Impulse Response, Convolution, and Difference Equations
Stub. The impulse response as a complete description of an LTI system, the convolution sum that lets it predict the output for any input, and linear constant-coefficient difference equations as the other standard way to specify a discrete-time system.
For an LTI system, knowing the response to a single impulse is enough to predict the response to any input, via superposition over shifted, scaled impulses — the convolution sum. Difference equations give an alternative, often more compact, specification of the same system (an implicit recursive relation between input and output samples), and relating the two representations is the first step toward transform-domain analysis.
Outline (TODO — flesh out each)
- Impulse response of a discrete-time system
- Convolution — definition, properties (commutativity, associativity, distributivity)
- Linear constant-coefficient difference equations