Fourier Transform 4: z-transform (part 2) & Introduction to 2D Fourier Analysis

Transcription

1 VU Signal and Image Processing Fourier Transform 4: z-transform (part 2) & Introduction to 2D Fourier Analysis Torsten Möller + Hrvoje Bogunović + Raphael Sahann torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at raphael.sahann@univie.ac.at vda.cs.univie.ac.at/teaching/sip/7s/ Raphael Sahann

2 Overview Sampling & Impulse Train Fourier Transform (D) properties convolution theorem sampling in the Fourier space z-transform inverse z-transform 2D Fourier Transform Raphael Sahann 2

3 Check Yourself Two signals and two regions of convergence. ROC z-plane x[n] =! 7 8 " n u[n] n z z y[n] =! 7 8 " n u[ n] z-plane n z z 7 8 ROC Prof. Dennis Freeman, MIT, 20 3

4 Properties of Z Transforms The use of Z Transforms to solve di erential equations depends on several important properties. Property x[n] X(z) ROC Linearity ax [n]+ bx 2 [n] ax (z)+ bx 2 (z) (R fl R 2 ) Delay x[n ] z X(z) R Multiply by n nx[n] dx(z) z dz R Convolve in n ÿœÿ x [m]x 2 [n m] X (z)x 2 (z) (R fl R 2 ) m= Œ 45 Prof. Dennis Freeman, MIT, 20 4

5 Relationship between the z- Transform and the Fourier Transform or Raphael Sahann 5

6 Relationship between the z- Transform and the Fourier Transform This z-transform can be interpreted as the Fourier transform of the product of the original sequence x[n] and the exponential sequence r -n r = reduces to the Fourier transform of x[n]. Raphael Sahann 6

7 Relationship between the z- Transform and the Fourier Transform image source: Raphael Sahann 7

8 Relationship between the z- Transform and the Fourier Transform Interpreting Fourier transform as the z-transform on the unit circle in the z-plane corresponds to wrapping the frequency axis around the unit circle. Inherent periodicity in frequency is captured naturally, since a change of angle of 2π radians in the unit circle corresponds to traversing the unit circle once and returning to the same point. Raphael Sahann 8

9 Relationship between the z- Transform and the Fourier Transform If the Region of Convergence (ROC) includes the unit circle, the Fourier transform and all its derivatives with respect to must be continuous functions of. Raphael Sahann 9

10 Example 3.5 Consider the sequence with a = -/3 we obtain and using a = /2 yields Raphael Sahann 0

11 Example 3.5 Consider the sequence with a = -/3 we obtain and using a = /2 yields Raphael Sahann

12 Example 3.5 by the linearity of the z-transform image source: Raphael Sahann 2

13 Example 3.7 n u[n] n u[ n ] x[n] = X(z) = 2 2 z {z } z > z {z } z < 3 Since there is no overlap between z >/2 and z </3, x[n] has no z-transform (nor Fourier transform) representation. Raphael Sahann 3

14 Example 3.7 n u[n] n u[ n ] x[n] = X(z) = 2 2 z {z } z > z {z } z < 3 Since there is no overlap between z >/2 and z </3, x[n] has no z-transform (nor Fourier transform) representation. Raphael Sahann 4

15 Overview Sampling & Impulse Train Fourier Transform (D) z-transform inverse z-transform 2D Fourier Transform Raphael Sahann 5

16 Inverse z-transform x[n] = 2 j I C X(z)z n dz The inverse z-transform is a complex contour integral, where C represents a closed contour within the ROC of the z-transform. Too complex for a typical task in this domain, so we use less formal procedures Raphael Sahann 6

17 Inspection Method We inspect a z-transform by looking up its transform in a table of common z-transforms. Find the inverse z-transform for: X(z) = 2 z, z > 2 Raphael Sahann 7

18 Common z-transform pairs image source: Raphael Sahann 8

19 Inspection Method Find the inverse z-transform for: X(z) = 2 z, z > 2 n u[n] right-sided x[n] = 2 Raphael Sahann 9

20 Inverse z-transform Inspection Method Partial Fractions > build partial fractions until they can be interpreted by the inspection method Power Series Expansion > Taylor series expansion of z-transform, interpret result with inspection method 20

21 Properties of Z Transforms The use of Z Transforms to solve di erential equations depends on several important properties. Property x[n] X(z) ROC Linearity ax [n]+ bx 2 [n] ax (z)+ bx 2 (z) (R fl R 2 ) Delay x[n ] z X(z) R Multiply by n nx[n] dx(z) z dz R Convolve in n ÿœÿ x [m]x 2 [n m] X (z)x 2 (z) (R fl R 2 ) m= Œ 45 Prof. Dennis Freeman, MIT, 20 2

22 Overview Sampling & Impulse Train Fourier Transform (D) z-transform 2D Fourier Transform Raphael Sahann 22

23 How to represent an image? An image is made of pixels (=picture elements) the coordinate values are discretized Laurent Condat / Torsten Möller 23

24 What is a Fourier Transform? (D) Let s go back to (spatial) representation of functions: f(t) f(t) X n= X n= Z f(t) (t n T )dt c[n] (t n T ) (t n T ) Fourier series into Frequency Domain: c n = T f(t) Z T/2 X n= T/2 f(t)e j 2 n T c n e j 2 n T t t dt 24

25 What is a Fourier Transform? (2D) Let s go back to (spatial) representation of functions: f(x) f(x) X n= X n= Z f(s) (s n X)ds (x n X) c[n] (x n X) X = x 0 0 y Fourier series into Frequency Domain: c n = T x T y f(x) X n= Z T/2 T/2 c n e j 2 n T x f(x)e j 2 n T x dx 25

26 There are 4 Fourier Transforms! (D) Recall Fourier series: c n = T f(t) f(t) is periodic with period T! Z T/2 General Fourier Transform requires no X n= T/2 f(t)e j 2 n T c n e j 2 n T t t periodicity: Z F (!) = f(t)e j2!t dt Z f(t) = F (!)e j2!t dt 26

27 There are 4 Fourier Transforms! (2D) Recall Fourier series: f(x) is periodic with period T! c n = T x T y f(x) General Fourier Transform requires no X n= Z T/2 T/2 c n e j 2 n T x f(x)e j 2 n T x dx periodicity: Z F (!) = f(x)e j2! x dx Z f(x) = F (!)e j2! x dx 27

28 DFT the most important one (D) Discrete Fourier Transform (DFT) requires periodicity in both transform pairs F m = M X n=0 f n e j2 mn/m f n = M M X m=0 F m e j2 mn/m 28

29 DFT the most important one (2D) Discrete Fourier Transform (DFT) requires periodicity in both transform pairs F ab = M X N X f mn e j2 (am/m+bn/n) m=0 n=0 f mn = M X N X F ab e j2 (am/m+bn/n) MN a=0 b=0 29

30 All Fourier Transforms (D) Spatial Domain Frequency Domain FT FS Fourier Series DFT Discrete FT DTFT Discrete Time FT f(t) = f(t) = Z F (!)e j2!t dt continuous X n= c n e j 2 n T continuous + periodic f n = M f n = 2 MX m=0 Z t F m e j2 mn/m discrete + periodic discrete F (!) = c n = T F m = F (!)e j!n d! F (!) = Z Z T/2 MX n=0 T/2 f(t)e j2!t dt continuous discrete f n e j2 mn/m X n= f(t)e j 2 n T discrete + periodic t dt f n e j!n d! continuous + periodic 30

31 All Fourier Transforms (nd) Spatial Domain Frequency Domain FT continuous continuous FS Fourier Series continuous + periodic discrete DFT Discrete FT discrete + periodic discrete + periodic DTFT Discrete Time FT discrete continuous + periodic 3

32 What happens to an impulse? (D) it is basically a constant! c n = T Z T/2 T/2 (t)e j 2 n T t c n = T e0 c n = T 32

33 What happens to an impulse? (2D) it is basically a constant! c n = T x T y Z T/2 T/2 (x)e j 2 n T x dx c n = T x T y e 0 c n = T x T y 33

34 What about a shifted impulse? (D) the shifts remain as frequencies c n = T Z T/2 T/2 (t t 0 )e j 2 n T t c n = T e j 2 n T t 0 34

35 What about a shifted impulse? (2D) the shifts remain as frequencies c n = T x T y Z T/2 T/2 (x x 0 )e j 2 n T x dx c n = T x T y e j 2 n T x 0 35

36 What happens to an impulse train? Impulse train is periodic apply Fourier series, will not do the math here, see book: X s T (t) = (t n T ) X T n S T (!) = n= (! T ) distance between impulses grows inversely 36

37 What happens to an impulse train? X = x 0 0 y Impulse train is periodic apply Fourier series, will not do the math here, see book: X s X (x) = (x Xn) n= S X (!) = X X (! X n) n= distance between impulses grows inversely 37

38 What happens to a box? it is the well-known sinc function sinc(t) = sin t t 38

39 What happens to a box? it is the well-known sinc function sinc(tt,zz)= sin(t t) t sin(z z) z 39

40 What is the Fourier Transform of a convolution? convolution <==> multiplication: f h(x) () F (!)H(!) multiplication <==> convolution: f(x)h(x) () F (!) H(!) 40

41 What is sampling in 2D? (mathematically speaking) modeled through an impulse if x = y =0 (x, y) = 0 if t 6= 0 not really a function, but a distribution: Z Z (x, y)dxdy = 4

42 The sifting property picking a value off from f: more general: Z Z f(x) (x)dt = f(0) f(x) (x x 0 )dx = f(x 0 ) 42

43 What is sampling? f(t) s T (t) = X f(n T ) (t n T ) X f[n] (t n T ) 43

44 The 2D impulse train pick up multiple values of f at once: 44

45 Sampling in the Fourier Domain (2D) T x > 2 max T y > 2µ max 45

46 Aliasing 46

47 Aliasing 47

48 Moire patterns 48

49 Moire patterns 49

50 Moire patterns Raphael Sahann 50

51 Periodic patterns Laurent Condat / Torsten Möller 5

52 Fourier Spectrum 52

53 Moire patterns 53