If X(z) is the z-transform of the signal x(n), then what is the z-transform of the signal x(-n)?Group of answer choicesX(-z)X(z^-1)X^-1(z)NoneNext

To determine the z-transform of the signal $x(-n)$, we can analyze the relationship between a signal and its z-transform.

Break Down the Problem

1. Start with the definition of the z-transform.
2. Apply the definition to the reversed signal $x(-n)$.
3. Explore the options provided in the question regarding the z-transform of $x(-n)$.

Relevant Concepts

The z-transform $X(z)$ of a discrete-time signal $x(n)$ is defined as: $X(z) = \sum_{n=-\infty}^{\infty} x(n) z^{-n}$ For the signal $x(-n)$, the z-transform can be defined as: $Y(z) = \sum_{n=-\infty}^{\infty} x(-n) z^{-n}$ By changing the index of summation to $m = -n$ (thus $n = -m$), we can reformulate the z-transform of $x(-n)$: $Y(z) = \sum_{m=-\infty}^{\infty} x(m) z^{m} = X(z^{-1})$

Analysis and Detail

Thus, we have derived:

1. The z-transform of the signal $x(-n)$ is the same as the z-transform of $x(n)$ evaluated at $z^{-1}$: $Y(z) = X(z^{-1})$

Verify and Summarize

The transformation shows that reversing a signal in time corresponds to evaluating its z-transform at the reciprocal of $z$. Therefore, the correct answer is encapsulated in the options provided.

Final Answer

The z-transform of the signal $x(-n)$ is $X(z^{-1})$.