_2(N - m + 1);
end
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for m = N: 1: (2*N - 1);
r_xx_w_2_expanded(m) = r_xx_w_2(m + 1 - N);
end
%************************************************************
% Plot the Results
%************************************************************
figure
plot(index_expanded,r_xx_w_2_expanded)
title(['Auto Correlation Function of 0.1 w.n Input'])
RESULT:
Figure : Auto Correlation function of 0.1 White noise input
Questions:1. Define noise.2. How the signals are affected by the noise?3. What are the differences between random noise and white
noise?
PMS_PSKR: 34
Department of Electronics and Communication EngineeringBasic Simulation Lab
Expt No: 13 Date:Impulse response of a raised cosine filter
AIM: write a MATLAB code to find the Impulse Response of a Raised Cosine Filter.
Program:
% Script for plotting the time domain and frequency domain representation% of raised cosine filters for various values of alphaclear allfs = 10;% defining the sinc filtersincNum = sin(pi*[-fs:1/fs:fs]); % numerator of the sinc functionsincDen = (pi*[-fs:1/fs:fs]); % denominator of the sinc functionsincDenZero = find(abs(sincDen) < 10^-10);sincOp = sincNum./sincDen;sincOp(sincDenZero) = 1; % sin(pix/(pix) =1 for x =0
alpha = 0;cosNum = cos(alpha*pi*[-fs:1/fs:fs]);cosDen = (1-(2*alpha*[-fs:1/fs:fs]).^2);cosDenZero = find(abs(cosDen)<10^-10);cosOp = cosNum./cosDen;cosOp(cosDenZero) = pi/4;gt_alpha0 = sincOp.*cosOp;GF_alpha0 = fft(gt_alpha0,1024);
close allfigureplot([-fs:1/fs:fs],[gt_alpha0],'b','LineWidth',2)
grid onxlabel('time, t')ylabel('amplitude, g(t)')title('Time domain waveform of raised cosine pulse shaping filters')
figureplot([-512:511]/1024*fs, abs(fftshift(GF_alpha0)),'b','LineWidth',2);axis([-2 2 0 14])grid onxlabel('frequency, f')ylabel('amplitude, |G(f)|')title('Frequency domain representation of raised cosine pulse shaping filters')
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Department of Electronics and Communication EngineeringBasic Simulation Lab
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