Convolution
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Convolution. 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]. Convolution methods:. 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]

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Convolution

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Convolution

Convolution

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]


Convolution methods

Convolution methods:

  • 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


Convolution methods1

Convolution methods:

  • 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!)


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