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. Commutative: 2. Superposition: If each generates multiple outputs, Then the addition of inputs generates an addition of outputs. 4

Decomposition - Synthesis 5

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 6

Response of Linear System • Impulse response h[t]  output of the system to the input δ [t]. 7

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 8

Convolution – Input side i=1 i=2 i=3 Input a1 a2 a3 a4 a5 a6 a7 Kernel k1 k2 k3 k1 k2 k3 � � � � � � � � � � � � � � � � Output � � � � � � � � � � � � � � � � 9

Convolution – Output side Input a1 a2 a3 a4 a5 a6 a7 Kernel k3 k3 k2 k2 k1 k1 � � � � � � � � � � � � Output � � � � � � � � � � � � � � � � � � � � i=3 i=1 i=2 10

Convolution s[0].h[n] s[m] s[1].h[n-1] h[m] s[2].h[n-2] 11

Visualization of Convolution 12

2D Convolution 13

Properties of Convolution • All pass system • Amplifier (k>0) / attenuator (k<0) • Delay 14

Properties of Convolution • Conducive • Associative • Distributive 15

Properties of Convolution • Cascading convolutions • Combination of parallel convolutions 16