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DSP-CIS Chapter 10: Cosine-Modulated Filter Banks & Special Topics

DSP-CIS Chapter 10: Cosine-Modulated Filter Banks & Special Topics. Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be www.esat.kuleuven.be / stadius /. Part-II : Filter Banks. : Preliminaries Filter bank set-up and applications

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DSP-CIS Chapter 10: Cosine-Modulated Filter Banks & Special Topics

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  1. DSP-CISChapter 10: Cosine-Modulated Filter Banks & Special Topics Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be www.esat.kuleuven.be/stadius/

  2. Part-II : Filter Banks : Preliminaries • Filter bank set-up and applications • `Perfect reconstruction’ problem + 1st example (DFT/IDFT) • Multi-rate systems review (10 slides) : Maximally decimated FBs • Perfect reconstruction filter banks (PR FBs) • Paraunitary PR FBs : Modulated FBs • Maximally decimated DFT-modulated FBs • Oversampled DFT-modulated FBs : Cosine-modulated FBs & Special topics • Cosine-modulated FBs • Time-frequency analysis & Wavelets • Frequency domain filtering Chapter-7 Chapter-8 Chapter-9 Chapter-10

  3. Cosine-Modulated Filter Banks Motivation : Cosine-modulated FBs offer an alternative to DFT-modulated FBs… • Similar to DFT-modulated FBs, cosine-modulated FBs offer economy in design- and implementation complexity • Unlike DFT-modulated FBs, cosine-modulated FBs can be PR/FIR/paraunitary under maximal decimation (with design flexibility).

  4. H0 H1 H2 H3 P0 Ho H1 Cosine-Modulated Filter Banks • Uniform DFT-modulated filter banks: Ho(z) is prototype lowpass filter, cutoff at for N filters • Cosine-modulated filter banks : Po(z) is prototype lowpass filter, cutoff at for N filters Then... etc...

  5. Cosine-Modulated Filter Banks • Cosine-modulated filter banks : - if Po(z) is prototype FIRlowpass filter with real coefficients po[k], k=0,1,…,L then i.e. `cosine modulation’ (with real coefficients) instead of `exponential modulation’ (with complex coeffs, see DFT-modulated FBs Chapter-9) - if Po(z) = `good’lowpass filter, then Hk(z)’s = `good’bandpass filters

  6. Cosine-Modulated Filter Banks Realization based on polyphase decomposition (analysis): - if Po(z) has 2N-fold polyphase expansion (ps: 2N-fold for N filters!!!) then... u[k] : :

  7. Cosine-Modulated Filter Banks Realization based on polyphase decomposition (continued): - if Po(z) has L+1=m.2N taps, and m is even (similar formulas for m odd) (i.e. `m’ is the number of taps in each polyphase component) then... With Skip this slide ignore all details here !!!!!!!!!!!!!!!

  8. u[k] : : Cosine-Modulated Filter Banks Realization based on polyphase decomposition (continued): - Note that C (the only dense matrix here) is NxN DCT-matrix (`Type 4’) hence fast implementation (=fast matrix-vector product) based on fast discrete cosine transform (DCT) procedure, with complexity O(N.logN) Modulated filter bank gives economy in * design (only prototype Po(z) ) * implementation (prototype + modulation (DCT)) Similar structure for synthesis bank Skip this slide

  9. u[k] N u[k] = N N N : : N N Cosine-Modulated Filter Banks Maximally decimated cosine modulated (analysis) bank :

  10. Cosine-Modulated Filter Banks Question:How do we obtain Maximal Decimation + PR/FIR/Paraunitariness? Theorem: (proof omitted) -If prototype Po(z) is a real-coefficient (L+1)-taps FIR filter, (L+1)=2N.m for integer m and po[k]=po[L-k] (linear phase), with polyphase components En(z), n=0,1,…2N-1, -then the (FIR) cosine-modulated analysis bank is PARAUNITARY if and only if (n=0..N-1) are power complementary, i.e. form a lossless 1 input/2 output system FIR synthesis bank (for PR) can then be obtained by paraconjugation !!! = great result… Skip this slide ..this is the hard part…

  11. Cosine-Modulated Filter Banks Perfect Reconstruction (continued) Design procedure: Parameterize lossless systems for n=0,1..,N-1 Optimize all parameters in this parametrization so that the prototype Po(z) based on these polyphase components is a linear-phase lowpass filter that satisfies the given specifications Example parameterization: Parameterize lossless systems for n=0,1..,N-1, -> lattice structure (see Part-II), where parameters are rotation angles Skip this slide

  12. Cosine-Modulated Filter Banks lossless 1-in/2-out PS: Linear phase property for po[n] implies that only half of the power complementary pairs have to be designed. The other pairs are then defined by symmetry properties. u[k] p.9 = N : : N :

  13. Cosine-Modulated Filter Banks PS: Cosine versus DFT modulation In a maximally decimated cosine-modulated (analysis) filter bank 2 polyphase components of the prototype filter, , actually take the place of only 1 polyphase component in the DFT- modulated case. For paraunitariness (hence FIR-PR) in a cosine-modulated bank, each such pair of polyphase filters should form a power complementary pair, i.e. represent a lossless system. In the DFT-modulated case, imposing paraunitariness is equivalent to imposing losslessness for each polyphase component separately, i.e. each polyphase component should be an `allpass’ transfer function. Allpass functions are always IIR, except for trivial cases (pure delays). Hence all FIR paraunitary DFT-modulated banks (with maximal decimation) are trivial modifications of the DFT bank. provides flexibility for FIR-design no FIR-design flexibility

  14. Time-Frequency Analysis & Wavelets Starting point is discrete-time Fourier transform: = infinitely long sequence u[k] is evaluated at infinitely many frequencies Inversion/reconstruction/synthesis (=filter bank jargon) is.. = sequence u[k] is represented as weighted sum of basis functions

  15. Time-Frequency Analysis & Wavelets • `uncertainty principle’ says that if u[k] has a narrow support (i.e. is localized), then U(.) has a wide support (i.e. is non- localized), and vice versa • Hence notion of `frequency that varies with time’ not accommodated (e.g. `short lived sine’ will correspond to non-localized spectrum)

  16. Time-Frequency Analysis & Wavelets Tool to fill this need is `short-time Fourier transform’(STFT) where w[k] is your favorite window function (typically with `compact support’ (=FIR) ) • Window slides past the data. For each window position k, compute discrete-time Fourier transform PS: If w[k]=1 (all k) then result is goodold discrete-time FT for all window positions • In following slides, will provide a filter bank version of STFT, also leading to simple inversion formula

  17. Time-Frequency Analysis & Wavelets Rewrite STFT formula as… or similarly…(swap k and k-bar) • If we forget about the fase factor up front (=modulate window i.o input), then this corresponds to performing a convolution of u[k] with a filter (see Chapter-2, p.19) In practice, will compute this for a discrete set of (N) frequencies… leading to a set of filters… • This is a DFT-modulated analysis bank, prototype filter = window function

  18. u[k] u[k] Time-Frequency Analysis & Wavelets • If window length>N, can use efficient implementation based on polyphase decomposition of prototype Ho + DFT-modulation • Often window length=N, hence 1-tap polyphase components window length/N freq.resolution N

  19. Time-Frequency Analysis & Wavelets In practice, will consider ‘window shift’ >1 (decimation D>1) • If maximally decimated (D=N), then analysis FB operation is xn[k] = decimated subband signals = `STFT-coefficients’ = infinitely long sequence u[k] is evaluated at N frequencies, infinitelymany times (i.e. for infinitely many window positions) ...to be compared to page 14 u[k] 4 4 4 4

  20. 4 4 4 + + + 4 Time-Frequency Analysis & Wavelets • Corresponding (PR) synthesis filter bank (also DFT-modulated) is i.e. synthesis prototype filter fo[k] = ‘synthesis window’ = • Reconstruction/synthesis formula (=inverse STFT) is ..to be compared to page 14

  21. Time-Frequency Analysis & Wavelets • If oversampled (D=N/d), then analysis FB operation is Example: d=2 (‘window shift’= N/2) u[k] 2 u[k] = 2 2 2 2 2 B(z)

  22. Time-Frequency Analysis & Wavelets • Corresponding (PR) synthesis filter bank (also DFT-modulated) with PR condition for synthesis window v_k • Can easily generalize this for other oversampling factors d, leading to PR condition explain… (and try examples with Hann windows) • This is referred to as Weighted OverLap-Add (WOLA)

  23. H0 H3 H1 H2 8 H0(z) u[k] 8 H1(z) 4 H2(z) 2 H3(z) Time-Frequency Analysis & Wavelets Now, for some applications (e.g. audio) would like to have a non-uniform filter bank, hence also with non-uniform (maximum) decimation, for instance • non-uniform filters = low frequency resolution at high frequencies, high frequency resolution at low frequencies (as human hearing) • non-uniform decimation = high time resolution at high frequencies, low time resolution at low frequencies

  24. 2 2 2 2 2 2 u[k] Time-Frequency Analysis & Wavelets This can be built as a tree-structure, based on a 2-channel filter bank with H1 H0 H3 H2

  25. Time-Frequency Analysis & Wavelets Note that may be viewed as a prototypefilter, from which a series of filters is derived The lowpass filters are then needed to turn these multi-band filters into bandpass filters (i.e. remove images)

  26. 2 2 2 + 2 2 + 2 + Time-Frequency Analysis & Wavelets Similar synthesis bank can be constructed with • If and form a PR FB, then the complete analysis/synthesis structure is PR (why?)

  27. Time-Frequency Analysis & Wavelets • Analysis bank corresponds to `discrete-time wavelet transform’ (DTWT) (xk[n] = `DTWT-coefficients’) • With a corresponding (PR) synthesis filter bank, the reconstruction/synthesis formula (inverse DTWT) is ..to be compared to page 14 & 20

  28. Time-Frequency Analysis & Wavelets • Reconstruction formula may be viewed as an expansion of u[k], using a set of basis functions (infinitely many) • If the 2-channel filter bank is paraunitary, then this basis is orthonormal (which is a desirable property) : =`orthonormal wavelet basis’

  29. Time-Frequency Analysis & Wavelets • Example : `Haar’ wavelet (after Alfred Haar) • Compare to 2-channel DFT/IDFT bank • Derive formulas for Ho, H1, H2, H3, … Derive formulas for Fo, F1, F2, F3, … Paraunitary FB (orthonormal wavelet basis) ?

  30. Time-Frequency Analysis & Wavelets Not treated here : • `continuous wavelet transform’ (CWT) of a continuous-time function u(t) h(t)=prototype p,q are real-valued continuous variables p introduces `dilation’ of prototype, q introduces `shift’ of prototype • `discrete wavelet transform’ (DWT) is CWT with discretized p,q T is sampling interval p-bar, q-bar are real-valued integer variables mostly a=2 Skip this slide

  31. Time-Frequency Analysis & Wavelets Not treated here : • Theory - multiresolution analysis - wavelet packets - 2D transforms - etc … • Applications : - audio: de-noising, … - communications : wavelet modulation - image : image coding Skip this slide

  32. Frequency Domain Filtering Chapter 6! • See DSP-I : cheap FIR realization based on frequency domain processing (`time domain convolution equivalent to component-wise multiplication in the frequency domain’), cfr. `overlap-add’ & `overlap-save’ procedures • This can be cast in the subband processing setting, as a non-critically downsampled (2-fold oversampled) DFT-modulated filter bank based operation! • Leads to more general approach to performance/delay trade-off PS: formulae given for N=4, for conciseness (but without loss of generality)

  33. Frequency Domain Filtering Have to know a theorem from linear algebra here: • A `circulant’ matrix is a matrix where each row is obtained from the previous row using a right-shift (by 1 position), the rightmost element which spills over is circulated back to become the leftmost element • The eigenvalue decomposition of a `circulant’ matrix is trivial…. example (4x4): with F the NxN DFT-matrix, this means that the eigenvectors are equal to the column-vectors of the IDFT-matrix, and that then eigenvalues are obtained as the DFT of the first column of the circulant matrix (proof by Matlab)

  34. T(z)*u[k-3] u[k] 4 4 4 4 + 4 4 4 4 Frequency Domain Filtering Starting point is this (see Chapter-8) : meaning that a filtering with can be realized in a multirate structure, based on a pseudo-circulant matrix N*N filters, L/N taps each at an N-fold lower rate(N=4)

  35. Frequency Domain Filtering Now some matrix manipulation… :

  36. y[k] u[k] 4 4 4 4 + 4 4 4 4 Frequency Domain Filtering • An (8-channel) filter bank representation of this is... Analysis bank: Subband processing: Synthesis bank: This is a 2N-channel filter bank, with N-fold downsampling. The analysis FB is a 2N-channel uniform DFT filter bank (see Chapter 9, p.30 !). The synthesis FB is matched to the analysis bank, for PR: 2N filters, L/N taps each at an N-fold lower rate || N/2-fold complexity reduction

  37. `block’ `previous block’ `throw away’ `save’ Frequency Domain Filtering • This is known as an `overlap-save’ realization : • Analysis bank: performs 2N-point DFT (FFT) of a block of (N=4) samples, together with the previous block of (N) samples (hence `overlap’) • Synthesis bank: performs 2N-point IDFT (IFFT), throws away the first half of the result, saves the second half (hence `save’) • Subband processing corresponds to `frequency domain’ operation

  38. Frequency Domain Filtering `Overlap-add’ can be similarly derived :

  39. `overlap’ `add’ Frequency Domain Filtering • This is known as an `overlap-add’ realization : • Analysis bank: performs 2N-point DFT (FFT) of a block of (N=4) samples, padded with N zero samples • Synthesis bank: performs 2N-point IDFT (IFFT), adds second half of the result to first half of previous IDFT (hence `add’) • Subband processing corresponds to `frequency domain’ operation `block’ `zero padding’

  40. Frequency Domain Filtering • Standard `Overlap-add’ and `overlap-save’ realizations are derived when 0th order poly-phase components are used in the above derivation, i.e. each poly-phase component represents 1 tap of an L-tap filter T(z). (N=L) The corresponding 0th ordersubband processing (H) then corresponds to what is usually referred to as the `component-wise multiplication’ in the frequency domain. Note that for an L-tap filter, with large L, this leads to a cheap realization based on FFT/IFFTs instead of DFT/IDFTs. However, for large L, as 2L-point FFT/IFFTs are needed, this may also lead to an unacceptably large processing delay (latency) between filter input and output.

  41. Frequency Domain Filtering • In the more general case, with higher-order polyphase components (hence N smaller than the filter length L), a smaller complexity reduction is achieved, but the processing delay is also smaller. • This provides an interesting trade-off between complexity reduction and latency !!

  42. Conclusions • Great (=FIR/paraunitary) perfect reconstruction FB designs based on `modulation’: • Oversampled DFT-modulated FBs (Chapter-9) • Maximally decimated (and oversampled (not treated here)) cosine-modulated FBs • `Perfect reconstruction’ concept provides framework for time-frequency analysis of signals • Filter bank concept provides framework for frequency domain realization of long FIR filters

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