Convolution

Outline• Linear SystemsProperties

• Response of Linear System• ConvolutionProperties

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Properties of Linear System1. Homogeneity:

2. Additivity:

3. Shift Invariance:

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Other Properties of Linear Systems1. Commutative:

2. Superposition: If each generates multiple outputs, Then the addition of inputs generates an addition of outputs.

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Decomposition - Synthesis

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Response of Linear System• Impulse: Signal with only one non-zero sample. • Delta (δ[t]) is an impulse with non-zero sample at t = 0

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Response of Linear System• Impulse response h[t] output of the system to the input δ[t].

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Response of Linear System• Impulse response h[t] output of the system to the input δ[t].

• Convolution: Response of a linear system with impulse response, h, to a general signal

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Convolution – Input side

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a1 a2 a3 a4 a5 a6 a7

i=1

Input

Kernel

Output

k1 k2 k3

i=2

k1 k2 k3

i=3

Convolution – Output side

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a1 a2 a3 a4 a5 a6 a7Input

Kernel

Output

k3 k2 k1

i=1 i=2

k3 k2 k1

i=3

Convolution

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s[m]

h[m]

s[0].h[n]

s[1].h[n-1]

s[2].h[n-2]

Visualization of Convolution

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2D Convolution

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Properties of Convolution• All pass system

• Amplifier (k>0) / attenuator (k<0)

• Delay

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Properties of Convolution• Conducive

• Associative

• Distributive

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Properties of Convolution• Cascading convolutions

• Combination of parallel convolutions

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