(i) Memory less

(ii) Causal

(iii) Linear

(iv) Time invariant

$Y[n] = X [n^2]$

Mumbai University > EXTC > Sem 4 > Signals and Systems

Marks : 05

Year : DEC 2014

(i) If present output depends on either past or future input, then system is called as Memory or Dynamic system. If present output depends only on present input, then system is called as Memory-less or Static system.

In the above system if n=2, then y[n] = x [4]. Thus the output depends on future inputs.

∴ It is not a Memory less system.

(ii) If present output depends only on present input or past input, then system is called as Causal system. If present output depends on future input, then system is called as Non-Causal system.

In the above system if n=2, then y[n] = x [4]. Thus the output depends on future inputs.

∴ It is not a Causal system.

(iii) Response of the system to two inputs $x_1 [n]$ and $x_2 [n]$ is,

$$y_1 [n]= f(x_1 [n])=x_1 [n^2] $$

$y_2 [n]= f(x_2 [n])=x_2 [n^2]$

By using linear combination of o/p,

$y_n [n]=ay_1 [n]+by_2 [n] \\ =ax_1 [n^2 ]+bx_2 [n^2] $

Now by using linear combination of i/p,

$y_{n}' [n]=f(ay_1 [n]+by_2 [n]) \\ =ax_1 [n^2 ]+bx_2 [n^2] \\ ∴y_n [n] = y_n' [n]$

thus it is a Linear system.

(iv) If input x[n] is delayed, then output becomes

$y [n, k] =f(x[n^2-k]) \\ = x [n^2-k ] …………… (1)$

Replace n by n-k, then output becomes

$y [n-k] = x [(n-k)^2] ………….... (2)$

Since $y [n, k] ≠ y [n-k],$ System is Time Variant.