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Effect of noise on output signal of a system

Linear System:

It is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case.

Application:

-automatic control theory

-signal processing

-telecommunications

A general deterministic system can be described by operator, H, that maps an input, x(t), as a function of t to an output, y(t), a type of black box description.

Linear systems satisfy the properties of superposition and scaling. Given two valid inputs and as well as their respective outputs and then a linear system must satisfy For any scalar values and .

A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience.

Linear Time-invariant System:

The system which is linear and time-invariant to an arbitrary input signal is known as Linear Time-invariant (LTI) system.

Application:

-NMR spectroscopy

-seismology

-circuits

-signal processing

-control theory, and other technical areas

These systems are also called linear translation-invariant system.

The defining properties of any LTI system are linearity and time invariance.

Ø Linearity means that the relationship between the input and the output of the system is a linear map: If input produces response and input produces response then the scaled and summed input produces the scaled and summed response where a1 and a2 are real scalars.

Ø Time invariance means that whether we apply an input to the system now or T seconds from now, the output will be identical except for a time delay of the T seconds. That is, if the output due to input x(t) is y(t), then the output due to input x(t − T) is y(t − T). Hence, the system is time invariant because the output does not depend on the particular time the input is applied.

For time-invariant systems this is the basis of the impulse response or the frequency response methods (see LTI system theory), which describe a general input function x(t) in terms of unit impulses or frequency components.

Typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case (especially in computer implementations).

Motivation:

Matlab Program for showing the distortion in output signal because of noise using different commands:

Ø Using "awgn" command: This command adds real type of noise in the form of Gaussian noise.

>> % this program shows the change in original output signal

>> % by adding real noise in the form of Gaussian noise and also

>> % by changing SNR

>> t = linspace(0,2*pi,50);

>> x = sin(t); % where x is the input signal

>> h = sin(t/2); % where h is the system

>> o = conv(x,h);% where o is the original output without noise

>> SNR=2; % where SNR is the signal to noise ratio

>> y = o + awgn(o,SNR); % where y is the output after adding Gaussian noise

>> subplot (221), plot(o)

>> title ('o/p without noise');

>> xlabel('t --->');

>> ylabel('o --->');

>> subplot (222), plot(y)

>> xlabel('t --->');

>> ylabel('y --->');

>> title ('o/p with noise where SNR=2');

>> SNR=0.5;

>> y = o + awgn(o,SNR); % where y is the output after adding Gaussian noise

>> subplot(223), plot(y)

>> xlabel('t --->');

>> ylabel('y --->');

>> title('o/p with noise for SNR=0.5');

>> SNR=0.1;

>> y = o + awgn(o,SNR); % where y is the output after adding Gaussian noise

>> subplot(224), plot(y)

>> xlabel('t --->');

>> ylabel('y --->');

>> title('o/p with noise where SNR=0.1');

Ø Using "wgn" command: This command adds white Gaussian noise in the signal.

>> t=0:6;

>> x = [1 -2 3 4]; % where x is the input signal

>> h = [2 1 -2 3]; % where h is the system

>> o = conv(x,h); % where o is the original output without noise

>> subplot(221), stem(n,o);

>> xlabel('n--->');

>> ylabel('o --->');

>> title('o/p without noise');

>> y = o +wgn(1,1,5); % where y is the output with noise of power = 5 decibels

>> subplot(222), stem(n,y);

>> xlabel('n--->');

>> ylabel('y--->');

>> title('o/p with noise of power 5 dB');

>> y = o + wgn(1,1,15); % where y is the output with noise of power = 15 decibels

>> subplot(223), stem(n,y);

>> xlabel('n--->');

>> ylabel('y--->');

>> title('o/p with noise of power 15 dB');

>> y = o + wgn(1,1,20); % where y is the output with noise of power = 20 decibels

>> subplot(224), stem(n,y);

>> xlabel('n--->');

>> ylabel('y--->');

>> title('o/p with noise of power 20 dB');

Ø Using "randn" command: This command adds random noise in the signal.

>> t = linspace(0,0.5*pi,40);

>> x = cos(0.75*t); % where x is the input signal

>> h = cos(t); % where h is the system

>> o = conv(x,h); % where o is the output without noise

>> subplot(121), plot(o)

>> xlabel('t --->');

>> ylabel('o --->');

>> title('o/p without noise');

>> y = o + randn(size(o)); % where y is the output with random noise

>> subplot(122), plot(y)

>> xlabel('t --->');

>> ylabel('y --->');

>> title('o/p with random noise');

Conclusion:

The change in output signal of system due to noise can be easily represented with the use of software known as Matlab. In the above programs, the first program shows the distortion in signal by adding Gaussian noise with different Signal to noise ratios with the use of ' awgn' command, here it is observed that decrease in the value of SNR increases the distortion in the output signal. Now, the second program shows the change due to noise added by command 'wgn' with different powers of the noise and here it is observed that as the power of noise increases; distortion of output signal increases. At last, the last program shows the change by using command 'randn' which adds random noise in the output signal.

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