Question

Question 2 (a) Determine whether the discrete time system which has an output y[n] 2*x[n] over the nterval 010 is linear or not by determining the response yi[n] to the input signalxj[n]- sin( (2*pi / 10 ) * n ) and the response y2[n] to the input signal x2[n] = cos( (2*pi/10 ) * n ). Determine the response y3[n] to the input signal x1n] = xi [n] + x2[n] and compare it with y4[n] = [n] + y2[n] . Plot (using stem) İn one graph all the input signals and their corresponding Out- put signals. Use the disp command to output whether the system has ‘outputs consistent with a linear system ' or 'not linear'. (b) Design an experiment to test whether the systems (ii).vlnl-2x[n] +5δ[n] 14 are linear and time-invariant. Use the input data xn] [0,1]. Next, using a larger set of values, for your choice of the input data, repeat the experiment and analyze and interpret the results obtained with this new data set. Do the new results validate or invalidate the original results obtained with x[n] = [0.1] as choice of input data? Explain how a choice ofdata used may impact the results obtained.

Answer 1

a)MATLAB:

clc;close all;clear all;

n=0:1:10
%y1(n) to input x1(n)=sin(2*pi*n/10)
x1=sin(2*pi*n/10)
y1=2*x1
subplot(421)
stem(n,x1,'r');xlabel('n');ylabel('x1(n)');title('x1(n)=sin(2*pi*n/10)')
subplot(422)
stem(n,y1,'b');xlabel('n');ylabel('y1(n)');title('y1(n)=2*x1(n)')

%y2(n) to input x2(n)=cos(2*pi*n/10)
x2=cos(2*pi*n/10)
y2=2*x2
subplot(423)
stem(n,x2,'g');xlabel('n');ylabel('x2(n)');title('x2(n)=cos(2*pi*n/10)')
subplot(424)
stem(n,y2,'b');xlabel('n');ylabel('y2(n)');title('y2(n)=2*x2(n)')

%y3(n) to input x3(n)=x1(n)+x2(n)
x3=x1.+x2
y3=2*x3
subplot(425)
stem(n,x3,'m');xlabel('n');ylabel('x3(n)');title('x3(n)=x1(n)+x2(n)')
subplot(426)
stem(n,y3,'b');xlabel('n');ylabel('y3(n)');title('y3(n)=2*x3(n)')

%y3(n) to input x3(n)=x1(n)+x2(n)
y4=y1+y2
subplot(427)
stem(n,y4,'b');xlabel('n');ylabel('y4(n)');title('y4(n)=y1(n)+y2(n)')

if(y3==y4)
disp('outputs consistent with a linear system')
else
disp('Not linear')
end

y1 (n)-2"x1(n) x1(n)-sin(2*pi*n/10) E 0.5 10 y2(n)-2x2(n) x2(n)-cos(2 pi*n/10) 10 10 y3(n)-2*x3(n) x3(n)-x1(n)+x2(n) 10 4 10 y4(n)-y1 (n)+y2(n)

Command window:

y3 =

Columns 1 through 9:

2.00000 2.79360 2.52015 1.28408 -0.44246 -2.00000 -2.79360 -2.52015 -1.28408

Columns 10 and 11:

0.44246 2.00000

y4 =

Columns 1 through 9:

2.00000 2.79360 2.52015 1.28408 -0.44246 -2.00000 -2.79360 -2.52015 -1.28408

Columns 10 and 11:

0.44246 2.00000

outputs consistent with a linear system

____________________________________________________

b) i) Linearity for x1=[0,1],x2=n

MATLAB:

clc;close all;clear all;

n=0:1:10
%y1(n) to input x1(n)=[0,1]

x1=[0,1,zeros(1,length(n)-2)]
y1=x1.^2
subplot(421)
stem(n,x1,'r');xlabel('n');ylabel('x1(n)');title('x1(n)')
subplot(422)
stem(n,y1,'b');xlabel('n');ylabel('y1(n)');title('y1(n)=x1(n).^2')

%y2(n) to input x2(n)=n
x2=n
y2=x2.^2
subplot(423)
stem(n,x2,'g');xlabel('n');ylabel('x2(n)');title('x2(n)')
subplot(424)
stem(n,y2,'b');xlabel('n');ylabel('y2(n)');title('y2(n)=x2(n).^2')

%y3(n) to input x3(n)=x1(n)+x2(n)
x3=x1.+x2
y3=x3.^2
subplot(425)
stem(n,x3,'m');xlabel('n');ylabel('x3(n)');title('x3(n)=x1(n)+x2(n)')
subplot(426)
stem(n,y3,'b');xlabel('n');ylabel('y3(n)');title('y3(n)=x3(n).^2')

%y3(n) to input x3(n)=x1(n)+x2(n)
y4=y1+y2
subplot(427)
stem(n,y4,'b');xlabel('n');ylabel('y4(n)');title('y4(n)=y1(n)+y2(n)')

if(y3==y4)
disp('outputs consistent with a linear system')
else
disp('Not linear')
end

y1 (n)-x1(n). x1 (n) 10 10 x2(n) 10 10 y3(n)-x3(n) x3(n)-x1(n)+x2(n) 100 2 10 4 10 y4(n)-y1 (n)+y2(n) 100

Command window:

y3 =

0 4 4 9 16 25 36 49 64 81 100

y4 =

0 2 4 9 16 25 36 49 64 81 100

Not linear

ii)Linearity:

clc;close all;clear all;

n=0:1:10
%y1(n) to input x1(n)=[0,1]

x1=[0,1,zeros(1,length(n)-2)]
y1=x1.^2
subplot(421)
stem(n,x1,'r');xlabel('n');ylabel('x1(n)');title('x1(n)=sin(2*pi*n/10)')
subplot(422)
stem(n,y1,'b');xlabel('n');ylabel('y1(n)');title('y1(n)=x1(n).^2')

%y2(n) to input x2(n)=n
x2=n
y2=x2.^2
subplot(423)
stem(n,x2,'g');xlabel('n');ylabel('x2(n)');title('x2(n)=cos(2*pi*n/10)')
subplot(424)
stem(n,y2,'b');xlabel('n');ylabel('y2(n)');title('y2(n)=x2(n).^2')

%y3(n) to input x3(n)=x1(n)+x2(n)
x3=x1.+x2
y3=x3.^2
subplot(425)
stem(n,x3,'m');xlabel('n');ylabel('x3(n)');title('x3(n)=x1(n)+x2(n)')
subplot(426)
stem(n,y3,'b');xlabel('n');ylabel('y3(n)');title('y3(n)=x3(n).^2')

%y3(n) to input x3(n)=x1(n)+x2(n)
y4=y1+y2
subplot(427)
stem(n,y4,'b');xlabel('n');ylabel('y4(n)');title('y4(n)=y1(n)+y2(n)')

if(y3==y4)
disp('outputs consistent with a linear system')
else
disp('Not linear')
end

y1 (n)-2x1(n)+5del(n) x1(n) 0.8 0.6 0.2 10 10 y2(n)-2x2(n)+5del(n) x2(n) 20 10 10 y3 (n)-2x3(n)+5del(n) x3(n)-x1(n)+x2(n) 20 10 10 10 4 10 y4(n)-y1 (n)+y2(n) 20

Command window:

y3 =

5 4 4 6 8 10 12 14 16 18 20

y4 =

10 4 4 6 8 10 12 14 16 18 20

Not linear

b(i) Time invariant:

clc;close all;clear all;

n=0:1:4
%y1(n) to input x1(n)=[0,1]

x=[0,1,zeros(1,length(n)-2)]
subplot(221)
stem(n,x,'r');xlabel('n');ylabel('x(n)');title('x(n)')
n0=2
%plot y(n-k)
y=x.^2
for k=1:length(n)
y1(k+n0)=y(k)
end
m=0:1:max(n)+n0
subplot(222)
stem(m,y1,'b');xlabel('n');ylabel('y(n-k)');title('y(n-k)')

for k=1:length(n)
x1(k+n0)=x(k)
end
subplot(223)
stem(m,x1,'r');xlabel('n');ylabel('x(n-no)');title('x(n-no)')
y=x1.^2
subplot(224)
stem(m,y,'b');xlabel('n');ylabel('y(n,k)');title('y(n,k)')

if(y1==y)
disp('the system is time invariant')
else
disp('the system is time variant')
end

x(n) y(n-k) 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 3 4 x(n-no) Y(n,k) 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2

Command window:

the system is time invariant
>> y
y =

0 0 0 1 0 0 0

>> y1
y1 =

0 0 0 1 0 0 0

ii)time invariant:

clc;close all;clear all;

n=0:1:4
%y1(n) to input x1(n)=[0,1]

x=[0,1,zeros(1,length(n)-2)]
subplot(221)
stem(n,x,'r');xlabel('n');ylabel('x(n)');title('x(n)')
n0=2
%plot y(n-k)
y=(2*x)+(5*(n==0))
for k=1:length(n)
y1(k+n0)=y(k)
end
m=0:1:max(n)+n0
subplot(222)
stem(m,y1,'b');xlabel('n');ylabel('y(n-k)');title('y(n-k)')

for k=1:length(n)
x1(k+n0)=x(k)
end
subplot(223)
stem(m,x1,'r');xlabel('n');ylabel('x(n-no)');title('x(n-no)')
y=(2*x1)+(5*(m==0))
subplot(224)
stem(m,y,'b');xlabel('n');ylabel('y(n,k)');title('y(n,k)')

if(y1==y)
disp('the system is time invariant')
else
disp('the system is time variant')
end

x(n) y(n-k) 0.8 4 0.6 0.4 0.2 3 x(n-no) Y(n,k) 0.8 4 0.6 0.4 0.2

the system is time variant
>> y
y =

5 0 0 2 0 0 0

>> y1
y1 =

0 0 5 2 0 0 0