Question

What is the z-transform of the signal x(n)=[3(2^n)-4(3^n)]u(n)?

(a) \(\frac{3}{1-2z^{-1}}-\frac{4}{1-3z^{-1}}\)

(b) \(\frac{3}{1-2z^{-1}}-\frac{4}{1+3z^{-1}}\)

(c) \(\frac{3}{1-2z}-\frac{4}{1-3z}\)

(d) None of the mentioned

This question was addressed to me by my college professor while I was bunking the class.

I would like to ask this question from Properties of Z Transform in portion Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

Correct choice is (a) \(\frac{3}{1-2z^{-1}}-\frac{4}{1-3z^{-1}}\)

The explanation is: Let us divide the given x(n) into x1(n)=3(2^n)u(n) and x2(n)= 4(3^n)u(n)

and x(n)=x1(n)-x2(n)

From the definition of z-transform X1(z)=\(\frac{3}{1-2z^{-1}}\) and X2(z)=\(\frac{4}{1-3z^{-1}}\)

So, from the linearity property of z-transform

X(z)=X1(z)-X2(z)

=> X(z)=\(\frac{3}{1-2z^{-1}}-\frac{4}{1-3z^{-1}}\).