The discrete-time signal
$x [n] ⟷ X (z) = \sum_{n = 0}^{\infty} \frac{3^{n}}{2 + n} z^{2 n}$
where $⟷$ denotes a transform-pair relationship, is orthogonal to the signal

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Solution

$x (n) ⇌ X (z) = \sum_{n = 0}^{\infty} \frac{3^{n}}{2 + n} z^{2 n}$

In X(z), power of z are even. Therefore, samples in x(n) are available at even instants of time.

By observing all the options. Option (b):
$y_{2} (n) ⇌ Y_{2} (z) = \sum_{n = 0}^{\infty} (5^{n} - n) z^{- (2 n + 1)}$

In $Y_{2} (z)$ , powers of z are odd. Therefore, samples in $y_{2} (n)$ are available only at odd instants of time.

Hence, $\sum_{n = - \infty}^{\infty} x (n) . y_{2} (n) = 0$

Thus. x(n) is orthogonal to $y_{2} (n)$ .

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The discrete-time signal x[n]⟷X(z)=∑∞n=03n2+nz2n where ⟷ denotes a transform-pair relationship, is orthogonal to the signal

The discrete-time signal
$x [n] ⟷ X (z) = \sum_{n = 0}^{\infty} \frac{3^{n}}{2 + n} z^{2 n}$
where $⟷$ denotes a transform-pair relationship, is orthogonal to the signal