Question

Find the Z-transform of y(n) = x(n+2)u(n).

(a) z^2 X(Z) – z^2 x(0) – zx(1)

(b) z^2 X(Z) + z^2 x(0) – zx(1)

(c) z^2 X(Z) – z^2 x(0) + zx(1)

(d) z^2 X(Z) + z^2 x(0) + zx(1)

The question was posed to me during a job interview.

Asked question is from The Z-Transform topic in chapter Z-Transform and Digital Filtering of Signals and Systems

Answer 1

Correct choice is (a) z^2 X(Z) – z^2 x(0) – zx(1)

To explain I would say: Given y(n) = x(n+2)u(n)

Y(z) = Z[y(n)] = Z[x(n+2)u(n)] = \(\sum\limits_{n=0}^{∞} x(n+2)u(n) z^{-n} = \sum\limits_{n=0}^{∞} x(n+2)z^{-n}\)

Let n + 2 = p,i.e.n = p – 2

Y(z) = \(∑_{p=2}^∞ x(p)z^{-(p-2)} = z^2 ∑_{p=2}^∞ x(p)z^{-p} = z^2 ∑_{p=0}^∞ x(p)z^{-p} – x(0) – x(1) z^{-1}\)

=z^2 X(Z) – z^2 x(0) – zx(1).