Appendix 02 Linear Systems Timeinvariant systems ft Linear
- Slides: 31
Appendix 02 Linear Systems - Time-invariant systems f(t) Linear System g(t) 1
Linear System A linear system is a system that has the following two properties: Homogeneity: Scaling: The two properties together are referred to as superposition. 2
Time-invariant System A time-invariant system is a system that has the property that the shape of the response (output) of this system does not depend on the time at which the input was applied. If the input f is delayed by some interval T, the output g will be delayed by the same amount. 3
Harmonic Input Function Linear time-invariant systems have a very interesting (and useful) response when the input is a harmonic. If the input to a linear time-invariant system is a harmonic of a certain frequency , then the output is also a harmonic of the same frequency that has been scaled and delayed: 4
Transfer Function H( ) The response of a shift-invariant linear system to a harmonic input is simply that input multiplied by a frequency-dependent complex number (the transferfunction H( )). A harmonic input always produces a harmonic output 5 at the same frequency in a shift-invariant linear system.
Transfer Function Convolution F( ) H( ) G( ) f(t) h(t) g(t) 6
Convolution f(t) h(t) g(t) 7
Impulse Response [1/4] 8
Impulse Response [2/4] f(t) h(t) g(t) F( ) H( ) G( ) 9
Impulse Response [3/4] Convolution g(t) t 10
Impulse Response [4/4] Convolution = * 11
Convolution Rules 12
Some Useful Functions A B a/2 b 13
The Impulse Function [1/2] The impulse is the identity function under convolution 14
The Impulse Function [2/2] 15
Step Function [1/3] b b 16
Step Function [2/3] b b 17
Step Function [3/3] b 18
Smoothing a function by convolution b 19
Edge enhancement by convolution b 20
Discrete 1 -Dim Convolution [1/5] Matrix 21
Discrete 1 -Dim Convolution [2/5] Example 22
Discrete 1 -Dim Convolution [3/5] Discrete operation 23
Discrete 1 -Dim Convolution [4/5] Graph - Continuous / Discrete 24
Discrete 1 -Dim Convolution [5/5] Wrapping h index array 25
Two-Dimensional Convolution 26
Discrete Two-Dimensional Convolution [1/3] 27
Discrete Two-Dimensional Convolution [2/3] 28
Discrete Two-Dimensional Convolution [3/3] Kernel matrix Input image Array of products Output image Summer x. C Scaling factor Output pixel 29
Linear System - Fourier Transform Impulse respons Input function f(t) Spectrum of input function F( ) h(t) H( ) g(t) Output function G( ) Spectrum of output function Transfer function 30
End 31
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