Skip to main content

Plot sequence; properties of LTI system

Short answer

The sequence values are x[0]=x[1]=x[2]=1, x[3]=0, x[4]=5, x[5]=0, and x[n]=n for n≥6. LTI properties include linearity, time invariance, convolution description, commutativity, associativity, distributivity, causality iff h[n]=0 for n<0, BIBO stability iff sum |h[n]| is finite, memoryless iff h[n]=K delta[n], invertibility, and eigenfunction property.

Plot the sequence x[n] = u[n] − u[n−3] + 5δ[n−4] + n u[n−6]. List out the properties of LTI system.

Plot of x[n] = u[n] − u[n−3] + 5δ[n−4] + n u[n−6]

Term by term:

  • u[n] − u[n − 3]: a rectangle, value 1 at n = 0, 1, 2.
  • 5δ[n − 4]: a single sample of height 5 at n = 4.
  • n u[n − 6]: a ramp, value n for n ≥ 6 (6, 7, 8, …).

So x[n] = 1 (n = 0, 1, 2); 0 (n = 3); 5 (n = 4); 0 (n = 5); n (n ≥ 6); 0 for n < 0.

n012345678
x[n]111050678
 x[n]
  8 |                          |
  7 |                       |
  6 |                    |
  5 |             |
  1 | |  |  |
  0 +-+--+--+--+--+--+--+--+--+--  n
    0  1  2  3  4  5  6  7  8 ...

Properties of LTI systems

  1. Linearity and time-invariance: superposition holds and a shift in input shifts the output equally. The system is described by h[n]: y[n] = x[n] * h[n].
  2. Commutative: x * h = h * x.
  3. Associative: (x * h₁) * h₂ = x * (h₁ * h₂), so cascaded systems can be interchanged (h = h₁ * h₂).
  4. Distributive: x * (h₁ + h₂) = x * h₁ + x * h₂, giving parallel combination.
  5. Causality: causal if and only if h[n] = 0 for n < 0.
  6. Stability (BIBO): stable if and only if Σ |h[n]| < ∞.
  7. Memoryless: only if h[n] = 0 for n ≠ 0 (h[n] = Kδ[n]).
  8. Invertibility: an inverse system exists if h * h_inv = δ[n].
  9. Identity: x * δ[n] = x[n]; shift: x * δ[n − n₀] = x[n − n₀].
  10. Complex exponentials are eigenfunctions: e^(jωn) → H(e^(jω)) e^(jωn).

Related questions

All Digital Signal Analysis and Processing chapter 1 questions →