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3. Realize the system which generates the output

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Answer #1

MATLAB code is given below in bold letters.

clc;
close all;
clear all;

% Define w as follows
w = [0 pi/6 3*pi/2 1.9*pi/2];

n = 0:10;
for k = 1:length(w)

y1_x1 = 1/2*(sin(w(k)*n)+sin(w(k)*(-n)));
y2_x1 = 1/2*(sin(w(k)*n)-sin(w(k)*(-n)));
disp('For w = ')
disp(w(k))
disp('For input x1(n), y1(n) = '),y1_x1
disp('For input x1(n), y2(n) = '),y2_x1

y1_x2 = 1/2*(cos(w(k)*n)+cos(w(k)*(-n)));
y2_x2 = 1/2*(cos(w(k)*n)-cos(w(k)*(-n)));
disp('For w = ')
disp(w(k))
disp('For input x2(n), y1(n) = ');y1_x2
disp('For input x2(n), y2(n) = '),y2_x2

end

RESULTS:

For w =
0

For input x1(n), y1(n) =

y1_x1 =

0 0 0 0 0 0 0 0 0 0 0

For input x1(n), y2(n) =

y2_x1 =

0 0 0 0 0 0 0 0 0 0 0

For w =
0

For input x2(n), y1(n) =

y1_x2 =

1 1 1 1 1 1 1 1 1 1 1

For input x2(n), y2(n) =

y2_x2 =

0 0 0 0 0 0 0 0 0 0 0

For w =
0.5236

For input x1(n), y1(n) =

y1_x1 =

0 0 0 0 0 0 0 0 0 0 0

For input x1(n), y2(n) =

y2_x1 =

Columns 1 through 8

0 0.5000 0.8660 1.0000 0.8660 0.5000 0.0000 -0.5000

Columns 9 through 11

-0.8660 -1.0000 -0.8660

For w =
0.5236

For input x2(n), y1(n) =

y1_x2 =

Columns 1 through 8

1.0000 0.8660 0.5000 0.0000 -0.5000 -0.8660 -1.0000 -0.8660

Columns 9 through 11

-0.5000 -0.0000 0.5000

For input x2(n), y2(n) =

y2_x2 =

0 0 0 0 0 0 0 0 0 0 0

For w =
4.7124

For input x1(n), y1(n) =

y1_x1 =

0 0 0 0 0 0 0 0 0 0 0

For input x1(n), y2(n) =

y2_x1 =

Columns 1 through 8

0 -1.0000 0.0000 1.0000 -0.0000 -1.0000 0.0000 1.0000

Columns 9 through 11

-0.0000 -1.0000 0.0000

For w =
4.7124

For input x2(n), y1(n) =

y1_x2 =

Columns 1 through 8

1.0000 -0.0000 -1.0000 0.0000 1.0000 -0.0000 -1.0000 -0.0000

Columns 9 through 11

1.0000 -0.0000 -1.0000

For input x2(n), y2(n) =

y2_x2 =

0 0 0 0 0 0 0 0 0 0 0

For w =
2.9845

For input x1(n), y1(n) =

y1_x1 =

0 0 0 0 0 0 0 0 0 0 0

For input x1(n), y2(n) =

y2_x1 =

Columns 1 through 8

0 0.1564 -0.3090 0.4540 -0.5878 0.7071 -0.8090 0.8910

Columns 9 through 11

-0.9511 0.9877 -1.0000

For w =
2.9845

For input x2(n), y1(n) =

y1_x2 =

Columns 1 through 8

1.0000 -0.9877 0.9511 -0.8910 0.8090 -0.7071 0.5878 -0.4540

Columns 9 through 11

0.3090 -0.1564 -0.0000

For input x2(n), y2(n) =

y2_x2 =

0 0 0 0 0 0 0 0 0 0 0

From the above results, it is observed that the odd signal x1[n] has resulted in zero response when passed through filter 1 Likewise, the even signal x2[n] has resulted in zero response when passed through filter 2.

It is observed that the filters 1 and 2 are odd and even parts of a signal.

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Answer 1

MATLAB code is given below in bold letters.

clc;
close all;
clear all;

% Define w as follows
w = [0 pi/6 3*pi/2 1.9*pi/2];

n = 0:10;
for k = 1:length(w)

y1_x1 = 1/2*(sin(w(k)*n)+sin(w(k)*(-n)));
y2_x1 = 1/2*(sin(w(k)*n)-sin(w(k)*(-n)));
disp('For w = ')
disp(w(k))
disp('For input x1(n), y1(n) = '),y1_x1
disp('For input x1(n), y2(n) = '),y2_x1

y1_x2 = 1/2*(cos(w(k)*n)+cos(w(k)*(-n)));
y2_x2 = 1/2*(cos(w(k)*n)-cos(w(k)*(-n)));
disp('For w = ')
disp(w(k))
disp('For input x2(n), y1(n) = ');y1_x2
disp('For input x2(n), y2(n) = '),y2_x2

end

RESULTS:

For w =
0

For input x1(n), y1(n) =

y1_x1 =

0 0 0 0 0 0 0 0 0 0 0

For input x1(n), y2(n) =

y2_x1 =

0 0 0 0 0 0 0 0 0 0 0

For w =
0

For input x2(n), y1(n) =

y1_x2 =

1 1 1 1 1 1 1 1 1 1 1

For input x2(n), y2(n) =

y2_x2 =

0 0 0 0 0 0 0 0 0 0 0

For w =
0.5236

For input x1(n), y1(n) =

y1_x1 =

0 0 0 0 0 0 0 0 0 0 0

For input x1(n), y2(n) =

y2_x1 =

Columns 1 through 8

0 0.5000 0.8660 1.0000 0.8660 0.5000 0.0000 -0.5000

Columns 9 through 11

-0.8660 -1.0000 -0.8660

For w =
0.5236

For input x2(n), y1(n) =

y1_x2 =

Columns 1 through 8

1.0000 0.8660 0.5000 0.0000 -0.5000 -0.8660 -1.0000 -0.8660

Columns 9 through 11

-0.5000 -0.0000 0.5000

For input x2(n), y2(n) =

y2_x2 =

0 0 0 0 0 0 0 0 0 0 0

For w =
4.7124

For input x1(n), y1(n) =

y1_x1 =

0 0 0 0 0 0 0 0 0 0 0

For input x1(n), y2(n) =

y2_x1 =

Columns 1 through 8

0 -1.0000 0.0000 1.0000 -0.0000 -1.0000 0.0000 1.0000

Columns 9 through 11

-0.0000 -1.0000 0.0000

For w =
4.7124

For input x2(n), y1(n) =

y1_x2 =

Columns 1 through 8

1.0000 -0.0000 -1.0000 0.0000 1.0000 -0.0000 -1.0000 -0.0000

Columns 9 through 11

1.0000 -0.0000 -1.0000

For input x2(n), y2(n) =

y2_x2 =

0 0 0 0 0 0 0 0 0 0 0

For w =
2.9845

For input x1(n), y1(n) =

y1_x1 =

0 0 0 0 0 0 0 0 0 0 0

For input x1(n), y2(n) =

y2_x1 =

Columns 1 through 8

0 0.1564 -0.3090 0.4540 -0.5878 0.7071 -0.8090 0.8910

Columns 9 through 11

-0.9511 0.9877 -1.0000

For w =
2.9845

For input x2(n), y1(n) =

y1_x2 =

Columns 1 through 8

1.0000 -0.9877 0.9511 -0.8910 0.8090 -0.7071 0.5878 -0.4540

Columns 9 through 11

0.3090 -0.1564 -0.0000

For input x2(n), y2(n) =

y2_x2 =

0 0 0 0 0 0 0 0 0 0 0

From the above results, it is observed that the odd signal x1[n] has resulted in zero response when passed through filter 1 Likewise, the even signal x2[n] has resulted in zero response when passed through filter 2.

It is observed that the filters 1 and 2 are odd and even parts of a signal.