signals and systems lab manual print

LAB MANUAL

SUBJECT: SIGNALS AND SYSTEMSLABORATORY

PAPER CODE: EC-209

DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING

DELHI TECHNOLOGICAL UNIVERSITY, DELHI-110042

SIGNALS AND SYSTEMS LABORATORY EC-209

LIST OF EXPERIMENTS

1. Write a MATLAB code to generate different types of Signals.2. Write a MATLAB code to perform an Arithmetic operation on signal’s

Addition, Subtraction, Multiplication and Division.3. Write a MATLAB code to perform signal processing operations: Time

shifting, Time scaling, and Time inversion.4. Write a MATLAB code to perform Convolution of two signals.5. Write a MATLAB code to determine response of LTI system for a given

input signal.6. Write a MATLAB code to determine Fourier transform of a given input

signal and plot its Magnitude and Phase spectra.

7. Write a MATLAB code to generate Frequency Spectrum of a Periodicsignal.

8.Write a MATLAB code to generate Inverse Fourier transform of a signal.

9. Write a MATLAB code to determine Frequency response of FIR system.

10. Write a MATLAB code to determine Frequency response of Infinite ImpulseResponse system.

11. Write a MATLAB code to determine Z-transform and plot its poles andzeros location in z-plane.


EXPERIMENT NO.:-1

OBJECTIVE: - Write a Matlab code to generate different types of basic Signals.

HARDWARE /SOFTWARE REQUIRED:-

1) MATLAB 7.6 Version2) Computer3) PrinterTheory:-Basic idea of different types of basic signals i.e.

i. Unit impulse function

0 ; t ≠ 0

ii. Unit step function

0; t<0


III Ramp function

IV exponential function0 ; t<0

There can be different cases in which ‘a’ value differences.

• ‘a’ can be real (i) real (ii) negative• ‘a’ can be a purely imaginary• ‘a’ can be complex with both real and imaginary part.

Program Code:

t = -20:20;t1 = -5:0.01:5;t2 = -5:5;unit_imp = [zeros(1,20),1,zeros(1,20)];unit_step = [zeros(1,20),ones(1,21)];i=1;for k=-20:20

if k>0ramp(i) = k;

elseramp(i) = 0;

endi = i+1;

endexpo = exp(-t2);sinu = sin(t1);subplot(3,2,1);stem(t,unit_imp);


xlabel('Time');ylabel('Unit Impulse signal');grid on;

subplot(3,2,2);stem(t,unit_step);xlabel('Time');ylabel('Unit Step signal');grid on;

subplot(3,2,3);stem(t,ramp);xlabel('Time');ylabel('Ramp signal');grid on;

subplot(3,2,4);stem(t2,expo);xlabel('Time');ylabel('Exponential signal')grid on;Observation/Plot

Result:-Practical result is closely matched with experimental result.


EXPERIMENT NO.:-2

OBJECTIVE:-Write a Matlab code to perform arithmetic operations for a signal such asaddition, subtraction, multiplication and division.HARDWARE/SOFTWARE REQUIRED:-1) MATLAB version7.62) Computer3) PrinterTheory:- The theoretical results of various arithmetic operations such as addition,multiplications, subtraction of two signals (say sine wave and ramp signal).Let us consider two signalsx(t)=sin(wt)y(t)-cos(wt)

Signal processing operation on dependent variable is of the following types:-1) Addition

z(t)=x(t)+y(t)=sin(wt)+cos(wt)

2)Subtractionz(t) =x(t)-y(t)

=sin(wt)-cos(wt)3) Multiplication

z(t)=x(t)*y(t)=sin(wt)*cos(wt)

4) Divisionz(t)=x(t)/y(t);y(t)≠0

=sin(wt)/cos(wt)Program Code:t = -5:0.1:5;x = sin(t);y= cos(t);x_addition = x+y;x_division = x./y;x_multiplication = x.*y;x_subtraction = x-y;grid on;

subplot(2,3,1);plot(t,x);grid on;xlabel('Time');ylabel('Original Signal');subplot(2,3,2);plot(t,x_addition);


grid on;xlabel('Time');ylabel('Added Signal');

subplot(2,3,3);plot(t,x_division);grid on;xlabel('Time');ylabel('Divided Signal');

subplot(2,3,4);plot(t,x_multiplication);grid on;xlabel('Time');ylabel('Multiplied Signal');subplot(2,3,5);plot(t,x_subtraction);grid on;xlabel('Time');ylabel('subtracted Signal');

Observation/Plot



EXPERIMENT NO.:-3

OBJECTIVE: -Write a MATLAB code to perform signal processing operations time scalingtime shifting time inversion.Hardware/Software required:-1) MATLAB version7.62) Computer3) PrinterTheory:-Various signal processing operation that can be performed on a signal are:-

x(t)=

0 3

• Time inversion: -if x (t) is the original signal then x (-t) is the time inverted signal.

-3 0• Time scaling: - if x (t) is a given signal then x(t) is time scaled by ‘a’ is given below;

0 1• Time shifting:- if x(t) is a given signal then x(t) is time shifted by t0 can be obtained

• Delayed x(t-t0 )

0 t0 3-to


• Advanced x(t+t0)

t0+2 0 1

Program Code: (Example)f = input('Enter frequency of the signal -> ');w = 2*pi*f;t = 0:1/(1000*f):1/f;X = sin(w*t);

%Time Scalingk = input('Enter the value by which you want to scale the signal (compress or elongate) -> ');Xsc = sin(k*w*t);%Time Shiftingt1 = input('Enter the value of time by which you want to advance (enter positive value) OR delay(enter negative value) the signal ->');Xsh = sin(w*(t-t1));%Time InversionXin = sin(-w*t);

%Plotting the transformed and original signalshold on;plot(t,X,'-k','Linewidth',2); grid on;plot(t,Xsc,':r','Linewidth',2); grid on;plot(t,Xsh,'-g','Linewidth',2); grid on;plot(t,Xin,'-.b','Linewidth',2); grid on;xlabel('Time');ylabel('Amplitude of the signal');


Observation/Plot



Experiment NO.:-4

OBJECTIVE:- Write a program to find the Convolution of two signals.Hardware/Software required:-1) MATLAB version 7.62) Computer3) PrinterTheory:-The below equation defines the integral convolution of 2 time signals x (t) and h (t) denoted by

Or

Program Code:

a=[1 2 3];b=[4 5 6];i=length(a);j=length(b);A=[a,zeros(1,i)];B=[b,zeros(1,j)];for k=1:i+j-1y(k)=0;for l=1:i

if (k-l+1>0)y(k)=y(k)+A(l)*B(k-l+1);

elseend

endend;stem (y);

ylabel('Y[k]');xlabel('----->n');title('convolution of two signals');


Observation/Plot



Experiment NO.:-5

OBJECTIVE:-Write a matlab code to determine the LTI response of a given Input signal.

Hardware/Software required:-

1) Matlab version 7.6

2) Computer

3) Printer

Theory: - LTI system or linear time invariant system

LTI response of the signal y(t)

i.e convolution of two signal x(t) and h(t) is given as

m=length (x); n=length (h) the vector y is of length m+n-1, where kth element is

Program Code: (Example)

t = 0:0.01:10;x = exp(-t);h = [ones(1,1001)];Lx = length(x);Lh = length(h);Ly = Lx + Lh -1;for p=1:Ly

sum=0;for k=1:p

if(k<=Lx && k>=(p+1-Lh))sum = sum + x(k).*h(p+1-k);


endendy(p) = sum;

endt1 = 0:0.01:(Ly-1)*0.01;

%Input Signalsubplot(2,2,1);plot(t,x);title('Input Signal');xlabel('Time'); ylabel('Amplitude of the signal');

%Impulse Responsesubplot(2,2,2);plot(t,h);title('Impulse Response');xlabel('Time'); ylabel('Amplitude of the signal');

%Output of the LTI Systemsubplot(2,2,[3:4]);plot(t1,y);title('Output of the LTI System');xlabel('Time'); ylabel('Amplitude of the signal');

Observation/Plot



Experiment No:-6

OBJECTIVE:- Write a matlab code to generate Fourier transform of a given Input signal andplot its Magnitude and Phase spectra.Hardware required/software required:-1) MATLAB version 7.62) Computer3) PrinterTheory:-Fourier transform is divided into 2 parts

• Continuous time Fourier transformThis is applicable for aperiodic function and periodic functions

And the corresponding function isX(t)= Fourier transform of a signal can be obtained by using FFT function in matlab.The plot of X(W) vs w is called as magnitude spectraAnd angle of X(W) vs w is known as phase spectra.• Discrete time Fourier transform

It is divided for periodic function.


x =[1,1,1,1,zeros(1,4)];n=8;X= fft(x,n);magx= abs(X);phase=angle(X)*180/pi;subplot(2,2,1);plot(x,'k');xlabel('n');ylabel('Signal');title('Input signal');


subplot(2,2,2);plot(magx);grid;xlabel('n');ylabel('Magnitude');title('Magnitude plot');subplot(2,2,3);plot(phase);grid;xlabel('n');ylabel('degrees');title('Phase plot')

Observation/Plot



Experiment No:-7

OBJECTIVE: - Write a matlab code to generate frequency spectrum of a periodic signal.Hardware/Software required:-1) MATLAB version 7.62) Computer3) PrinterTheory:-A continuous time signal x(t) to be periodic if there is a positive non zero value of t forwhich

All t ----- (1)

The fundamental period T0 of x (t) is the smallest positive value of T for which equation (1) isnot satisfied and 1/t0=f0 is referred as fundamental frequency.Two basic example of periodic signal are the real sinusoidal signal

The complex exponential Fourier series representation of a periodic signal x(t) with To asfundamental period

wo=2π/To

Where are the complex Fourier Coefficients


t = 0:0.01:10;%Train of Pulsesx = [ones(1,201),zeros(1,99),ones(1,201),zeros(1,99),ones(1,201),zeros(1,99),ones(1,101)];T = 3;w = 2*pi/T;


dtau = 0.01; %Difference between two time intervals as taken in matlab;for k=-10:10

sum = 0;i=1;for tau=0:dtau:T

exp_part = exp(-j*w*k*tau)*dtau;sum = sum + exp_part.*x(i);i=i+1;

enda(k+11) = sum;

end

for i=1:21mag(i) = abs(a(i));phase(i) = angle(a(i));

endk=-10:10;

%Original Signalsubplot(2,2,[1:2]);plot(t,x);title('Input Signal (A Pulse Train)');xlabel('Time');ylabel('Amplitude of the signal');%Magnitude Spectrasubplot(2,2,3);stem(k,mag);title('Magnitude Spectra');xlabel('k \rightarrow');ylabel('Magnitude');%Phase Spectrasubplot(2,2,4);stem(k,phase,'Linewidth',1.2)title('Phase spectra');xlabel('k \rightarrow');ylabel('Phase');


Observation/Plot



Experiment No.:– 8

OBJECTIVE:-Write a matlab code to generate inverse Fourier transform of a signal.

Hardware/Software Requirements:1) MATLAB version 7.62) Computer/Laptop2) PrinterTheory:- Inverse Fourier transform is used to convert a signal in frequency domain to a signal intime domain.If X(w) is the Fourier transform of a signal then, the original signal x(t) or inverse Fouriertransform of X(w) is given by:-

In MATLAB, we can use a direct command to evaluate the inverse Fourier transform i.e.d=ifft(x);Program Code: (Example)w=-100:.1:100;X = 1./(1+w.^2)dw = 0.1;m=1;for t=0:0.01:10

i=1;sum=0;for w=-100:dw:100

sum = sum + X(i)*exp(j*w*t)*dw;i=i+1;

endx(m) = sum;m=m+1;

endw=-100:.1:100;t=0:.01:10;

%given signalsubplot(2,1,1);plot(w,X);title('Given signal in Frequency domain');xlabel('Frequency \rightarrow');ylabel('X(w)');grid on;

%signal in time domainsubplot(2,1,2);


plot(t,x);title('Signal obtained via Inverse FT');xlabel('Time \rightarrow');ylabel('x(t)');grid on;

Observation/Plot



Experiment No.:-9

OBJECTIVE:-Write a Matlab code to perform and plot the Frequency response and Phaseresponse of FIR system.Hardware/Software Required:-1) MATLAB version 7.62) Computer3) PrinterTheory:- [h,p]=-freqz(ba,a,w) returns the frequency response vector h and the correspondingangular frequency vector for the digital filter whose transfer function is determined by the (realor complex) numerator and denominator polynomial represented by the vector b and arespectively. The vector h and p are both of length w has values ranging from 0 to pi per sample.

Freqz generally uses an fft algorithm to compute the frequency response.Program Code: (Example)h=[1,2,3,4];a=[1 2 3 4];b=[1];w=-pi:.01:pi;H=freqz(b,a,h);% plot of the impulse response of the systemsubplot(3,1,1);plot(h);grid on;xlabel('time');ylabel('h[n]');title('Impulse Response');% plot of the magnitude of frequency response of the systemsubplot(3,1,2)plot(w,abs(H));grid on;title('plot of the magnitude response ');xlabel('w');ylabel('|H[w]|');

% plot of the phase of frequency response of the systemsubplot(3,1,3)plot(w,angle(H));grid on;title('plot of the magnitude response ');xlabel('w');


ylabel('<H[w]');

Observation/Plot



Experiment No.:-10

OBJECTIVE:-Write a Matlab code to perform and plot the frequency response and phaseresponse of IIR system.Hardware/Software Required:-1) MATLAB version 7.62) Computer3) PrinterTheory:- a system whose impulse response or no. of non zero values of the output is infinite iscalled infinite impulse response IIR system.Let x (t) be the input signal. Let h(t) be the impulse response &Let y(t) be the response of thesystem.Taking Fourier transform, of the three. We get X (w), Y(w) &h(w)Y (t)=x(t)*h(t)Taking Fourier transformY(w)=X(w)H(w)H (w)=Y(w)/X(w)H (w) is called frequency response of system.Program Code: (Example)h1=(0.5)^n;a=[1,-0.5];b=[1];w=-pi:.01:pi;[h,p]=freqz(b,a,h);

% plot of the magnitude of frequency response of the systemsubplot(2,1,1)plot(w,h);grid on;title('plot of the magnitude response ');xlabel('w');ylabel('|H[w]|');

% plot of the phase of frequency response of the systemsubplot(2,1,2)plot(p);grid on;title('Phase response ');xlabel('w');ylabel('<H[w]');

Observation/Plot


Experiment No.:-11

OBJECTIVE:-Write a Matlab code to plot poles &zeros of the given Z transform of signal.Hardware/Software Required:-1) MATLAB version 7.62) Computer3) Printer

THEORY:- A pole-zero plot shows the location in the complex plane of the poles and zeros ofthe transfer function of a dynamic system, such as a controller, compensator, sensor, equalizer,filter, or communications channel. By convention, the poles of the system are indicated in theplot by an X while the zeroes are indicated by a circle or O.A pole-zero plot can represent eithera continuous-time (CT) or a discrete-time (DT) system. For a CT system, the plane in which thepoles and zeros appear is the s plane of the Laplace transform. In this context, theparameter s represents the complex angular frequency, which is the domain of the CT transferfunction. For a DT system, the plane is the z plane, where z represents the domain of the Z-transform. This function displays the poles and zeros of discrete-time systems.zplane(b,a) where b and a are row vectors, first uses roots to find the zeros and poles of thetransfer function represented by numerator coefficients b and denominator coefficients a. Thetransfer function is defined in terms of z-1:

Zplane (b,a) plots the zeros specified in column vector b and the poles specified in column vectora in the current figure window. The symbol 'o' represents a zero and the symbol 'x' represents apole. The plot includes the unit circle for reference. If z and p are arrays, zplane plots the polesand zeros in the columns of z and p in different colors.Program Code: (Example)b= [1 0 0 -8];a= [1 0 -1];Zplane (b, a);Observation/Plot

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