Jean Baptiste Joseph Fourier meets Stephen Hawking

Transcription

1 Jean Baptiste Joseph Fourier meets Stephen awking Presenter: Dr. Bingo Wing-Kuen ing incoln School of Engineering, University of incoln Postal address: Brayford Pool, incoln, incolnshire, N6 7S, UK. address: elephone number: 44 ()

2 y Research Focus Filter banks and wavelets (signal processing branch) Perfect reconstruction of nonuniform filter banks. Filter banks with block sampling structures. Compressions with applications to scalable JPEG image coding schemes. Denoising with applications to image and biomedical signal processing such as ECG signal and ultra sound image denoising. Edge detection and edge linking with applications to image processing such as cancer cell image diagnosis and bad potato diagnosis. Signal separations such as audio signal separations for digital audio hearing aids applications and ECG/EG signal separations, pattern recognitions such as gait recognitions for military applications and fault analysis such as machine fault detections.

3 y Research Focus Optimization (signal processing) Semi-infinite programming and functional inequality constrained optimizations with applications to filter, filter bank, wavelet kernel, sigma delta modulator and transport system designs. Nonsmooth optimizations with applications to motion estimations as well as filter, filter bank, wavelet kernel and transport system designs. Nonconvex optimizations with applications to spectral allocations for wireless communication networks as well as filter, filter bank, wavelet kernel and transport system designs. Real-time optimizations with applications to filter, filter bank and wavelet kernel designs. 3

4 y Research Focus Symbolic dynamics, fractal and chaos (control branch) Digital filters with two s complement arithmetic and saturation nonlinearity with applications to computer cryptography. Sigma delta modulators with applications to analog-to-digital conversions. Perceptron training algorithms with applications to pattern recognitions. Random early detection mechanisms with applications to internet traffic control. DC/DC converters with applications to industrial and consumer electronic products. Road traffic light signaling with applications to road traffic system control. Nano-particle quantum effect analysis with applications to nano-device fabrications. 4

5 y Research Focus Control theories (control branch) Fuzzy control with applications to time delay feedback systems, sample data control systems and chaos synchronization systems. Optimal switching control with applications to DC/DC converters and transport systems. Impulsive control with applications to sigma delta modulators. Chaos control with applications to CPIP networks, IV model systems and avian influenza model systems. 5

6 Introduction Jean Baptiste Joseph Fourier ( arch ay 83) Invention of the Fourier transform Representation of the frequency Stephen awking (8 January 94-now) Invention of a new concept of time Representation of the time What happens if time meets frequency? 6

7 Outline Frequency analysis ime frequency analysis Conclusions Q&A Session 7

8 Frequency Analysis Understanding of time and frequency x( t) sin( 5t) Example : ime range: (-,+ ).5 x(t) ime t 8

9 Frequency Analysis Understanding of time and frequency X ( ω) ( δ ( ω 5) δ ( ω + 5) ) Example : j Angular frequency 5 radians per second Frequency range (Bandwidth): x 4 8 X(ω) Frequency ω 9

10 Frequency Analysis Understanding of time and frequency t x() t otherwise ime range: (-.5,+.5) Example :.8.6 x(t) ime t

11 Frequency Analysis Understanding of time and frequency Example : X ω sin ω ω ( ) Frequency range: (-,+ )..8.6 because X ( ω) x( t) + jωt e dt X(ω) Frequency ω

12 Frequency Analysis Understanding of time and frequency Representation with a finite duration in time will result to a representation with an infinite bandwidth in frequency. hat means good localization in time will result to a bad localization in frequency. Representation with a finite bandwidth in frequency will result to a representation with an infinite duration in time. hat means good localization in frequency will result to a bad localization in time. Are there any representations that are good localizations in both time and frequency?

13 3 Frequency Analysis Backgrounds on linear time invariant filters Definition A system which is linear. A system which is time invariant. () () N i i i N i i i t y t x α α ( ) ( ) ( ) t t y t t x R t ( ) ( ) N i i i N i i i n y n x α α ( ) ( ) ( ) n n y n n x Z n

14 4 Frequency Analysis Backgrounds on linear time invariant filters Systems represented by linear time invariant filters any practical systems can be represented by linear time invariant filters. ( ) ( ) m m m m N n n n n dt t x d b dt t y d a ( ) ( ) q q N p p q n x b p n y a

15 Frequency Analysis Backgrounds on linear time invariant filters Impulse response () ( ( )) ( ) ( ( )) h t δ t h n δ n A system is linear and time invariant if and only if the input output relationship in the time domain is governed by the convolution and that in the frequency domain is governed by the multiplication. hat is y () t + x( τ ) h( t τ ) dτ y( n) x( m) h( n m) n Y ( ω) X ( ω) ( ω) + 5

16 Frequency Analysis Backgrounds on linear time invariant filters y Frequency response jωt ( ω ) h( t) e dt ( ω) h( n) Frequency response is the eigen function of linear time invariant filters. ence, if the inputs of linear time invariant filters consist of monotonic frequency only, then the outputs of linear time invariant filters also consist of the same monotonic frequency only with the gains equal to ( )and the phase shifts equal to ( ). ω + + n e jωn jω ( tτ ) jωt jω ( nm) jωn () t h( τ ) e dτ e ( ω ) y( n) h( m) e e ( ω ) Y + ( ω) ( ω) δ ( ω ω ) ( ω ) δ ( ω ) ω + n ω 6

17 Frequency Analysis Applications Denoising Apply a linear time invariant filter to a noisy signal with the bandwidth of the filter equal to the bandwidth of the uncorrupted signal. y x () t h( t) y() t () t + ( x( τ ) + n( τ )) h( t τ ) dτ ( ) + p ( x( m) + n( m) ) h( p m) m ( ω) ( ω) ( X ( ω) N( ω) ) y n( t) Y + 7

18 Frequency Analysis Applications Sampling Shannon sampling theorem x d ( n) x () t x ( t) s c h( t) xˆ ( t ) Suppose that is bandlimited by, that is ω B X c ( ω) ( ω) for. he signal can be reconstructed from the sampled sequence x d ( n) via an ideal lowpass filter with the impulse response π ( t ) if the sampling frequency is higher than or equal to twice of. hat is, if, then + π ( tn ). x c n δ () t x ( n) n ( t n ) d Impulse to sequence h () t sin π ( ) ( tn ) sin ( ) π () t B B X c π B 8

19 xˆ s d Frequency Analysis x d ( n) x ( t) x ( t) s c h() t xˆ ( t ) x X x X X () t x () t δ ( t n ) x ( n ) δ ( t n ) s + + jωt jωt ( Ω) x ( t) e dt x ( n ) δ ( t n ) e dt x ( n ) ( Ω) ( n) x ( n ) d d X π jωn ( ω) x ( n ) e X ( Ω) ( ω) c + ωω c c () t x ( n) h( t n ) x ( n ) n d n δ + n s π + k ( t n ) + + k c n X c Impulse to sequence δ Ω + c n ω πk + n + c n πk ωω c s + k sin π π ( tn ) ( ) ( tn ) X c Ω + k πk X c Ω + c n πk e jωn 9

20 Frequency Analysis Applications Oversampled analog-to-digital conversion u(k) F(z) y(k) Q s(k) (z) x(k) By modeling the quantizer as an additive noise source n(k), we have S( z) F( z) S( z) S( z) lim lim U z + F F z ( z ) ± U ( z) F ( z ) U ( z) ( ) ( ) ( ) ( ) S z S z lim lim N( z) + F( z) F ( z ) ± N( z) F ( z ) Signal and noise can be separated. S N ( z) ( z)

21 Applications Frequency Analysis Oversampled analog-to-digital conversion Since F(z) has to be unstable, stability is an issue. In order to design oversampled analog-to-digital converter, F(z) has to force the state variables to go to the infinity, but the quantizer has to force the state variables to go back to the origin. Chaotic behaviours occur. inear system theory is not applied. Set theory is employed for the analysis.

22 Frequency Analysis Applications Oversampled analog-to-digital conversion

23 Frequency Analysis Applications Oversampled analog-to-digital conversion 3

24 Applications Frequency Analysis Oversampled analog-to-digital conversion ( R ) R ( R ) R for k k R k U i R k I R k R j N k Ø for i j is an invariant map and R is an invariant set. :R N R N is surjective, but not injective. As there are two quantizer levels, we have - (R - )(R )R. his implies that R is attractive. In other words, the system is locally stable. o understand why the system is globally stable, as is surjective, the set R N R k is also an invariant set. owever, the invariant set has to be near the origin. ence, R N R k is an empty set. In other words, R k forms 4 the partition of R N and the system is globally stable.

25 Applications Frequency Analysis Oversampled analog-to-digital conversion Open problems Global stability of multibit high order oversampled analog-todigital converters is general unknown. Design of oversampled analog-to-digital converters subject to the global stability condition is not available yet. Could the results be applied to similar symbolic dynamical systems? 5

26 Frequency Analysis Applications x () t Amplitude modulated radio systems ransmission h( t) xˆ ( t) N () t e jω t e jω t ( t) h x ˆ N ( t) jωn e t jωn e t S xˆ N () t x () t n n + () jωnτ t S( τ ) e h( t τ ) n e jω t n dτ 6

27 Fundamental building blocks in signal processing Conventional upsampler Input output relationship of conventional upsampler x[n] x[n] y[n] x[] n x[] v [n] x[] n 7

28 Fundamental building blocks in signal processing Conventional upsampler Input output relationship of conventional upsampler n x y[] n n is integer multiple of otherwise Y Y + jωn jωn ( ω) y( n) e x( n) e X ( ω) n + n n ( ) ( ) ( ) ( z y n z x n z X z ) n n + + n No loss of information. 8

29 Fundamental building blocks in signal processing Conventional downsampler Input output relationship of conventional downsampler x[n] x[n] y[n] v [n] + 3 n x[] x[] x[3] x[] 3 n 9

30 3 ime Frequency Analysis Fundamental building blocks in signal processing Conventional downsampler Input output relationship of conventional downsampler where here is a lost of information and aliasing occurs. j e W π ( ) k z W k X z Y [ ] [ ] n x n y ( ) k k X Y π ω ω [] [] [ ] + k k n n x n y δ ( ) ( ) + + k k k X d k X Y π ω θ π ω θ δ π θ π ω π π

31 Filter banks Filter banks are systems that contain banks of filters and conventional samplers. s [n] h [n] f [n] x[n] s [n] h [n] f [n] s h N- [n] N- [n] N- f N- [n] N- y[n] 3

32 Filter banks ime localization of filter banks s j ( n) x( m) h ( n m) + m j j For finite impulse response h j n, it becomes a finite summation of mand hence it only captures a finite time duration information of x( n). hat means, filter banks have a good time localization property. ( ) 3

33 Filter banks Frequency localization of filter banks As the filters has finite bandwidths, the filtered signals have finite bandwidths. ence, filter banks have a good frequency localization property. 33

34 ree structure filter banks ree structure filter banks is the most common approach for implementing discrete-time wavelet transforms if the filter bank in each tree level is paraunitary. x[n] h [n] h [n] h [n] h [n] f [n] f [n] f [n] f [n] y[n] 34

35 ree structure filter banks Paraunitary condition E ~ E i k ( ) ( z z E z ) ( z) k E E,, n ( z) E( z) z I i, k ( z) E ( z) O, ( z) E ( z) ( jω ) E e, For each ω [-π,π], is constrained to be a unitary matrix. In other words, it is constrained to be in a high dimensional ball. owever, there are infinite number of ω in [-π,π]. ence, this problem is an infinite number of high dimensional ball constrained problem. 35

36 ree structure filter banks attice structure design of paraunitary filter banks he problem involves products of many unitary matrices, which is a highly nonlinear and nonconvex optimization problem. It could be difficult to find the gradient vector of the objective function of the optimization problem. ence, there is no efficient algorithm for the design. 36

37 37 ime Frequency Analysis ree structure filter banks Real-time design of paraunitary filter banks ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) p p p N p N p N p p p N p p k n m p N h N h h h p n m p k h k h I I I I δ δ δ ˆ ˆ O

38 ree structure filter banks Real-time design of paraunitary filter banks U D U U D [ U U ],, D, ( N ) D V, V, R R N N N Diagonal elements of D, are either or -, V could be arbitrary unitary matrix. ( ) D Θ diag UΘ,, U D Θ Θ U where is a unitary matrix. satisfies the paraunitary condition. Θ 38

39 39 ime Frequency Analysis ree structure filter banks Real-time design of paraunitary filter banks Define hen ( ) ( ) tr Θ Θ I V V V U D V U D V ˆ ˆ subject to ˆ ˆ ˆ ˆ min,, ˆ B B B V D U B D U B Θ ˆ, B B B B B V D V λ V U V ˆ

40 4 ime Frequency Analysis ree structure filter banks Real-time design of paraunitary filter banks Define ( ) ( ) tr Θ Θ Θ Θ Θ I U U V U D V U D U subject to ˆ ˆ ˆ ˆ min,, [ ] B B B A A A m N n m n m n m N n m n i m n i ˆ ˆ ˆ,,,,, ˆ ˆ ˆ ˆ ˆ V D U B V D U A Ω Ω A b b B Ω b h [ ] [ ] N m m m m n m n m n N m m m,,,,,,,,,, ˆ ˆ ˆ ˆ ˆ Ω Ω Ω Ω Ω Ω V U h h h h h h

41 ree structure filter banks Real-time design of paraunitary filter banks hen U λ Θ V Bˆ V U Bˆ Bˆ ( ) D U AU V Bˆ Bˆ Bˆ Bˆ Implication: Numerical optimization computer aided design tools are not required to find a locally optimal solution. ence, we could design adaptive real-time wavelet kernels. 4

42 Basis of ilbert space { xi} i J { x i } i J is a basis of ilbert space E if are linearly independent. A, B > such that A y xi, y B y. N Example 3: R, c x, y for et c c X y y N i A B x [ ] c c N ( X ) c i, y min max c ( ( eig XX ) ( eig( XX ) i J E i i i,, N and X x x XX c [ ] N 4

43 Frames ime Frequency Analysis { } J is a frame of ilbert space E if xi i A, B > such that A y xi, y B y. N i J xi R i,, N et E R, ci x i, y for i,, c [ c c ] and X [ x x ] Assume rank( ) ~ X N. etx X XX ~ x and ~ ~ c ~ Xy X ~ c XXy XX ( XX ) y y ~ ~ ~ ~ ~ x, y c c y X Xy Example 4: for, where i A y { x~i } i J i i ~ x i, y > X ( ) [ ] B is the dual frame to X ~ A min eig X ~ and B max ( ( ) y { x i } i J X ~ ~ ( ( X ) eig N ~ x 43

44 Basis of filter banks Define the inner product of signals as ( n) g( n) x( m) g( m) x, + m { ( )} jm j,, N, n Z { h ( )} j jn m j,, N, n Z Suppose that f j n of. is a dual frame 44

45 45 ime Frequency Analysis Perfect reconstruction of filter banks where d is the delay of the system, c is gain of system and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) z X cz zw zw X z F z X cz z F zw zw X z F W z W z X z F z z X z Y d N j k k j k j j j d N j j k k j j k j j N j j k k j j k j j N j j j j j j j j j j j j j j j j j j j j j i W j e π

46 Perfect reconstruction of filter banks If { f ( )} j n jm j,, N, n Z and { h ( )} j jn m j,, N, n Z are biorthogonal, then the filter banks achieve perfect reconstruction. 46

47 Incompatible nonuniform filter banks Consider a nonuniform filter bank with, 3 and 6. h [n] f [n] x[n] h [n] 3 3 f [n] y[n] h [n] 6 f [n] 6 47

48 48 ime Frequency Analysis Incompatible nonuniform filter banks We have where ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) z X cz z F z F z F zw zw zw zw zw zw zw zw z z z zw X z X d iπ e W

49 49 ime Frequency Analysis Incompatible nonuniform filter banks If perfect reconstruction could be achieved for arbitrary bounded inputs, then Impossible to achieve perfect reconstruction because aliasing cannot be cancelled. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d cz z F z F z F zw zw zw zw zw zw zw zw z z z

50 Incompatible nonuniform filter banks Open problem ow to design the filters and the samplers such that perfect reconstruction can be achieved? P. Q. oang and P. P. Vaidyanathan, Non-uniform multirate filter banks: theory and design, International Symposium on Circuits and Systems, ISCAS, pp , 989. (Open problem seeking for general solutions) ongwen Chen, i Qiu and Er-Wei Bai, General multirate building structures with application to nonuniform filter banks, IEEE ransactions on Circuits and Systems-II: Analog and Digital Signal Processing, vol. 45, no. 8, pp , 998. (Solution based on time varying filters) Sony Akkarakaran and P. P. Vaidyanathan, New results and open problems on nonuniform filter-banks, International Conference on Acoustics, Speeches and Signal Processing, ICASSP, pp. 5-54, 999. (Open problem seeking for time invariant solutions) Charlotte Yuk-Fan o, Bingo Wing-Kuen ing and Peter Kong-Shun am, Representations of linear dual-rate system via single SISO I filter, conventional sampler and block sampler, IEEE ransactions on Circuits and Systems-II: Express Briefs, vol. 55, no., pp. 68-7, 5 8. (Solution based on time invariant filters)

51 Incompatible nonuniform filter banks ime varying solution By converting the nonuniform filter bank to a uniform filter bank, we have (z) z (z) 6 3 X(z) z 4 (z) 6 3 (z) 6 z 3 (z) 6 (z) 6 z - z - z - z z z 6 6 F (z) z - F (z) z -4 F (z) Y(z) F (z) z -3 F (z) F (z) he dash rectangle is an identity system, so this nonuniform filter bank becomes a uniform filter bank. 5

52 Incompatible nonuniform filter banks ime varying solution he dash block is a time varying system. 3 6 G (z) (z) z 3 6 G (z) z 3 6 G (z) Y(z) 6 G 3 (z) X(z) (z) 3 z 6 G 4 (z) (z) 6 G 5 (z) 6 5

53 Basic property of upsampler and downsampler he upsampler and the downsampler is in general not commutative. Upsampler (upsampled by ) and downsampler (downsampled by ) is commutative if and only if and is co-prime. 53

54 r[n] r[ +] r[ +-] r[ +] r[+] r[] - n r[ ] - r[ -] r[ ] r[] r[] r[ -] r[ --] r[ -] r[ - ] r[ +- ] r[-] r[- ] r[-] r[- ] r[+- ] r[- ] - r[ - ] r[ -- ] and are co-prime. r[] r[-] r[] r[-] r[ - ] r[- ] r[- ] k n r[a +] r[] - r[a ] r[] r[a -] r[a - ] r[] r[] r[-] r[- ] r[-a ] r[-a ] a- a k a a 54

55 55 ime Frequency Analysis - r[a ] r[+a ] r[ +a ] a a r[ a -] r[ a ] r[] r[] r[ ] r[a -] r[a ] a- a r[-a ] r[-a ] r[ -a ] r[-] r[] k n - r[] r[ ] r[3 ] r[] r[] r[ ] r[-] r[] r[] k n

56 inear time invariant solution Block decimators (decimation ratio and block length ) [ k + j] x[ k j] y + for j,,,- and k Z. x[n] x[n] (,) y[n] - x[-] y[n] + +- x[+-] n x[] x[] x[] x[+] n 56

57 inear time invariant solution Block expanders (expansion ratio and block length ) y [ k] k mod x ( k, ) + mod ( k, ) k mod( k, ) k < k mod( k, ) k mod + ( k, ) + k < k + mod( k, ) x[n] x[] x[-] x[n] (,) y[n] x[+] n x[] x[-] y[n] x[] n

58 inear time invariant solution m,n Z + (no matter m and n are co-prime or not), all linear dual rate systems with shifting input by n samples resulting to shifting an output by m samples can be represented via a series cascade of m, followed by a linear time invariant filter with an impulse response f[k], and then followed by (n,m). 58

59 inear time invariant solution he input output relationship of all linear dual rate systems is y[ km + i] + g[ i, l kn]u[ l], k,l Z, m,n Z + l - and i,,,m-. he input output relationship of the proposed representation is y[ km + i] f[ kmn ml + i]u[ l], k,l Z, m,n Z + l and i,,,m-. k,l Z, m,n Z + and i,,,m-, the mapping from {,,,m-}xz to Z, where [i,l-kn] {,,,m- }xz and kmn-ml+i Z is bijective. 59

60 inear time invariant solution ence, k,l Z, m,n Z + and i,,,m-, there exists a unique time index kmn-ml+i corresponding to the time index [i,l-kn]. As a result, there exists a linear time invariant filter with an impulse response f[k] satisfying f[kmn-ml+i]g[i,l-kn], k,l Z, m,n Z + and i,,,m-, that the linear dual rate systems and the proposed representation are input output equivalent. 6

61 inear time invariant solution Consequently, the incompatible nonuniform filter bank can achieve perfect reconstruction via the following structure. (z) 6 F (z) (3,6) (z) 3 6 F (z) (,6) (z) 6 6 F (z) 6

62 inear time invariant solution m,n Z + (no matter m and n are co-prime or not), all linear dual rate systems with shifting input by n samples resulting to shifting an output by m samples can be represented via a series cascade of (m,n), followed by a linear time invariant filter with an impulse response f[k], and then followed by n. 6

63 inear time invariant solution he input output relationship of all linear dual rate + n y[ k] g[ k, nl + i]u[ nl + i] systems is, k,l Z, m,n Z + l - i and i,,,n-. he input output relationship of the proposed + n y[ k] representation is f[ kn mnl i]u[ nl + i], k,l Z, m,n Z + l - i and i,,,n-. l Z, m,n Z +, k {,,,m-} and i {,,,n-}, the mapping from {,,,m-}xz to Z, where [k,nl+i] {,,,m-}xz and kn-mnl-i Z is bijective. 63

64 inear time invariant solution ence, l Z, m,n Z +, k {,,,m-} and i {,,,n-}, there exists a unique time index kn-mnl-i corresponding to the time index [k,nl+i]. As a result, there exists a linear time invariant filter with an impulse response f[k] satisfying f[knmnl-i]g[k,nl+i], k,l Z, m,n Z + and i,,,n-, that the linear dual rate systems and the proposed representation are input output equivalent. 64

65 inear time invariant solution Consequently, the incompatible nonuniform filter bank can achieve perfect reconstruction via the following structure. (z) (6,3) F (z) 3 (z) 3 (6,) F (z) (z) 6 6 F (z) 65

66 inear time invariant solution Implication: We could have arbitrarily time localization and frequency localization. 66

67 Application of nonuniform filter banks Nonuniform transmultiplexers x [n] f [n] h [n] y [n] x [n] f [n] h [n] y [n] x N- [n] N- f N- [n] Perfect reconstruction y i [ n] c x [ n d ] i i i h N- [n] N- y N- [n] 67

68 Application of nonuniform filter banks Nonuniform transmultiplexers Example 4: F (z), F (z)z -4 +z -5, F (z)z -3, then R (z)-z +z 5 +z -z 3, R (z)z +z 3 and R (z)z 3. x [n] F (z) 3 R (z) (,3) 6 r y [n] x [n] 3 F (z) R (z) (,) 6 r y [n] x [n] 3 F (z) R (z) 6 y [n] 68

69 Application of nonuniform filter banks Nonuniform image coding Existing successive approximation technique B/ 3 B/4 B B/4 B/ 3 B/4 where B is the maximum absolute value of the wavelet coefficients. B 69

70 Application of nonuniform filter banks Nonuniform image coding Absolute values of wavelet coefficients follows aplacian distribution approximately. 8 log of number of wavelet coefficients log of absolute values of wavelet coefficients

71 Application of nonuniform filter banks Nonuniform image coding Probability of assigning the symbol is greater than that of the symbol. Uniform distribution of the symbols gives maximum entropy. So the existing successive approximation technique is not optimal. A higher coding gain curve may be achieved by means of non-uniform successive approximation. 7

72 Application of nonuniform filter banks Nonuniform image coding Number of wavelet coefficients B/ 3 B/4 Absolute values of wavelet coefficients B 7

73 Application of nonuniform filter banks Nonuniform image coding Set the thresholds j at B/p j, where p>. et a and b are the boundaries in the region, c be the coded value and f(x) be the distribution of the wavelet coefficients, then the error introduced in the quantization is: b E () c ( x c) f () x a x ( ) k where f x A e d E() c dc b k e b k c k e dx a k ( ) e ( a k ) ( b k a k e ) 73

74 Application of nonuniform filter banks Nonuniform image coding B/a c B B/a c B/a c c c 3 B 74

75 Application of nonuniform filter banks Nonuniform image coding Non-uniform successive approximation 3 Uniform successive approximation 8 PSNR Bit per pixel

76 Filter window banks and Fractional Fourier transform Downsampling first and then upsampling is equivalent to a sampling window function. What happens if we have general window functions? h [n] w [n] f [n] x[n] h [n] w [n] f [n] y[n] h N- [n] w N- [n] f N- [n] 76

77 Filter window banks and Fractional Fourier transform Fractional Fourier transform is to rotate the time frequency plane u cosθ v sinθ sinθ t cosθ ω By designing a set of windows and filters as well as applying the fractional Fourier transform to rotate the time frequency plane, the signals could be extracted out precisely. 77

78 Filter window banks and Fractional Fourier transform 78

79 Filter window banks and Fractional Fourier transform Open problems: ow to guarantee the perfect reconstruction? ow to design the globally optimal set of filters and the windows such that the filters have good frequency selectivities? Filtering and windowing could be understood as the multiplication in certain particular domains, such as in the frequency domain and in the time domain. In fact, these domains are obtained by certain particular unitary transforms. For example, frequency domain is obtained by applying the DF transform which is a unitary transform and the time domain is obtained by applying the IDF transform which is also a unitary transform. What happens if the transform is generalized to arbitrarily unitary transforms and how to determine such optimal transform? ow to apply these results to some practical problems, such as denoising problems, signal separation problems, pattern recognition problems and fault detection problems? 79

80 Applications ECG signal denoising Ideal signal Noisy signal Filtered Signal - Signal levels ime index k 8

81 Applications ime Frequency Analysis Audio signal denoising for digital audio hearing aids Original signal Noisy signal Denoised signal ime t x ime t x ime t 8 x 4

82 Applications - Signal separation orignal male speech - separated male speech x 4 orignal flute x 4 separated flute x x 4 8

83 Applications achine fault analysis requency ω ime t

84 Applications achine fault analysis Frequency ω ime t 84

85 - 4 ime Frequency Analysis Applications achine fault analysis 3 - Frequency ω ime t 85

86 Applications achine fault analysis Frequency ω ime t

87 Applications achine fault analysis x Frequency ω ime t

88 Conclusions any applications, such as denoising, sampling, analog-to-digital conversions and amplitude modulation schemes, are derived based on frequency domain approaches. Further applications, such as denoising, signal separations, fault analysis, could be derived based on time frequency domain approaches. 88

89 Q&A Session hank you! et me think Bingo 89