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Question

# Consider a system S with input x[n] and output y[n] related by y[n]=x[n]{g[n]+g[nâˆ’1]} Select the correct statement about system S

A
If g[n] = 1 for all n, then S is time invariant.
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B
If g[n] = n for all n, then S is time variant.
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C
If g[n] = δ[n], then S is time variant.
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D
If g[n] = 1 + (-1)n, then S is time invariant.
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Solution

## The correct option is D If g[n] = 1 + (-1)n, then S is time invariant.(a) g[n] = 1 g[n - 1] = 1 So, y[n] = x[n] [1 + 1] = 2 x[n] Thus system is time invariant (b) g[n] = n So, g[n - 1] = (n - 1) Now, y[n] = x[n] [n + n -1] x[n] [2n - 1] which is time variant. (c) g[n]=1+(−1)n g[n−1]=1+(−1)n−1 y[n]=x[n][1+(−1)n+1+(−1)n−1] =x[n][2] = 2x [n] which is time invariant (d) g[n] = δ[n] g[n-1] = δ[n−1] g[n]=x[n][δ[n]+δ[n−1]] which is time variant.

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