Question

Determine the output y(n) of a LTI system with impulse response h(n)=a^nu(n), |a|<1with the input sequence x(n)=u(n).

(a) \(\frac{1-a^{n+1}}{1-a}\)

(b) \(\frac{1-a^{n-1}}{1-a}\)

(c) \(\frac{1+a^{n+1}}{1+a}\)

(d) None of the mentioned

The question was posed to me in final exam.

Origin of the question is Analysis of Discrete time LTI Systems topic in section Discrete Time Signals and Systems of Digital Signal Processing

Answer 1

Correct option is (a) \(\frac{1-a^{n+1}}{1-a}\)

Easiest explanation: Now fold the signal x(n) and shift it by one unit at a time and sum as follows

y(0)=x(0)h(0)=1

y(1)=h(0)x(1)+h(1)x(0)=1.1+a.1=1+a

y(2)=h(0)x(2)+h(1)x(1)+h(2)x(0)=1.1+a.1+a^2.1=1+a+a^2

Similarly, y(n)=1+a+a^2+….a^n=\(\frac{1-a^{n+1}}{1-a}\).