Basic Simulation Lab

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SREENIVASA INSTITUTE OF TECHNOLOGY AND MANAGEMENT STUDIES

Dept.of ECE 1

BASIC SIMULATION LAB

(Using MATLAB)

Ex.No. Name of the Experiment

1. Basic operations on Matrices2. Generation Of Various Signals and Sequence ( periodic and aperiodic)3. Operations on signals and sequence4. Even and Odd parts of signal or sequence5. Convolution Between Signals6. Convolution Between Sequences7. Auto correlation and Cross correlation between signals and sequences8. Computation of Unit sample, Unit step and sinusoidal response of LTI

system

9. Reconstruction of Periodic Signal by its Fourier Series10. Locating Zeros and Poles on S-Plane and Z-Plane11. Sampling Theorem

ADDITIONAL EXPERIMENTS

1. Removal of noise by Auto correlation / Cross Correlation

2. Impulse response of a raised cosine filter


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Dept.of ECE 2

EXPERIMENT-1

BASIC OPERATIONS ON MATRICES

AIM:To perform the different operations on Matrices.

Requirements: PC with MATLAB R2008a

THEORY:

What Is MATLAB?

MATLAB is a high-performance language for technical computing. It integrates

computation, visualization, and programming in an easy-to-use environment where problems

and solutions are expressed in familiar mathematical notation. Typical uses include Math andcomputation Algorithm development Data acquisition Modeling, simulation, and prototyping

Data analysis, exploration, and visualization Scientific and engineering graphics Application

development, including graphical user interface building .

MATLAB is an interactive system whose basic data element is an array that does not

require dimensioning. This allows you to solve many technical computing problems,

especially those with matrix and vector formulations, in a fraction of the time it would take to

write a program in a scalar noninteractive language such as C or Fortran.

The name MATLAB stands for matrix laboratory. MATLAB was originally writtento provide easy access to matrix software developed by the LINPACK and EISPACK

projects. Today, MATLAB engines incorporate the LAPACK and BLAS libraries,

embedding the state of the art in software for matrix computation.

MATLAB has evolved over a period of years with input from many users. In

university environments, it is the standard instructional tool for introductory and advanced

courses in mathematics, engineering, and science. In industry, MATLAB is the tool of choice

for high-productivity research, development, and analysis.

MATLAB features a family of add-on application-specific solutions called toolboxes.

Very important to most users of MATLAB, toolboxes allow you to learn and apply

specialized technology. Toolboxes are comprehensive collections of MATLAB functions

(M-files) that extend the MATLAB environment to solve particular classes of problems.

Areas in which toolboxes are available include signal processing, control systems, neural

networks, fuzzy logic, wavelets, simulation, and many others.

Basic Matrix Operations:

This is a demonstration of some aspects of the MATLAB language.

First, let's create a simple vector with 9 elements called a.

a = [1 2 3 4 6 4 3 4 5]

a =

1 2 3 4 6 4 3 4 5

Now let's add 2 to each element of our vector, a, and store the result in a new vector.

Notice how MATLAB requires no special handling of vector or matrix math.

b = a + 2


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Dept.of ECE 3

b = 3 4 5 6 8 6 5 6 7

Creating a matrix is as easy as making a vector, using semicolons (;) to separate the

rows of a matrix.

A = [1 2 0; 2 5 -1; 4 10 -1]

A =

1 2 0

2 5 -1

4 10 -1

We can easily find the transpose of the matrix A.

B = A'

B =

1 2 4

2 5 10

0 -1 -1

Now let's multiply these two matrices together.

Note again that MATLAB doesn't require you to deal with matrices as a collection of

numbers. MATLAB knows when you are dealing with matrices and adjusts your calculations

accordingly.

C = A * B

C =

5 12 24

12 30 59

24 59 117

Instead of doing a matrix multiply, we can multiply the corresponding elements of

two matrices or vectors using the .* operator.

C = A .* B

C =

1 4 0

4 25 -100 -10 1

Let's find the inverse of a matrix ...

X = inv(A)

X =

5 2 -2

-2 -1 1

0 -2 1

... and then illustrate the fact that a matrix times its inverse is the identity matrix.

I = inv(A) * A

I =

1 0 0

0 1 0

0 0 1

MATLAB has functions for nearly every type of common matrix calculation.


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Dept.of ECE 4

There are functions to obtain eigenvalues ...

eig(A)

ans =

3.7321

0.2679

1.0000

... as well as the singular value decomposition.

svd(A)

ans =

12.3171

0.51490.1577

The "poly" function generates a vector containing the coefficients of the characteristic

polynomial.

The characteristic polynomial of a matrix A is

p = round(poly(A))

p =

1 -5 5 -1

We can easily find the roots of a polynomial using the roots function.

These are actually the eigenvalues of the original matrix.

roots(p)

ans =

3.7321

1.0000

0.2679

MATLAB has many applications beyond just matrix computation.

To convolve two vectors ...

q = conv(p,p)

q =

1 -10 35 -52 35 -10 1

r = conv(p,q)

r =1 -15 90 -278 480 -480 278 -90 15 -1

At any time, we can get a listing of the variables we have stored in memory using the

who or whos command.

whos

Name Size Bytes Class

A 3x3 72 double array

B 3x3 72 double array

C 3x3 72 double array


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Dept.of ECE 5

I 3x3 72 double array

X 3x3 72 double array

a 1x9 72 double array

ans 3x1 24 double arrayb 1x9 72 double array

opts 1x1 522 struct array

p 1x4 32 double array

q 1x7 56 double array

r 1x10 80 double array

Grand total is 162 elements using 1218 bytes

You can get the value of a particular variable by typing its name.

A

A =

1 2 0

2 5 -1

4 10 -1

You can have more than one statement on a single line by separating each statement

with commas or semicolons.

If you don't assign a variable to store the result of an operation, the result is stored in

a temporary variable called ans.

sqrt(-1)

ans =

0 + 1.0000i

RESULT:Thus the different basic operations on Matrices and vectors were verified using

MATLAB.

QUESTIONS:

1. What is MATLAB ?2. What are the applications of MATLAB?3. What are the advantages of MATLAB over other software?4. What are the different operators in MATLAB?5. What are the different windows in MATLAB?


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Dept.of ECE 6

EXPERIMENT-2

GENERATION OF VARIOUS SIGNALS AND SEQUENCES

(PERIODIC AND APERIODIC)

AIM:To generate various signals and sequences, such as Unit Impulse, Unit Step, Square,Sawtooth, Triangular, Sinusoidal, Ramp and Sinc functions


MATLAB CODE:

%Unit impluse signaln=-10:2:10;

X1=1; X2=0;X=X1.*(n==0)+X2.*(n~=0);

subplot(4,3,1); stem(n,X);title('unit impluse');

xlabel('n'); ylabel('X(n)');

%Unit step signaln=-10:2:10;

X1=1; X2=0;X=X1.*(n>=0)+X2.*(n


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Dept.of ECE 7

xlabel('n'); ylabel('X(n)');

%increasing ramp

n=-10:2:10;X1=n; X2=0;X=X1.*(n>=0)+X2.*(n


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Dept.of ECE 8

title('sinc signal');

Expected Waveforms:Discrete Time Signals

Continuous Time Signals

RESULT: Thus the various signals in discrete and continuous form were generated by using

MATLAB.

-10 0 100

0.51

unit impluse

n

X(n)

-1

0 0 100

0.51

step signal

n

X(n)

0 1 20

0.51

time base

t

X(t)

-10 0

10012

increasing exponential

n

X(n)

-10 0

10012

decreasing exponential

n

X(n)

-10 0 10-101

sine function

n

X(n)

-1

0 0 1005

10increasing ramp

n

X(n)

-1

0 0 10-10-50

decreasing ramp

n

X(n)

0 0.5 1-101

sawsooth

t

X(t)

0 0.5

1-101

triangular singal

t

x(t)

-10 0

10-101

square wave form

n

X(n)

-10 0 10-101rectangular wave form

n

X(n)

-1

0 0 1000.5

1unit impluse

n

X

(n)

-1

0 0 1000.5

1step signal

n

X

(n)

0 1 200.5

1time base

t

X

(t)

-1

0 0 10012

increasing exponential

n

X(n)

-1

0 0 10012

decreasing exponential

n

X(n)

-1

0 0 10-101

sine function

n

X(n)

-1

0 0 1005

10increasing ramp

n

X(n)

-1

0 0 10-10-50

decreasing ramp

n

X(n)

0 0.5 1-101

sawsooth

t

X(t)

0 0.5

1-101

triangular singal

t

x(t)

-10 0 10-101

square wave form

n

X(n)

-10 0 10-101rectangular wave form

n

X(n)


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Dept.of ECE 9

QUESTIONS:

1.Define a signal?

A.A signal is defined as a single valued function of one or more independent variables whichcontains some information.

2.Define unit ramp function?

A.The continuous time unit ramp function r(t) is defined asr(t)=t for t>=0;

0 for t=0;

0 for n


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Dept.of ECE 10

EXPERIMENT-3

OPERATIONS ON SIGNALS AND SEQUENCES

AIM:To perform the different operations on signals and sequences such as addition,multiplication, scaling, shifting, folding, computation of energy and average power.


MATLAB Code: a) Operations on signals

%operations on signalsN=128;

f1=150; f2=450; fs=8000;

n=0:N-1; %specify the range of n

x1=sin(2*pi*(f1/fs)*n);x2=(1/3)*sin(2*pi*(f2/fs)*n);

subplot(2,2,1);

plot(n,x1); %plot the generated signal x1(n)grid;

xlabel('t');ylabel('x(t)');

title('signal,x1(t)');subplot(2,2,2);

plot(n,x2); %plot the generated signal x2(n)

grid;

xlabel('t');ylabel('x(t)');title('signal,x2(t)');

%signal delayx1d=[zeros(1,20),x1(1:N-20)];

subplot(2,2,3);

plot(n,x1d); %plot the generated delayed signalgrid;


title('delayed x1(t),[x1(t-20)]');

%signal additionxadd=x1+x2; %generate signal addition

subplot(2,2,4);plot(n,xadd); %plot the generated signal

grid;

xlabel('t');ylabel('x(t)');title('x1(t)+x2(t)');


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Dept.of ECE 11

Expected Waveforms: Addition and Delay Operation

%signal multiplication

N=128;f1=150;

f3=1500;

fs=8000;

n=0:N-1; %specify the range of nx1=sin(2*pi*(f1/fs)*n);

x3=sin(2*pi*(f3/fs)*n);

subplot(2,2,1);

plot(n,x1); %plot the generated signal x1(n)grid;


subplot(2,2,2);plot(n,x3); %plot the generated signal x2(n)

grid;


xmult=x1.*x3;

subplot(2,2,3);plot(n,xmult); %plot the generated signal

grid;

xlabel('t');ylabel('x(t)');title('x1*x3');

0 50 100 150-1

-0.5

0

0.5

1

t

x(t)

signal,x1(t)

0 50 100 150-0.4

-0.2

0

0.2

0.4

t

x(t)

signal,x2(t)

0 50 100 150-1

-0.5

0

0.5

1

t

x(t)

delayed x1(t),[x1(t-20)]

0 50 100 150-1

-0.5

0

0.5

1

t

x(t)

x1(t)+x2(t)


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Dept.of ECE 12

Expected Waveforms: Multiplication operation

%amplitudeand frequency scalingN=128;f1=150;

f2=450;

fs=8000;

n=0:N-1;x1=sin(2*pi*(f1/fs)*n);

x4=5*sin(2*pi*(f1/fs)*n);

%amplitude scalingsubplot(2,2,1);

plot(n,x1);

grid;title('original signal');

subplot(2,2,2);

plot(n,x2);grid;

title('amplitude scaling of a signal');

%frequency scalingf2=450;

x5=sin(2*pi*(f2/fs)*n);subplot(2,2,3);plot(n,x5);

grid;


title('frequency scaling');

0 50 100 150-1

-0.5

0

0.5

1

t

x(t)

signal,x1(t)

0 50 100 150-1

-0.5

0

0.5

1

t

x(t)

signal,x3(t)

0 50 100 150-1

-0.5

0

0.5

1x1*x3


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Dept.of ECE 13

Expected Waveforms: Amplitude scaling and Frequency Scaling

%time folding

t=0:0.1:10;x=0.5*t;

lx=length(x);

nx=0:lx-1;

xf=fliplr(x);nf=-fliplr(x);

subplot(1,2,1);stem(nx,x);

xlabel('n(x)');ylabel('x');

title('original signal');

subplot(1,2,2);

stem(nf,xf);

xlabel('nf');ylabel('xf');title('folding signal');

Expected Waveforms: Signal folding operation

0 50 100 150-1

-0.5

0

0.5

1original signal

0 50 100 150-0.4

-0.2

0

0.2

0.4amplitude scaling of a signal

0 50 100 150-1

-0.5

0

0.5

1

t

x(t)

frequency scaling


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Dept.of ECE 14

OPERATIONS ON SEQUENCE

%operations on sequence

N=5;

n=0:N-1;x1=[1,2,3,-2,7];

x2=[0,-1,4,3,-5];

subplot(2,2,1);stem(n,x1); grid;xlabel('n');ylabel('x(n)');

title('sequence,x1(n)');

subplot(2,2,2);stem(n,x2); grid;

xlabel('n');ylabel('x(n)');title('sequence,x2(n)');

%sequence additionxadd=x1+x2;

subplot(2,2,3);

stem(n,xadd);grid;

xlabel('n');ylabel('x(n)');

title('x1(n)+x2(n)');

%sequence multiplicationxmult=x1.*x2;

subplot(2,2,4);stem(n,xmult);grid;


title('x1(n)*x2(n)');

Expected Waveforms: Addition and Multiplication Oprations

0 5 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

n(x)

x

original signal

-6 -4 -2 00

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

nf

xf

folding signal


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Dept.of ECE 15

%operations on sequence

N=5;n=0:N-1;

x1=[1,2,3,-2,7];

x2=[0,-1,4,3,-5];%sequence subtraction

xsub=x1-x2;

subplot(2,2,1);

stem(n,xsub); grid;xlabel('n');ylabel('x(n)');

title('x1(n)-x2(n)');

subplot(2,2,2);stem(n,x1); grid;xlabel('n');ylabel('x(n)');

title('sequence,x1(n)');

%sequence delayx3=[zeros(1,2),x1(1:N-2)];

subplot(2,2,3);

stem(n,x3); grid;xlabel('n');ylabel('x(n)');

title('delayed x1(n),[x1(n-3)]');

%amplitude scaling of sequencex4=5*x1;

subplot(2,2,4);

stem(n,x4); grid;xlabel('n');ylabel('x(n)');

title('amplitude scaling of sequence,x1(n)');

Expected Waveforms: Subtraction , delay and scaling operation

0 1 2 3 4-5

0

5

10

n

x(n)

sequence,x1(n)

0 1 2 3 4-5

0

5

n

x(n)

sequence,x2(n)

0 1 2 3 40

2

4

6

8

n

x(n)

x1(n)+x2(n)

0 1 2 3 4-40

-20

0

20

n

x(n)

x1(n)*x2(n)


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Dept.of ECE 16

%time shiftingx=input('type sample');

n1=input('type time origin first sample');N=length(x);

n2=n1+N-1;

n=n1:n2;disp('the amount of shift');

disp('enter positive no. for delay,negative no. for advance');

pause;

d=input('enter desired amount of shift');nn=n+d;

xs=x;

subplot(2,1,1);stem(n,x);

xlabel('time index n(sec)');ylabel('x(n)');

title('original signal');subplot(2,1,2);

stem(nn,x);

xlabel('time index n(sec)');ylabel('x2(n)');title('time shifted signal');

Results in command window:

type sample [1 2 4 3 0]

type time origin first sample 1

the amount of shift

enter positive no. for delay, negative no. for advance

enter desired amount of shift 2

0 1 2 3 4-5

0

5

10

15

n

x(n)

x1(n)-x2(n)

0 1 2 3 4-5

0

5

10

n

x(n)

sequence,x1(n)

0 1 2 3 40

1

2

3

n

x(n)

delayed x1(n),[x1(n-3)]

0 1 2 3 4-20

0

20

40

n

x(n)

amplitude scaling of sequence,x1(n)


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Dept.of ECE 17

%energy sequence

n=0:1:5;

x1=(0.5).^n;disp('calculated energy E is ');

E=sum(abs(x).^2);

figuresubplot(1,2,1);stem(x1);axis([0 5 0 2]);


title('x1=(0.5).^n sequence');subplot(1,2,2);

stem(E);axis([0 5 0 3]);


title('energy sequence');

%power sequence

n=0:1:5;x2=0.8.^n;

disp('calculated power P is');

P=(sum(abs(x).^2))/length(x);figure

subplot(1,2,1);

stem(x2);axis([0 5 0 2]);xlabel('n');ylabel('x(n)');

title('x2=(0.8).^n sequence');

subplot(1,2,2);

stem(P);axis([0 5 0 0.5]);xlabel('n');ylabel('x(n)');

title('power sequence');

Results in command window:

calculated energy E is E = 2.5869

calculated power P is P = 0.4311

Expected Waveforms: Computation of Energy and Power

1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

time index n(sec)

x(n)

original signal

3 3.5 4 4.5 5 5.5 6 6.5 70

1

2

3

4

time index n(sec)

x2(n)

time shifted signal


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Dept.of ECE 18

RESULT:Thus the different operations on signals and sequences were performed by usingMATLAB.

QUESTIONS:

1.What are the basic operations on signals?

The basic set of operations on signals are:

A)Time shifting b)Time reversal c)Time scaling d)Amplitude scaling e)Signal additionf)Signal multiplication

2.What is an analog signal?A.Continuous time signals are also called as analog signals.

3.What are digital signals?

A.The signals that are discrete in time and quantized in amplitude are called digital signals.

4.Distinguish between energy and power signals?

A.An energy signal is one whose total energy E=finite value and whose average power P=0.

Power signal is the one whose average power P=finite value and total energy E =infinity.

5.Distinguish between deterministic and random signals?

A.A deterministic signal is a signal exhibiting no uncertainity of its magnitude and phase at anygiven instant of time.It can be represented by a mathematical equation.

A random signal is a signal characterized by uncertainity about its occurrence.It cannot be

represented by a mathematical equation.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

n

x(n)

x1=(0.5).nsequence

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

n

x(n)

energy sequence

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

n

x(n)

x2=(0.8).nsequence

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

n

x(n)

power sequence


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Dept.of ECE 19

EXPERIMENT-4

EVEN AND ODD PARTS OF SIGNALS AND SEQUENCES

AIM:To find the even and odd parts of signals


MATLAB Code:

%finding the even and odd part of the sequence x(n)=0.8^n

n=-5:1:5; %specify the range of n

A=0.8;x1=A.^n; %generate the given sequence

x2=A.^(-n); %generate the folded sequence

if(x2==x1)disp('the given sequence is even sequence');

elseif(x2==(-x1))

disp('the given sequence is odd sequence');else

disp('the given sequence is neither even nor odd signal');

end


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Dept.of ECE 20

xe=(x1+x2)/2; %compute even partxo=(x1-x2)/2; %compute odd part

subplot(2,2,1);stem(n,x1); %plot the generated sequencexlabel('n');ylabel('x1(n)');title('sequence,x(n)');

subplot(2,2,2);stem(n,x2); %plot the generated folded sequencexlabel('n');ylabel('x2(n)');

title('sequence,x(-n)');

subplot(2,2,3);stem(n,xe); %plot the generated even part of sequence

xlabel('n');ylabel('xe(n)');

title('even part of x(n)');

subplot(2,2,4);stem(n,xo); %plot the generated odd part of sequence

xlabel('n');ylabel('xo(n)');

title('odd part of x(n)');

Expected Waveforms: Even and odd parts of signal x(n)=0.8^n

%finding real and imaginary parts of signalt=[-0.5:0.01:0.5];

w=20;

y=exp(i*pi*w*t/2);

-5 0 50

1

2

3

4

n

x1(n)

sequence,x(n)

-5 0 50

1

2

3

4

n

x2(n)

sequence,x(-n)

-5 0 50

0.5

1

1.5

2

n

xe(n)

even part of x(n)

-5 0 5-2

-1

0

1

2

n

xo(n)

odd part of x(n)


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Dept.of ECE 21

a=real(y);

subplot(2,2,1);plot(t,a);


title('real part of signal');

b=imag(y);

subplot(2,2,2);plot(t,b);


title('imaginary part of signal');

c=abs(y);

subplot(2,2,3);plot(t,c);

axis([-0.5 0.5 0 2]);


title('absolute value');

d=angle(y);

subplot(2,2,4);plot(t,d);


title('phase angle');

Expected Waveforms: Real and Imaginary parts of signal and its magnitude & phase


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Dept.of ECE 22

RESULT: Thus the even and odd parts of real signals and complex signals were calculated

using MATLAB.

QUESTIONS:

1.Distinguish between even and odd signals?A.A continuous time signal x(t) is said to be even (symmetric) signal,if it satisfies the condition

X(-t)=x(t) for all t.If it is said to be odd (anti symmetric)signal if it satisfies the condition

X(-t)=-x(t) for all t.

A discrete time signal x(n) is said to bee even signal if it satisfies the conditionX(-n)=x(n) for all n.If it is said to be odd signal if it satisfies the condition

X(-n)=-x(n) for all n.

2.Can every signal be decomposed into even and odd parts?A.Yes,every signal can be decomposed into even and odd parts.

3.Write the expression for even and odd parts of signal?A.The even and odd parts of a continuous time signal are given by

Xe(t)=1/2[x(t)+x(-t)]

Xo(t)=1/2[x(t)-x(-t)]The even and odd parts of a discrete time signal are given by

Xe(n)=1/2[x(n)+x(-n)]

Xo(n)=1/2[x(n)-x(-n)]

4.Do all the signals correspond to either even or odd type?A.No ,All the signals need not necessarily belong to either even or odd type.There are signals

which are neither even nor odd.

5.Distinguish between causal and non causal signals?

A.A continuous time signal x(t) is said to be causal,if x(t)=o for t


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Dept.of ECE 23

EXPERIMENT-5

CONVOLUTION BETWEEN SIGNALS AND SEQUENCES

AIM:The convolution between signals and sequences


MATLAB Code:

% convolution between two signals

tmin=0;

tmax=4;dt=0.01;

t=tmin:dt:tmax;

x1=1.*(t>=0&t=0&t=1&t


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Dept.of ECE 24

%CONVOLUTION between two sequences

n= 0:1:4;n1=0:1:3;

x1=[2 -4 3 -1 5];

x2=[-1 2 -3 6];N1=length(x1);N2=length(x2);

x3=conv(x1,x2)

n2=0:1:N1+N2-2;

subplot(3,1,1); stem(n,x1);

xlabel('n'); ylabel('x1(n)');title('sequence x1(n)');

subplot(3,1,2); stem(n1,x2);

xlabel('n'); ylabel('x2(n)');title('sequence x2(n)');

subplot(3,1,3); stem(n2,x3);

xlabel('n'); ylabel('x3(n)');title('sequence,x3(n)=x1(n)*x2(n)');

Result in Command Window:x3 = -2 8 -17 31 -40 31 -21 30

Expected Waveforms:

%CONVOLUTION of x(t)=1/(1+t^2) itself

syms ttau;x=1/(1+t^2);

z=int(subs(x,tau)*subs(x,t-tau),tau,-inf,inf);

z1=simplify(z);pretty(z1);

figure;subplot(211); ezplot(x);

subplot(212); ezplot(z1);

0 0.5 1 1.5 2 2.5 3 3.5 4-5

0

5

n

x1(n)

sequence x1(n)

0 0.5 1 1.5 2 2.5 3-10

0

10

n

x2(n)

sequence x2(n)

0 1 2 3 4 5 6 7-50

0

50

n

x3(n)

sequence,x3(n)=x1(n)*x2(n)


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Dept.of ECE 25

Expected Waveforms:

Results:z1 = 2*i*pi/(t-2*i)/(-2+t*i)

Ans:

2 i pi

--------------------(t - 2 i) (-2 + t i)

%CONVOLUTION of x(t)=exp(-t^2);h(t)=3*t^2;

syms ttau;x=exp(-t^2);

h=3*t^2;

z=int(subs(x,tau)*subs(h,t-tau),tau,-inf,inf);

z1=simplify(y)figure();

subplot(211);ezplot(x);

subplot(212);ezplot(h);figure;

ezplot(z);

Expected Waveforms: Input Signal x(t) and h(t)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0

0.5

1

t

original signal x (t)=1/(1+t2)

-6 -4 -2 0 2 4 60

0.5

1

1.5

t

Convolution of x(t)=1/(1+t2) and itself

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

t

exp(-t2)

-6 -4 -2 0 2 4 6

0

50

100

t

3 t2


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Dept.of ECE 26

Output: CONVOLUTION of x(t)=exp(-t^2);h(t)=3*t^2

z1 = 3*t^2*pi^(1/2)+3/2*pi^(1/2)

2 1/2 1/23 t pi + 3/2 pi

RESULT:Thus the convolution between signals and sequences were verified using MATLAB.

QUESTIONS:

1.What is meant by convolution?A.It is a mathematical way (multiplication and addition)if combininig two signals to form a thirdsignal.

2.What is the associative property of convolution?A. x1(t)*[x2(t)*x3(t)] = [x1(t)*x2(t)*]x3(t) where x1(t),x2(t) and x3(t) are three real signals.

3.What is the convolution of a signal with an impulse?

A.The convolution of a signal with an impulse is equal to that signal itself.

4.State time convolution theorem?

A.The time convolution theorem states that the convolution in time domain is equivalent tomultiplication of their spectra in frequency domain.mathematically,if x1(t)X1(w) andx2(t)X2(w)

Then x1(t)*x2(t)X1(w) X2(w)

5.State frequency convolution theorem?

A.It states that the multiplication of two functions in time domain is equal to convolution of their

spectra in frequency domain.Mathematically, if x1(t)X1(w) and x2(t)X2(w)Then x1(t) x2(t)1/2pi[X1(w)*X2(w)]

-6 -4 -2 0 2 4 6

0

50

100

150

200

t

3 t21/2+3/2 1/2


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Dept.of ECE 27

EXPERIMENT-6

AUTOCORRELATION AND CROSSCORRELATION

BETWEEN SIGNALS AND SEQUENCES

AIM:To find autocorrelation and cross correlation between signals and sequences.


MATLAB Code:

%CROSS CORRELATION:

N=1024;

F=1;Fs=200;

n=0:N-1;x=sin(2*pi*F*n/Fs);

y=x+10*randn(1,N);subplot(3,1,1); plot(x);

title('pure sine wave');grid;

subplot(3,1,2); plot(y); grid;

title('y(n), pure sine wave + noise');Rxy=xcorr(x,y);

subplot(3,1,3);

plot(Rxy); grid;title('cross correlation Rxy');

%AUTOCORRELATION:N=1024;

f1=1;

Fs=200;

n=0:N-1;x=sin(2*pi*f1*n/Fs);

t=[1:N]*(1*Fs);

subplot(2,1,1);plot(t,x); grid;

title('sin wave of frequency 1000 hz[Fs=8000 hz]');

xlabel('time,(s)'); ylabel('amplitude');Rxy=xcorr(x);subplot(2,1,2);

plot(Rxy); grid;

title('auto crrelation function of the sine wave');xlabel('lags'); ylabel('auto correlation');


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Dept.of ECE 28

Expected Waveforms:%CROSS CORRELATION:

%AUTO CORRELATION:

RESULT:Thus the cross correlation and auto correlation of different signals and sequences

were verified using MATLAB.

QUESTIONS:

1.What is meant by correlation?

A: Correlation is measure of the degree of the similarity between two signals.

2. What is auto correlation?

A. Auto correlation is the measure of similarity of a signal with its delayed version.

3. What is meant by cross correlation?

A. Correlation between the two different signals is called cross correlation.

4. What is the autocorrelation function at the origin?

A. Auto correlation at the origin equals to its energy.

5. What is meant by Parsevals theorem?

A. Total average power of a periodic signal x(t) is the sum of the squares of its fourier

coefficients.

0 200 400 600 800 1000 1200-1

0

1

time

x1

pure sine wave

0 200 400 600 800 1000 1200-1

0

1

time

cosine

y(n),cosine wave

-1500 -1000 -500 0 500 1000 1500-500

0

500

time

Rxy

cross correlation Rxy

0 0.5 1 1.5 2 2.5

x 105

-1

-0.5

0

0.5

1sin wave of frequency 1000 hz[Fs=8000 hz]

time,(s)

amplitude(x)

-1500 -1000 -500 0 500 1000 1500-500

0

500

1000auto crrelation function of the sine wave

time

autocorrelation(Rxx


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Dept.of ECE 29

EXPERIMENT-7

VERIFICATION OF LINEARITY AND TIME INVARIENCE PROPERTIES

AIM:To verify the linearity and time-invariance property Reconstruction of a periodic signal byFourier series.


MATLAB Code:

%LINEARITY:

x1=input('type the samples of x1');

x2=input('type the samples of x2');

if(length(x1)~=length(x2));disp('error:lengths of x1 & x2 are diffrent');return;

end;

h=input('type the samples of h');

N=length(x1)+length(x2)-1;disp('length of the output signal will be');

disp(N);

a1=input('the scale factor a1 is');a2=input('the scale factor a2 is');

x=a1*x1+a2*x2

y01=conv(x,h)y1=conv(x1,h)

ys1=a1*y1

y2=conv(x2,h)ys2=a2*y2

y02=ys1+ys2

disp('input signal x1 is');

disp(x1);disp('input signal x2 is');

disp(x2);

disp('output sequence y01 is');

disp(y01);disp('output sequence y02 is');

disp(y02);

if(y01==y02)disp('y01=y02,hence the LTI system is linear');

end;


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Dept.of ECE 30

COMMAND WINDOW:

a)Verification of Lineariy:

type the samples of x1[1 2 3 4]type the samples of x2[2 3 4 5]type the samples of h[1 2 3 4 5 6 7]

length of the output signal will be 7

the scale factor a1 is3the scale factor a2 is2

x =7 12 17 22

y01 = 7 26 62 120 178 236 294 296 251 154

y1 = 1 4 10 20 30 40 50 52 45 28

ys1 = 3 12 30 60 90 120 150 156 135 84

y2 = 2 7 16 30 44 58 72 70 58 35

ys2 = 4 14 32 60 88 116 144 140 116 70

y02 = 7 26 62 120 178 236 294 296 251 154

input signal x1 is 1 2 3 4

input signal x2 is 2 3 4 5

output sequence y01 is

7 26 62 120 178 236 294 296 251 154

output sequence y02 is

7 26 62 120 178 236 294 296 251 154

y01=y02,hence the LTI system is linear


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Dept.of ECE 31

%TIME INVARIENT:

x=input('type the sample of signal x(n)');

h=input('type the samples of signal h(n)');

y=conv(x,h);disp('enter a positive number for delay');

d=input('desired delay of the signals is');

xd=[zeros(1,d),x];nxd=0:length(xd)-1;

yd=conv(xd,h);

nyd=0:length(yd)-1;disp('original input signal x(n) is');

disp(x);

disp('delayed input signal xd(n) is');disp(xd);

disp('original output signal y(n) is');disp(y);

disp('delayed output signal yd(n) is');disp(yd);

xp=[x,zeros(1,d)];

figure

subplot(2,1,1); stem(nxd,xp);grid;xlabel('time index n'); ylabel('x(n)');

title('original input signal x(n)');

subplot(2,1,2); stem(nxd,xd); grid;xlabel('time index n'); ylabel('xd(n)');

title('delayed input signal xd(n)');yp=[y,zeros(1,d)];figure

subplot(2,1,1); stem(nyd,yp); grid;

xlabel('time index n'); ylabel('y(n)');title('original output signal y(n)');

subplot(2,1,2); stem(nyd,yd); grid;

xlabel('time index n'); ylabel('yd(n)');

title('delayed output signal yd(n)');

COMMAND WINDOW:

b)Verification of Time Invarience:

type the sample of signal x(n)[1 2 3 4]

type the samples of signal h(n)[2 3 4 5 ]

enter a positive number for delaydesired delay of the signals is 4

original input signal x(n) is 1 2 3 4

delayed input signal xd(n) is 0 0 0 0 1 2 3 4

original output signal y(n) is 2 7 16 30 34 31 20

delayed output signal yd(n) is 0 0 0 0 2 7 16 30 34 31 20


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Dept.of ECE 32

Expected Waveforms:

RESULT:Thus the linearity and timeinvarient properties of an LTI system were verified using

MATLAB.

QUESTIONS:

1.What i s meant by linear system?

A.A system is said to be linear if it satisfies the superposition principle.

2.What is meant by time-invariant system?A.A system is said to be time-invariant if the i/p-o/p relationship does not vary with time

3.What is meant by causal system?A.A system is said to be causal if the o/p depends only on the values of present i/p.

4.What is meant by stable system?A.A system is said to be stable if bounded i/p results in bounded o/p.

5.What is meant by order of a system?

A.The order of a continuous system corresponds to highest derivative of o/p signal which may

appear in i/po/p differentialequation.

0 1 2 3 4 5 6 70

1

2

3

4

time index n

x(n)

original input signal x(n)

0 1 2 3 4 5 6 70

1

2

3

4

time index n

xd(n)

delayed input signal xd(n)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

time index n

y(n)

original output signal y(n)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

time index n

yd(n)

delayed output signal yd(n)


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Dept.of ECE 33

EXPERIMENT-8

RESPONSE OF THE LTI SYSTEM

AIM:To compute unit sample, unit step response of the LTI systems.


MATLAB Code:

%IMPULSE RESPONSE OF LTI SYSTM:

clear all

syms zn

H=1/(1-0.8*(z^(-1))+0.16*(z^(-2)));disp('impulse response h(n) is');

h=iztrans(H)

simplify(h)

N=15;b=[1 0 0];

a=[1 -0.8 0.16];

[H,n]=impz(b,a,N);

stem(n,H);xlabel('n'); ylabel('h(n)');

title('Impulse response of a LTI system');

Expected Waveforms:

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n

h(n)

Impulse response of a LTI system


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Dept.of ECE 34

%UNIT STEP RESPONSE OF LTI SYSTM:

syms scomplex

H1=5/(s^2+25);

disp('step response of first order system is:');h1=ilaplace(H1)simplify(h1)

H2=1/((s-0.8)^2);

disp('step response of second order system');h2=ilaplace(H2)

simplify(h2)

s=tf('s');H1=1/(s+2);

H2=1/(s^2+2.5*s+25);

t1=0:0.005:5;

s1=step(H1,t1);s2=step(H2,t1);

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');

Expected Waveforms:

RESULT:Thus the step and impulse response of LTI system were verified using MATLAB.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

time in seconds

s1(t)

step response of first order system

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.02

0.04

0.06

time in seconds

s2(t)

step response of second order system


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Dept.of ECE 35

QUESTIONS:

1.What is meant by system?

A:A set of elements or functional blocks which are connected together and produces an output in

response to an input signal.

2.What is meant by continuous time system?

A:A continuous time system is one in which continuous time i/p signals are transformedcontinuous time o/p signals.

3. What is meant by discrete time system?A: A discrete time system is one in which discrete time i/p signals are transformed discrete time

o/p signals.

4.What is meant by step response of a discrete system?

A:The step response of a discrete-time LTI system is the running sum of its impulse response.

5.What is meant by an impulse response ?

A:The impulse response specifies the i/p-o/p behavior of an LTI system.


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Dept.of ECE 36

EXPERIMENT-9

RECONSTRUCTION OF PERIODIC SIGNAL BY FOURIER SERIES

OR GIBBS PHENOMENON

AIM:To Reconstruct a periodic signal by Fourier series.


MATLAB Code:

%gibbs phenomenon

N=input('type the total number of harmonics');t=0:0.001:1;

y=square(2*pi*t);

plot(t,y,'r','linewidth',2);

axis([0 1.5 -1.5 1.5]);

hold;

sq=zeros(size(t));

forn=1:2:N;

sq=sq+((4/(pi*n))*sin(2*pi*n*t));

end;

plot(t,sq);

grid;xlabel('t');

ylabel('sq(t)');

title('synthesized square wave using Fourier Series');legend('N=0','N=3');

COMMAND WINDOW:

type the total number of harmonics 3

Expected Waveforms:

0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

t

s

q(t)

synthesized square wave using Fourier Series

N=0

N=3


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Dept.of ECE 37

COMMAND WINDOW:

type the total number of harmonics 5

Expected Waveforms:

COMMAND WINDOW:

type the total number of harmonics111

Expected Waveforms:

RESULT: Thus the extraction of a periodic signal was performed successfully using

MATLAB.

0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

t

sq(t)


N=0

N=5

0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

t

sq(t)


N=0

N=111


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Dept.of ECE 38

QUESTIONS:

1.What do you mean by gibbs phenomenon?

A.Gibbs discovered that for a periodic signal with discontinuities, if the signal is reconstructed

by adding the fourier series ,overshoots appear around the edges.These overshoots decayoutwards in a damped oscillatory manner away from the edges.This is called Gibbs

Phenomenon.

2.What is fourier spectrum?

A.The fourier spectrum of a periodic signal is x(t) is a plot of its Fourier coefficients versus

frequency w.It is in two parts:a)The amplitude spectrum b)Phase spectrum

3.What is fourier series?

A.The representation of signals over a certain interval of time in terms of the linear combination

of orthogonal functions is called fourier series.

4.What are Dirichlets conditions?

A.The conditions under which a periodic signal can be represented by a Fourier series are known

as Dirichlets conditions.They are:

a)The function x(t) must be a single valued function

b)The function x(t) has only a finite number of maxima and minima

c)The function x(t) has a finite number of discontinuities.

d)The function x(t)is absolutely integrable over one period,that is integral(0,t)[l(x(t)ldt]


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Dept.of ECE 39

EXPERIMENT-10

FINDING FOURIER TRANSFORM

AIM: To find the Fourier Transform of a various signals such as cosine, sine, impulse, step,

square and rectangular signals


MATLAB Code:

%cosine signalclc;clear all;

f=100;fs=1000;

Fs=1/fs;

N=1024;n=[0:N-1]*Fs;

x=0.8*cos(2*pi*f*n);

figure;

plot(n,x); grid;axis([0 0.05 -1 1]);

title('cosine signal of frequency f');

xlabel('time t(sec)');ylabel('x(t)');

xk=fft(x,N);

k=0:N-1;figure;

xmag=abs(xk);subplot(2,1,1);

plot(k,xmag); grid;

title('magnitude of Fourier Transform');

xlabel('frequency index,k');ylabel('magnitude');

subplot(2,1,2);plot(k,angle(xk)); grid;

title('phase of fourier transform');

xlabel('frequency index,k');ylabel('phase');

Expected Waveforms:


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Dept.of ECE 40

%sine signal

clc;clear all;f=100;

fs=1000;

Fs=1/fs;N=1024;

n=[0:N-1]*Fs;

x=0.8*sin(2*pi*f*n);figure;

plot(n,x);

grid;axis([0 0.05 -1 1]);

title('sine signal of frequency f');xlabel('time t(sec)');ylabel('x(t)');

xk=fft(x,N);

k=0:N-1;

figure;

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1cosine signal of frequency f

time t(sec)

x(t)

0 200 400 600 800 1000 12000

100

200

300

400magnitude of Fourier Transform

frequency index,k

m

agnitude

0 200 400 600 800 1000 1200-2

-1

0

1

2phase of fourier transform

frequency index,k

phase


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Dept.of ECE 41

xmag=abs(xk);subplot(2,1,1);

plot(k,xmag);

grid;title('magnitude of Fourier Transform');xlabel('frequency index,k');ylabel('magnitude');

subplot(2,1,2);plot(k,angle(xk));

grid;

title('phase of fourier transform');xlabel('frequency index,k');ylabel('phase');

Expected Waveforms:

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1sine signal of frequency f

time t(sec)

x(t)

0 200 400 600 800 1000 12000

100

200

300


frequency index,k

magnitude

0 200 400 600 800 1000 1200-4

-2

0

2


frequency index,k

pha

se


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Dept.of ECE 42

%square signal

clc;clear all;f=100; fs=900;

Fs=1/fs;

N=1024;n=[0:N-1]*Fs;

x=0.5*square(2*pi*f*n);

figure;plot(n,x); grid; axis([0 0.05 -1 1]);

title('square signal of frequency f'); xlabel('time t(sec)');ylabel('x(t)');

xk=fft(x,N);

k=0:N-1;

figure;

xmag=abs(xk);

subplot(2,1,1);

plot(k,xmag); grid;

title('magnitude of Fourier Transform');xlabel('frequency index,k');ylabel('magnitude');

subplot(2,1,2); plot(k,angle(xk)); grid;title('phase of fourier transform');


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1square signal of frequency f

time t(sec)

x(t)

0 200 400 600 800 1000 12000

100

200

300

400

magnitude of Fourier Transform

frequency index,k

magnitude

0 200 400 600 800 1000 1200-4

-2

0

2

4

phase of fourier transform

frequency index,k

phase


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Dept.of ECE 43

%impulse signal

n=-20:2:20;N=1024;

x1=1;

x2=0;

x=x1.*(n==0)+x2.*(n~=0);

figure;

stem(n,x);grid;axis([-20 20 -1 1]);

title('impulse signal');

xlabel('time n(sec)');ylabel('x(n)');

xk=fft(x,N);

k=0:N-1;

figure;

xmag=abs(xk);

subplot(2,1,1);

plot(k,xmag); grid;title('magnitude of Fourier Transform');


subplot(2,1,2);

plot(k,angle(xk)); grid;



-20 -15 -10 -5 0 5 10 15 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1impulse signal

time n(sec)

x(n)

0 200 400 600 800 1000 12001

1

1


frequency index,k

magnitude

0 200 400 600 800 1000 1200-4

-2

0

2


frequency index,k

phase


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Dept.of ECE 44

%rectangular pulse

n=-20:2:20;t=-10:1:10;

f=1/20;

x=square(2*pi*f*t,75);figure;

stem(n,x);grid; axis([0 40 -1 1]);

title('rectangular pulse'); xlabel('time n(sec)');ylabel('x(n)');

xk=fft(x,N);

k=0:N-1;figure;

xmag=abs(xk);

subplot(2,1,1);plot(k,xmag); grid; title('magnitude of Fourier Transform');


subplot(2,1,2);plot(k,angle(xk)); grid;



0 5 10 15 20 25 30 35 40-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1rectangular pulse

time n(sec)

x(n)

0 200 400 600 800 1000 12000

5

10


frequency index,k

magnitude

0 200 400 600 800 1000 1200-4

-2

0

2


frequency index,k

phase


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Dept.of ECE 45

%step signal

n=-20:2:20;N=1024;

x1=1;

x2=0;

x=x1.*(n>=0)+x2.*(n


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Dept.of ECE 46

% Fourier Transform Rectangular Pulse

z = int(x*expw, omega,-2,2);

z=simplify(z);

figure();subplot(211);ezplot('2',[-2,2]);

subplot(212);ezplot(z);

Expected Waveforms:

RESULT: Thus the computation of Fourier transform of a given signal was done using

MATLAB.

QUESTIONS:

1.What is Fourier transform?

A.Fourier transform is a transformation technique which transforms signals from the continuous

time domain into frequency domain and vice versa.

2.Write the formulae for Fourier transform and inverse fourier transform?A.The formula for Fourier transform,also called the equation for direct transform or analysis

equation is:

X(w)=integral(-infinity,infinity)[x(t)e^(-jwt)dt]

The formulae for I.F.T also called the equation for inverse transform or the synthesis equation is:

x(t)=1/2pi[integral(-infinity,infinity)[X(W)e^(jwt)dw]]

3.What is frequency spectrum?

A.The plot of lX(w)l versus w is known as magnitude spectrum or amplitude spectrum,and theplot of LX(w) versus w is known as phase spectrum.The amplitude spectrum and phase spectrum

together is called the frequency spectrum.

4.What are the limitations of F.T?

A.a)It is less powerful than Laplace transformation.

b)There are many functions for which the laplace transform exists,but the fourier transform does

not exist.

5.What is CTFT?

A.CTFT(continuous time fourier transform) is the F.T applied to continuous time signals.In

general CTFT is called as Fourier Transform.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 21

1.5

2

2.5

3

x

2

-6 -4 -2 0 2 4 6

-2

0

2

4

6

8

t

4/t sin(2 t)


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Dept.of ECE 47

EXPERIMENT-11

FINDING THE LAPLACE TRANSFORM OF GIVEN FUNCTIONS

AIM: To write a MATLAB program to find the Laplace transform of a given

signal. F(t)=1.5-2.5te^-2t+1.25e^-3t


MATLAB Code:

syms ts;

f=1.5 -2.5 * t * exp(-2*t)+1.25 * exp(-3*t);f = laplace(f, t, s)

simplify(f)pretty(ans)

COMMAND WINDOW:

Expected Results:

3/2/s-5/2/(s+2)^2+5/4/(s+3)

ans =3/2/s-5/2/(s+2)^2+5/4/(s+3)

ans =1/4*(11*s^3+52*s^2+86*s+72)/s/(s+2)^2/(s+3)

3 2

11 s + 52 s + 86 s + 721/4 -------------------------

2

s (s + 2) (s + 3)

RESULT:Thus the computation of Laplace transform of a given signal was done using

MATLAB.


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Dept.of ECE 48

QUESTIONS:

1.How is laplace transform useful in the analysis of LTI system?

A.The laplace transform is a powerful mathematical technique used to convert the differential

equations in time domain into algebraic equations in s-domain.

2.What is the region of convergence(ROC)?

A.The set of points in the s plane for which the laplace transform of x(t) ,i.e. the function X(s)

converges is called the ROC.

3.What is the usual condition for x(t) to be laplace transformable?

A.The usual condition for x(t) to be laplace transformable is that it should be piece-wise

continuous and must be of exponential order.

4.What is the relation between F.T and L.T?

a.Laplace transform is a complex fourier transform.The Fourier transform of a function can be

obtained from its laplace transform by replacing s by jw,i.e. F.T is the L.T evaluated along the

imaginary axis of the s plane.

5.What is transient response?

A.The transient response is the response due to the poles of the system function H(s).It vanishesafter some time and hence is called the transient response of the system.


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Dept.of ECE 49

EXPERIMENT-12

LOCATING POLES AND ZEROS IN S-PLANE AND Z-PLANE

AIM: To write a MATLAB program to find the location of poles and zeros, and plotting thepole-zero maps in S-plane and Z-plane for given transfer function.


MATLAB Code:

%LOCATING POLES AND ZEROS ON S PLANE

syms s

num_coeff=[1 5 6];

disp('roots of numerator polynomial are');zeros=roots(num_coeff)

den_coeff=[1 9 20];

disp('roots of denominator polynomial are');

poles=roots(den_coeff)H=tf('s');

a=tf([num_coeff],[den_coeff]);

grid on;pzmap(a);

if(max(real(poles)))>=1

disp('all poles DO NOT LIE in the left Half of S-plane');

disp('The given LTI system is NOT a stable system');else

disp('ALL the POLES lie in the Left half of S-plane');

disp('The given LTI system is a STABLE system');end;


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Dept.of ECE 50

Command Window:

case(1):num_coeff = [1 5 6]den_coeff = [1 9 20]

roots of numerator polynomial are

zeros =

-3.0000

-2.0000

roots of denominator polynomial are

poles =

-5.0000

-4.0000

ALL the POLES lie in the Left half of S-plane

The given LTI system is a STABLE system

Expected Waveforms:

Pole Zero map of S-plane:

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

10.40.660.820.90.9450.974

0.99

0.997

0.40.660.820.90.9450.974

0.99

0.997

1234

Pole-Zero Map

Real Axis

ImaginaryAxis


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Dept.of ECE 51

COMMAND WINDOW:

Case(2):num_coeff = [1 5 6]den_coeff = [1 -9 20]


zeros =

-3.0000-2.0000


poles =

5.00004.0000

all poles DO NOT LIE in the left Half of S-plane

The given LTI system is NOT a stable system

Expected Waveforms:

-4 -3 -2 -1 0 1 2 3 4 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

10.60.860.930.966

0.982

0.992

0.997

0.999

0.60.860.930.966

0.982

0.992

0.997

0.999

0.511.522.533.5

Pole-Zero Map

Real Axis

Ima

ginaryAxis


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Dept.of ECE 52

%Z-PLANE:

syms z

num_coeff=[1 3 5];

disp('roots of numerator polynomial are');zeros=roots(num_coeff)den_coeff=[1 1 0.16];

disp('roots of denominator polynomial are');

poles=roots(den_coeff)H=tf('z');

a=tf([num_coeff],[den_coeff]);

radpoles=abs(poles);grid on;

pzmap(a);

ifmax(radpoles)


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Dept.of ECE 53

Case(2):num_coeff=[1 3 5];

den_coeff=[1 0.5 -1.25];


zeros =

-1.5000 + 1.6583i

-1.5000 - 1.6583i


poles =

-1.3956

0.8956

all the poles donot lie with in the unit circle

the given LTI system is not a stable system

>>

RESULT:Thus the poles and zeros in S-Plane and Z-Plane were located and plotted using

MATLAB.

-2 -1.5 -1 -0.5 0 0.5 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Pole-Zero Map

Real Axis

ImaginaryAxis


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Dept.of ECE 54

QUESTIONS:

1.What are the advantages and limitations of Z-Transforms?A.Advantages of Z-Transform:

1.They convert the difference equations of a discrete time system into linear algebraic equations

so that the analysis become easy and simple.

2.Convolution in time domain is converted into multiplication in Z-domain.

3.Z-transform exists for most of the signals for which DTFT does not exist.

Limitation:Frequency domain response cannot be achieved and cannot be plotted

2.What is ROC of Z-Transform?

A.The range of values of lzl for which X(z) converges is called ROC of X(z).

3.What is the ROC of an infinite duration causal sequences?

A.The ROC of an infinite duration causal sequences is lzl>alpha,i.e. it is the exterior of a circle

of radius alpha where z=alpha is the largest pole in X(z).

4.What is the ROC of an infinite duration non causal sequences?

A.The ROC of an infinite duration non causal sequences is lzlX2(z)

then ax1(n)+bx2(n)---->aX1(Z)+bX2(z)


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Dept.of ECE 55

EXPERIMENT-13

RANDOM NOISE SIGNAL GENERATION

AIM: To write a MATLAB program to generate the random noise signal.


MATLAB Code:

clear all;

x1=randn(1,5000);x2=randn(1,5000);

figure;plot(x1,x2,'.')

title('scatter plot of gaussian distributed random numbers');

x1=rand(1,5000);

x2=rand(1,5000);

figure;plot(x1,x2,'.')

title('scatter plot of uniform distributed random numbers');

x3=rand(1,100000);

figure;subplot(2,1,1);hist(x3)title('uniform distribution')

y=randn(1,100000);

subplot(2,1,2);hist(y)

title('gaussian distribution')

ymu=mean(y);

ymsq=sum(y.^2)/length(y);ysigma=std(y);

yvar=var(y);

yskew=skewness(y);ykurt=kurtosis(y);

EXPECTED GRAPH :

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4scatter plot of gaussian distributed random numbers


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Dept.of ECE 56

RESULT:Thus the different characteristics of a random noise signal were verified using

MATLAB.

QUESTIONS:

1. What is meant by Mean of a signal?Ans: Mean is the average value of a signal and is a natural measure of center of distribution.

It is obtained by adding all the values xi of a signal and div iding by N, the total number ofvalues, as

sum(xi)/N from i=0 to N-1

2. What is meant by Mean squqre of a signal?Ans:The summation of a squared sample values of the signal and its average.

3. What is meant by standard deviation of a signal?Ans:the standard deviation is a measure of how far the signal fluctuates from the mean.

4.What is meant by varience?

Ans:The square of the standard deviation is called the varience of the signal.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1scatter plot of uniform distributed random numbers

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

10

5000

10000

15000uniform distribution

-5 -4 -3 -2 -1 0 1 2 3 4 50

1

2

3

4x 104 gaussian distribution


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Dept.of ECE 57

EXPERIMENT-14

VERIFICATION OF SAMPLING THEOREM

AIM: To write a MATLAB code for verification of Sampling theorem.


MATLAB Code:

% MATLAB Code for Verification of Sampling Theorem

t = -5 : 0.0001 : 5;

F1 = 3; F2 = 23;

x = cos(2*pi*F1*t) + cos (2*pi*F2*t);

figure(1); plot(t,x);axis([-0.4 0.4 -2 2])

xlabel('Time t(sec)'); ylabel('x(t)');title('Continuous Time Signal : x(t) = cos(2*pi*F1*t) + cos (2*pi*F2*t)');

%CASE 1

Fs1 = 1.4*F2; ts1 = 1/Fs1; %set the sampling freq to be 1.4 times Fmax=F2

n1 = -0.4 : ts1 : 0.4;xs1 = cos(2*pi*F1*n1) + cos (2*pi*F2*n1);

figure(2); stem(n1,xs1);

hold on; plot(t, x, 'r:');

axis([-0.4 0.4 -2 2]); hold off;xlabel('Time samples (n)'); ylabel('Amplitude');

title('Discrete Time Signal'); legend('Fs < 2Fmax');

%CASE 2

Fs2 = 2*F2; ts2 = 1/Fs2; %set the sampling freq to be 1.4 times Fmax=F2





title('Discrete Time Signal'); legend('Fs = 2Fmax');

%CASE 3

Fs3 = 7*F2; ts3 = 1/Fs3; %set the sampling freq to be 1.4 times Fmax=F2





title('Discrete Time Signal'); legend('Fs > 2Fmax');


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Expected Waveforms:

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time t(sec)

x(t)

Continuous Time Signal : x(t) = cos(2*pi*F1*t) + cos (2*pi*F2*t)

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time samples (n)

Amplitude

Discrete Time Signal

Fs < 2Fmax


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RESULT:Thus the Sampling theorem was verified successfully using MATLAB.

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time samples (n)

Amp

litude


Fs = 2Fmax

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time samples (n)

Amplitude


Fs > 2Fmax


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Dept.of ECE 60

QUESTIONS:

1.What is sampling?

A.Sampling is the process of converting a continuous time signal into a discrete time signal.

2.What is sampling period?

A.Sampling period or a sampling interval is the time interval between successive samples.

3.What is Nyquist rate?

A.Nyquist rate is the theoretical minimum sampling rate at which a signal can be sampled and

still be rteconstructed from its samples without any distortion.

4.What is aliasing?

A.Aliasing is defined as the phenomenon in which a high frequency component in the frequency

spectrum of signal takes identity of a lower frequency component in the spectrum of the samplesignal.

5.State sampling theorem?

A.The sampling theorem also called uniform sampling theorem or low pass sampling theorem

states thatA band limited x(t) with X(w)=0 for lwl>=wm can be represented intoand uniquely

determined from its samples x(nT), if the sampling frequency fs>=2fm,where fm is the highestfrequency component present in it.


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Dept.of ECE 61

EXPERIMENT-15

NOISE REMOVAL USING CORRELATION

AIM: To Write a MATLAB Code for Noise Removal using cross correlation or auto correlation.

Requirements:PC with MATLAB R2008a

MATLAB Code:clear all;

N=100;

n=0:N-1;dsnr=input('type desired SNR in dB');

x=sqrt(2)*sin((pi/5)*n);

figure(1);stem(n,x);grid

%axis([0 50 -1.5 1.5])xlabel('n');ylabel('x(n)');title('sinusoidal signal x(n)')

px=var(x) ;an=sqrt(px*(10^(-1*dsnr/10)));

w=sqrt(12)*(rand(1,N)-0.5);

w=w*an;

pn=var(w);disp('The calculated SNR');

SNRdB=10*log10(px/pn);

figure(3);stem(n,w);grid

axis([0 50 min(w) max(w)])xlabel('n');ylabel('w(n)');title('Random Noise Signal w(n)');

y=x+w;

figure(6);subplot(2,1,1);

Stem(n,y);grid

axis([0 50 min(y) max(y)])xlabel('n');ylabel('y(n)=x(n)+w(n)');

title('Sinusoidal Signal Corrupted With Random NOise')

[ryy,lag]=xcorr(y,y,'unbiased');subplot(2,1,2);stem(lag,ryy);grid

axis([0 50 -1.5 1.5])

xlabel('lag index 1');ylabel('R_y_y(1)');title('Auto Correlation R_y_y(1)')

[rxx,lag]=xcorr(x,x);figure(2);stem(lag,rxx);grid

axis([-20 20 min(rxx) max(rxx)])

xlabel('lag index 1');ylabel(R_x_x(1)');

title('Auto Correlation R_x_x(1)')

[rxw,lag]=xcorr(x,w);

figure(5);stem(lag,rxw);gridaxis([-20 20 min(rxw) max(rxw)])

xlabel('lag index 1');ylabel(R_x_w(1)');

title('Cross Correlation Between x(n) and w(n)')

[rww,lag]=xcorr(w,w);figure(4);stem(lag,rww);grid

axis([-20 20 min(rww) max(rww)])

xlabel('lag index 1');ylabel(R_w_w(1)');

title('Auto Correlation R_w_w(1)')


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Expected Waveforms:

0 10 20 30 40 50 60 70 80 90 100-1.5

-1

-0.5

0

0.5

1

1.5

n

x(n)

sinusoidal signal x(n)

-2

0 -15 -10 -5 0 5 10 15 20

-80

-60

-40

-20

0

20

40

60

80

100

lag index 1

0 5 10 15 20 25 30 35 40 45 50

-0.8

-0.6

-0.4-0.2

0

0.2

0.4

0.6

0.8

n

w(n)

Random Noise Signal w(n)


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Dept.of ECE 63

RESULT:Thus the removal of noise by correlation was erformed using MATLAB.

QUESTIONS:

1. What are the applications in which the removal of noise is required( correlation)?Ans: Detection of RADAR and SONAR signals , ocean wave analysis and Geo Physics.

2. What is the procedure to remove noise signal?Ans: A signal is masked by noise can be detected either by correlation technique or by

filtering.

3.What is the signal to noise ratio?A.Signal to noise ratio is measure of signal power to noise power.

SNR=10 log10(Ps/Pn)

4.What is the auto correlation of a periodic signal?

N-1

A:Rxx(l)=1/N x(n)x(n-l)n=0

5. What is the cross correlation of a periodic signal?

N-1

A:Rxy(l)=1/N x(n)y(n-l)n=0

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2

n

y(n)

=x(n)+w(n) Sinusoidal Signal Corrupted With Random NOise

0 5 10 15 20 25 30 35 40 45 50

-1

0

1

lag index 1

Ryy

(1)

Auto Correlation Ryy(1)


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Dept.of ECE 64

EXPERIMENT-16

IMPULSE RESPONSE OF RAISED COSINE FILTER

AIM: To write a MATLAB program to plot the impulse response of raised cosine filter.


MATLAB Code:

clc;

clear all;t=linspace(-30,30,1000);

b=0.2;

T=1;

h1=(sin(pi*t/T))./(pi*t/T);

h2=(cos(pi*b*t/T))./(1-(2*b*t/T).^2);

h=h1.*h2;

figure(1);

plot(t,h);

Expected Waveforms:

Impulse response of raised cosine filter:

RESULT:Thus the periodic signal extraction BY correlation was performed successfully using

MATLAB.


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QUESTIONS:

1. What is meant by raised cosine filter?Ans: A raised cosine filter is a low pass filter which is commonly used in digital

communication systems, such as digital data modems, especially for pulse shaping and toavoid inter symbol interference(ISI).

2. What is meant by ISI?Ans: The ISI is a form of distortion of a signal in which one digital symbol interferes which

subsequent symbols.

3. What are the three different frequency bands of a filter?Ans: Pass band, Transition band, Stop band.

4. What is meant by transition band?Ans: The region between fp(pass band edge frequency) and fs(stop band edge frequency).

5. What is the Ideal value of transition band?Ans: Zero


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Dept.of ECE 66

EXPERIMENT-17

WINERKHINTCHINE RELATIONS

AIM: To verify WinerKhintchine Relations


MATLAB Code:

clc; clear all; close all;

fs=100;

t=0:1/fs:10;x=sin(2*pi*15*t);

N=512;

X=fft(x,N);

f=fs*(0:N-1)/N;power=X.*conj(X)/N;

figure(1)

plot(f,power);title('power spectrum through fourier transform');

xlabel('frequency');

ylabel('power');

figure(2)

rxx=xcorr(x,x);

sxx=fft(rxx,512);plot(f,abs(sxx));

title('fourier transform of autocorrelation function');xlabel('frequency');

ylabel('abs(sxx)');

Expected Waveforms:

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120power spectrum through fourier transform

frequency

power


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Dept.of ECE 67

RESULTS: Thus power spectrum, auto correlation of a signal using fourier transform wereplotted.

QUESTIONS:

1. What is the use of winer-khintchine theorem?Ans: The important properties of a signal like correlation and spectral density can be analysied using

wiener-khitchine theorem.

2. What is meant by wiener-khitchine theorem?

Ans:It states that the Energy spectral density of an energy signal is the fourier transform of its auto

correlation.

3. Which of the two parameters contains the same information about a signal?Ans: Auto correlations and Energy spectral density.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5x 10

4 fourier transform of autocorrelation function

frequency

abs(sxx)


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Dept.of ECE 68

EXPERIMENT-1

GENERATION OF COEFFICIENTS FOR ANALOG LOWPASS FILTER

%%generation of coefficients for analog lowpass filter%%

r=1000;c=0.000001;w1=1/(r*c);

h1=[];

w2=[];theta1=[];

forw=10:10:10000

h=(1/sqrt(1+(w/w1)^2));theta= -atan(w/w1);

theta1=[theta1,theta];

deg=rad2deg(theta1);

h1=[h1,h];w2=[w2,w];

end

subplot(2,1,1); semilogx(w2,h1,'k-'); grid on;

xlabel('angular frequency'); ylabel('gain');title('Magnitude plot for lowpass filter')

subplot(2,1,2); plot(w2,theta1,'k-'); grid on;

xlabel('frequency'); ylabel('phase angle');title('Phase plot for lowpass filter')

RESULTS: Thus the spectrum were plotted for low pass filter.

101

102

103

104

0

0.5

1

angular frequency

gain

Magnitude plot for lowpass filter

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1.5

-1

-0.5

0

frequency

phaseangle

Phase plot for lowpass filter


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SREENIVASA INSTITUTE OF TECHNOLOGY AND MANAGEMENT STUDIESEXPERIMENT-2

GENERATION OF COEFFICIENTS FOR ANALOG HIGHPASS FILTER

AIM: To Write a MATLAB Code for generation of analog lowpass filter.

Requirements:PC with MATLAB R2008a

MATLAB Code:

%%generation of coefficients for analog highpass filter%%r=10000;

c=0.000001;

w1=1/(r*c);h1=[];

w2=[];theta1=[];

forw=10:10:10000h=(1/sqrt(1+(w1/w)^2));

theta= atan(w1/w);

theta1=[theta1,theta];

deg=rad2deg(theta1);h1=[h1,h];

w2=[w2,w];

endsubplot(2,1,1); semilogx(w2,h1,'k-'); grid on;

xlabel('angular frequency'); ylabel('gain');title('Magnitude for highpass filter')subplot(2,1,2); plot(w2,theta1,'k-'); grid on;

xlabel('frequency'); ylabel('phase angle');

title('Phase plot for highpass filter')

101

102

103

104

0

0.5

1

angular frequency

gain

Magnitude for highpass filter

1

1.5

seangle

Phase plot for highpass filter

Basic Simulation Lab

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Transcript of Basic Simulation Lab