Question

If an LTI system is described by the difference equation y(n)=ay(n-1)+bx(n), 0<a<1, then what is the output of the system when input x(n)=\(5+12sin\frac{π}{2}n-20cos(πn+\frac{π}{4})\)?(Given a=0.9 and b=0.1)

(a) \(5+0.888sin(\frac{π}{2}n-420)-1.06cos(πn-\frac{π}{4})\)

(b) \(5+0.888sin(\frac{π}{2}n-420)+1.06cos(πn+\frac{π}{4})\)

(c) \(5+0.888sin(\frac{π}{2}n-420)-1.06cos(πn+\frac{π}{4})\)

(d) \(5+0.888sin(\frac{π}{2}n+420)-1.06cos(πn+\frac{π}{4})\)

The question was posed to me in an internship interview.

My question is from Frequency Domain Characteristics of LTI System in portion Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct choice is (c) \(5+0.888sin(\frac{π}{2}n-420)-1.06cos(πn+\frac{π}{4})\)

To explain I would say: From the given difference equation, we obtain

|H(ω)|=\(\frac{|b|}{\sqrt{1-2acosω+a^2}}\)

We get |H(0)|=1, |H(π/2)|=0.074 and |H(π)|=0.053

θ(0)=0, θ(π/2)=-420 and θ(π)=0 and we know that y(n)=H(ω)x(n)

=>y(n)=\(5+0.888sin(\frac{π}{2}n-42^0)-1.06cos(πn+\frac{π}{4})\)