SLIDE 1
Outline
- Linear Systems
Properties
- Response of Linear System
- Convolution
Properties
Convolution Outline Linear Systems Properties Response of Linear - - PowerPoint PPT Presentation
Convolution Outline Linear Systems Properties Response of Linear - - PowerPoint PPT Presentation
Convolution Outline Linear Systems Properties Response of Linear System Convolution Properties 2 Properties of Linear System 1. Homogeneity: 2. Additivity: 3. Shift Invariance: 3 Other Properties of Linear Systems 1.
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Properties of Linear System
- 1. Homogeneity:
- 2. Additivity:
- 3. Shift Invariance:
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Other Properties of Linear Systems
- 1. 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
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Convolution – Output side
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a1 a2 a3 a4 a5 a6 a7
Input Kernel Output
k3 k2 k1
- i=1
i=2
- k3
k2 k1
i=3
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Convolution
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s[m] h[m] s[0].h[n] s[1].h[n-1] s[2].h[n-2]
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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