Define LTI; properties of DT LTI systems

A linear time-invariant (LTI) system is a system that is both linear (obeys superposition: $a x_1+b x_2 \rightarrow a y_1+b y_2$) and time-invariant (a shift of the input only shifts the output: $x[n-n_0]\rightarrow y[n-n_0]$). A discrete-time LTI system is completely described by its impulse response $h[n]$, and its output is the convolution sum

$$y[n]=x[n]*h[n]=\sum_{k=-\infty}^{\infty}x[k]\,h[n-k]$$

Properties of discrete-time LTI systems

1. Commutative: $x[n]*h[n]=h[n]*x[n]$. Input and impulse response can be interchanged.
2. Associative: $\{x[n]*h_1[n]\}*h_2[n]=x[n]*\{h_1[n]*h_2[n]\}$. Two systems in cascade are equal to one system with $h=h_1*h_2$, and the order of the cascade does not matter.
3. Distributive: $x[n]*\{h_1[n]+h_2[n]\}=x*h_1+x*h_2$. Two systems in parallel equal one system with $h=h_1+h_2$.
4. Memory: the system is memoryless only if $h[n]=K\delta[n]$, i.e. $y[n]=Kx[n]$. Otherwise it has memory.
5. Causality: causal if and only if $h[n]=0$ for $n<0$. Then $y[n]=\sum_{k=0}^{\infty}h[k]x[n-k]$ uses only present and past inputs.
6. Stability (BIBO): stable if and only if $\sum_{n}|h[n]|<\infty$.
7. Invertibility: invertible if an inverse system $h_i[n]$ exists with $h[n]*h_i[n]=\delta[n]$. Example: accumulator $h=u[n]$ and first difference $h_i=\delta[n]-\delta[n-1]$.
8. Unit step response: $s[n]=\sum_{k=-\infty}^{n}h[k]$ and $h[n]=s[n]-s[n-1]$.
9. Eigenfunction property: complex exponentials pass through unchanged in shape: $z^n \rightarrow H(z)z^n$, $e^{j\omega n}\rightarrow H(e^{j\omega})e^{j\omega n}$. This is why Fourier and z-transforms are used for LTI systems.

Example: $h[n]=(0.5)^n u[n]$ is causal (zero for $n<0$), has memory, and is stable since $\sum (0.5)^n = 2<\infty$.