Question

What is the particular solution of the first order difference equation y(n)+ay(n-1)=x(n) where |a|<1, when the input of the system x(n)=u(n)?

(a) $\frac{1}{1+a}$ u(n)

(b) $\frac{1}{1-a}$ u(n)

(c) $\frac{1}{1+a}$

(d) $\frac{1}{1-a}$

This question was addressed to me during an interview.

I want to ask this question from Discrete Time Systems Described by Difference Equations topic in section Discrete Time Signals and Systems of Digital Signal Processing

Answer 1

Correct choice is (a) $\frac{1}{1+a}$ u(n)

To explain I would say: The assumed solution of the difference equation to the forcing equation x(n), called the particular solution of the difference equation is

yp(n)=Kx(n)=Ku(n) (where K is a scale factor)

Substitute the above equation in the given equation

=>Ku(n)+aKu(n-1)=u(n)

To determine K we must evaluate the above equation for any n>=1, so that no term vanishes.

=> K+aK=1

=>K=$\frac{1}{1+a}$

Therefore the particular solution is yp(n)=$\frac{1}{1+a}$ u(n).