Check for the following system is, i) Static or Dynamic, ii) Causal or Non-Causal, iii) Linear or Non-Linear, iv) Time Variant or Time Invariant, v) Stable or Unstable

a) y(n) = 2x(n-1) + x(2n)

1. Static or Dynamic
Since the system requires a previous value (n-1) to obtain the current value it needs a memory to store the previous values. Hence, a system is Dynamic.
2. Causal or Non-Causal
Since the system requires a future value x(2n) to obtain the current value, the system is Non-Causal.
3. Linear or Non-Linear
The system of linear is,
T[a₁x₁(n) + a₂x₂(n)] = a₁T[x₁(n)] + a₂T[x₂(n)]
Y₃ = Y₂ + Y₁
Y₁ = 2x₁(n-1) + x₁(2n)
Y₂ = 2x₂(n-1) + x₂(2n)
L.H.S = 2[a₁x₁(n-1) + a₂x₂(n-1)] + a₁x₁(2n) + a₂x₂(2n)
R.H.S = a₁[2x₁(n-1) + x₁(2n)] + a₂[2x₂(n-1) + x₂(2n)]

= 2a₁x₁(n-1) + 2a₂x₂(n-1) + a₁x₁(2n) + a₂x₂(2n)
= L.H.S
Therefore, system is Linear.
4. Time Variant or Time Invariant
y(n, k) = 2x(n-1-k) + x(2n - k)
y(n-k) = 2x(n-k-1) + x(2(n-k))

= 2x(n-k-1) + x(2n -2k)
y(n, k) ≠ y(n-k)
Therefore, System is Time Variant
5. Stable or Unstable
If n → ∞, input signal always remains finite thus input remains finite.
The given system remains finite, ∴ System is stable

b) y(n) = a[x(n)]² + bx(n)

1. Static or Dynamic
Since the system depends only on the present value it does not need any memory to store value. Therefore, the system is Static.
2. Causal or Non-Causal
Since the system depends only on present values it does not depend on any future value. Therefore, the system is Causal.
3. Linear or Non-Linear
The system of linear is,
T[a₁x₁(n) + a₂x₂(n)] = a₁T[x₁(n)] + a₂T[x₂(n)]
Y₃ = Y₂ + Y₁
Y₁ = a[x₁(n)]² + bx₁(n)
Y₂ = a[x₂(n)]² + bx₂(n)
L.H.S = a{[a₁[x₁(n)] + a₂[x₂(n)]}²
R.H.S = a₁{a[x₁(n)]² + bx₁} + a₂{a[x₂(n)]² + bx₂}

=aa₁[x₁(n)]² + bx₁(n) + aa₂[x₂(n)]² + bx₂(n)
= a{a₁[x₁(n)]² + a₂[x₂(n)]²} + b[a₁x₁(n) + a₂x₂(n)]
≠ L.H.S
Therefore, system is Non-Linear.
4. Time Variant or Time Invariant
y(n, k) = a[x(n-k)]² + bx(n-k)
y(n-k) = a[x(n-k)]² + bx(n-k)
y(n, k) = y(n-k)
Therefore, System is Time Invariant.
5. Stable or Unstable
If n → ∞, input signal always remains finite and input remains finite too.
The given system remains finite, ∴ System is stable