Taking z-transform
a)
w=-pi:0.001:pi;
z=exp(1i.*w);
H=(0.207+0.413.*z.^-2+0.207.*z.^-4)./(1+0.9051.*z.^-1+0.598.*z.^-2+0.29.*z.^-3+0.1958.*z.^-4);
subplot(211)
plot(w,abs(H));
xlabel('\omega');
ylabel('|H(e^{j\omega})|');
subplot(212)
plot(w,angle(H));
xlabel('\omega');
ylabel('\angle H(e^{j\omega})');
---------------
b)
N=100;
n1=0:N;
w1=0.25.*pi;
w2=0.45.*pi;
x1=4.*cos(w1.*n1-pi./6).*sin(w2.*n1+0.25.*pi);
x=[0 0 0 0 x1];
y=zeros(size(x));
for k=5:length(y)
y(k)=-0.9051.*y(k-1)-0.598.*y(k-2)-0.29.*y(k-3)-0.1958.*y(k-4)+0.207.*x(k)+0.413.*x(k-2)+0.207.*x(k-4);
end
n=-4:N;
stem(n,y);
hold on
stem(n,x);
hold off
xlabel('n');
legend('y(n)','x(n)');
--------------
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c)
Now
and
therefore the steady state output for x(n) is
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N=100;
n1=0:N;
w1=0.25.*pi;
w2=0.45.*pi;
x1=4.*cos(w1.*n1-pi./6).*sin(w2.*n1+0.25.*pi);
x=[0 0 0 0 x1];
y=zeros(size(x));
for k=5:length(y)
y(k)=-0.9051.*y(k-1)-0.598.*y(k-2)-0.29.*y(k-3)-0.1958.*y(k-4)+0.207.*x(k)+0.413.*x(k-2)+0.207.*x(k-4);
end
n=-4:N;
y1=1.05.*sin(0.7.*pi.*n1+2.9)+0.484.*sin(0.2.*pi.*n1+0.78);
ys=[0 0 0 0 y1];
stem(n,(y-ys))
hold on
stem(n,ys);
hold off
xlabel('n');
ylabel('y(n)=y_{s}(n)');
legend('transient response','steady state response');
-------------------
(5) For the system described by the following difference equation y(n)= 0.9051y(n 1) 0.598y(n 2) -0.29y(n...
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