Design of Maximally Decimated Near-Perfect-Reconstruction DFT Filter Banks with Allpass-Based Analysis Filters

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Dana Ramsey
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1 O Copyright 1 EEE Published in the Proceedings of the Thirty-ifth silomar Conference on ignals ystems and Computers pages November Pacific Grove C U esign of aximally ecimated Near-Perfect-Reconstruction T ilter anks with llpass-ased nalysis ilters Enisa Galijašević Jörg Kliewer University of Kiel nstitute for Circuits and ystems Theory 414 Kiel Germany eg@tfuni-kielde jkl@tfuni-kielde bstract n this paper we address a imally decimated R T filter bank where all polyphase components are replaced by allpass filters Generally a perfect reconstruction PR) solution leads to a synthesis filter bank with unstable subband filters s a novelty we show that by using closed-form R phase compensation filters as polyphase components in the synthesis filter bank we obtain a stable near-pr analysissynthesis system urthermore the remaining linear and aliasing distortions only depend on the overall system delay and can be made arbitrarily small 1 ntroduction Critically subsampled uniform channel filter banks can be used in many applications for example in image and audio compression where most design techniques deal with R-based systems However especially in realtime applications computationally efficient solutions are often reasonable also for the filter bank system or instance a very low-complexity alternative for performing an -channel subband decomposition can be obtained by a complex-modulated R filter bank where all polyphase components are replaced by allpass filters [1] However an overall perfect reconstruction PR) system can generally only be achieved with unstable synthesis subband filters This problem can be solved by using anticausal filtering [] and employing a double buffering schema as in [ 4] for the processing of infinite-length signals Unfortunately this method requires extra information which grows with the order of the allpass filters in the analysis since initial conditions for the synthesis allpasses have to be additionally transmitted n this paper we propose a novel design for the synthe- This work was supported in part by the German Research oundation eutsche orschungsgemeinschaft G Graduate College No 57) sis filter bank which avoids anticausal filtering and leads to a near-pr solution for the critically subsampled analysissynthesis system The synthesis subband filters are of Rtype and can be derived from simple closed-form expressions urthermore all distortions in the reconstructed signal can be made arbitrarily small and depend only on the overall system delay of the filter bank llpass-based -band T R ilter anks n the following we consider imally decimated - band T filter banks with where the corresponding analysis bank is shown in ig 1 Herein de- "!$# igure 1 T R analysis filter bank in polyphase notation notes the T matrix with % '&)+-/ : ;=?> CC+ The polyphase components H JLKN are allpass transfer functions leading to an R prototype filter LKNP RQ :TVU K H WLK ince the analysis filters X- 8KY are modulated versions of the lowpass prototype O LKN they are also R filters and 1)

2 ; O 5 can be obtained from 1) according to X LKNP 8KY P K :TVU : H 8K or the sake of simplicity we restrict ourselves in the following derivations to first-order allpass filters defined as H JLKNP K RQ However later on we will also briefly address the higher order cases Normalized magnitude d) Normalized frequency ω /π) igure Normalized magnitude frequency re- sponse for the lowpass first-order allpasses) ) ) s an example a lowpass prototype is designed for and a passband edge frequency by using the Remez-type optimization approach from [5] where the normalized magnitude frequency response is depicted in ig The side-lobes are inevitable [5] but they do not introduce any additional error into the reconstructed signal Compensation of the Phase istortion We now derive a closed-form expression for R filters which approximately compensate the phase distortions introduced by first-order allpass filters up to a certain error [6] Let us first consider the polynomial factorization relation K LK RQ K : 4) TVU with Exploiting this relation R filters P LKN can be defined according to LKNP RQ K K : 5) TVU where it can be shown from ) 4) and 5) that H LKN LKN GK GK 6) we can select the order N of the filter LKN such that the error N can be made arbitrarily small at the expense of additional delay Then 6) can be approximated as H JLKN JLKN!K " 4 ynthesis ilter ank esign 41 asic idea ased on the result from the last section we are now able to construct a synthesis filter bank where the phasecompensation filters W8KY from 5) are used as type-) polyphase components The resulting critically subsampled analysis-synthesis filter bank is shown in ig with CC+ When an ap- $#&%' Y ; proximately equal phase-compensation ; error N in each polyphase branch is desired it is necessary to introduce the extra delay blocks K " due to different coefficients The synthesis filters ) 8KY in ig are modulated versions of the synthesis prototype LKN +TVU and can be written as K : ) LKNP LKN P K : LK +TVU Note that due to + L 4- + LK W : JK " K 7) + H LKN has almost the same normalized magnitude frequency response as the analysis prototype which will be demonstrated with a design example in ection 5 4 Linear and aliasing distortions n the following we derive expressions for the linear and aliasing transfer functions for the proposed analysissynthesis system in ig The analysis polyphase matrix 8KY can be given as LKNP VC / diag % H U LKN H Q LKN H RQ 8KY & 8) whereas the synthesis polyphase matrix 1 8KY is written as 1 LKN diag % U 8KYJK 4 Q LKNJK "65 VCC CC LKNJK " 987 & 9)

3 1 5 & % % ' # $ +- + )! # $ % "! # $ & igure Critically subsampled -band analysis-synthesis R / R T filter bank / With 8) 9) and / 8KYP 1 LKN LKNP diag % H U 8KY U 8KYWK " C CC H LKN RQ 8KYJK 9 7 the product LKN finally yields the diagonal matrix 1) where the corresponding simplified system is depicted in ig 4 f all diagonal elements in 8KY are equal which is a special case of a pseudocirculant matrix) it is a well known fact that the filter bank is free from aliasing [7] urthermore if those elements are equal to a delay the filter bank is a PR system However since all diagonal elements of LKN in 1) are almost equal and represent approximate delays we can say that the filter bank is close to PR or with other words it represents a near-pr system Using 1) the input-output relation can now be obtained as LKNP H 8K :TVU :TVU TVU TVU 8K 8K LKN LKN JK " K 11) The linear distortion transfer function 4 lin 8KY can be derived from 11) for according to 4 linlknp K 76 Q K 6 Q K " 1) 8 :TVU 9: ; T=?> lin@ E GH C C C J J J H LJ LJ L J C C C K E GH igure 4 implified system from ig where N lin 8KY denotes the transfer function of the remaining linear distortion error y appropriate selection of the parameters N the linear distortion error tends to zero ie esides we can see from 11) that 4 linlkn has approximately linear phase and the overall system delay is O The transfer function P QJCC+ LKN 5 of the 5 -th aliasing component LKN can be likewise obtained from 11) as P LKNP :TRU 6 Q K 1) t is evident that the amount of aliasing distortion only depends on the parameters and the appropriate choice of the delays This choice is arbitrary and we can select the such that the aliasing frequency responses approach zero at all frequencies Thus all distortions in the reconstructed signal are due to the phase-compensation error in 6) mplitude and phase distortion as well as aliasing can be made arbitrarily small and can be exchanged for an increase of the overall system delay and vice versa The resulting distortions can be further reduced when we

4 ; use the imal order for all R phase-compensation filters ie ; CC: for n this case eq 6) writes with + H LKN LKNP K " 8 9: ; + + T N + urthermore the extra delay blocks in the synthesis filter bank ig ) vanish 4 Higher order allpasses The proposed phase-compensation approach can be extended to higher-order allpass filters n -th order allpass function can be in general written as a product of first order allpass functions [7] Therefore we now choose the -th polyphase component of the analysis filter bank as an allpass filter of -th order according to H 8KYP K where is someq constant and denotes the coefficient 14) of the ; -th first order allpass filter in the -th polyphase branch or simplicity reasons we restrict ourselves to real valued coefficients y cascading the corresponding filters from 5) in the synthesis filter bank we derive an R filter LKN of order LKNP C N according to LKN 15) with N denoting the order of the individual R filters 8KY y multiplying 14) and 15) we finally get H LKN LKNP K 5 Herein Q CCC H 8KY LKN Q CCC CCC Y Q N 16) CC+ Y Q Y specifies the error caused by the non-ideal phase compensation The expressions for the linear distortion and aliasing transfer functions can be derived in a straightforward way as in the firstorder case discussed above 5 esign Example n the following design example we use the first-order allpass-based analysis prototype! from ection with pa- and the magnitude fre- rameters and '@ quency response from ig y choosing the synthesis polyphase components according to 5) with we obtain the results depicted in ig 5 ig 5a) shows the overall magnitude amplitude distortion of the resulting analysis-synthesis system ig 5b) the peak aliasing distortion N P RQ + P and ig 5c) the group delay respectively We can see that the linear distortion transfer function has almost linear phase with an average system delay of samples n this example the choice of the W reflects a good a) b) c) 1 T lin z) liasing distortion d) Group delay samples) 4 6 x Normalized frequency ω /π) Normalized frequency ω /π) Normalized frequency ω /π) igure 5 Overall analysis-synthesis system ): a) amplitude distortion b) peak aliasing distortion c) group delay compromise between low complexity on the one hand and sufficient suppression of aliasing and linear distortions on the other hand The normalized magnitude frequency response of the lowpass synthesis prototype LKN is shown in ig 6 Note that this frequency response is strongly similar to the one of the analysis prototype in ig n the second example displayed in ig 7 the same allpass-based analysis prototype as in the example from ig 5 is used but the synthesis polyphase components are now obtained from 5) with J!" ; for all t can be observed that all distortions are reduced whereas the overall delay remains the same ll aliasing magnitude frequency responses + P are constant functions which

5 @ 1 a) x 1 4 Normalized magnitude d) Normalized frequency ω /π) igure 6 Normalized magnitude frequency response for the lowpass synthesis prototype $ ) can be shown from 1) in a straightforward way 6 Conclusion We have proposed a new near-pr design approach for the allpass- / R-based critically subsampled T filter bank leading to stable subband filters The novel synthesis filter bank only employs R polyphase components which almost eliminate the phase distortion introduced by the allpass polyphase components on the analysis side urthermore the reconstruction error only depends on the imally allowable overall system delay The proposed compensation technique can also be extended to higher order allpass filters in the analysis bank by concatenating allpassfilters on the analysis and the corresponding R phasecompensation filters on the synthesis side respectively nother advantage is that by exchanging the role of the analysis and synthesis filter bank our phase-compensation approach can also be applied to the analysis side We then obtain a filter bank system having an efficient allpass-based synthesis filter bank which is especially well suited for lowcomplexity decoder-only applications References [1] J H Husøy and T Ramstad pplication of an efficient parallel R filter bank to image subband coding ignal Processing 4):79 9 ugust 199 [] T Chen and P P Vaidyanathan General theory of time-reversed inversion for perfect reconstruction filter banks n Proc 6th silomar Conference on ignals ystems and Computers pages Pacific Grove C U October 199 b) c) 1 T lin z) liasing distortion d) Group delay samples) Normalized frequency ω /π) Normalized frequency ω /π) Normalized frequency ω /π) igure 7 Overall analysis-synthesis system with N " ; C: : a) amplitude distortion b) peak aliasing distortion c) group delay [4] K itra C Creusere and H abic novel implementation of perfect reconstruction Q banks using R filters for infinite length signals n Proc EEE nt ympos Circuits and ystems pages 1 15 an iego U ay 199 [5] Renfors and T aramäki Recursive Nth-band digital filters-part : esign and properties EEE Trans on Circuits and ystems C-41):4 7 January 1987 [6] E Galijašević and J Kliewer Non-uniform nearperfect-reconstruction oversampled T filter banks based on allpass-transforms n Proc Ninth EEE P Workshop P ) Hunt TX U October [7] P P Vaidyanathan ultirate ystems and ilter anks Prentice Hall Englewood Cliffs 199 [] T Ramstad R filterbank for subband coding of images n Proc EEE nt ympos Circuits and ystems pages 87 8 Helsinki inland June 1988

Copyright S. K. Mitra

Copyright S. K. Mitra 1 In many applications, a discrete-time signal x[n] is split into a number of subband signals by means of an analysis filter bank The subband signals are then processed Finally, the processed subband signals