convolution - ics.uci.edumajumder/vc/new-lectures/convolution.pdf · •convolution: response of a...
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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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