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EXPRIMENT NO. 1(A)

AIM :: TO GENERATE BASIC CONTINEOUS TIME SIGNALS ( SINE, UNIT STEP AND

UNIT RAMP).

THEORY ::

A continuous signal or a continuous-time signal is a varying quantity (a signal)

whose domain, which is often time, is a continuum (e.g., a connected interval of

the reals). That is, the function's domain is an uncountable set. The function itself

need not be continuous. To contrast, a discrete time signal has

a countable domain, like the natural numbers.

A signal of continuous amplitude and time is known as a continuous time signal or

an analog signal. This (a signal) will have some value at every instant of time. The

electrical signals derived in proportion with the physical quantities such as

temperature, pressure, sound etc. are generally continuous signals. The other

examples of continuous signals are sine wave, cosine wave, triangular wave etc.

Some of the continuous signals.

MATLAB CODE ::

clc;

clear all;

close all;

t=0:0.01:5;

f=2;

y=sin(2*pi*f*t);

subplot(3,1,1);

plot(t,y);

title('sin wave');

ylabel('y(t)');

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xlabel('t');

z= ones(1,501);

subplot(3,1,2);

plot(t,z);

title('unit step');

ylabel('z(t)');

xlabel('t');

x= t;

subplot(3,1,3);

plot(t,x);

title('ramp signal');

ylabel('x(t)');

xlabel('t');

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OUTPUT WAVEFORM ::

RESULT :: the program for BASIC SIGNAL GENERATION is written using MATLAB

and verified.

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EXPRIMENT NO. 1(B)

AIM :: TO GENERATE BASIC DISCRETE TIME SIGNALS ( SINE, UNIT STEP AND

UNIT RAMP).

THEORY ::

A discrete signal or discrete-time signal is a time series consisting of

a sequence of qualities. In other words, it is a type series that is a function over

a domain of discrete integral.

Unlike a continuous-time signal, a discrete-time signal is a function of a

continuous argument; however, it may have been obtained by sampling from a

continuous-time signal, and then each value in the sequence is called a sample.

When a discrete-time signal obtained by sampling a sequence corresponding to

uniformly spaced times, it has an associated sampling rate; the sampling rate is

not apparent in the data sequence, and so needs to be associated as a separate

data item.

MATLAB CODE ::

clc;clear all;

close all;

t=0:0.1:5;

f=2;

y=sin(2*pi*f*t);

subplot(3,1,1);

stem(t,y);

title('sin wave');

ylabel('y(n)');

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xlabel('n');

z= ones(1,51);

subplot(3,1,2);

stem(t,z);

title('unit step');

ylabel('z(n)');

xlabel('n');

x= t;

subplot(3,1,3);

stem(t,x);

title('ramp signal');

ylabel('x(n)');

xlabel('n');

OUTPUT WAVEFORM ::

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stem(n,y);

xlabel=('Time');

ylabel=('Amplitude');

title=('Impulse response');

grid on;

SAMPLE INPUT-

Enter the required legth of impulse response N=40

enter the coefficients of x(n),b=[0.8 -0.44 0.36 0.02]

enter the coefficients of y(n),a=[1 0.7 -0.45 -0.6]

FIGURE-

RESULT-The program for impulse response of an LTI system is written using

Matlab and verified.

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EXPRIMENT NO. 3

LINEAR CONVOLUTION

THEORY:

MATLAB CODE-

>> clear all;

>> close all;

>> a=input('enter the starting point of x[n]=');

enter the statring point of x[n]=0

>> b=input('enter the starting point of h[n]=');

enter the starting point of h[n]=-1

>> x=input('enter the co-efficients of x[n]=');

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enter the co-efficients of x[n]=[1 2 3]

>> h=input('enter the co-efficients of h[n]=');

enter the co-efficients of h[n]=[1 1 1]

>> y=conv(x,h);

>> subplot(3,1,1);

>> p=a:(a+length(x)-1);

>> stem(p,x);

>> grid on;

>> xlabel('Time');

>> ylabel('Amplitude');

>> title('INPUT x(n)');

>> subplot(3,1,2);

>> q=b:(b+length(h)-1);

>> stem(q,h);

>> grid on;

>> xlabel('Time');


>> title('IMPULSE RESPONSE h(n)');

>> subplot(3,1,3);

>> n=a+b:length(y)+a+b-1;

>> stem(n,y);

>> grid on;

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>> disp(y);

1 3 6 5 3

>> xlabel('Time');


>> title('linear convolution');

SAMPLE INPUT-

Enter the starting point of x(n)=0

Enter the starting point of h(n)=-1

Enter the co-efficients of x(n)=[1 2 3]

Enter the co-efficients of h[n]=[1 1 1]

FIGURE

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RESULT-

The program for linear convolution is written and verified.

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EXPRIMENT NO. 4

CIRCULAR CONVOLUTION

THEORY:

USER DEFINED FUNCTION

function y=crconc(x,h);

n1=length(x);

n2=length(h);

N=max(n1,n2);

x=[x,zeros(1,N-n1)];

h=[h,zeros(1,N-n2)];

for n=0:N-1;

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y(n+1)=0;

for k= 0:N-1;

j=mod(n-k,N);

y(n+1)=y(n+1)+x(k+1)*h(j+1);

end

end

MATLAB CODE-

clc;

clear all;

close all;

x=input('enter the co-efficients of x1[n]=');

h=input('enter the co-efficients of x2[n]=');

y=crconc(x,h);

subplot(3,1,1);

n=0:(length(x)-1)

stem(n,x);

grid on;

xlabel=('Time');

ylabel=('amplitude');

title=('x1[n]');

subplot(3,1,2);

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n=0:(length(h)-1)

stem(n,h);

grid on;

xlabel=('Time');


title=('x2[n]');

subplot(3,1,3);

n=0:(length(y)-1)

stem(n,y);

grid on;

disp(y);

xlabel=('Time');


title=('OUTPUT x3[n]');

SAMPLE INPUT

enter the co-efficients of x1[n]=[1 2 3]

enter the co-efficients of x2[n]=[1 2]

n =

0 1 2

n =

0 1

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1n =

0 1 2

7 4 7

FIGURE-

RESULT- Program for circular convolution is written using Matlab and verified.

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EXPRIMENT NO. 5

AIM :: TO WRITE A PROGRAME TO COMPUTE DISCRETE FOURIER TRANSFORM.

THEORY ::

In mathematics, the discrete Fourier transform (DFT) converts a finite list of

equally-spaced samples of a function into the list of coefficients of a finite

combination of complex sinusoids, ordered by their frequencies, that has those

same sample values. It can be said to convert the sampled function from its

original domain (oftentime or position along a line) to the frequency domain.

The input samples are complex numbers (in practice, usually real numbers), and

the output coefficients are complex too. The frequencies of the output sinusoidsare integer multiples of a fundamental frequency, whose corresponding period is

the length of the sampling interval. The combination of sinusoids obtained

through the DFT is therefore periodic with that same period. The DFT differs from

the discrete-time Fourier transform (DTFT) in that its input and output sequences

are both finite; it is therefore said to be the Fourier analysis of finite-domain (or

periodic) discrete-time functions.

The DFT is the most important discrete transform, used to perform Fourier

analysis in many practical applications. In digital signal processing, the function is

any quantity or signal that varies over time, such as the pressure of a sound wave,

a radio signal, or daily temperature readings, sampled over a finite time interval

(often defined by a window function). In image processing, the samples can be

the values of pixels along a row or column of a raster image. The DFT is also used

to efficiently solve partial differential equations, and to perform other operations

such as convolutions or multiplying large integers.

Since it deals with a finite amount of data, it can be implemented

in computers by numerical algorithms or even dedicated hardware. These

implementations usually employ efficient fast Fourier transform (FFT)algorithms; so much so that the terms "FFT" and "DFT" are often used

interchangeably. The terminology is further blurred by the (now rare)

synonym finite Fourier transform for the DFT, which apparently predates the term

"fast Fourier transform" but has the same initialism.

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MATLAB CODE ::

clc;

clear all;

close all;

N=input('Enter the value of N');x=input('Enter the input sequence X(n):');

t=0:N-1;

subplot(2,1,1);

stem(t,x);

xlabel('TIME');

ylabel('AMPLITUDE');

title('INPUT SIGNAL');

grid on;

y=fft(x,N)subplot(2,1,2);

stem(t,y);

xlabel('TIME');

ylabel('AMPLITUDE');

title('OUTPUT SIGNAL');

grid on;

OUTPUT ::

Enter the value of N5Enter the input sequence X(n):[1 3 5 7 9]

y =

25.0000 -5.0000 + 6.8819i -5.0000 + 1.6246i -5.0000 - 1.6246i -5.0000 - 6.8819i

WAVEFORM ::

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RESULT :: the program for DISCRETE FOURIER TRANSFORM is written using

MATLAB and verified.

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EXPRIMENT NO. 6

INVERSE DISCRETE FOURIER TRANSFORM

THEORY-

Given a sequence ofN samplesf(n), indexed by n = 0..N-1, the Discrete Fourier

Transform (DFT) is defined as F(k), where k=0..N-1:

F(k) are often called the 'Fourier Coefficients' or 'Harmonics'.

The sequencef(n) can be calculated from F(k) using the Inverse Discrete Fourier

Transform (IDFT):

In general, bothf(n) and F(k) are complex.

MATLAB CODE-

clc;

clear all;

close all;

N=input('enter the value of N=');

y=input('enter the sequence y[n]=');

t=0:N-1;

subplot(2,2,1);

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stem(t,y);

xlabel('TIME');

ylabel('APMLITUDE');

title('input signal');

grid on;

x=ifft(y,N);

subplot(2,1,2);

stem(t,x);

xlabel('TIME');

ylabel('APMLITUDE');

title('input signal');

grid on;

SAMPLE INPUT-

enter the value of N=4

enter the sequence y[n]=[10 -2+2i 2 -2-2i]

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FIGURE-

RESULT-

The program for IDFT is written using Matlab and verified.

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EXPERIMENT NO.7

AIM :: TO FIND Z-TRANSFORM OF GIVEN FUNCTION.

THEORY :: The z-transform is the discrete-time counterpart of the Laplace

transform. The Z-transform, like many integral transforms, can be defined as

either a one-sidedor two-sidedtransform.

Bilateral Z-transform

The bilateralor two-sidedZ-transform of a discrete-time signalx[n] is the formal

power seriesX(z) defined as

where n is an integer and z is, in general, a complex number:

whereA is the magnitude ofz,jis the imaginary unit, and is the complex

argument(also referred to as angle orphase) in radians.

Unilateral Z-transform

Alternatively, in cases wherex[n] is defined only for n 0, the single-sidedor

unilateralZ-transform is defined as

In signal processing, this definition can be used to evaluate the Z-transform of the

unit impulse response of a discrete-time causal system.

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MATLAB CODE ::

k = input('Number of frequency points = ');

num = input('Numerator coefficients = ');

den = input('Denominator coefficients = ');

w = 0:pi/(k-1):pi;

h = freqz(num, den, w);

plot(w/pi,real(h));grid

title('Real part')

xlabel('\omega/\pi');

ylabel('Amplitude')

subplot(2,2,2)

plot(w/pi,imag(h));grid

title('Imaginary part')


ylabel('Amplitude')

subplot(2,2,3)

plot(w/pi,abs(h));grid

title('Magnitude Spectrum')


ylabel('Magnitude')

subplot(2,2,4)

plot(w/pi,angle(h));grid

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title('Phase Spectrum')


ylabel('Phase, radians')

SAMPLE INPUT-

enter the value of N=4

enter the sequence y[n]=[10 -2+2i 2 -2-2i]

Number of frequency points = 4

Numerator coefficients = [1 2]

Denominator coefficients = [1 5 6]

FIGURE-

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RESULT-

The ztransform is plotted and verified using Matlab.

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EXPERIMENT NO.8

AIM :: TO GENERATE POLE-ZERO PLOT OF GIVEN TRANSFER FUNCTION.

THEORY::

In DSP, a polezero 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 phaseA pole-zero plot shows the location in the complex plane of the poles and zeros of

the transfer function of a dynamic system, such as a controller, compensator,

sensor, equalizer, filter, or communications channel. By convention, the poles of

the system are indicated in the plot by an X while the zeroes are indicated by a

circle or O.

A pole-zero plot can represent either a continuous-time (CT) or a discrete-time(DT) system. For a CT system, the plane in which the poles and zeros appear is

the s plane of the Laplace transform. In this context, the parameter s represents

the complex angular frequency, which is the domain of the CT transfer function.

For a DT system, the plane is the z plane, where z represents the domain of the Z-

transform.

MATLAB CODE-

clear all;

b=input('enter the numerator coefficients');

a=input('enter the denominator coefficients');
http://en.wikipedia.org/wiki/Rational_functionhttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/BIBO_stabilityhttp://en.wikipedia.org/wiki/Causal_systemhttp://en.wikipedia.org/wiki/Anticausal_systemhttp://en.wikipedia.org/wiki/Radius_of_convergencehttp://en.wikipedia.org/wiki/Minimum_phasehttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Dynamic_systemhttp://en.wikipedia.org/wiki/S_planehttp://en.wikipedia.org/wiki/Laplace_transformhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Z-transformhttp://en.wikipedia.org/wiki/Z-transformhttp://en.wikipedia.org/wiki/Z-transformhttp://en.wikipedia.org/wiki/Z-transformhttp://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Laplace_transformhttp://en.wikipedia.org/wiki/S_planehttp://en.wikipedia.org/wiki/Dynamic_systemhttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Minimum_phasehttp://en.wikipedia.org/wiki/Radius_of_convergencehttp://en.wikipedia.org/wiki/Anticausal_systemhttp://en.wikipedia.org/wiki/Causal_systemhttp://en.wikipedia.org/wiki/BIBO_stabilityhttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Rational_function

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zplane(b,a);

SAMPLE INPUT-

enter the numerator coefficients[1 2]

enter the denominator coefficients[1 5 6]

FIGURE-

RESULT-The pole zero plot is drawn using Matlab.

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EXPRIMENT NO. 9(A)

DIGITAL BUTTERWORTH LOW PASS FILTER

THEORY-

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

a frequency response as possible in the passband. It is also referred to as

a maximally flat magnitude filter. It was first described in 1930 by the

British engineer Stephen Butterworth.

Butterworth stated that:

"An ideal electrical filter should not only completely reject the unwanted

frequencies but should also have uniform sensitivity for the wanted frequencies".

Such an ideal filter cannot be achieved but Butterworth showed that successively

closer approximations were obtained with increasing numbers of filter elements

of the right values. At the time, filters generated substantial ripple in the

passband, and the choice of component values was highly interactive.

Butterworth showed that a low pass filter could be designed whose cutoff

frequency was normalized to 1 radian per second and whose frequency response

(gain) was

where is the angular frequency in radians per second and n is the number

ofpoles in the filterequal to the number of reactive elements in a passive

filter.

MATLAB CODE-

>>clc;

>> clear all;

>> close all;

>> rp=input('enter the passband attenuation:');
http://en.wikipedia.org/wiki/Low_pass_filterhttp://en.wikipedia.org/wiki/Gainhttp://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Pole_(complex_analysis)http://en.wikipedia.org/wiki/Pole_(complex_analysis)http://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Gainhttp://en.wikipedia.org/wiki/Low_pass_filter

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enter the passband attenuation:0.4

>> rs=input('enter the stopband attenuation:');

enter the stopband attenuation:30

>> wp=input('enter the passband frequency:');

enter the passband frequency:0.2*pi

>> ws=input('enter the stopband frequency:');

enter the stopband frequency:0.4*pi

>> [N,wn]=buttord(wp/pi,ws/pi,rp,rs);

>> [b,a]=butter(N,wn);

>> freqz(b,a);

SAMPLE INPUT

Enter the passband attenuation- 0.4

Enter the stopband attenuation- 30

Enter the passband frequency- 0.2*pi

Enter the stopband frequency- 0.4*pi

FIGURE-

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RESULT-

Magnitude and phase response of butterworth filter are verified.

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EXPRIMENT NO. 9(B)

BUTTERWORTH HIGH PASS FILTER

MATLAB CODE-

clc;

clear all;

close all;

rp=input('enter the passband attenuation:');

rs=input('enter the stopband attenuation:');

wp=input('enter the passband frequency:');

ws=input('enter the stopband frequency:');

[N,wn]=buttord(wp/pi,ws/pi,rp,rs);

[b,a]=butter(N,wn);

freqz(b,a);

SAMPLE INPUT-



enter the passband frequency:0.6*PI

enter the passband frequency:0.6*pi

enter the stopband frequency:0.2*pi

FIGURE

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RESULT-

Magnitude and phase response of butterworth high pass filter is verified.

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EXPRIMENT NO. 9(C)

BUTTERWORTH BAND STOP FILTER

MATLAB CODE-

clc;

clear all;

close all;






*b,a+=butter(N,wn,stop);

freqz(b,a);

SAMPLE INPUT-



enter the passband frequency:[0.1*pi,0.5*pi]

enter the stopband frequency:[0.2*pi,0.4*pi]

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FIGURE

RESULT-

Magnitude and phase response of butterworth band stop filter is verified.

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EXPRIMENT NO. 9(D)

BUTTERWORTH BAND PASS FILTER

MATLAB CODE-

clc;

clear all;

close all;






[b,a]=butter(N,wn);

freqz(b,a);

SAMPLE INPUT-

enter the passband attenuation:.2


enter the passband frequency:[0.2*pi,0.4*pi]

enter the stopband frequency:[0.1*pi,0.5*pi]

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FIGURE

RESULT-

Magnitude and phase response of butterworth band pass filter is verified.

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