Filter Banks II. Prof. Dr.-Ing Gerald Schuller. Fraunhofer IDMT & Ilmenau Technical University Ilmenau, Germany

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1 Filter Banks II Prof. Dr.-Ing Gerald Schuller Fraunhofer IDMT & Ilmenau Technical University Ilmenau, Germany Prof. Dr.-Ing. G. Schuller, Page

2 Modulated Filter Banks Extending the DCT The DCT IV transform can be seen as modulated filter bank with window hn: π yk m = x m + n cos k+ n+ n 2 2 = π yk m = x m + n h n cos k+ n+ n= 2 2 with a simple window function independent of k! defined as n =... h n = else Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 2

3 Modulated Filter Banks In general: y m = x m + n h n Φ n k n= k hn Φ k n window function not nec. limited in length modulation function, for instance the cosine function frequency index Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 3

4 Modulated Filter Banks The resulting filters of this cosine modulated filter bank are: h k π n = h n cos k +.5 n +.5 With the cosine modulation, the resulting frequency responses of the filters of the filter bank are: π π k ω = ω δ ω k δ ω k Mult in time becomes Delta functions from cosine term convolution in frequency Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 4

5 Prof. Dr.-Ing. G. Schuller, Page 5 Modulated Filter Banks ence: Modulated filter banks obtain their filters by shifting a baseband filter hn in frequency. pi < omega < pi = 2 2 k k π ω π ω Shift in frequency

6 Modulated Filter Banks: Frequency Shifts The subbands of the filter bank are frequency shifted versions of window frequency response: Place passband in frequency, depending on the modulation function, for Subbands k and k+. Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 6

7 Modulated Filter Banks: The Window Function Frequency response of the rectangular window function of the DCT: ideal stopband Prof. Dr.-Ing. G. Schuller, Page 7

8 Modulated Filter Banks Improve filter banks: - make window longer - different window shape Examples all have the same principle: - TDAC time domain aliasing cancellation Princen and Bradley 986&987 - LOT lapped orthogonal transform Malvar MDCT modified DCT Bernd Edler 988 Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 8

9 Prof. Dr.-Ing. G. Schuller, Page 9 Analysis Polyphase Matrix Remember: the analysis polyphase matrix is: with the analysis polyphase components =,,,,, O M L m m k n k m n h = + =,

10 Prof. Dr.-Ing. G. Schuller, Page MDCT The analysis polyphase matrix of the MDCT can be written with for n=,,2-; k=,,- as observe: this has length 2, and is more general than the rectangular window not just or is composed of st order polynomials Goal: find good hn + + = cos n k n h n h k π n h k = 2 h h h h h h h h h h O M K Still square matrix, still invertible, x

11 MDCT h = 2 h 2 h Fortunately the MDCT Polyphase matrix can be decomposed into a product of matrices, hence easier to invert to obtain perfect reconstruction: h h.5 h.5 {T O O h h Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page Fa, real valued = 4 9 O 2 3 O matrix inverse is inverse of each entry! Delay matrix D DCT4-Matrix ½ / 3 =

12 MDCT Observe the diamond shaped form of the matrix F a, and the sparse structure Beneficial for an efficient implementation Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 2

13 Prof. Dr.-Ing. G. Schuller, Page 3 MDCT synthesis The MDCT synthesis Polyphase matrix can be similarly decomposed into a product of matrices. eeds to be the inverse and a delay for Perfect Reconstruction PR O O F s g g g g g g g g : Delay by one time step past : Looking into the future non-causal not practical hence mult. with - delay! cause of signal delay O O = D T G

14 Graphical Interpretation of Analysis Matrix F a time h h O O h h [] { x = h h 2 O O h h [ DCT ] [{ y 4 ] subbands F a - - Folding the upper and lower quarter of the signal into a length block aliasing components Invertible by matrix inversion containing overlap add Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 4

15 Prof. Dr.-Ing. G. Schuller, Page 5 MDCT DCT matrix T and the delay matrix D are easily invertible for perfect reconstruction. System Delay results from making inverse of D causal one block, and the blocking delay of -. F a is also easily invertible, with some simple matrix algebra: 2 n h n h n h n h n h n g + + = 2 n h n h n h n h n h n g = + with n=,,-

16 MDCT Filter Banks Modified Discrete Cosine Transform MDCT: gn=hn Denominator= Example which fulfills this condition: Sine window π h n = sin n for n=,,2- System delay=2-=23 for =52 from the delay matrices, block of, and the blocking delay of - Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 6

17 Sine-Window Frequency Response sinusodial funct. Modulation of a rectangular window of length 2 sine window length 2 Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 7 Better attenuation outside of passband than with simple sinc

18 MDCT, Advantages Improved frequency responses, higher stopband attenuation Easy to design filter banks with many subbands for instance =24 for audio coding Efficient implementation with the shown sparse matrices and a fast DCT. Important for large number of subbands, as in audio coding. Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 8

19 MDCT Filter Banks 2 Examples: filter impulse responses h 7 n, h 5 n, =24 bands, sine window. Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 9

20 MDCT Filter Banks 3 Magnitude response 7th band Magnitude response 5th band indeed better filters! Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 2

21 Extending the Length of the MDCT Longer filters are obtained with higher order polynomials in the polyphase matrix Approach to obtain easily invertible polyphase matrices multiply MDCT polyphase matrix with more easily invertible matrices with polynomials of st order To control the resulting system delay: design different matrices with different needs for delay to make them causal Prof. Dr.-Ing. G. Schuller, Page 2

22 Extending the Length Take the MDCT Polyphase matrix with a general window function hn not nec. Sine window: MDCT This matrix contains polynomials of first order. Multiply it with another matrix with polynomials of first order Schuller, 996, 2: L MDCT Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 22

23 Extending the Length This matrix needs to have a form such that again a modulated filter bank results. Diamond shaped form needs to be maintained Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 23

24 Prof. Dr.-Ing. G. Schuller, Page 24 Extending the Length, Zero-Delay Matrix This matrix fulfills the conditions Zero-Delay Matrix: = 2 / O O l l L

25 Prof. Dr.-Ing. G. Schuller, Page 25 Extending the Length, Zero-Delay Matrix Its inverse is causal, hence does not need a delay to make it causal: = 2 / l l L O O still increases filter length!

26 Extending the Length, Zero-Delay Matrix Observe: Since the matrix has a causal inverse, it can increase the filter length of the resulting filter bank without increasing the system delay! ence adds eros inside unit circle The coefficients hn and don t affect the n delay or the PR property, but the frequency response of the resulting filter bank l Coefficients need to be found by numerical optimiation. Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 26

27 Extending the Length, Maximum-Delay Matrix Consider the following matrix Maximum-Delay Matrix: = L Its inverse and delay for causality is = L 2 Observe: This matrix and its inverse need a delay of 2 blocks to make it causal. ence adds eros outside the unit circle Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 27

28 Extending the Length, Design Method Determine the total number of Zero-Delay Matrices and Maximum-Delay Matrices according to the desired filter length Determine the number of Maximum-Delay Matrices according to the desired system Delay Determine the coefficients of the matrices with numerical optimiation to optimie the frequency response Prof. Dr.-Ing. G. Schuller, Page 28

29 Example Comparison for 28 subbands. Dashed line: Orthogonal filter bank, filter length 256, system delay 255 samples. Solid line: Low delay filter bank, length 52, delay 255 Prof. Dr.-Ing. G. Schuller, Page 29

30 Block Switching Problem: In audio coding, Pre-echoes appear before transients - reason: blocks too long too many bands Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 3

31 Block Switching Approach: for fast changing signals use block switching to lower number of subbands less noise spread in time! Prof. Dr.-Ing. G. Schuller, Page 3

32 Accommodate Overlap-Add for Block Switching aliasing cancellation perfect reconstruction window Analysis switching bad frequency resolution Prof. Dr.-Ing. G. Schuller, Page 32

33 Block Switching Sequence of windows for switching the number of sub-bands window value hn both, analysis and synthesis Prof. Dr.-Ing. G. Schuller, Page 33

34 Wavelets, QMF Filter Banks Iterate 2-band system See also: Wavelet Packets more general Problem: Aliasing propagation reduces frequency selectivity! Important in image coding, but no big role in Audio- Coding h n d, n c, n igh-pass h n d2, n h n c, n igh-pass iterate Low-Pass h n c2, n Low-Pass Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 34

35 ow to Obtain a Two Band Filter Bank Application: QMF filter banks, Wavelets, Analysis polyphase for a 2-band filter bank: =,,,, Observe:, contains the even coefficients of the low pass filter, and, its odd coefficients. Accordingly for the high pass filter Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 35

36 ow to Obtain a Two Band Filter Bank Given the analysis filters, the synthesis filters can be obtained by inverting the analysis polyphase matrix, = Det,,,, Observe: If the analysis filters have a finite impulse response FIR, and the synthesis is desired to also be FIR, the determinant of the polyphase matrix needs to be a constant or a delay! Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 36

37 ow to Obtain a Two Band Filter Bank = det,,,, = const or a delay Observe: This is the output of the lower band of the filter bank if the input signal is ence the determinant can be formulated as a convolution This input is the high band filter coefficients, with the sign of the even coefficients flipped and switched places with the odd coefficients. Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 37 [, ] x =,,

38 ow to Obtain a Two Band Filter Bank Since this represents a critically sampled filter bank, the result represents every second sample of the convolution of the low band filter with the correspondingly modified high band filter. This modified high band filter is a low band filter every second sample sign flipped. The desired output of this downsampled convolution is a single pulse corresponding to a constant or a delay, hence flat in frequency Another interpretation: correlation of the 2 signals, where the even lags that appear after downsampling are ero, except for the one pulse Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 38

39 QMF Quadrature Mirror Filter analysis FB high pass: synth. FB low pass: This suggests a simple design strategy: Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 39 Design a low pass filter for analysis and synthesis Obtain the high pass filters by flipping the low pass filters every second coefficient n h n = h n n =,,..., g n = h n g n = h n This is an early two band filter bank: QMF, Quadrature Mirror Filter Croisier, Esteban, Galand, 976 For more than 2 bands: GQMF Cox, 986, PQMF

40 QMF 2 Sign flipping to obtain the high band filter leads to the polyphase components:,, =, =, The resulting determinant is: det = 2, =,,,,, Observe: This cannot be made a constant or delay for finite polynomials of order or greater, hence no PR for finite length filters! Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 4

41 QMF 3 The QMF accounts for the sign flipping in the determinant equation. But not for the trading places of even and odd coefficients ence: o Perfect Reconstruction only for simple aar and IIR filters igh stopband attenuation needed to keep reconstruction error small umerical optimiation, to obtain 2 2 e jω + e jω Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 4

42 QMF 4 lower band upper band f s 4 f Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 42

43 QMF: Example with Impulse Response of Length 96 Low-Pass igh-pass Prof. Dr.-Ing. G. Schuller, Page 43

44 CQMF : Conjugate QMF To also accommodate for the trading places of odd and even coefficients, a natural choice is to also reverse the temporal order of the synthesis filter. h n = h L n n With L: filter length, and g g n = h n = h n n Introduced e.g. by Smith, Barnwell, 984 Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 44

45 CQMF 2 For the polyphase components this means,, = = L / 2 L / 2 And the input for our determinant calculation is,, [, ] x = L / 2,, This corresponds exactly to the time reversed low band filter! Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 45

46 CQMF Lets define A The determinant is now det = =,, L / 2 = A +, A Observe: This can be a constant if all even coefficients of A are ero, except for the center coefficient!,,, Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 46

47 CQMF 3 Remember: the determinant was the output of the low band with this input ence: Every second sample of the convolution of the low band filter with its time reversed version. This is equal to every second value of the autocorrelation function of the low band filter! Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 47 Determinant a constant or a delay: only one sample of this downsampled autocorrelation function all even coefficients can be unequal ero most even coefficients are ero

48 CQMF 4 The Determinant is a constant means: The eroth autocorrelation coefficient is a constant unequal, and all other even coefficient must be ero. Called yquist filter property -> Design method Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 48

49 CQMF 5 -transform of ACF of low pass filter -> power spectrum In other terms: Define P as the -transform of this autocorrelation function, the Power Spectrum: P : = Then all nonero coefficients of P are the eroth coefficient and the odd coefficients. As a result: P + P = Frequency reversal This is also called the halfband filter property. Design approach: Design P accordingly, then The odd coefficients cancel const Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 49

50 Pseudo-QMF PQMF So far we only had 2 subband QMF filter banks Only for the 2-band case we get perfect reconstruction in the CQMF case The PQMF extents the QMF approach to >2 subbands But it has only ear Perfect Reconstruction, meaning a reconstruction error by the filter bank It is modulated filter band like the MDCT, using a baseband prototype filter hn a lowpass Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 5

51 PQMF Its analysis filters are given by the impulse responses h k π n = cos k +.5 n +.5 L + 2 π 4 It is an orthogonal filter bank, which means that the synthesis filter impulse responses are simply the time inverses of the analysis impulse responses, f k n = h L n k k Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 5

52 PQMF Its baseband prototype filters hn are now designed such that aliasing cancels between adjacent neighbouring bands, e jω 2 j / 2 π π Ω + e =,for < Ω < 2 beyond the adjacent bands, the attenuation should go towards infinity, e j Ω 2 π =,for Ω > Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 52

53 PQMF The PQMF filter bank is used in MPEG/2 Layer I and II. There it has =32 subbands and filter length L=52 Also used in MPEG 4 for so-called SBR Spectral Band Replication and for parametric sourround coding. There it has =32 or =64 subbands, and filter length L=32 or L=64 Prof. Dr.-Ing. G. Schuller, shl@idmt.fraunhofer.de Page 53

54 PQMF used in MPEG4 Impulse response of the baseband prototype the window, with =64 and L=64 Prof. Dr.-Ing. G. Schuller, Page 54

55 PQMF used in MPEG4 Frequency response of the baseband prototype the window, Prof. Dr.-Ing. G. Schuller, Page 55

Filter Banks II. Prof. Dr.-Ing. G. Schuller. Fraunhofer IDMT & Ilmenau University of Technology Ilmenau, Germany

Filter Banks II. Prof. Dr.-Ing. G. Schuller. Fraunhofer IDMT & Ilmenau University of Technology Ilmenau, Germany Filter Banks II Prof. Dr.-Ing. G. Schuller Fraunhofer IDMT & Ilmenau University of Technology Ilmenau, Germany Page Modulated Filter Banks Extending the DCT The DCT IV transform can be seen as modulated