signals and systems using matlab

Systems Lab

Dept. Of ECE, SJCET, Palai 1

DATE: 08-02-2011

EXPERIMENT NO.1

FAMILIARISATION OF MATLAB

AIM

To familiarize with MATLAB and its functions.

THEORY

MATLAB is an ideal software tool for studying digital signal processing. The plotting capacity

of MATLAB makes it possible to view the result of processing and gain understanding into

complicated operations. The tool box support a wide range of signal processing from waveform

generation to filter design and implementation, parametric modelling and spectral analysis. There are

two categories of tools Signal processing functions Graphical interaction tools. The first category of

tools is made of functions that we can call from the common line or from your own application. Many

of the functions are MATLAB statements that implement specialized signal processing algorithm.

(a) DEFINTION OF VARIABLES

Variables are assigned numerical values by typing the expression directly

Eg: a=1+2 yields a=3

The answer will not be displayed when semicolon is put at the end of an expression

Eg: a=1+2

MATLAB utilizes the following arithmetic operations

+ Addition

- Subtraction

* Multiplication

/ Division

ˆpower operation

„Transpose

A variable can be assigned using a formula that utilizes these operations and either numbers or

previously defined variables.

Eg: since „a‟ was defined previously the following expression is valid, B=z*a.To determine the value

of a previously defined quantity type the quantity by b=6. If your expression does not fill on the line,

use an ellipse and continue on the next line.

C=1+2+3+. . . . . . . . . . . . ..+5+6+7. These are certain predefined variables which can be used at any

time, in the same manner as user defined variables.

i=sqrt(-1); j=sqrt(-1)

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pi=3.1416

Eg: y=2*(1+4*j)yields y=2.000+8.000y

These are also a number of predefined functions that can be used when defining a variable.

Some common functions that are used in this text are Abs- magnitude of a number Angle- angle of a

complex numberCos- cosine function, assume arguments in radian.

Exp-exponential functions

For example with y defined as aboveE= abs(y), yields e=8.2462, c=angle(y) yields c=1.3258

with a=3 as defined previously c=cos(a)yields, c= -0.9900c = exp(a), yields c=20.0855.Note that exp

can be used on complex numbers example with y = 2 + 8i as defined above

c = -1.0751 + 7.3104i which can be verified by using Euler‟s formula

c = c2cos(8) + jc2sin(8).

(b) DEFINTION OF MATRICES

MATLAB is based on matrix and vector algebra. Even scalars are treated 1x1 matrix.

Therefore, vector and matrix operation are simple as common calculator operations. Vectors can be

defined in two ways. The first method is used for arbitrary elements, v=[1 3 5 7] creates 1x4 vector

elements with elements 1 3 5 &7. Note that commas would have been used in the place of spaces to

separate the elements. Additional elements can be added to the vector v(5)=8 yields the vector v =

[1357]. Previously defined vectors can be used to define a new vectors. For example with we defined

above a= [910]; b = [v a]; creates the vector b = [1357910]. The second method is used for creating

vector with equally spaced elements t=0:0.1:10; creates 1x101 vector with elements0,0.1,0.2. . . . . 10.

Note that the middle number defines the increments is set to a default of 1k=0,10 creates 1x11 vector

with the elements 0,1,2. . . .10. Matrices are defined by entering the element row by row. M = [124;

368] creates the matrix M= 1 2 43 6 8 There are number of special matrices that can be defined Null

matrix: [ ];

Nxm matrixes of zero: M=zeros (m,m);

Nxm matrix of ones: M= ones (n,m);

Nxn matrix of identity: M=eye (n)

A particular elements of matrix can e assigned M(1,2)=5 place the number 5 in the first

row,2nd column. Operations and functions that were defined for scalars in the previous section can be

used on vectors and matrices. For example a=[1 2 3]; b=[4 5 6]; c=a+b yield c=579. Functions are

applied element by element. For example t=0:„0; x=cos(2*t) creates a vector „x‟ with elements equal

to cos(2t) for t=0,1,2. . . ..10

(c) GENERAL INFORMATION

MATLAB is case sensitive. So „a‟ and „A‟ are two different names. Comment statements are

preceded by a %.

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1) M-files M-files are macros of MATLAB commands that are stored as ordinary text file

with the extension „in‟ that is „filename.m‟. An m-file can be either a function with input and output

variables or a set of commands. MATLAB requires that the m-file must be stored either in the

working directory or in a directory that is specified in the MATLAB path list. The following

commands typed from within MATLAB demonstrate how this m-file is used. X=2,y=3,z=y plusx(y,x)

MATLAB m-files are most efficient when written in a way that utilizes matrix or vector operations,

loops and if statements are available, but should be used sparingly since they are computationally

inefficient. An examples is For k=1:10 x(k)=cos(k) end; This creates a 1x10vector „x‟ containing the

cosine of the positive integers from 1 to 10. This operation is performed more efficiently with the

commands k=1:10 x=cos(k) which utilizes a function of a vector instead of a for loop. An if statement

can be used to define combinational statement.

(d) REPRESENTATION OF SIGNALS

A signal can be defined as function that conveys information, generally about or behaviour of

physical system; signals are represented mathematically as functions of one or more independent

variables. The independent variables is the mathematical expression of a signal may be either

continuous is discrete. Continuous time signals are defined along a number of times and thus they

are represented by a continuous independent variable. TheMATLAB is case sensitive. Discrete time

signals are represented mathematically.Continuoustimesignals are often referred to as analog

signals. Discrete time signals are defined at discrete times and thus the independent variable has

discrete values. In a sequence of numbers x, the nthnumber in the sequence is denoted as x[n]. The

basic signals used in digital signal processing are the unit impulse signal [U+0260](n), exponential of

the form au[n], sine waves and their generalization to complex exponentials. Since the only numerical

data type in MATLAB is the m x nmatrix, signals must be represented as vectors either mx1 matrix if

column vector 1xn matrices if row vectors. A constant will be treated as 1x1 matrix. The signals are

sampling frequency Fs,which is greater than twice the maximum frequency of the signal. In time

domain, the signals are represented as time versus amplitude. In MATLAB you can generate time

base for given signals T=t start: 1/Fs:t-stop,tstart is the starting time of the signal t-sto is the stop time

of the signal and Fs sampling frequency. 1/Fs is the time period between the two samples. In signals

like speech if the data secured length is too long, plotting of the whole signal will not be clean view.

In that case we can use step function.

RESULT

The familiarization of MATLAB was completed.

Systems Lab


DATE: 08-02-2011

EXPERIMENT NO: 2

BASIC DISCRETE TIME SIGNALS

AIM

To plot the following waveforms

1. Step Function

2. Impulse Function

3. Exponential Function

4. Ramp Function

5. Sine Function

THEORY

A digital signal can be either deterministic signal that can be predicted with certainty or

random signal that is unpredictable. Due to ease in signal generation and need for predictability,

deterministic signal can be used for system simulation studies.Standard forms of some deterministic

signal that is frequently used in DSP are discussed below:

1. Impulse:the simplest signal is the unit impulse signal which is defined as

δ (n) = 1 for n = 0

= 0 for n ≠ 0

2. Step:the general form of step function is

u (n) = 1 for n ≥ 0

= 0 for n< 0

3. Exponential:the decaying exponential is a basic signal in DSP.Their general form is given by

x (n) = a n for all n.

4. Ramp:signal is given by

ur (n) = n for n ≥ 0

= 0 for n < 0

5. Sine:A sinusoidal sequence is defined as x(n) = sin(n).

ALGORITHM

1. Start

2. Generate the time axis

3. Define A as sine function

4. Plot the sine wave

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5. Define B as exponential signal

6. Plot the exponential signal

7. Define r(n) as ramp signal

8. Plot the ramp signal

9. Define unit impulse signal

10. Plot the signal

11. Define unit step signal

12. Plot the signal

13.Stop

MATLAB FUNCTIONS USED

• CLC: clc clears all input and output from the Command Window display, giving you a ”clean

screen. “After using clc, you cannot use the scroll bar to see the history of functions, but you still can

use the up arrow to recall statements from the command history.

• CLEAR ALL: clear removes all variables from the workspace. This frees up system memory.

• CLOSE ALL: Closes all the open figure windows

• SUBPLOT: subplot divides the current figure into rectangular panes that are numberedrowwise.

Each pane contains an axes object. Subsequent plots are output to the current pane.

• STEM: A two-dimensional stem plot displays data as lines extending from a baseline along the x-

axis. A circle (the default) or other marker whose y-position represents the data value terminates each

stem.

• TITLE: Each axes graphics object can have one title. The title is located at the top and in the centre

of the axes.

• XLABEL: Each axes graphics object can have one label for the x-, y-, and z-axis. The label appears

beneath its respective axis in a two-dimensional plot and to the side or beneath the axis in a three-

dimensional plot.

• YLABEL: Each axes graphics object can have one label for the x-, y-, and z-axis. The label appears

beneath its respective axis in a two-dimensional plot and to the side or beneath the axis in a three-

dimensional plot.

• ZEROS: Create array of all zeros

• ONES: Create array of all ones

• SIN: The sin function operates element-wise on arrays. The function‟s domains and ranges include

complex values. All angles are in radians.

• EXP: The exp function is an elementary function that operates element-wise on arrays. Its domain

includes complex numbers.

Systems Lab


PROGRAM

%Generation of Basic Discrete Signals

clear all

clc

close all

%Generation of Sinusoidal signal

n =0:0.5:10;

y=sin(n);

subplot(3,2,1);

stem(n,y);

xlabel(' Samples ');

ylabel(' Amplitude');

title (' Sine');

%Generation of Exponential signal

y=exp(n);

subplot(3,2,2);

stem(n,y);

xlabel (' Samples ');


title (' Exponential');

%Generation of Ramp signal

n =1:10;

y=n;

subplot(3,2,3);

stem(n,y);



title (' Ramp');

%Generation of Unit Impulse signal

n =-10:10;

y=[zeros(1,10) 1 zeros(1,10)];

subplot(3,2,4);

stem(n,y);


ylabel (' Amplitude');

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title(' Impulse');

%Generation of Unit Step signal

n =-10:10;

y=[zeros(1,10) 1 ones(1,10)];

subplot(3,2,5);

stem(n,y);



title (' Step');

OUTPUT

RESULT The MATLAB program to generate basic discrete signals is executed and output

waveforms are obtained.

Systems Lab


DATE: 17-02-2011

EXPERIMENT NO: 3

IMPULSE RESPONSE & STEP RESPONSE

AIM

Write a MATLAB program to find the impulse response and step response of the system given

by y(n) = x(n) + x(n-1) – 0.7y(n-1) at -20 ≤ n ≤ 100.

THEORY

The impulse response of a given system is its response to impulse function input. We know

that y[n]=impulse response h[n],when input x[n] is unit impulse function Step response of a system is

its output for step function.

ALGORITHM

1. Start

2. Input the coefficients of x(n).

3. Generate impulse signal.

4. Input the coefficients of y(n).

5. Obtain the impulse response using filter function

6. Plot the impulse response.

7. Generate step signal.

8. Obtain the step response using filter function.

9. Plot the step response.

10. Stop.


FILTER: One dimensional digital filter. Y=FILTER (B, A, X) filter the data in vector X with the filter

described by vectors A and B to create the filtered data Y. The filter is a “Direct Form II Transposed”

implementation of the standard difference equation: a(1) * y(n) = b(1) * x(n) +b(2) * x(n - 1) + .... +

b(nb + 1) * x(n - nb) - a(2) * y(n - 1) - . . . a(na + 1) * y(n - na) If a(1) is not equal to 1, filter

normalizes the filter coefficients by a(1).

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PROGRAM

clc

clear all

close all

x = [ 1 1];

y = [1 0.7];

n = -20:1:10;

imp = [zeros(1,20) 1 zeros(1,10)];

h = filter (x, y, imp);

subplot (2, 1, 1);

stem (n,h);

title(' Impulse Response ');


ylabel (' Amplitude ');

stp = [zeros(1,10 1 ones(1,20)];

h = filter(x,y,stp);

subplot ( 2, 1, 2);

stem(n,h);

title (' Step Response ');



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OUTPUT

RESULT

The MATLAB program to find the impulse response and step response of the given system is

executed and output waveform is obtained.

Systems Lab


DATE: 24-02-2011

EXPERIMENT NO: 4

LINEARITY, STABILITY & CAUSALITY

AIM

Write a MATLAB program to find whether the given systems are linear, stable and causal.

1. y(n) = x(n) – 0.9y(n-1)

2. y(n) = exp x(n)

THEORY

A system is linear if and only if T [a1x1[n] + a2x2n]] = a1T [x1[n]] + a2T [x2[n]], for any

arbitrary constants a1 and a2.

Let x[n] be bounded input sequence and h[n] be impulse response of the system and y[n], the

output sequence. Necessary and sufficient boundary condition for stability is8n = -8|h[n]| < 8.

Causal system is a system where the output depends on past or current inputs but not on the

future inputs. The necessary condition is h[n] = 0; < 0, where h[n] is the impulse response.

ALGORITHM

1. Start

2. Input the coefficients of x(n) and y(n)

3. Generate random signals x1 and x2.

4. Check for linearity using filter function of the given system and display

5. Generate impulse signal.

6. Check for causality using filter function of the given system and display

7. Obtain the absolute value of impulse response and check for the stability of the system and display.

8. Stop


• RAND: Uniformly distributed pseudo random numbers. R=RAND (N) returns an N-by-N matrix

containing pseudo-random values drawn from a uniform distribution on the unit interval. RAND (M ,

N)or RAND([M, N]) returns an M-by-N matrix.

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• DISP: Display array. DISP(X) displays the array, without printing the array name. In all other ways

it‟s the same as leaving the semicolon of an expression except that empty arrays don‟t display.

• ABS: Absolute value. ABS(X) is the absolute value of the elements of X.

• SUM: Sum of elements. S=SUM(X) is the sum of the elements of the vector X.

PROGRAM

%Program to find linearity of system 1

clc

clear all

close all

% System 1

% Linearity

b = [1 0];

a = [1 0.9];

x1 = rand (1,10);

x2 = rand (1,10);

y2 = filter (b, a, 2.*x1);

y5 = filter (b, a,2.* x2);

y = filter (b, a, x1);

y0 = filter (b, a, x2);

y6 = y2+y5;

y7 = y+y0;

if (y6-y7 ~= 0)

disp (' Linear ')

else

disp (' Non Linear ')

end;

% Causality--------%%%

n= -10:1:10;

x = [zeros(1,10) 1 zeros(1,10)];

y1 = filter (a, b, x);

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subplot (2, 1, 1);

stem (n1,y1);



% Stability

T = abs (y1);

t = sum (T);

if (t < 1000)

disp (' Stable ')

else

disp (' Unstable ')

end;

%%%%%%%%%% % System 2%%%%%%%%%%%%%%%%%%%%%

% 1 % Linearity

y8 = exp (x2);

y9 = exp ( x3);

y10 = 5*y8+5*y9;

y11 = exp(5*x2+5*x3);

if (y11-y10~=0)

disp (' 2Linear ')

else

disp (' 2Non Linear ')

end;

% 2 % Causality

Y8= exp (x);

subplot (2, 1, 2);

stem (n1,y8);



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% 3 % Stability

T1= abs (y8);

t1 = sum (T1);

if (t1 < 1000)

disp (' 2Stable ')

else

disp (' 2Unstable ')

end;

OUTPUT

System 1

Linear.

Stable.

System 2

Non Linear.

Stable.

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RESULT

The MATLAB program to find the linearity, causality, and stability of 2 systems given is

executed and output is obtained

System 1

System 2

Linear Non Linear

Stable Stable

Causal Non Causal

Systems Lab


DATE: 15-03-2011

EXPERIMENT NO: 5

CONVOLUTION

AIM

Write a MATLAB program to obtain linear and circular convolution of the following sequences

1. x(n) = [1 2 3 4]

h(n) = [1 2 3]

2. x(n) = u(n)

h(n) = u(n-2)

THEORY

Convolution is used to find the output response of a digital system. The linear convolution of

two continuous time signals x[n] and h[n] is defined by y[n] = x[n] * h[n].

Circular Convolution of two sequences x1[n] and x2[n], each of length N is given by y[n] = If

the length of sequence is not equal, zero padding is done. Convolution operation will result in the

convolved signal be zero outside of the range n= 0,1,...,N-1.

ALGORITHM

1. Start

2. Input the sequences

3. Input the length of sequences

4. Calculate the linear convolution.

5. Calculate the circular convolution.

6. Stop


• CONV: Convolution and polynomial multiplication. C = CONV (A,B) convolves vectors A and B.

The resulting vector is length LENGTH(A) + LENGTH(B) – 1 .If A and Bare vectors of polynomial

coefficients, convolving them is equivalent to multiplying the two polynomials.

• LENGTH: Length of vector. LENGTH(X) returns the length of vector X.

Systems Lab


• CCONV: Convolution and polynomial multiplication. C = CONV (A,B) convolves vectors A and B.

The resulting vector is length LENGTH(A) + LENGTH(B) – 1 .If A and Bare vectors of polynomial

coefficients, convolving them is equivalent to multiplying the two polynomials.

PROGRAM

%program to find convolution of two sequences

clc

close all

clear all

n = -10:10;

x1 = [1 2 3 4];

subplot(4,2,1);

stem(x1);

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

h1 = [1,2,3];

subplot(4,2,3);

stem(h1);

title('-----h1(n)----');

%%%%%linear convolution%%%%%%

l1 = conv(x1,h1)

subplot(4,2,5);

stem(l1);

title('-----linear convolution-1--');

%%%%%%%%%% circular convolution %%%%%%%%%%%5

x1 = [1 2 3 4];

h1 = [1 2 3 0];

c1 = cconv(x1,h1,4)

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subplot(4,2,7);

stem(c1);

title('-----circular convolution-1---');

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%

x2 = [zeros(1,10) ones(1,11)];

subplot(4,2,2);

stem(n,x2);

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

h2 = [zeros(1,12) ones(1,9)];

subplot(4,2,4);

stem(n,h2);

title('-----h2(n)----');

%%%%%linear convolution%%%%%%

l2= conv(x2,h2)

subplot(4,2,6);

stem(l2);

title('-----linear convolution--2--');

%%%%%%%%%% circular convolution %%%%%%%%%%%

c2 = cconv(x2,h2,8)

subplot(4,2,8);

stem(c2);

title('-----circular convolution--2----');

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OUTPUT

RESULT

The MATLAB program to find the linear convolution and circular convolution of given

sequences is executed and output is obtained.

Systems Lab


DATE: 15-03-2011

EXPERIMENT NO: 6

CONVOLUTION USING DFT

AIM

1. Write a MATLAB program to obtain linear convolution of

x(n) = [1 2 3 4]

h(n) = [1 2 2 -1] using DFT.

2. Write a MATLAB program to obtain circular convolution of

x(n) = u(n), h(n) = u(n-3)

THEORY

Convolution is used to find the output response of a digital system. The linear convolution of

two continuous time signals x[n] and h[n] is defined by y[n] = x[n] * h[n] = Let X(k) and H(k) be the

Fourier transforms of x[n] and h[n] respectively. Then by DFT method, convolution of x[n] and h[n]

is given by: x[n]*h[n]= Inverse Fourier transform of X(k).H(k).

Circular convolution of two sequences x1[n] and x2[n], each of length N is given by y[n] = If

the length of sequence is not equal, zero padding is done. Convolution operation will result in the

convolved signal be zero outside of the range n= 0,1,...,N-1.Then by DFT method, convolution of x[n]

and h[n] is given by: x[n]*h[n]= Inverse Fourier transform of X(k).H(k).

ALGORITHM

1. Start


3. Input the length of the sequences.

4. Find the FFT of the sequences

5. Multiply the FFT of the sequences.

6. Calculate the IFFT

7. Stop

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• FFT: Discrete Fourier Transform. FFT(X) is the discrete Fourier transform(DFT) of vectorX.

FFT(X,N) is the N-point FFT, padded with zeros if X has less than N points and truncated if it has

more.

• IFFT: Inverse discrete Fourier transform. IFFT(X) is the inverse discrete Fourier transform of X.

IIF(X,N) is the N-point inverse transform.

PROGRAM

clc; close all; clear all;

x1 = [1 2 3 4];

subplot(4,2,1);

stem(x1);

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

h1 = [1 2 2 -1];

subplot(4,2,3);

stem(h1);

title('-----h1(n)----');

l = conv(x1,h1)

subplot(4,2,5);

stem(l);

title('-----linear convolution----')

dx1 = fft(x1,7);

dh1 = fft(h1,7);

dl = dx1.*dh1;

k1 = ifft(dl)

subplot(4,2,7);

stem(k1);

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title('-----linear convolution--dft--');

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

x2 = ones(1,6);

subplot(4,2,2);

stem(x2);

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

h2 = [zeros(1,3) ones(1,3)];

subplot(4,2,4);

stem(h2);

title('-----h2(n)----');

c = cconv(x2,h2,6)

subplot(4,2,6);

stem(c);

title('-----circular convolution------');

dx2 = fft(x2,6);

dh2 = fft(h2,6);

dc = dx2.*dh2;

k2 = ifft(dc)

subplot(4,2,8);

stem(k2);

title('-----circular convolution--dft----');

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OUTPUT

RESULT

The MATLAB program to find linear convolution and circular convolution of given

sequences using DFT is executed and output is obtained.

Systems Lab


DATE: 22-03-2011

EXPERIMENT NO: 7

CORRELATION

AIM

a) Write a MATLAB program to compute autocorrelation of sequence x(n) = 0.9 n , n= 0 to 20 and

verify the property.

b) Compute cross-correlation of sequences x(n) = {1,2,3,4,5,6} and y(n) = x(n-3) and verify the

property.

THEORY

Autocorrelation is the cross-correlation of a signal with itself. The autocorrelation sequence of

x(n) is given by

RXX[l] = ∑ ( ) ( )

Cross correlation is a measure of similarity of two signals, commonly used to find features in

an unknown signal by comparing it to a known one. The correlation sequence of x(n) and y(n) is

given by

RXY[l] = ∑ ( ) ( ) l=0,±1,±2,…

ALGORITHM

1. Start


3. Find the autocorrelation of the sequence

4. Check for the symmetry property of the sequence

5. Find the cross correlation of the sequences

6. Check for the symmetry property of the sequence

7. Stop

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XCORR(A): when A is a vector, is the auto-correlation sequence.XCORR(A), when A is an M-by-N

matrix, is a large matrix with2*M-1 rows whose N^2 columns contain the cross-correlation sequences

for all combinations of the columns of A.

XCORR(A,B): where A and B are length M vectors (M>1), returns the length 2*M-1 cross-

correlation sequence C. If A and B are of different length, the shortest one is zero-padded.

FLIPLR: Flip matrix in left/right direction. FLIPLR(X) returns X with row preserved and columns

flipped in the left/right direction.

PROGRAM

clc;

clear all;

close all;

%autocorrelation of x(n)

n=0:20;

x=0.9.^n; % input sequence

y =xcorr(x); % autocorrelation

subplot(2,2,1);

stem (y);

xlabel (' Samples ')

ylabel ('Amplitude');

title(' Autocorrelation of x(n)');

%autocorrelation ofx(-n)

z =fliplr (y); % flip the matrix y

subplot(2,2,2);

stem (z);



title (' Autocorrelation of x(-n)');

%cross correlation of x(n) & y(n)

x1= [1 2 3 4 5 6 0 0 0]; %input

h1 = [0 0 0 1 2 3 4 5 6]; % input

y1 =xcorr (x1, h1); % cross correlation of e and f

subplot(2,2,3);

stem (y1);

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xlabel (' Samples ') ;ylabel(' Amplitude');

title(' Cross correlation of x(n) and y(n)');

%cross correlation property

z1= fliplr (y1);

subplot(2,2,4); stem (z1);

xlabel (' Samples '); ylabel(' Amplitude');

title (' Cross correlation of x(-n) and h(-n)');

OUTPUT

RESULT

The MATLAB program to compute autocorrelation and cross correlation of the given

sequences is executed and output is obtained.

Systems Lab


DATE: 22-03-2011

EXPERIMENT NO: 8

SAMPLING THEOREM

AIM

Write a MATLAB program to generate x(n) = sin 0.5πn for n to 50. Verify sampling theorem.

THEORY

Sampling Theorem: A band limited signal can be reconstructed exactly if it is sampled at a

rate at least twice the maximum frequency component in it.” Figure 1 shows a signal g(t) that is band

limited.

Figure 1: Spectrum of band limited signal g(t)

The maximum frequency component of g(t) is fm. To recover the signal g(t) exactly from its

samples it has to be sampled at a rate fs=2fm. The minimum required sampling rate fs = 2fm iscalled

Nyquist rate.

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ALGORITHM

1. Start

2. Generate the sine wave

3. Resample the signal at various values.

4. Obtain the output waveforms

5. Stop


RESAMPLE : Change the sampling rate of a signal.

Y = RESAMPLE(X,P,Q) resamples the sequence in vector X at P/Q times the original sample rate

using a polyphase implementation. Y is P/Q times the length of X (or the ceiling of this if P/Q is not

an integer). P and Q must be positive integers.

PROGRAM

% Verification of sampling theorem

clc;

clear all;

close all;

n = 0:0.5:50;

x = sin(0.5*pi*n);

subplot(4,1,1);

stem(x);

title('original');

y = resample(x,2,1);

subplot(4,1,2);

stem(y);

title('fs = 2fm');


subplot(4,1,3);

stem(y);

title('fs > 2fm');


subplot(4,1,4);

stem(y);

title('fs<2fm');

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OUTPUT

RESULT

The MATLAB program to generate sine wave and to verify sampling theorem is executed

and output is obtained.

Systems Lab


DATE: 22-03-2011

EXPERIMENT NO: 9

POLE-ZERO PLOT

AIM

Write a MATLAB program to find poles and zeros of the system given by

y(n) = x(n) + 2x(n-1) – 0.9y(n-1)

THEORY

In mathematics, signal processing and control theory, a pole–zero plot is a graphical

representation of a rational transfer function in the complex plane which helps to convey certain

properties of the system such as:

Stability

Causal system / anticausal system

Region of convergence (ROC)

Minimum phase / non minimum phase

In general, a rational transfer function for a discrete LTI system has the form:

Where

zi such that P(zi) = 0 are the zeros of the system

zj such that Q(zj) = 0 are the poles of the system

In the plot, the poles of the system are indicated by an x while the zeroes are indicated by an o.

ALGORITHM

1. Start

2. Input the coefficients of the system

http://en.wikipedia.org/wiki/Anticausal_system

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3. Plot the zeros and poles of the system

4. Stop


ZPLANE: Z-plane zero-pole plot. ZPLANE (Z,P) plots the zeros Z and poles P (in column vectors)

with the unit circle for reference. Each zero is represented with a 'o' and each pole with a 'x' on the

plot. Multiple zeros and poles are indicated by the multiplicity number shown to the upper right of

the zero or pole.

PROGRAM

clc ;close all;clear all;

b = [1 2]; % coefficient of x

a = [1 0.9]; % coefficient of y

zplane (b, a);

OUTPUT

RESULT

The MATLAB program to find the poles and zeros of given system was executed and

output obtained correctly.

Systems Lab


DATE: 22-03-2011

EXPERIMENT NO: 10

FIR FILTER DESIGN

AIM

1. Write a MATLAB program to design FIR filter and to plot magnitude and phase response for the

given specification using Kaiser window.

a. Pass band Ripple = 0.087 dB

b. Stop band attenuation = 60 dB

c. Pass band edge frequency = 0.4π

d. Stop band edge frequency = 0.6π

2. Design FIR filter using rectangular, hamming and hanning window



c. Pass band edge frequency = 3000Hz

d. Stop band edge frequency = 2000Hz

e. Sampling Frequency = 8000 Hz

THEORY

The basic idea behind the window design is to choose a proper ideal frequency selective filter

(which always has a non-causal, infinite duration impulse response), and then truncate ( or window)

its impulse response to obtain a linear phase and causal FIR filter. One possible way of obtaining FIR

filter is to truncate the Fourier series at n= ±( )

, where N is the length of the desired sequence but

abrupt truncation of the Fourier series results in oscillation in the pass band and stop band. These

oscillations are due to slow convergence of the Fourier series and this effect is known as Gibbs

phenomenon. To reduce this oscillations the Fourier coefficients of the filter are modified by

multiplying the infinite impulse response with a finite weighing sequence w(n) called a window.

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Rectangular window is given by

Wr (n) = 1; for n=±( )

= 0; otherwise

Frequency response: H( )

∫ (

) (

( ))

The convolution of the desired response and the rectangular window response gives rise to

ripples in both passband and stop band. This Gibbs phenomenon can be reduced by using a less abrupt

truncation of filter coefficients. This can be achieved using a window function that tapers smoothly

towards zero at both ends.

Hamming window is given by,

Wh(n) = 0.54+0.046cos(2πn/N-1); for n=±( )

= 0; otherwise

Frequency response: H( )

∫ (

) ( ( ))

This window generates less oscillation in the side lobes than the hamming window. At higher

frequencies the stop band attenuation is low when compared to the hanning window.

Hanning window is given by,

WHn(n) = .5+.5cos(2πn/N-1);for n=±( )

= 0; otherwise

This window results in smaller ripples in both passband and stop band of the filter. At higher

frequencies the stop band attenuation is even greater.

Kaiser window is given by,

WK(n) = ( (

( ))

( ) ; for n=±

( )

= 0; otherwise

This is one of the most useful and optimum window. It provides large main lobe width for the

given stop band attenuation, which implies the sharpest transition width. Α is the parameter that

depends on N (window length) and that can be chosen to yield the various transmission widths and

near optimum stop band attenuation.

ALGORITHM

1. Start

2. Input the design parameters of the system.

3. Find the order of the

4. Design the FIR filter using Kaiser Window

5. Plot the magnitude and phase response.

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6. Design the FIR filter using Hanning window

7. Plot the magnitude and phase response

8. Design the FIR filter using Hamming window


10. Design the FIR filter using Rectangular window


12. Stop


FIR1: FIR filters design using the window method. B = FIR1(N,Wn) designs an N'th order

low pass FIR digital filter and returns the filter coefficients in length N+1 vector B

FREQZ: Digital filter frequency response.

[H, W] = FREQZ(B,A,N) returns the N-point complex frequency response vector H and the

N-point frequency vector W in radians/sample of the filter

ANGLE: Phase angle.

ANGLE(H) returns the phase angles, in radians, of a matrix with complex elements.

HANN: Hanning window.

HANN(N) returns the N-point symmetric Hanning window in a column vector.

GRID: Grid lines.

GRID ON: adds major grid lines to the current axes.

HAMMING: Hamming window.

HAMMING(N) returns the N-point symmetric Hamming window in a column vector.

RECTWIN: Rectangular window.

W = RECTWIN(N) returns the N-point rectangular window.

KAISERORD: FIR order estimator (lowpass, highpass, bandpass, and multiband).

[N,Wn,BTA,FILTYPE] = KAISERORD(F,A,DEV,Fs) is the approximate order N,

normalized frequency band edges Wn, Kaiser window beta parameter BTA and filter type

FILTYPE to be used by the FIR1 function: B = FIR1(N, Wn, FILTYPE, kaiser( N+1,BTA ),

'noscale' ), F vector of band edge frequencies in Hz in ascending order between 0 and half the

sampling frequency Fs, DEV is the vector of maximum deviation or ripples allowable for

each band.

PLOT: Linear plot. PLOT(X, Y) plots vector Y versus vector X. If X or Y is a matrix then

the vector is plotted versus the rows or columns of the matrix, whichever line up.

PROGRAM

clear all;

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close all;

clc ;

% Kaiser Window

%pass band ripple = 0.087dB

%stop band attenuation = 60dB

%pass band edge frequency = 0.4π rad/sec

%stop band edge frequency = 0.6π rad/sec

%sampling frequency = 100 rad/sec

fs = 100; %rad/sec, sampling frequency

pf = 0.4*pi; %rad/sec, pass band frequency

sf = 0.6*pi; %rad/sec, stop band frequency

fsamp = fs/(2*pi); % fs into hertz

pf1 = pf/(2*pi); % pf into hetz

sf1 = sf/(2*pi); % sf into hertz

d1 = 10^(-0.05*60); %stop band ripple

d2 = (10^(0.05*0.087)-1)/(10^(0.05*0.087)+1); %pass band ripple

fcuts = [pf1 sf1]; %pass and stop band cut off frequency

mags = [1 0]; %magnitude assigning

%d3 = min(d1,d2);

devs = [d2 d1]; %setting deviation

w = 0:0.01:pi;

%to get the order n, kaiser window beta parameter BTA and filter type

[n,Wn,beta,ftype] = kaiserord(fcuts,mags,devs,fs);

%filter design using windows

h = fir1(n,Wn,ftype,kaiser(n+1,beta),'noscale');

[H,f] = freqz(h,1,w); % frequency response and frequency vector

gain = 20*log10(abs(H));

a = angle(H);

%PLOTTING THE GRAPHS

subplot(2,1,1);

plot(f/pi, gain); grid on;

title(' Magnitude Response of Kaiser Window');

xlabel(' Normalised Frequency');

ylabel (' Gain in DB');

subplot(2,1,2);

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plot(f/pi, a); grid on;

title(„ Phase Response of Kaiser Window „);


ylabel(' Angle');

%hanning,hamming and rectangular window



%pass band edge frequency = 3000Hz

%stop band edge frequency = 2000Hz

%sampling frequency = 8000Hz

fsamp = 8000; %sampling frequency

fcuts = [2000 3000]; %pass band and stopband edge frequency

mags = [0 1]; %magnitude

d1 = 10^(-0.05*44); %stop band ripple

d2 = (10^(0.05*0.1)-1)/(10^(0.05*0.1)+1); %pass band ripple

devs = [d2 d1]; %setting deviation

W = 0:0.01:pi;

[n,Wn,beta,ftype] = kaiserord(fcuts,mags,devs,fsamp); %to get the order

% hanning window

h = hann(n+1);

b = fir1(n,Wn,h);

w = 0:0.01:pi;

[H,f] = freqz(b,1,w);


an = angle(H);

%plotting magnitude and phase response of hanning window

figure;

subplot(2,1,1);


title('Magnitude Response Hanning Window');

xlabel('Normalised Frequency');

ylabel('Gain in dB');

subplot(2,1,2);

plot(f/pi, an); grid on;

title('Phase Response of Hanning Window');

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ylabel(' Angle');

% HAMMING WINDOW

h = hamming(n+1);

b = fir1(n,Wn,h);

w = 0:0.01:pi;



an = angle(H);


figure;

subplot(2,1,1);


title (' Magnitude Response of Hamming Window ');


ylabel (' Gain in dB');

subplot(2,1,2);

plot(f/pi, an); grid on;

title(' Phase Response of Hamming Window ');

xlabel (' Normalised Frequency');

ylabel(' Angle');

%RECTANGULAR WINDOW

h = rectwin(n+1);

b = fir1(n,Wn,h);

w = 0:0.01:pi;



an = angle(H);


figure;

subplot(2,1,1);

plot(f/pi,gain); grid on;

title (' Magnitude Response of Rectangular Window ');

xlabel (' Normalised Frequency');

ylabel (' Gain in dB');

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subplot(2,1,2);

plot (f/pi, an); grid on;

title ('Phase Response of Rectangular Window');


ylabel(' Angle');

OUTPUT

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RESULT

The MATLAB program to design FIR filter and to plot the responses are executed and

output was verified.

Systems Lab


DATE: 22-03-2011

EXPERIMENT NO: 11

IIR FILTER DESIGN

AIM

1. Write a MATLAB program to design Butterworth BPF with given specification and plot its

response



c. Pass band edge frequency = 800 Hz < f< 1000 Hz

d. Stop band edge frequency = 0 to 400 Hz and 2000Hz to infinity


2. Write a MATLAB program to design Type -1 Chebyshev filter with given specification and plot its

response



c. Pass band edge frequency = 800Hz

d. Stop band edge frequency = 300Hz


THEORY

The Butterworth filter is a type of signal processing filter designed to have as flat a frequency

response as possible in the pass band so that it is also termed a maximally flat magnitude filter.The

frequency response of the Butterworth filter is maximally flat (has no ripples) in the pass band and

rolls off towards zero in the stop band. When viewed on a logarithmic Bode plot the response slopes

off linearly towards negative infinity.

A first-order filter's response rolls off at −6 dB per octave (−20 dB per decade) (all first-order

low pass filters have the same normalized frequency response). A second-order filter decreases at −12

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dB per octave, a third-order at −18 dB and so on. Butterworth filters have a monotonically changing

magnitude function with ω, unlike other filter types that have non-monotonic ripple in the pass band

and/or the stop band.

Chebyshev filters are analog or digital filters having a steeper roll-off and more pass band

ripple (type I) or stop band ripple (type II) than Butterworth filters. Chebyshev filters have the

property that they minimize the error between the idealized and the actual filter characteristic over the

range of the filter, but with ripples in the pass band. Because of the pass band ripple inherent in

Chebyshev filters, filters which have a smoother response in the passband but a more irregular

response in the stop band are preferred for some applications.

These are the most common Chebyshev filters. The gain (or amplitude) response as a function of

angular frequency ω of the nth order low pass filter is

Where is the ripple factor, ω0 is the cut-off frequency and Tn() is a Chebyshev polynomial of

the nth order.

ALGORITHM

1. Start

2. Input the system parameters

3. Calculate the cut-off and order.

4. Design the Butterworth filter


6. Design the Type-1 Chebyshev filter


8. Stop


BUTTORD Butterworth filter order selection.

[N, Wn] = BUTTORD (Wp, Ws, Rp, Rs) returns the order N of the lowest order digital

Butterworth filter that loses no more than Rp dB in the passband and has at least Rs dB of

attenuation in the stop band. Wp and Ws are the passband and stop band edge frequencies,

normalized from 0 to 1 (where 1 corresponds to pi radians/sample)

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BUTTER Butterworth digital and analog filter design.

[B, A] = BUTTER (N, Wn) designs an Nth order low pass digital Butterworth filter and returns

the filter coefficients in length N+1 vectors B (numerator) and A (denominator). The coefficients

are listed in descending powers of z. The cutoff frequency Wn must be 0.0 < Wn < 1.0, with 1.0

corresponding to half the sample rate.

CHEBY1 Chebyshev Type I digital and analog filter design.

[B, A] = CHEBY1(N, R, Wp) designs an Nth order lowpass digital Chebyshev filter with R

decibels of peak-to-peak ripple in the passband. CHEBY1returns the filter coefficients in length

N+1 vectors B (numerator) andA (denominator). The passband-edge frequency Wp must be 0.0 <

Wp < 1.0 with 1.0 corresponding to half the sample rate.

CHEB1ORD Chebyshev Type I filter order selection.

[N, Wp] = CHEB1ORD (Wp, Ws, Rp, Rs) returns the order N of the lowest order digital

Chebyshev Type I filter that loses no more than Rp dB in the passband and has at least Rs dB of

attenuation in the stopband. Wp and Ws are the passband and stopband edge frequencies,

normalized from 0 to 1 (where 1 corresponds to pi radians/sample).

PROGRAM

%BUTTERWORTH BAND PASS FILTER



%pass band frequency = 800Hz<= f <=1000Hz

%stop band frequency = 0 <= f <= 400Hz and 2000Hz <= f <= infinity


clear all

close all

clc

rp = 1.5; % passband ripple

rs = 20; %stop band ripple

ft = 8000; %sampling frequency

wp1 = 2* 800/ft; %normalize passband edge frequencies

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wp2 = 2* 1000/ft;

wp = [wp1 wp2]; %pass band corner frequencies

ws1 = 2* 400/ft; %normalize stop band edge frequencies

ws2 = 2* 2000/ft;

ws = [ws1 ws2]; %stop band corner frequency

[n,Wn] = buttord(wp, ws, rp,rs); %to calculate order and cut off

[b,a] = butter (n, Wn); % to design bandpass filter coefficients

w = 0:0.01:pi;

freqz(b,a,w); % to plot magnitude and phase response

%CHEBYSHEV HIGHPASS FILTER



%pass band edge frequency = 800Hz

%stop band edge frequency = 300Hz


rp = 0.5; % passband ripple

rs = 20; %stop band ripple

ft = 3000; %sampling frequency

wp = 2* 800/ft; %normalize passband frequency

ws = 2* 300/ft; %normalize stop band frequency

[n,Wn] = cheb1ord(wp,ws,rp,rs); %returns order N and cut off

[b,a] = cheby1(n,rp,Wn,'high'); %designs an nth order high pass digital Chebyshev

filter

w = 0:0.01:pi;

figure;

freqz(b,a,w); % to plot magnitude and phase response

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OUTPUT

Butterworth filter

Chebyshev filter

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RESULT

The MATLAB program to design a Butterworth band pass filter and type 1 Chebyshev

high pass filter has been executed and the filter responses are plotted.

Systems Lab


DATE: 30-03-2011

EXPERIMENT NO: 12

FAMILIARIZATION OF VHDL

AIM

To familiarize the VHDL programming.

THEORY

VHDL is a hardware description language. It describes the behaviour of an electronic circuit

or system from which the physical circuit or system can then be attained (implemented). VHDL

stands for VHSIC Hardware Description Language. VHSIC is itself an abbreviation for Very High

Speed Integrated Circuits, an initiative funded by the United States Department of Defence in the

1980s that led to the creation of VHDL. Its first version was VHDL 87, later upgraded to the so called

VHDL 93.vHDL was the original and first hardware description language to be standardized by the

institute of electrical and Electronics Engineers through the IEEE 1076 standard. An additional

standard, the IEEE 1164, was later added to introduce a multi-valued logic system.

VHDL is intended for circuit synthesis as well as circuit simulation. However, though VHDL

is, fully simulateable, not all constructs are synthesizable.

A fundamental motivation to use VHDL (or its competitor, verilog) is that VHDL is a

standard, technology/vendor independent language, and is therefore portable and reusable. The two

main immediate applications of VHDL are in the field of programmable Logic Devices (including

CPLDs and FPGAs) and in the field of ASICs (Application Specific Integrated Circuits). Once the

VHDL code has been written, it can be used either to implement the circuit in a programmable device

(from Atera, Xilinx, Atmel etc) or can be submitted to a foundry for fabrication of an ASIC chip.

Currently, many complex commercial chips (microcontrollers, for example) are designed using such

an approach.

A final note regarding VHDL is that, contrary to regular computer programs which are

sequential, its statements are inherently concurrent parallel 0. for that reason, VHDL is usually

referred to as code rather than program. In VHDL, only statements placed inside a PROCESS,

FUNCTION, or PROCEDURE are executed sequentially.

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DESIGN FLOW

We start the design by writing the VHDL code, which is saved in a file with the extension

.vhd and the same name as its ENTITY's name. The first step in the synthesis process is compilation.

Compilation is the conversion of the high-level VHDL language, which describes the circuit at the

register Transfer level (RTL) into a netlist at the gate.

VHDL is a hardware description language that can be used to model a digital system. The

digital system can be as simple as a logic gate or as complex as a complete electronic system. A

hardware abstraction of this digital system is called an entity. This model specifies the external view

of the device and one or more internal views. The internal view of the device specifies the

functionality or structure, while the external view specifies the interface of the device through which

it communicates with the other models in its environment. Figure I.I shows the hardware device and

the corresponding software model. Figure 1.2 shows the VHDL view of a hardware device that has

multiple device models, with each device model representing one entity.

Netlist

(gate level)

(Gate

level) Optimized Netlist

(gatelevel)

Computation

Optimization

Simulation

Stimulation Physical Device

VHDL

enter (RTD Level)

( RTD level)

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To describe an entity, VHDL provides five different types of primary constructs, called"

design units. They are

1. Entity declaration

2. Architecture body

3. Configuration declaration

4. Package declaration

5. Package body

Entity declaration

An entity is modelled using an entity declaration and at least one architecture body. The entity

declaration describes the external view of the entity, for example, the input and output signal names.

The entity' declaration specifies the name of the entity being modelled and lists the set of interface

ports. Ports are signals through which the entity communicates with the other models in its external

environment.

Architecture body

The architecture body contains the internal description of the entity, for example, as a set of

interconnected components that represents the structure of the entity, or as a set of concurrent or

sequential statements that represents the behavior of the entity. Each style of representation can be

specified in a different architecture body or mixed within a single architecture body. The internal

details of an entity are specified by an architecture body using any of the following modeling styles:

1. As a set of interconnected components (to represent structure),

2. As a set of concurrent assignment statements (to represent dataflow),

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3. As a set of sequential assignment statements (to represent behavior),

4. Any combination of the above three.

The structural style of modelling describes only an interconnection of components without

implying any behaviour of the components themselves, nor of the entity that they collectively

represent. In data flow style of modelling , the flow of data through the entity is expressed primarily

using concurrent signal assignment statements. The structure of the entity is not explicitly specified in

this modelling style, but it can be implicitly deduced. The behavioural style of modelling specifies the

behaviour of an entity as a set of statements that are executed sequentially in the specified order. This

set of sequential statements, that are specified inside a process statement, do not explicitly specify the

structure of the entity but merely specifies its functionality. A process statement is a concurrent

statement that can appear within an architecture body. It is possible to mix the three modelling styles

that we have seen so far in a single architecture body. That is, within an architecture body, we could

use component instantiation statements (that represent structure), concurrent signal assignment

statements (that represent dataflow), and process statements (that represent behaviour).

Configuration declaration

A configuration declaration is used to create a configuration for an entity. It specifies the

binding of one architecture body from the many architecture bodies that may be associated with the

entity. It may also specify the bindings of components used in the selected architecture body to other

entities. An entity may have any number of different configurations.

Package declaration

A package declaration encapsulates a set of related declarations such as type declarations,

subtype declarations, and subprogram declarations that can be shared across two or more design units.

Package body

A package body is primarily used to store the definitions of functions and procedures that

were declared in the corresponding package declaration, and also the complete constant declarations

for any deferred constants that appear in the package declaration. Therefore, a package body is always

associated with a package declaration; furthermore, a package declaration can have at most one

package body associated with it. Contrast this with an architecture body and an entity declaration

where multiple architecture bodies may be associated with a single entity declaration.

Library declaration

In Library, the various definition group to use in VHDL is declared.

Generally, the declaration of the standard logic which is prescribed in the IEEE is specified.

There is various declaration but it seems OK if declaring the following 2 lines.

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library ieee;

use ieee.std_logic_1164.all;

The above declares all definitions which are prescribed in STD_LOGIC_1164 of the IEEE.

When using Xilinx Project Navigator, these declare statements are already written when making a

source file newly. When you design the more complicated circuit, more declaration must be done.

When using an arithmetic operation in the processing STD_LOGIC_UNSIGNED must be

specified.

use ieee.std_logic_unsigned.all;

Entity statements

In Entity, a port declaration, an attribute declaration, a generic declaration and so on are

written. When Entity ends, it writes END. The entity name behind end can be omitted.

Below, I will explain a port declaration and an attribute declaration.

Entity and architecture declaration

entity entity-name is

port (signal-names : mode signal-type;

signal-names : mode signal-type;

…

signal-names : mode signal-type);

end entity-names;

architecture architecture-name of entity-name is

type declarations

signal declarations

constant declarations

function definitions

procedure definitions

component declarations

begin

concurrent-statement

…

concurrent-statement

end architecture-name;

RESULT

The VHDL programming is familiarized.

Systems Lab


DATE: 30-03-2011

EXPERIMENT NO: 13

LOGIC GATES

AIM

Write a VHDL program for generating waveforms of various logic gates.

THEORY

AND gate

An AND gate may have two or more inputs but only one output. The output of an AND gate

is 1 if and only if all the inputs are 1. Its logical equation is given by

C = A . B

Truth Table

OR gate

An OR gate may have two or more inputs but only one output. The output of an OR gate is

one if and only if one or more inputs are 1. Its logical equation is given by

C = A + B

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Truth Table

NOT gate

A NOT gate, also called an inverter, has only one input and only one output. The

output of a NOT gate assumes the logic 1 state when its input is in logic 0 state and assumes the logic

0 state when its input is in logic 1 state.

Truth Table

NAND gate

NAND gate is a combination of an AND gate and a NOT gate. The output is a logic 0 level,

only when each of the inputs assumes logic 1 level. For any other combination of inputs, the output is

logic 1 level.

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Truth Table

NOR gate

NOR gate is a combination of an OR gate and a NOT gate. The output is logic 1

level, only when each of its inputs assumes a logic 0 level. For any other combination of inputs, the

output is logic 0 level.

Truth Table

XOR gate

The XOR operation is widely used in digital circuits. The output of XOR gate is logic 0 when

both the inputs are same and the output is logic 1 when the inputs are not same.

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Truth Table

XNOR gate

An XNOR gate is a combination of an XOR gate and a NOT gate. The XNOR gate is a two

input, one output logic, whose output assumes a 1 state only when both the inputs assumes a logic 0

state and the other a 1 state.

Truth Table

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PROCEDURE

1. Create a new work space.

2. Create new file

3. Choose vhdl source

4. Enter name of source file and name of entity

5. Enter inputs and outputs.

6. Enter the program

7. Compile

8. Initialize simulation

9. Add simulators and generate waveform

10. End simulation

PROGRAM

library IEEE;

use IEEE.STD_LOGIC_1164.all;

entityentity_logicgate is

port(

a : in STD_LOGIC;

b : in STD_LOGIC;

c_or : out STD_LOGIC;

c_and : out STD_LOGIC;

c_nor : out STD_LOGIC;

c_nand : out STD_LOGIC;

c_xor : out STD_LOGIC;

c_not : out STD_LOGIC

);

endentity_logicgate;

architecturearch_logicgate of entity_logicgate is

begin

c_or <= a or b;

c_and <= a and b;

c_nor <= a nor b;

c_nand <= a nand b;

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c_xor <= a xor b;

c_not <= not b;

endarch_logicgate;

WAVEFORMS:

RESULT

The VHDL programs for logic gates are executed correctly and output waveforms are plotted.

Systems Lab


DATE: 30-03-2011

EXPERIMENT NO: 14

HALF ADDER AND FULL ADDER

AIM

To write a program for half adder and full adder in VHDL.

THEORY

1. HALF ADDER

A half adder is a circuit that adds two binary digits, giving a sum bit and a carry bit.

Sum, S=A ̅+B ̅=A B

Carry, C =AB

Truth table

OUTPUTS

A B SUM CARRY

0 0 0 0

0 1 1 0

1 0 1 0

1 1 1 1

2. FULL ADDER

A full adder is an arithmetic circuit that adds two bits and outputs a sum bit and a carry bit.

When adding two binary numbers each having two or more bits the LSBs can be added using a half

adder. The carry resulted from the addition of the LSBs is carried to the next significant column and

added to the two bits in that column. The full adder adds the bits A and B and the carry from the

previous column called carry_in, Cin and outputs the sum bit S and the carry bit called the carry_out

Cout.

Sum=A B C

Carry, Cout=AB+ACin+BCin

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Truth table

OUTPUTS

A B C SUM CARRY

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

PROCEDURE

1. Create new workspace

2. Create new file

3. Choose vhdl source


5. Enter inputs and outputs


7. Compile



10. End simulation

PROGRAM

library IEEE;


entity adder is

port(

a : in STD_LOGIC;

b : in STD_LOGIC;

c : in STD_LOGIC;

sum_half : out STD_LOGIC;

sum_full : out STD_LOGIC;

c_half : out STD_LOGIC;

c_full : out STD_LOGIC

);

Systems Lab


end adder;

architecture behaviour of adder is

begin

sum_half <= a xor b;

c_half <= a and b;

sum_full <= a xor b xor c ;

c_full <= (a and b) or ( a and c) or (b and c);

end behaviour;

WAVEFORMS

RESULT

Half adder and full adder are implemented using VHDL and output waveforms are plotted.

Systems Lab


DATE: 30-03-2011

EXPERIMENT NO.: 15

FLIP FLOPS

AIM

To write VHDL program for implementing the following flip flops.

D flip flop

T flip flop

JK flip flop

SR flip flop

THEORY

Flipflop, which is made up of an assembly of logic gates, is a memory element. A logic gate

by itself has no storage capability, but when connected together in ways permits the information to be

stored. A flipflop known as bistable multivibrator has two states. It can remain in either of the twp

states indefinitely. Its state can be changed by applying the proper triggering signal.

SR Flipflop

S-R flipflop is the simplest type of flip flop. It has two outputs labelled Q and ̅ and

two inputs labelled S and R. The state of the flipflop corresponds to the level of Q (high or low, 1 or

0) and ̅ the compliment of the state. Q0 represents the state of the flipflop before applying the inputs.

The name S-R or SET- RESET is derived from the names of its inputs.

Truth Table

INPUTS OUTPUT COMMENTS

S R Q

0 0 Q0 No change

0 1 0 RESET

1 0 1 SET

1 1 U Invalid

JK Flipflop

JK flipflop is very versatile and most widely used. The functioning of JK is similar to that of

SR flipflop, except that it has no invalid state as in SR When J=K=1, the flipflop toggles.

Systems Lab


Truth table

INPUTS OUTPUT COMMENTS

J K Q

0 0 Q0 No Change

0 1 0 RESET

1 0 1 SET

1 1 ̅0 Toggle

D Flipflop

The edge triggered D flip-flop has only one input terminal. The D flipflop may be

obtained from SR by just putting one inverter between the S and R terminals. The single input to the

D flipflop is called the data input. The level present at D will be stored in the flipflop at the instant the

positive going transition occurs.

Truth Table

INPUT OUTPUT COMMENTS

D Q

0 0 RESET

1 1 SET

T Flipflop

A T flipflop has a single control input, labelled T for toggle. When T is HIGH, the

flip-flop toggles on every new clock pulse. When T is LOW, the flip flop remains in whatever state it

was before. T flipflop is obtained by connecting the J and K inputs to JK flipflop together.

Truth Table

INPUT OUTPUT COMMENTS

T Q

0 Q0 No Change

1 ̅0 Toggle

Systems Lab


PROCEDURE

1. Create new workspace

2. Create new file

3. Choose VHDL source


5. Enter inputs and outputs


7. Compile



10. End simulation

PROGRAM

library IEEE;


entityff is

port(

d : in STD_LOGIC;

t : in STD_LOGIC;

s : in STD_LOGIC;

r : in STD_LOGIC;

j : in STD_LOGIC;

k : in STD_LOGIC;

clk : in STD_LOGIC;

rst : in STD_LOGIC;

dout : out STD_LOGIC;

tout : inout STD_LOGIC;

srout : inout STD_LOGIC;

jkout : inout STD_LOGIC

);

endff;

Systems Lab


architecture behaviour of ff is

begin

process(clk,d,rst)

begin

caserst is

when '0' =>

if (clk 'event and clk = '1') then

dout<= d;

end if;

when '1' => dout <= '0';

when others => null;

end case;

end process;

-- T FLIPFLOP

process (t, clk)

begin

caserst is

when '0' =>

if(clk' event and clk = '1' and t = '0')then

tout<= tout;

elsif ( clk' event and clk = '1' and t = '1') then

tout<= not tout;

end if;

when '1' => tout <= '0';


end case;

end process;

-- J K FLIPFLOP

Process (j, k, clk,rst)

begin

caserst is

when '0' =>

if(clk' event and clk = '1' and j = '0' and k = '0') then

jkout<= jkout;

elsif(clk' event and clk = '1' and j = '0' and k = '1') then

Systems Lab


jkout<= '0';


jkout<= '1';


jkout<= not jkout;

end if;

when '1' => jkout <= '0';


end case;

end process;

-- S R FLIPFLOP

process (s, r, clk,rst)

begin

caserst is

when '0' =>

if (clk' event and clk = '1' and s = '0' and r = '0') then

srout<= srout;

elsif (clk' event and clk = '1' and s = '0' and r = '1') then

srout<= '0';

elsif (clk' event and clk = '1' and s = '1' and r = '0') then

srout<= '1';

elsif(clk' event and clk = '1' and s = '1' and r = '1') then

srout<= '1';

end if;

when '1' => srout <= '0';


end case;

end process;

end behaviour;

Systems Lab


WAVEFORMS

RESULT

All the flipflops are implemented using VHDL and output waveforms are plotted.

Systems Lab


DATE: 30-03-2011

EXPERIMENT NO: 16

COMPARATOR

AIM

To design a comparator using VHDL

THEORY

A comparator is a logic circuit used to compare the magnitude of two binary numbers.

Depending on the design, it may either simply provide an output that is active when the two numbers

are equal or additionally provide outputs that signify which of the numbers is greater when equality

does not hold. Two binary numbers are equal, if and only if all their corresponding bits coincide. For

example, two 4-bit numbers, A3A2A1A0 and B3B2B1B0 are equal, if and only if A3= B3, A2= B2,

A1= B1 and A0= B0.

PROCEDURE

1. Create new workspace.

2. Create new file.

3. Choose vhdl source.

4. Enter name of source file and name of entity.

5. Enter inputs and outputs.

6. Enter the program.

7. Compile.

8. Initialize simulation.

9. Add simulators and generate waveform.

10. End simulation.

PROGRAM

library IEEE;


entity comparator is

port(

a : in STD_LOGIC_VECTOR(2 downto 0);

b : in STD_LOGIC_VECTOR(2 downto 0);

Systems Lab


c : out STD_LOGIC_VECTOR(1 downto 0)

);

end comparator;

architecture behaviour of comparator is

begin

process(a,b)

begin

if(a>b)then

c<="00";

elsif(a<b)then

c<="01";

elsif(a=b)then

c<="11";

end if;

end process;

end behaviour;

WAVEFORM

RESULT

The VHDL program for comparator is executed correctly and the waveform obtained while

simulation is plotted.

signals and systems using matlab

Documents

maximum frequency

axes graphics

discrete fourier

band limited

label appears

unit impulse

continuous

discrete time