Question

5) An N-point moving-average (running-average) system has the following input-output relation ship: N-1 a) Is the N-point moving-average system causal? b) Obtain the expression for the impulse response h[n] and sketch hin. c) Given the input sequence [n] below with 100 elements where the values of the index n change between 0 and 99: r-[21 22 22 21 18 19 21 20 19 23 23 22 23 25 27 30 31.5 32 33 32 28 29 28 29 30 32 32 24 24.5 24 28 29 30 31 31 32 33 35 40 43.2 45 44 47 50 47 47.5 48 52 52 53 56 54.5 56 59 62 63 62 63 59 60 62 61 57 59 59.6 63 62 54 54.5 61 63 61.5 63 62 63 70 71 62 55 54 49 46 43 41 41.5 45 43 42 41.5 40 40.5 38 34 38 37 36 35 34.5 35.5 34.5] Plot the sequence x[n] using MATLAB. Also plot the output sequence yin] of the N-point moving averager to the input sequence given above when i)N -2, ii)N 5, and iiN-10. For each case, explicitly indicate the range of indices for y[n]. Attach your MATLAB code to the plots. d) Explain what effect(s) N has on the shape of the output signal n

Answer 1

a)

Yes This is a causal system.For N=2 ,y(n)=[x(n)+x(n-1)]/2

It depends on the present and past values of the input signal x(n).This is independent of the future values of x(n).

present value > (x(n)) and past value> (x(n-1)

Hence the system is causal.

b)Impulse response,h(n)=(1/N)*ones(1,N)

For N=2, h(n)=[1 1]/2

c)MATLAB:

N=2
h=(1/N)*(ones(1,N))
x=[21 22 22 21 18 19 21 20 19 23 23 22 23 25 27 30 31.5 32 33 32 28 29 28 29 30 32 32 24 24.5 24 28 29 30 31 31 32 33 35 40 43.2 45 44 47 50 47 47.5 48 52 52 53 56 54.5 56 59 62 63 62 63 59 60 62 61 57 59 59.6 63 62 54 54.5 61 63 61.5 63 62 63 70 71 62 55 54 49 46 43 41 41.5 45 43 42 41.5 40 40.5 38 34 38 37 36 35 34.5 35.5 34.5]
y=conv(x,h)
subplot(311)
n=0:1:99
stem(n,x,'m')
xlabel('n')
ylabel('x(n)')
title('Input x(n)')

subplot(312)
nh=0:1:N-1
stem(nh,h,'g')
xlabel('n')
ylabel('h(n)')
title('Impulse response h(n)')

subplot(313)
M=length(x)+length(h)-1
m=0:1:M-1
stem(m,y,'r')
xlabel('n')
ylabel('y(n)')
title('Output y(n) for N=2')

Input x (n) 80 г 60 S 40 20 20 40 60 80 100 Impulse response h (n) 0.5G 0.4 2 0.3 0.2 0.1 F 0.2 0.4 0.6 0.8 Output y(n) for N-2 80 60 S 40 20 20 40 60 80 100

N=5
h=(1/N)*(ones(1,N))
x=[21 22 22 21 18 19 21 20 19 23 23 22 23 25 27 30 31.5 32 33 32 28 29 28 29 30 32 32 24 24.5 24 28 29 30 31 31 32 33 35 40 43.2 45 44 47 50 47 47.5 48 52 52 53 56 54.5 56 59 62 63 62 63 59 60 62 61 57 59 59.6 63 62 54 54.5 61 63 61.5 63 62 63 70 71 62 55 54 49 46 43 41 41.5 45 43 42 41.5 40 40.5 38 34 38 37 36 35 34.5 35.5 34.5]
y=conv(x,h)
subplot(311)
n=0:1:99
stem(n,x,'m')
xlabel('n')
ylabel('x(n)')
title('Input x(n)')

subplot(312)
nh=0:1:N-1
stem(nh,h,'g')
xlabel('n')
ylabel('h(n)')
title('Impulse response h(n)')

subplot(313)
M=length(x)+length(h)-1
m=0:1:M-1
stem(m,y,'r')
xlabel('n')
ylabel('y(n)')
title('Output y(n) for N=5')
xlim([0,M-1])

Input x (n) 80 г 60 S 40 20 20 40 60 80 100 Impulse response h (n) 0.2 б 0.15 S 0.1 0.05 2 3 4 Output y(n) for N-5 70 г 60 50 F 40 30 20 10 20 40 60 80 100

N=10
h=(1/N)*(ones(1,N))
x=[21 22 22 21 18 19 21 20 19 23 23 22 23 25 27 30 31.5 32 33 32 28 29 28 29 30 32 32 24 24.5 24 28 29 30 31 31 32 33 35 40 43.2 45 44 47 50 47 47.5 48 52 52 53 56 54.5 56 59 62 63 62 63 59 60 62 61 57 59 59.6 63 62 54 54.5 61 63 61.5 63 62 63 70 71 62 55 54 49 46 43 41 41.5 45 43 42 41.5 40 40.5 38 34 38 37 36 35 34.5 35.5 34.5]
y=conv(x,h)
subplot(311)
n=0:1:99
stem(n,x,'m')
xlabel('n')
ylabel('x(n)')
title('Input x(n)')

subplot(312)
nh=0:1:N-1
stem(nh,h,'g')
xlabel('n')
ylabel('h(n)')
title('Impulse response h(n)')

subplot(313)
M=length(x)+length(h)-1
m=0:1:M-1
stem(m,y,'r')
xlabel('n')
ylabel('y(n)')
title('Output y(n) for N=10')
xlim([0,M-1])

Input x (n) 80 г 60 S 40 20 40 60 80 100 20 Impulse response h (n) 0.1G 0.08 0.06 0.04 0.02 ︶ 2 Output y(n) for N-10 70 Γ 60 50 F 40 20 10 40 60 80 100 20

d)The filter averaging operation of N=10 is better than the cases N=2 and N=5.(Look at the responses.)