Question

Given X = (a, b, c, d) as the input, a linear time invariant system produces an output y = [x, x, x, x, ......, < repeated N-times >]. The impulse response of the system is

• A. $\sum _{i=0}^{N-1}\delta (n-4i)]$
• B. $\sum _{i=0}^{N-1}\delta (n-i)$
• C.
  $u\left[n\right]-u[n-N-1]$
• D.
  $u\left[n\right]-u[n-N]$

Answer 1

The correct option is A

$\sum _{i=0}^{N-1}\delta (n-4i)]$

$\text{Given,}$
$\text{x = [a, b, c, d] input signal}$
$\text{y = [x, x, x, x,..........N times] output signal}$

That is clearly observed for different inputs, outputs are same periodic with fundamental period N = 4.


In general, this can be written as

$\sum _{i=0}^{N-1}\delta (n-4i)]$

So, the correct answer is (a).