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Signal Processing 1

1.)Acontinuous system has a continuous out put in response to a continuous input signal.E.g.-: Analogue electronics,

Physical Systems such as a car or a simple mass mounted on a spring.

And when the Continuous discrete systems on the other hand, have discrete inputs & discrete outputs.

Y (t)

X (t)

X (n) Y (n)

To be system linear it must have these properties

yHomogeneityyAdditive

Homogeneity

It means that a change in the input signals amplitude results in a corresponding change in the output signals amplitude.

In mathematical terms this means that if an input signal X(n) results in an output signal of Y(n), then an input signal KX(n)

results in an output of KY(n) where K is a constant.

X (n) Y (n) KX (n) KY (n)

Additive

An Example of an additive system is hearing two voices (inputs) through a telephone. A system is said to be additive if

added signals pass through it without interacting mathematically,

If X1(n) at input result in Y1(n) and X2(n) result in Y2(n),then X1(n) + X2(n) to the input result in Y1(n) Y2(n).

Continuous

System

Discrete

Continuous

System

System

System

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X1(n) Y1(n)

X1(n) + X2(n) Y1(n) + Y2(n)

X2(n) Y2(n)

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2)

a)

Even Component

1

0.5

-2 -1 0 1

Odd Component

1

0.5

-2 -1

-0.5

b)

Even Component

System

System

System

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c) Even Component

2 2

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

n -4 n

d) Even Component

2

2

-4 -3 -2 -1 1 2 3 4 n 4 -3 -2 -1 1

-2

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3) Take the period as T

Then

) So the signal is periodic

K= set of Integers

Fundamental Period =

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4) U (t) X(t)

1 1

t 0 1

a)

X(t).U(-t)

1

t

b) 4.0 X(t)(U(t+1)-U(t-2)

3.0

2.5

2.0

1.5

1

1 2 3 4

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c)

x(t)U(t - 4)

4

2

1 2 3 4 5

d)

X(t)(t 3)

1 2 3 4 5

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=====================================================================================

05)

X(t) h(t)

2

1

0 4 t 3 t

X(t) h(t) y(t)

t < 0

2

t-3 t

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t-3 0 t 4

0 t-3 t 4

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t-3 4 t

4 t-3 t

y(t) 0 3

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====================================================================================

6)

From the Eulers formula;

(a)

Substituting from the equation (4);

Finding the fundamental period of:

For to be periodic;

The fundamental period is when k = 1; therefore,

Since ;

(A)

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From the definition of complex Fourier representation:

Since ;

Comparing the coefficient of the equations (A) and (B);

Comparing the coefficient of the equations (A) and (B);

Comparing the coefficient of the equations (A) and (B);

Comparing the coefficient of the equations (A) and (B);

Comparing the coefficient of the equations (A) and (B);

(B)

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Therefore, the coefficients of the complex Fourier series representation:

(b)

From the Eulers formula

Substituting from the equation number (3);

From the definition of complex Fourier representation:

Since ;

Comparing the coefficient of the equations (A) and (B);

(D)

(C)

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Comparing the coefficient of the equations (A) and (B);

Comparing the coefficient of the equations (A) and (B);

Comparing the coefficient of the equations (A) and (B);

Comparing the coefficient of the equations (A) and (B);

Therefore, the coefficients of the complex Fourier series representation:

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7)

When the signal is even

Then the

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=====================================================================

8)

The period of signal = 6

x t

2

1

3

4 6-2-4-

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Therefore,

;

;

by

;

;

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Following program should be run to calculate and plot the truncated sum for this series.

For w0=pi/3c0=1;

t=0:0.01:15;x=c0*ones(size(t));

for (k=-10:1:-1);

ck=(j/(2*k*pi))*(exp(-j*k*pi)-1)+((3*j)/(2*k*pi))*(exp((-j*k*4*pi)/3)-

exp(-j*k*pi));x=ck*exp(j*k*w0*t)+x;

endfor (k=1:1:10);

ck=(j/(2*k*pi))*(exp(-j*k*pi)-1)+((3*j)/(2*k*pi))*(exp((-j*k*4*pi)/3)-

exp(-j*k*pi));x=ck*exp(j*k*w0*t)+x;

endplot(t,x)

Then the figure

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For w0=pi/3c0=1;

t=0:0.01:15;x=c0*ones(size(t));for (k=-100:1:-1);

ck=(j/(2*k*pi))*(exp(-j*k*pi)-1)+((3*j)/(2*k*pi))*(exp((-j*k*4*pi)/3)-

exp(-j*k*pi));x=ck*exp(j*k*w0*t)+x;

endfor (k=1:1:100);

ck=(j/(2*k*pi))*(exp(-j*k*pi)-1)+((3*j)/(2*k*pi))*(exp((-j*k*4*pi)/3)-

exp(-j*k*pi));x=ck*exp(j*k*w0*t)+x;

endplot(t,x)

Then the figure

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For w0=pi/3c0=1;t=0:0.01:15;x=c0*ones(size(t));for (k=-1000:1:-1);

ck=(j/(2*k*pi))*(exp(-j*k*pi)-1)+((3*j)/(2*k*pi))*(exp((-j*k*4*pi)/3)-

exp(-j*k*pi));x=ck*exp(j*k*w0*t)+x;

end

for (k=1:1:1000);

ck=(j/(2*k*pi))*(exp(-j*k*pi)-1)+((3*j)/(2*k*pi))*(exp((-j*k*4*pi)/3)-

exp(-j*k*pi));x=ck*exp(j*k*w0*t)+x;

endplot(t,x)

Then the figure

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For a real signal, the Fourier series in terms of complex exponentials is,

The Fourier series works because it is an infinite sum of a series of sine waves, each of particular amplitude,

frequency and starting phase which is practically not feasible. In practice, to gain the Fourier series we add

finite number of a series of sine waves. Then the sharpness of the periodic signal depends on the number of the

sinusoids we add.

In this case, when the number of sinusoids we add is 21. Therefore, shape does not appear as astraight line and ripples are visible. When sinusoids are increased as in the case 2 ( ) the addednumber of sinusoid are 201; then ripples reduces and signal becomes somewhat closer to a straight line. As

sinusoids further increases up to 2001, as in the case 3; ripples are hardly visible and almost looks like a straight

line.

This shows that when number of sinusoids approaches to infinity the shape appears as the periodic signal .