Lecture 10 - Notes

January 26, 2016

We are going over what happened last week again this week.

Convolution Properties

Commutativity

$$x * h = h * x$$

Associativity

$$x * (y * h) = (x * g) * h$$

Distributivity

$$x * (y + h) = (x * y) + (y * h)$$

Associativity with scalar multiplication

$$a (y * g) = (a f) * g$$

LTI Systems from their Unit Impulse

Causal

definition: A Linear time invariant system is causal if the unit impulse, $h[n]$, is zero for all $n$ greater than zero, i.e.,

$$h[n] = 0 \text{ for all } n > 0$$

Stability

definition: A Linear time invariant system is stable if and only if,

$$\sum_{n=-\infty}^{\infty} \left| h[n] \right| \lt \infty$$

and $h[n]$ is absolutely summable (a finite energy signal).

Invertibility

definition: A linear time invariant system is invertible if we can find another system that can perfectly reverse the transform of the system. In terms of $h[n]$ the system is invertible if there exist $h^\prime[n]$ such that,

$$h[n] * h^\prime[n] = \delta[n].$$