Question

X(z) is the z-transform of the signal x(n), then what is the z-transform of the signal nx(n)?

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

(b) \(z\frac{dX(z)}{dz}\)

(c) \(-z^{-1}\frac{dX(z)}{dz}\)

(d) \(z^{-1}\frac{dX(z)}{dz}\)

I got this question in an interview for internship.

Question is taken from Properties of Z Transform topic in division Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

The correct choice is (a) \(-z\frac{dX(z)}{dz}\)

The explanation: From the definition of z-transform, we have

X(z)=\(\sum_{n=-\infty}^{\infty} x(n) z^{-n}\)

On differentiating both sides, we have

\(\frac{dX(z)}{dz}=\sum_{n=-\infty}^{\infty} (-n) x(n) z^{-n-1}=-z^{-1} \sum_{n=-\infty}^{\infty}nx(n) z^{-n}=-z^{-1}Z\{nx(n)\}\)

Therefore, we get \(-z\frac{dX(z)}{dz}\) = Z{nx(n)}.