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August 2004Multirate DSP (Part 2/2)1 Multirate DSP Digital Filter Banks Filter Banks and Subband Processing Applications and Advantages Perfect Reconstruction FB 2-band Quadrature-Mirror Filter Bank K-band Filter Bank Case Uniform DFT Modulated Filter Banks

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August 2004Multirate DSP (Part 2/2)2 Digital Filter Banks So far we have mostly concentrated on the design, realisation and applications of single-input, single-output digital filters. In certain applications, it is desirable to separate a signal into a set of subband signals occupying, usually nonoverlapping, portions of the original frequency band. In other applications, it is necessary to combine many such subband signals into a single composite signal occupying the whole Nyquist range. To achieve the above two requirements, we need digital filter banks.

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August 2004Multirate DSP (Part 2/2)3 H0(z)H0(z) H1(z)H1(z) H K-1 (z)... v0[n]v0[n]x[n]x[n] v1[n]v1[n] v K-1 [n] F0(z)F0(z) F1(z)F1(z) F K-1 (z)... v 0 ’[n]y[n]y[n] v 1 ’[n] v K-1 ’[n] + + Analysis Filter BankSynthesis Filter Bank K-band analysis filter bank with the subfilters H k (z) known as the analysis filters. Decomposes input signal x[n] into a set of K subband signals v k [n] with each subband signal occupying a portion of the original frequency band. Synthesis filter bank - a set of subband signals v’ k [n] is combined into one signal y[n]. K-band synthesis bank where each filter F k (z) is called synthesis filter.

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August 2004Multirate DSP (Part 2/2)4 Decimation: decimator (down-sampler) example : u[k]: 1,2,3,4,5,6,7,8,9,… 2-fold down-sampling: 1,3,5,7,9,... Interpolation: expander (up-sampler) example : u[k]: 1,2,3,4,5,6,7,8,9,… 2-fold up-sampling: 1,0,2,0,3,0,4,0,5,0... L u[0], u[N], u[2N]... u[0],u[1],u[2]... M u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],... Down-sampler and up-sampler (Revisited)

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August 2004Multirate DSP (Part 2/2)5 General `subband processing’ set-up/overview: - signals split into frequency channels/subbands (`analysis bank’) - per-channel/subband processing - reconstruction (`synthesis bank’) - multi-rate structure: down-sampling / up-sampling Filter Banks and Subband Processing [1/6] subband processing 3 H1(z) subband processing 3 H2(z) subband processing 3 H3(z) 3 3 3 3 subband processing 3 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT

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August 2004Multirate DSP (Part 2/2)6 Step-1: Analysis filter bank - collection of M filters (`analysis filters’, `decimation filters’) with a common input signal - ideal (but non-practical) frequency responses = ideal bandpass filters - typical frequency responses (overlapping, marginally overlapping, non-overlapping) H1(z) H2(z) H3(z) H4(z) IN H1H4H3H2 H1H4H3H2 H1H4H3H2 K=4 Filter Banks and Subband Processing [2/6]

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August 2004Multirate DSP (Part 2/2)7 Step-2: Decimators (down-samplers) - subband sampling rate reduction by factor N critically decimated - critically decimated filter banks (= maximally down-sampled filter banks): N = K (where, K = number filters/subbands) this sounds like maximum efficiency, but aliasing problem arises! - over-sampled filter banks (= non-critically down-sampled filter banks): N < K Filter Banks and Subband Processing [3/6] H1(z) H2(z) H3(z) H4(z) IN 3 3 3 3 N=3K=4

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August 2004Multirate DSP (Part 2/2)8 Step-3: Subband processing - Example : coding (=compression) + (transmission or storage) + decoding - Filter bank design mostly assumes subband processing has `unit transfer function’ (output signals = input signals), i.e. mostly ignores presence of subband processing subband processing H1(z) subband processing H2(z) subband processing H3(z) 3 3 3 3 subband processing H4(z) IN N=3K=4 Filter Banks and Subband Processing [4/6]

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August 2004Multirate DSP (Part 2/2)9 Step-4: Expanders (up-samplers) - restore original fullband sampling rate by N-fold up-sampling (= insert N-1 zeros in between every two samples) Filter Banks and Subband Processing [5/6] subband processing 3 H1(z) subband processing 3 H2(z) subband processing 3 H3(z) 3 3 3 3 subband processing 3 H4(z) IN K=4N=3

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August 2004Multirate DSP (Part 2/2)10 Filter Banks and Subband Processing [6/6] Step-5: Synthesis filter bank - collection of K filters (`synthesis filters’, `interpolation filters’) with a `common’ (summed) output signal - frequency responses : preferably `matched’ to frequency responses of the analysis filters, e.g., to provide perfect reconstruction (see below) G1G4G3G2 G1G4G3G2 G1G4G3G2 G1(z) G2(z) G3(z) G4(z) + OUT K=4

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August 2004Multirate DSP (Part 2/2)11 Aliasing versus Perfect Reconstruction Assume subband processing does not modify subband signals (e.g. lossless coding/decoding) - The overall aim could be to have y[k]=u[k-d], i.e. that the output signal is equal to the input signal up to a certain delay - But: down-sampling introduces ALIASING, especially in maximally decimated (but even so in non-maximally decimated) filter banks - Question : Can y[k]=u[k-d] be achieved in the presence of aliasing? - Answer = YES, see below: PERFECT RECONSTRUCTION banks with synthesis bank designed to remove aliasing effects ! output=input 3 H1(z) 3 H2(z) 3 H3(z) 3 3 3 3 3 H4(z) u[k] G1(z) G2(z) G3(z) G4(z) + y[k]=u[k-d]? output=input

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August 2004Multirate DSP (Part 2/2)12 Multirate DSP Digital Filter Banks Filter Banks and Subband Processing Applications and Advantages Perfect Reconstruction FB 2-band Quadrature-Mirror Filter Bank K-band Filter Bank Case Uniform DFT Modulated Filter Banks

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August 2004Multirate DSP (Part 2/2)13 Filter Banks Applications [1/5] 1.Speech coding 2.Image compression 3.Adaptive equalization 4.Echo cancellation 5.Adaptive beamforming 6.Transmultiplexers (TDM) 7.Code division multiple access (CDMA) coding adaptive filtering

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August 2004Multirate DSP (Part 2/2)14 Filter Banks Applications [2/5] Subband coding : Coding = Fullband signal split into subbands & down-sampled subband signals separately encoded (e.g. subband with smaller energy content encoded with fewer bits) Decoding = reconstruction of subband signals, then fullband signal synthesis (expanders + synthesis filters) Example : Image coding (e.g. wavelet filter banks) Example : Audio coding e.g. digital compact cassette (DCC), MiniDisc, MPEG,... Filter bandwidths and bit allocations chosen to further exploit perceptual properties of human hearing (perceptual coding, masking, etc.)

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August 2004Multirate DSP (Part 2/2)15 Subband adaptive filtering : Example 1: Adaptive channel equalization Adaptive equalizer is employed to mitigate the effect of inter-symbol interference (ISI), that occurs due to limited bandwidth and multipath propagation. Example 2: Acoustic echo cancellation Adaptive filter models (time-varying) acoustic echo path and produces a copy of the echo, which is then subtracted from microphone signal. = difficult problem ! * long acoustic impulse responses * time-varying Filter Banks Applications [3/5]

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August 2004Multirate DSP (Part 2/2)16 Subband adaptive filtering (continued): Example 1: Adaptive channel equalization Adaptive equalizer is employed to mitigate the effect of inter-symbol interference (ISI), that occurs due to limited bandwidth and multipath propagation. Filter Banks Applications [4/5]

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August 2004Multirate DSP (Part 2/2)17 3 H1(z) 3 H2(z) 3 H3(z) 3 H4(z) 3 H1(z) 3 H2(z) 3 H3(z) 3 H4(z) + + + + 3 G1(z) 3 G2(z) 3 G3(z) 3 G4(z) OUT + ad.filter Subband adaptive filtering (continued): Example 2: Acoustic echo cancellation - Subband filtering = K subband modeling problems instead of one fullband modeling problem - Perfect reconstruction guarantees distortion-free desired near-end speech signal Filter Banks Applications [5/5]

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August 2004Multirate DSP (Part 2/2)18 General advantages of subband decomposition techniques 1.Allow parallel processing of signal will slower processor, as the sampling rate is reduces in each subband. 2.Computational complexity reduction. For instance in compression application where subband are compressed at different rates, e.g. speech processing and image compression. 3.Improve/increase convergence speed in adaptive filtering applications, e.g. echo cancellation and adaptive equalization.

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August 2004Multirate DSP (Part 2/2)19 - analysis bank + synthesis bank - multirate structure: down-sampling after analysis, up-sampling for synthesis - aliasing vs. “perfect reconstruction” - filter bank implementation (for N = decimation factor, K = number of subband) : 1.critically-sampled FB N = K 2.over-sampled FB N < K - applications: coding, (adaptive) filtering, transmultiplexers - advantages of subband processing subband processing 3 H1(z) subband processing 3 H2(z) subband processing 3 H3(z) 3 3 3 3 subband processing 3 H4(z) IN G1(z) G2(z) G3(z) G4(z) + OUT Conclusion I: General `subband processing’ set-up

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August 2004Multirate DSP (Part 2/2)20 Multirate DSP Digital Filter Banks Filter Banks and Subband Processing Applications and Advantages Perfect Reconstruction FB 2-band Quadrature-Mirror Filter Bank K-band Filter Bank Case Uniform DFT Modulated Filter Banks

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August 2004Multirate DSP (Part 2/2)21 Two design issues : 1.filter specifications, e.g. stopband attenuation, passband ripple, transition band, etc. (for each (analysis) filter!) 2.(nearly) perfect reconstruction property... Design Issues

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August 2004Multirate DSP (Part 2/2)22 Perfect Reconstruction: K = 2 subbands case H0(z) H1(z) 2 2 u[k]2 2 G0(z) G1(z) + y[k] It is proved that... (try it!) U(-z) represents aliased signals, hence the `alias transfer function’ A(z) should ideally be zero T(z) is referred to as `distortion function’ (amplitude & phase distortion). For perfect reconstruction, T(z) should ideally be a pure delay

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August 2004Multirate DSP (Part 2/2)23 Alias-free filter bank: Perfect reconstruction filter bank (= alias-free + distortion-free): i) ii) PS: if A(z)=0, then Y(z) = T(z).U(z), hence the complete filter bank behaves as a linear time invariant (LTI) system (despite up- & down-sampling)! Perfect Reconstruction: K = 2 subbands case H0(z) H1(z) 2 2 u[k]2 2 G0(z) G1(z) + y[k]

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August 2004Multirate DSP (Part 2/2)24 An initial choice is ….. : so that For the real coefficient case: which means the amplitude response of H1 is the mirror image of the amplitude response of H0 with respect to the quadrature frequency hence the name `quadrature mirror filter’ (QMF) Perfect Reconstruction: K = 2 subbands case H0(z) H1(z) 2 2 u[k]2 2 G0(z) G1(z) + y[k] ignore the details!

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August 2004Multirate DSP (Part 2/2)25 quadrature mirror filter (QMF): hence if H0 (=G0) is designed to be a good lowpass filter, then H1 (=-G1) is a good high-pass filter. Perfect Reconstruction: K = 2 subbands case H0(z) H1(z) 2 2 u[k]2 2 G0(z) G1(z) + y[k] H0H1

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August 2004Multirate DSP (Part 2/2)26 Perfect Reconstruction: K number subbands case H2(z) H3(z) 4 4 4 4 G2(z) G3(z) y[k] H0(z) H1(z) 4 4 u[k] 4 4 G0(z) G1(z) + It is proved that... (try it!) 2nd term represents aliased signals, hence all `alias transfer functions’ A l (z) should ideally be zero (for all l ) T(z) is referred to as `distortion function’ (amplitude & phase distortion). For perfect reconstruction, T(z) should ideally be a pure delay ignore the details!

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August 2004Multirate DSP (Part 2/2)27 Multirate DSP Digital Filter Banks Filter Banks and Subband Processing Applications and Advantages Perfect Reconstruction FB 2-band Quadrature-Mirror Filter Bank K-band Filter Bank Case Uniform DFT Modulated Filter Banks

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August 2004Multirate DSP (Part 2/2)28 Uniform versus non-uniform (analysis) filter bank: non-uniform: e.g. for speech & audio applications uniform filter bank: = frequency responses uniformly shifted over the unit circle H0(z) = `prototype’ filter, is the only filter that has to be designed Uniform DFT-Modulated Filter Banks H0(z) H1(z) H2(z) H3(z) IN H0H3 H2H1 H0H3H2H1 uniform non-uniform

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August 2004Multirate DSP (Part 2/2)29 Uniform DFT-Modulated Filter Banks Uniform filter banks can be implemented cheaply based on polyphase decompositions + FFT : 1.Uniform DFT-modulated analysis filter banks If then H0(z) H1(z) H2(z) H3(z) u[k]

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August 2004Multirate DSP (Part 2/2)30 Uniform DFT-Modulated Filter Banks where F is NxN DFT-matrix u[k] ignore the details!

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August 2004Multirate DSP (Part 2/2)31 Uniform DFT-Modulated Filter Banks Uniform DFT-modulated analysis FB + decimation (K=N) 4 4 4 4 u[k] 4 4 4 4 =

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August 2004Multirate DSP (Part 2/2)32 2.Uniform DFT-modulated synthesis filter banks Uniform DFT-Modulated Filter Banks + + + y[k]

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August 2004Multirate DSP (Part 2/2)33 where F is NxN DFT-matrix y[k] + + + ignore the details!

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August 2004Multirate DSP (Part 2/2)34 Uniform DFT-Modulated Filter Banks Expansion (K=N) + uniform DFT-modulated synthesis FB: y[k] 4 4 4 4 + + + + + + 4 4 4 4 =

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August 2004Multirate DSP (Part 2/2)35 Uniform DFT-Modulated Filter Banks For over-sampled case, with down-sampling factor (N) smaller that the number of subbands (K) [i.e. N < K], aliasing is expected to become a smaller problem, possibly negligible if N<

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August 2004Multirate DSP (Part 2/2)36 Design problem = all filter specs + alias cancellation (PR) Uniform versus non-uniform filter banks Critically-sampled and over-sampled PR filter banks Modulated PR filter banks DFT-modulated filter banks Other methods: GDFT-modulated filter banks and DCT- modulated filter banks (not considered here!) Economy in design (only prototype) & implementation Conclusion II:

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