Question

Please answer all the questions

Here is evenodd function:

function [xe, xo, m] = evenodd(x,n)
% Real signal decomposition into even and odd parts
% -------------------------------------------------
% [xe, xo, m] = evenodd(x,n)
%
if any(imag(x) ~= 0)
error('x is not a real sequence')
end
m = -fliplr(n);
m1 = min([m,n]); m2 = max([m,n]); m = m1:m2;
nm = n(1)-m(1); n1 = 1:length(n);
x1 = zeros(1,length(m));
x1(n1+nm) = x; x = x1;
xe = 0.5*(x + fliplr(x));
xo = 0.5*(x - fliplr(x));

Problem 2: a. Show that any signal can be decomposed into it an even and an odd component. Is the decomposition unique? b. Using the evenodd function, decompose the following sequences into their even and odd components Plot the original sequence and its components on one figure using subplot commands. ·n(n)-cos(0.2πn + π/4), -20 n-20 · T2(n) = e-0.05n sin(0.1πn + π/3), 0 n-100

Answer 1

a)

If x(-n)=-x(n), then the signal is called as odd signal.--------------(1)

If x(-n)=x(n), then the signal is called as even signal.----------------(2)

Any signal can be expressed as a sum of even and odd components.

x(n)=xe(n)+xo(n) -----(3)

where xe(n) is the even component of x(n) and xo(n) is the even component of x(n).

Replace n by -n in (3)

x(-n)=xe(-n)+xo(-n) ----------------(4)

From(2), xe(-n)=xe(n)

From(1), xo(-n)=-xo(n)

Rewrite the equation(4),

x(-n)=xe(n)-xo(n) -----------------(5)

Add (5) and (3),

x(n)+x(-n)=2 xe(n)  

xe(n)= [x(n)+x(-n)]/2 -------------(6)

subtract (5) from (3),

x(n)-x(-n)=2 xo(n)  

xo(n)= [x(n)-x(-n)]/2 -------------(7)

Hence any signal can be decomposed into an even component and an odd component.

_______________________________________________________________________

b)

Steps:

1.Run the function file with the name 'evenodd.m'. This execution will show 'error: 'x' undefined '. Just neglect this error.

2.Run the main files one by one to get the waveforms.

function [xe, xo, m] = evenodd(x,n)
% Real signal decomposition into even and odd parts
% -------------------------------------------------
% [xe, xo, m] = evenodd(x,n)
%
if any(imag(x) ~= 0)
error('x is not a real sequence')
end
m = -fliplr(n);
m1 = min([m,n]); m2 = max([m,n]); m = m1:m2;
nm = n(1)-m(1); n1 = 1:length(n);
x1 = zeros(1,length(m));
x1(n1+nm) = x; x = x1;
xe = 0.5*(x + fliplr(x));
xo = 0.5*(x - fliplr(x));
endfunction

__________________________________________________________________

MAIN FILE :1

clc;
clear all;close all;
n1=-20:1:20
x1=cos((0.2*pi*n1)+(pi/4))
[xe1, xo1, m1] = evenodd(x1,n1)
subplot(411)
stem(n1,x1)
title('x1(n)=cos((0.2*pi*n1)+(pi/4))')
xlabel('n')
ylabel('x1(n)')

subplot(412)
stem(m1,xe1,'r')
title('Even component of x1(n)')
xlabel('n')
ylabel('xe(n)')

subplot(413)
stem(m1,xo1,'g')
xlabel('n')
ylabel('x0(n)')
title('Odd component of x1(n)')

y=xe1+xo1
subplot(414)
stem(n1,y,'b')
xlabel('n')
ylabel('x(n)')
title(' x1(n)=even +odd')

x1(n)-cos((0.2"pi*n1)+(pi/4) E 0.5 20 10 -10 -20 Even component of x1(n) 20 10 -10 -20 Odd component of x1(n) 20 10 -10 -20 x1(n)-even todd 0.5 20 10 -20 -10

________________________________________________________________________________________

MAIN FILE :2

clc;
clear all;close all;
n2=-20:1:20
x2=(exp(-0.05*n2)).*(sin((0.1*pi*n2)+(pi/3)))
[xe2, xo2, m2] = evenodd(x2,n2)
subplot(221)
stem(n2,x2)
title('x2(n)=(exp(-0.05*n2)).*(sin((0.1*pi*n2)+(pi/3)))')
xlabel('n')
ylabel('x2(n)')

subplot(222)
stem(m2,xe2,'r')
title('Even component of x2(n)')
xlabel('n')
ylabel('xe(n)')

subplot(223)
stem(m2,xo2,'g')
xlabel('n')
ylabel('x0(n)')
title('Odd component of x2(n)')

y=xe2+xo2
subplot(224)
stem(n2,y,'b')
xlabel('n')
ylabel('x(n)')
title(' x2(n)=even +odd')

Even component of x2(n) x2(n)-(exp(-0.05"n2))."(sin((0.1 *pi"n2)-(pi/3))) 3 0.5 0.5 -10 10 20 2 -20 20 20 -10 x2(n)-even todd Odd component of x2(n) 3 2 0.5 1.5 20 -10 10 20 20 -20 -10

Observation:

The fourth plot of both problem show that the addition of even and odd components produces the same signal which is used for decomposition.