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NIRMAL KUMAR. P. Reg. No: 3135127

AIM:To Write a MATLAB Code for sampling & aliasing.

ALGORITHM: Get the sinusoidal wave as input. Get three different frequencies as per sampling theorem. Get three different output based on that three frequencies.

THEORY:

When a function is evaluated by numerical procedures, it is always

necessary to sample the function in some manner, because digital computers

cannot deal with analog, continuous functions

Sampling: If delta is the time interval between consecutive samples, then the

sampled time data can be represented as

Examples where sampled time domain data is used in engineering include

simple and complex vibration analysis of machinery, as well as measurements of

other variables such as boiler pressures, temperatures, flow rates and turbine

speeds and many other machine parameters. It is also used in areas where

computers and microcontrollers are used to automate processes and react to

input data from the processes.

Aliasing and the Nyquist Theorem

One would expect that if the signal has significant variation then Ts must be

small enough to provide an accurate approximation of the signalx(t). Significant

signal variation usually implies that high frequency components are present in the

signal. It could therefore be inferred that the higher the frequency of the

components present in the signal, the higher the sampling rate should be. If the

sampling rate is not high enough to sample the signal correctly then a

phenomenon called aliasing occurs.

EX.NO:1SAMPLING & ALIASING

06/ 02/ 12


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NIRMAL KUMAR. P. Reg. No: 3135127The term aliasing refers to the distortion that occurs when a continuous

time signal has frequencies larger than half of the sampling rate. The process of

aliasing describes the phenomenon in which components of the signal at high

frequencies are mistaken for components at lower frequencies.

The Nyquist Sampling Theorem states that to avoid aliasing occurring in the

sampling of a signal the sampling rate should be greater than or equal to twice

the highest frequency present in the signal. This is referred to as the Nyquistsampling rate

SAMPLING & ALIASING: MATLAB PROGRAM

clc; clear all; close all;

fm= 1/20; % frequency

t= -20 : 20; % time axis from -20,-19,-18.....0, 1,.....19,20,

g= sin ( 2* pi * fm * t ); % continuous sine wave

subplot( 2, 2, 1), plot ( t, g); % t- x-axis, g- y axis

title(' Continuous Sine wave');

f1= 1.3 * fm;

f2= 2 * fm;

f3= 5 * fm;

% for freq < 2* fm

t1= -4 : 4; % time scale x- axisg1= sin( 2* pi * fm/f1 * t1 ); % discrete signal - y axis

subplot( 2, 2, 2), stem ( t1, g1);

title(' when freq < 2 * fm');

% for freq = 2* fm

t2= -5 : 5; % time scale x- axis

g2= sin( 2* pi * fm/f2 * t2 ); % discrete signal - y axis

subplot( 2, 2, 3), stem ( t2, g2);

title(' when freq = 2 * fm');

% for freq > 2* fm

t3= -20 : 20; % time scale x- axis

g3= sin( 2* pi * fm/f3 * t3 ); % discrete signal - y axis

subplot( 2, 2, 4), stem ( t3, g3);

title(' when freq > 2 * fm');


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NIRMAL KUMAR. P. Reg. No: 3135127OUTPUT FOR SAMPLING & ALIASING

RESULT:

Thus the MATLAB code for sampling & aliasing was written and executed.


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AIM:To Write a MATLAB Code for up sampling & down sampling.

ALGORITHM:

UP SAMPLING:

Get the length of input sequence. Get the up sampling factor. Generate the output sequence.

DOWN SAMPLING:

Get the length of input sequence. Get the down sampling factor. Generate the output sequence.

THEORY:

Down sampling:

The process of reducing a sampling rate by an integer factor is referred to as

down sampling of a data sequence. We also refer to down sampling as

''decimation'' The term ''decimation'' used for the down sampling process has

been accepted and used in many textbooks and fields. To down sample a data

sequence x(n) by an integer factor of M, we use the following notation:

y(m) = x(mM), (12.1)Where y (m) is the down sampled sequence, obtained by taking a sample from

the data sequence x (n) for every M samples (discarding M 1 samples for every

M samples).

As an example, if the original sequence with a sampling period T = 0.1 second

(sampling rate = 10 samples per sec) is given by

x(n):8 7 4 8 9 6 4 2257764

EX.NO:2

UP SAMPLING & DOWN SAMPLING09/ 02/ 12


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NIRMAL KUMAR. P. Reg. No: 3135127and we down sample the data sequence by a factor of 3, we obtain the down

sampled sequence as

y(m):8 8 4 5 6 ,

with the resultant sampling period T = 3 0.1 = 0.3 second (the sampling rate now

is 3.33 samples per second). Although the example is straightforward, there is a

requirement to avoid aliasing noise.

From the Nyquist sampling theorem, it is known that aliasing can occur in the

down sampled signal due to the reduced sampling rate. After down sampling by afactor of M, the new sampling period becomes MT, and therefore the new

sampling frequency is

fsM = 1/(MT) = fs /M, (12.2)

Where fs is the original sampling rate.

Hence, the folding frequency after down sampling becomes

fsM/2 = fs/(2M). (12.3)

This tells us that after down sampling by a factor of M, the new folding frequency

will be decreased M times. If the signal to be down sampled has frequency

components larger than the new folding frequency, f > fs/(2M), aliasing noise willbe introduced into the down sampled data.

Up sampling

Increasing a sampling rate is a process of up sampling by an integer factor of L.

This process is described as follows:

y(m) = { x(m/L) m=nL,

0 otherwise

where n = 0, 1, 2, , x(n) is the sequence to be up sampled by a factor of L, and

y(m) is the up sampled sequence. As an example, suppose that the data sequence

is given as follows:

x(n):8 8 4 5 6

After up sampling the data sequence x(n) by a factor of 3 (adding L 1 zeros for

each sample), we have the up sampled data sequence w(m) as:

w(m): 8 0 0 8 0 0 4 0 0 5 0 0 6 0 0


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NIRMAL KUMAR. P. Reg. No: 3135127UP SAMPLING & DOWN SAMPLING

clc; clear all; close all;

% UP SAMPLING

N= 10; % no of samples

t= 1 : N; % time axis from 0 - 9,

g= sin( 2* pi * t/10 ) + sin( 2* pi * t/5 ) ; % continuous sine wave

subplot( 2, 2, 1), stem ( t, g); % n- x axis, g- y axis

xlabel('time'),ylabel('Amplitude');

title(' INPUT SEQUENCE ');

L= 3;

K= L* N; % 30 samples.g1= zeros ( 1, K); % a zero row- matrix- 30 samples

t1= 1: K;

g1( 1 : L : K ) = g ; % up sampling. saving ' g' into g1-matrix

subplot( 2, 2, 2), stem ( t1, g1);


title(' UP SAMPLING OF THE SEQUENCE');

% DOWN SAMPLING

N= 20; % no of samples

t= 1 : N; % time axis from 0 - 19,

g= sin( 2* pi * t/10 ) + sin( 2* pi * t/5 ) ; % input sequence

subplot( 2, 2, 3), stem ( t, g); % n- xaxis, g- y axis


title(' INPUT SEQUENCE ');

t2= 1 : N/2 ; % half the time scale x- axis

g2= g( 1: 2: N ) ; % sampling the signal 'y' 2 samples apart

subplot( 2, 2, 4), stem ( t2, g2);


title(' DOWN SAMPLED SEQUENCE');


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OUTPUT

RESULT

Thus the MATLAB code for UP sampling & DOWN sampling was written and

executed.


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AIM:To Write a MATLAB Code for power spectral density.

ALGORITHM: Get the frequencies of two sinusoidal waves Get sampling frequency Get length of sequence to be considered Get the two FFT lengths for comparing the corresponding power spectral

densities

THEORY:

The power spectral density (PSD) is intended for continuous spectra. The

integral of the PSD over a given frequency band computes the average power in

the signal over that frequency band. In contrast to the mean-squared spectrum,

the peaks in these spectra do not reflect the power at a given frequency.

A one-sided PSD contains the total power of the signal in the frequency intervalfrom DC to half of the Nyquist rate. A two-sided PSD contains the total power in

the frequency interval from DC to the Nyquist rate.

Power spectral density (PSD), which describes how the power of a signal or time

series is distributed with frequency. Here power can be the actual physical power,

or more often, for convenience with abstract signals, can be defined as the

squared value of the signal, that is, as the actual power dissipated in a load if the

signal were a voltage applied to it. This instantaneous power (the mean or

expected value of which is the average power) is then given by,

P(t) = s(t)2

for a signal s(t).

Since a signal with nonzero average power is not square integrable, the Fourier

transforms do not exist in this case. Fortunately, the WienerKhinchin theorem

provides a simple alternative. The PSD is the Fourier transform of the

autocorrelation function, R(), of the signal if the signal can be treated as a wide-

sense stationary random process

EX.NO:3

POWER SPECTRAL DENSITIES13/ 02/ 12


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NIRMAL KUMAR. P. Reg. No: 3135127OUTPUT FOR POWER SPECTRAL DENSITY

RESULT

Thus the MATLAB code for power spectral density was written and

executed.

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