Convolution

• An operator that takes a kernel $h(t)$ and a signal $x(t)$ and accumulates the interactions between both
  • For each $\tau$, the reflected kernel $h(-τ)$ is shifted by $t$ to become $h(t−τ)$. This shifted kernel is then multiplied with the signal $x(τ)$
  • Fancy multiplication; the kernel acts as a weight/filter that determines how much influence different parts of the signal have on the output
• Linear, distributive, associative, and commutative

LTI System Responses

• If fed a weighted sum of impulses and the impulse response is known, the output is the corresponding weighted sum of response output

• If fed a weighted sum of functions and the response for each of those functions is known, the output is the corresponding weighted sum

Numerical Convolution

• Responses can be approximated when integral convolution is not practical
• MATLAB uses conv to approximate convolution integrals with midpoint method

MATLAB Code

References

• Continuous-time system response (course handout)
• https://betterexplained.com/articles/intuitive-convolution/