Question

Consider a system S with input x[n] and output y[n] related by

$y\left[n\right]=x\left[n\right]\left\{g\right[n]+g[n-1\left]\right\}$

Select the correct statement about system S

• A. If g[n] = 1 for all n, then S is time invariant.
• B. If g[n] = n for all n, then S is time variant.
• C. If g[n] = $\delta \left[n\right]$, then S is time variant.
• D. If g[n] = 1 + (-1)${}^n$, then S is time invariant.

Answer 1

Answer: (A), (B), (C), (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) $g\left[n\right]=1+(-1{)}^n$

$g[n-1]=1+(-1{)}^{n-1}$

$y\left[n\right]=x\left[n\right][1+(-1{)}^n+1+(-1{)}^{n-1}]$

$=x\left[n\right]\left[2\right]$

= 2x [n]

which is time invariant

(d) g[n] = $\delta \left[n\right]$

g[n-1] = $\delta [n-1]$

$g\left[n\right]=x\left[n\right]\left[\delta \right[n]+\delta [n-1\left]\right]$


which is time variant.