Question

If x(n)=a^n, then what is one sided z-transform of x(n-2)?

(a) \(\frac{z^{-2}}{1-az^{-1}} + a^{-1}z^{-1} + a^{-2}\)

(b) \(\frac{z^{-2}}{1-az^{-1}} – a^{-1}z^{-1} + a^{-2}\)

(c) \(\frac{z^{-2}}{1-az^{-1}} + a^{-1}z^{-1} – a^{-2}\)

(d) \(\frac{z^{-2}}{1+az^{-1}} + a^{-1}z^{-1} + a^{-2}\)

I had been asked this question in class test.

My question is based upon One Sided Z Transform topic in portion Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

Right answer is (a) \(\frac{z^{-2}}{1-az^{-1}} + a^{-1}z^{-1} + a^{-2}\)

Explanation: Given x(n)=a^n=>X^+(z)=\(\frac{1}{1-az^{-1}}\)

We will apply the shifting property for k=2. Indeed we have

Z^+{x(n-2)}=z^-2[X^+(z)+x(-1)z+x(-2)z^2]

=z^-2 X^+(z)+x(-1)z^-1+x(-2)

Since x(-1)=a^-1 and x(-2)=a^-2, we obtain

X1+(z)=\(\frac{z^{-2}}{1-az^{-1}}\) + a^-1z^-1 + a^-2