Lecture 1: Zero-State Respone
The steps are similar to what we did before: introduce delta function, then the impulse response, then the convolution sum.
Discrete time impulse function:
(Plot). We have the discrete-time impulse response :
Then is the solution when initial conditions are all zero:
We can find a numerical solution by substitution.
Zero-state response : Observe that we can write
Now we add up the response of the system to each of these outputs:
Adding these up,
Same sorts of properties as we had before:
- Commutative
- Distributive
- Associative
- Shifting
- Convolution. with impulse
- Width:
- If has length points and has length points, then has length points. (Note length given in points .)
For causal systems with causal inputs
Total response = zero-input component + zero-state component.