Question

Find the Z-transform of x(n) = n[a^n u(n)].

(a) \(\frac{z}{z(z-a)}\)

(b) \(\frac{az}{z(z-a)}\)

(c) \(\frac{az}{z(z+a)}\)

(d) \(\frac{a}{z(z-a)^2}\)

I got this question in final exam.

The doubt is from Properties of Z-Transforms in chapter Z-Transform and Digital Filtering of Signals and Systems

Answer 1

Right answer is (d) \(\frac{a}{z(z-a)^2}\)

To explain: Given x(n) = n[a^n u(n)]

We know that a^n u(n) ↔ \(\frac{z}{z-a}\)

Time differentiation property states that

If x(n) ↔ X(z), then nx(n) ↔ -z \(\frac{dX(z)}{dz}\)

Z[x(n)] = Z{n[a^n u(n)]} = -z \(\frac{dX(z)}{dz} = -z \frac{d}{dz} \Big[\frac{z}{z-a}\Big] = \frac{a}{z(z-a)^2}\).