Question

If the output of discrete-time LTI system is always identical to the inpu
Signal, then the unit-impulse response h(n) is​

Answer 1

Answer:

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Answer 2

Answer:

As a result, $a^n$ is an eigenfunction of a discrete-time LTI system.

Explanation:

Step : 1Despite the previous two responses, I'll add one more since I believe your final paragraph has yet to be fully explained.

Step : 2A transfer function exists in a discrete-time LTI system with impulse response $h[n]$.

$H(z)=\sum_{n=-\infty}^{\infty} h[n] z^{-n}$..........(1)

Step : 3The convolution sum determines its output $y[n]$ for a given input sequence $x[n]$.

$y[n]=\sum_{k=-\infty}^{\infty} h[k] x[n-k]$.........(2)

For $x[n]=a^n$ we get from (2)

$y[n]=\sum_{k=-\infty}^{\infty} h[k] a^n a^{-k}=a^n \sum_{k=-\infty}^{\infty} h[k] a^{-k}=a^n H(a)$.......(3)

Step : 4In the last equality, Eq. (1) was applied. Eq. (3) shows that for $x[n]=a^n$, A discrete-time LTI system's output is just a weighted version of the input signal. As a result, $a^n$ is an eigenfunction of a discrete-time LTI system.

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