1 Appendix 02 Linear Systems - Time-invariant systems Linear System Linear System f(t) g(t)

2 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.

3 A time-invariant system is a system that has the property that the 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. Time-invariant System

4 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: Harmonic Input Function

5 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 at the same frequency in a shift-invariant linear system. Transfer Function H(  )

6 Transfer Function Convolution h(t) f(t) g(t) H(  ) F()F() F()F() G()G() G()G()

7 ConvolutionConvolution h(t) f(t) g(t)

8 Impulse Response [1/4]

9 Impulse Response [2/4] h(t) f(t) g(t) H(  ) F(  ) G(  )

10 Impulse Response [3/4] Convolution t g(t)

11 Impulse Response [4/4] Convolution =*

12 Convolution Rules

13 Some Useful Functions A a/2 B b

14 The Impulse Function [1/2] The impulse is the identity function under convolution

15 The Impulse Function [2/2]

16 Step Function [1/3] b b

17 Step Function [2/3] b b

18 Step Function [3/3] b

19 Smoothing a function by convolution b

20 b Edge enhancement by convolution

21 Discrete 1-Dim Convolution [1/5] Matrix

22 Discrete 1-Dim Convolution [2/5] Example

23 Discrete 1-Dim Convolution [3/5] Discrete operation

24 Discrete 1-Dim Convolution [4/5] Graph - Continuous / Discrete

25 Discrete 1-Dim Convolution [5/5] Wrapping h index array

26 Two-Dimensional Convolution

27 Discrete Two-Dimensional Convolution [1/3]

28 Discrete Two-Dimensional Convolution [2/3]

29 Discrete Two-Dimensional Convolution [3/3]   x C SummerScaling factor Kernel matrix Input image Output image Array of products Output pixel

30 Linear System - Fourier Transform h(t) H(  ) h(t) H(  ) f(t) g(t) F(  ) G(  ) Input function Spectrum of input function Output function Spectrum of output function Impulse respons Transfer function

31 End