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Convolution

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LTI:

h(t)

LTI:

h[n]

g(t)

g[n]

g(t) h(t)

g[n] h[n]

Example: g[n] = u[n] – u[3-n]

h[n] = [n] + [n-1]

- Method 1: “running sum”
- Plot x and h vs. m
- Flip h over vertical axis to get h[-m]
- Shift h[-m] to obtain h[n-m]
- Multiply to obtain x[m]h[n-m]
- Sum on m, m x[m] h[n-m]
- Increment n and repeat steps 3~6

- Method 2: Superposition method
- We know [n] h[n] and the system is LTI.
- Therefore, a0[n-n0] a0h[n-n0]
- a0[n-n0] + a1[n-n1] + … a0h[n-n0] + a1h[n-n1] …
- Since the input can be expressed as a sum of shifted unit samples:
x[n] = … x[-2][n+2] + x[-1][n+1] + x[0][n] + x[1][n-1] + …

- The out is then a sum of shifted unit sample responses:
x[n0][n-n0] + x[n1][n-n1] + … x[n0]h[n-n0] + x[n1]h[n-n1] …

1k=-1x[k]h[n-k]

(The Convolution Sum!)