Properties of DT LTI systems

A discrete-time LTI system is linear and time invariant. It is completely described by its impulse response $h[n]$, and its output is the convolution sum $y[n] = \sum_k x[k]\,h[n-k] = x[n]*h[n]$. Its main properties are listed below, with the test in terms of $h[n]$.

1. Commutative property

$x[n]*h[n] = h[n]*x[n]$: input and impulse response can be interchanged. Example: $\{1,1\}*\{1,2\} = \{1,2\}*\{1,1\} = \{1,3,2\}$.

2. Associative property (cascade)

$\left(x*h_1\right)*h_2 = x*\left(h_1*h_2\right)$. Two systems in cascade equal one system with $h[n] = h_1[n]*h_2[n]$, and the order of the cascade does not matter.

x -->[h1]-->[h2]--> y  ==  x -->[h1*h2]--> y

3. Distributive property (parallel)

$x*(h_1 + h_2) = x*h_1 + x*h_2$. Two systems in parallel equal one system with $h[n] = h_1[n] + h_2[n]$.

4. Memory

The system is memoryless only if $h[n] = K\delta[n]$, i.e. $y[n] = Kx[n]$. Otherwise it has memory. Example: $h[n] = \delta[n] + \delta[n-1]$ ($y[n] = x[n] + x[n-1]$) has memory.

5. Causality

Causal if $h[n] = 0$ for $n < 0$, so the output uses no future inputs. Example: $h[n] = (0.5)^n u[n]$ is causal; $h[n] = \delta[n+1]$ ($y[n] = x[n+1]$) is not.

6. Stability (BIBO)

Stable if the impulse response is absolutely summable:

$$\sum_{n=-\infty}^{\infty} |h[n]| < \infty$$

Example: $h[n] = (0.5)^n u[n]$ gives $\sum = 2$, stable; $h[n] = u[n]$ (accumulator) gives an infinite sum, unstable. Every FIR system is stable.

7. Invertibility

Invertible if an inverse system $h_i[n]$ exists with $h[n]*h_i[n] = \delta[n]$. Example: the accumulator $h[n] = u[n]$ has inverse $h_i[n] = \delta[n] - \delta[n-1]$ (first difference).

8. Identity and unit step response

$x[n]*\delta[n] = x[n]$ and $x[n]*\delta[n-k] = x[n-k]$. The step response is the running sum of $h[n]$: $s[n] = \sum_{k=-\infty}^{n} h[k]$.