A FAST WINDOWING TECHNIQUE FOR DESIGNING DISCRETE WAVELET MULTITONE TRANSCEIVERS EXPLOITING SPLINE FUNCTIONS

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1 A FAST WINDOWING TECNIQUE FOR DESIGNING DISCRETE WAVELET ULTITONE TRANSCEIVERS EXPLOITING SPLINE FUNCTIONS Fernando Cruz-Roldán, Pilar artín-artín, anuel Blanco-Velasco, Taio Sarämai Ұ Deartamento Teoría de la Señal y Comunicaciones, Universidad de Alcalá, Camus de la Universidad de Alcalá, 887, Alcalá de enares, Sain fernando.cruz@ieee.org, web: htt://msc.tsc.uah.es/~fcruz Ұ Institute of Signal Processing, Tamere University of Technology, P. O. Box 553, FIN-33, Finland ts@cs.tut.fi ABSTRACT A very fast technique for designing discrete wavelet multitone (DWT) transceivers without using time-consuming nonlinear otimization is introduced. In this method, the filters in both the transmitting and receiving filter bans are generated based on the use of a single linear-hase finiteimulse resonse rototye filter and a cosine-modulation scheme and the rototye filter is otimized by using the windowing technique. The novelty of the roosed technique lies in exloiting sline functions in the transition band of the filter, instead of using the conventional bric-wall filter. In this aroach, a simle line search is used for finding the assband edge of the filter for minimizing a redetermined cost function. The resulting DWT transceivers closely satisfy the erfect reconstruction roerty, as is illustrated by means of examles. INTRODUCTION Cosine-odulated Filter Bans (CFBs) with alications to communications are generally referred to as discrete wavelet multitone (DWT) or discrete subband multicarrier (DSB) transceivers []-[3]. The use of CFBs is a very attractive way of generating these transceivers for the following reasons. First, high-selectivity and highdiscrimination systems can be easily designed. Second, the resulting transmitting and receiving filters can be generated based on the use of a single linear-hase finite-imulse resonse (FIR) rototye filter. Third, in articular cases, fast algorithms can be alied to efficiently imlementing the sub-carrier modulators utilizing arallel rocessing structures [4, 9]. This wor describes a very efficient technique for designing rototye filters for DWT systems as shown in Fig.. This design scheme is an imroved version of that described in [5, 6], the design of rototye filters for nearly erfect-reconstruction (NPR) CFBs is based on the use of the windowing technique in such a manner that the cut-off frequency of the filter the 3-dB oint of the magnitude resonse of the rototye filter is aroximately located at ω /(). As illustrated in [5, 6], this technique results in imroved analysis-synthesis (or receivingtransmitting) filter bans when comared with those achieved using other existing similar design methods [7, 8]. Figure - Critically-samled -channel DWT transceiver over a noisy channel. In order to achieve even better overall filter ban erformances, this wor incororates a sline transition function in the low-ass filter definition as suggested in [] for designing least-squared-error linear-hase FIR filters. Several examles are included illustrating the sueriority of the roosed aroach comared to those using bricwall filters. This wor is organized as follows: Section briefly reviews how to construct cosine-modulated DWT transceivers. Section 3 describes the roosed technique for designing the rototye filter. Section 4 considers roer quality measures for evaluating the erformance of the DWT systems resulting when using the roosed technique comared with the erfect-reconstruction (PR) systems. Due to the close relations between the synthesisanalysis and analysis-synthesis filter bans, similar measures are also introduced for the latter bans. In Section 5, examles are included for both illustrating the benefits of the roosed design scheme and on how to roerly select the window function and the sline function for the transition band shaing of the filter. Finally, concluding remars are drawn in Section 6.

2 . COSINE-ODULATED DWT TRANSCEIVERS This section briefly reviews how to construct criticallysamled cosine-modulated DWT systems as shown in Fig., the signal is transmitted over a noisy channel. For such a system, based on an Nth-order linear-hase FIR rototye transfer function given by h () z h [] n n z [] n h [ N n] for n,, K, N, N n the imulse resonse coefficients of all the transmitting and receiving sub-channel filters, denoted by f [n] and h [n] for,,,, resectively, are generated as follows: N [] n h [] n cos ( + ) n ( ) f, (a) 4 and N h [ n] h[ n] cos ( + ) n + ( ) 4, (b) for n,,, N. Comared to the conventional construction of the f [n] s (see, e.g., [4]), an additional constant is included in equation (a). This is because of the following reason. For the rototye filter as well as for the transmitting and receiving sub-channel filters resulting when alying the roosed technique, the maximum amlitude value in the assband is aroximately equal to unity. Therefore, this constant is needed in order to reserve the signal energy when interolating by a factor of before using the transmitting sub-channel filters. Deending on how to synthesize the rototye filter, the above cosine-modulation scheme results in either PR or NPR DWT transceivers. This wor concentrates on generating NPR systems. 3. PROPOSED DESIGN SCEE FOR TE PROTOTYPE FILTER This section describes the roosed design scheme for designing rototye filters based on alying the windowing technique to filters, the transition band is shaed by means of a sline function. 3. FIR filter design based on the use of transition-band sline functions When generating an Nth-order linear-hase FIR rototye filter using the roosed technique, first a roer window function w[n] is selected in such a way that it is nonzero only for n N and satisfies w[n n] w[n] for n,,, N. Second, the filter with the assband and stoband edges being located at ω ω and ω ω s, resectively, is generated as follows: h [] n {[ n N ]( ω + ω ) }, [ n N ] () sin s φ[], n (3a) {[ n N ]( ω ωs ) ( ψ )} [ n N ]( ω ωs ) ( ψ ) sin φ[] n (3b) is the ψth-degree sline function for shaing the transition band. The frequency resonse of the resulting infinite-duration filter is exressible as jnω ( e ) e A ( ω), (4) A (ω) in the assband region [, ω ] and A (ω) in the stoband region [ω s, ], resectively, as in the transition band (ω, ω s ) the erformance of A (ω) deends on ψ, the order of the sline function selected for shaing the transition band (for more details, see []). After nowing both the window function and the imulse-resonse coefficients of the filter, the imulseresonse coefficients of the resulting rototye filter are given by [ ] [ ] [ ] h n w n h n (5) for n,,, N, as in the case of using a conventional bric-wall filter with ω ω s. 3. ain benefits of using the sline function for the transition band shaing The roosed method basically follows the same technique as described in [5, 6], the imulse resonse of the rototye filter, as given by equation (5), is determined in the case of bric-wall filters. The use of the ψthorder sline function in the transition band in the transition band (ω, ω s ) results in the following benefits. First, the stoband edge of the rototye filter can be fixed in terms of the roll-off factor, denoted by ρ, as s ( ) ( ). ω + ρ (6) Second, by roerly selecting the degree of the sline function for the transition band shaing gives an extra freedom in arriving at a rototye filter giving rise to a better overall transceiver erformance comared to the case of using the bric-wall filter. 3.3 Proosed Design Technique Given N, the order of the rototye filter,, the number of channels, ρ secifying the stoband edge according to equation (6), the window function w[n] being nonzero for n N, and ψ, the degree of the sline function, the roosed design scheme carried out in the following stes: Ste : Fix ω s (+ρ) /(). Select the initial value for ω to be ω ( ρ) /(). Ste : Generate the imulse resonse of the filter h [n] according to equations (3a) and (3b). Ste 3: Generate the imulse resonse h [n] of the rototye filter according to equation (5). Ste 4: Determine the value of ω so that the magnitude resonse of the rototye achieves the value of ψ

3 at ω ( ), that is, j ( ) ( ) e. In this method, the otimized value of ω can be found by using a simle line search algorithm, thereby maing the overall synthesis extremely fast. What is left is to select the window function and the degree of the sline function in such a manner that the resulting NPR DWT transceiver closely aroximates the PR one in the case, the channel in Fig. is and there is no noise. Various alternatives for generating such a system will be considered in Section 5 by means of illustrative examles. 4. PERFORANCE EVALUATION This section considers roer quantities for measuring the distortions for evaluating the erformance of NPR DWT systems resulting when using the roosed technique comared to the PR ones. In addition, because of the close connections between the analysis-synthesis and synthesisanalysis filter bans, roer distortion measures are introduced for the corresonding NPR cosine-modulated filter ban (CFB) as shown in Fig.. 4. Quality easures for CFB Systems When omitting the effect of the rocessing unit and using definitions of the imulse-resonses of the F (z) s and (z) s, as given by equations (a) and (b), resectively, the relation between the outut signal xˆ [n] and the inut signal x [n] is exressible in the z-domain as Xˆ T X + Tl l X F j l ( ze ) (7a) T (7b) is the distortion transfer function determining the distortion caused by the overall system for the un-aliased comonent X(z) and T l F j l ( ze ) (7c) for l,,, - are called the alias transfer functions and determine how well the aliased comonents j l X ( ze ) of the inut signal are attenuated. For the PR condition, it is required that T (z) z N and T l (z) for l,,, -. Based on this, the quality measurements should concentrate on both the distortion on the un-aliased comonent and the aliasing distortion. A good measure for the un-aliased distortion is the following quantity: δ max { T ( e )} min { T ( e )}, (8) ω [, ] ω [, ] as a good measure for the overall aliasing distortion is the ea aliasing error as given by [4] agnitude(db) Fig.. Critically-samled -channel CFB system. ψ ψ Ref. [5,6] Normalized Frequency Fig. 3. Prototye filter magnitude resonses (e ) designed using the several techniques with the aid of the Kaiser window in the β 9 case. E alias j( ω l ) max F ( e ) e ω [, ] l ( ). 4. Quality easures for DWT Transceivers The erformance of the DWT transceivers as shown in Fig. can be evaluated using the following two main signal-to-interference ratios (SIRs) in the case there exist no noise. The first SIR, denoted by SIR ICI for later use, is the ratio between the ower of the received signal in the lth sub-channel due to the lth inut signal and that due to the other inut signals, called the inter-channel interference (ICI). This SIR ICI is given by SIR ICI () l ( ll ( ) ), l (9) T e dω, (a) Tl ( e ) dω j( ω m) / j( ω m) / j( ω m) / l ( ) l ( ) ( ) m T e e C e F e ( ) (b) is the frequency resonse between the lth outut and the th inut in the noisy-free transceiver of Fig..

4 TABLE I COPARISON BETWEEN VARIOUS TECNIQUES FOR DESIGNING 3-CANNEL CFBS AND DWT TRANSCEIVERS WIT TE SUB-CANNEL FILTERS OF LENGT 5 (N 5). Prototye Filter Design Technique δ E alias (db) SIR ICI (db) SIR ISI (db) Blacman window (ref. [5, 6]) Blacman window (roosed with ψ ) Blacman window (roosed with ψ ) amming window (ref. [5, 6]) amming window (roosed with ψ ) amming window (roosed with ψ ) Kaiser window (ref. [5, 6]) Kaiser window (roosed with ψ, β 9) Kaiser window (roosed with ψ, β 9) Ref. [7] Ref. [8] with β The second SIR, denoted by SIR ISI, is due to the intersymbol interference (ISI). This SIR is defined as the ratio between the ower of the received signal in the lth subchannel due to the lth inut signal and the reduction in this ower due to ISI. This SIR is given by Tll ( e ) dω SIRISI () l. () Tll ( e ) dω 5. EXAPLE DESIGNS In order to illustrate the benefits of the roosed design scheme, several 3-channel CFB and DWT systems ( 3) with 5-length rototye filters (N 5) have been designed for ρ, that is, ω s /. The comarisons are summarized in Table I, considering an environment in which there is an channel ( C( z ) ). The best results with resect to the above-defined quality measures, that is, δ, E alias, SIR ICI and SIR ISI, are indicated by the boldface letters. As seen from this table, the use of the Kaiser window with β 9 and ψ results in significant global imrovements and is a good choice for the 3 and N 5 case. The effect of the arameter ψ on the resulting rototye filter has also been studied. As an examle, the resulting rototye filter resonses designed using the Kaiser window in the β 9 for ψ and ψ as well as that synthesized using the earlier technique described in [5, 6] are shown in Fig. 3. It is seen from this figure, the integer value of ψ has an influence on the transition band behavior of the rototye filter and thus also on the overla factor between the adjacent channels. The erformance of transceivers with the rototye filter designed using various windowing techniques has also been tested inside a communication system. For comarison uroses, several bit-error rate (BER) curves based on onte- Carlo simulations are shown in Figs. 4 and 5. For these figures, the roosed transceivers has been designed using the Kaiser window with β 9 and ψ. It is assumed that the inut signals x [n] for the system of Fig. are indeendently chosen binary white data sequences. In addition, it is assumed that the transmission channel is distorted with both additive white Gaussian noise and a narrow-band interference shown in Fig. 6. The detection is based on simle threshold detectors. As can be seen, all the simulated DWT transceivers erform in almost the same manner including the low signal-to-noise ratios (SNRs). According to the measurement results (shown in Figs. 4 and 5), the DWT transceiver designed with the roosed technique (for ψ ) shows a good erformance, jointly with that designed through technique roosed in [5, 6]. 6. CONCLUSIONS An imroved windowing technique for designing rototye filter for cosine-modulated synthesis-analysis and analysis-synthesis multirate filter bans has been roosed. The ey idea in this technique is to include a sline function for the transition band shaing of the filter, instead of using a conventional bric-wall filter. Simulations results have shown the benefits of this aroach.

5 When designing such multirate filter bans for ractical alications, the main benefit of the roosed technique is that it enables one to design very quicly, without timeconsuming non-linear otimization algorithms, systems with very many channels and very long rototye filters. Furthermore, by roerly selecting the window function and the sline function for the transition band shaing of the filter, the erformance of the resulting NPR multirate systems aroach the PR ones. Future wor is devoted to further develoing this technique by both roerly selecting the degree of the sline function for the transition band shaing of the filter and finding or generating new window functions. BER Ref. [5, 6] Proosed (ψ) Ref. [7] Ref. [8] SNR Fig. 4. BER results for various 3-channel DWT transceivers..6.7 BER.8 Ref. [5, 6] Proosed (ψ) Ref. [7] Ref. [8] SNR Fig. 5. Details of the BER results for various 3-channel DWT transceivers. ACKNOLEDGENT This wor was suorted in art by Comunidad Autónoma de adrid and Universidad de Alcalá through roject CA- UA5/. REFERENCES [] S. D. Sandberg and. A. Tzannes, "Overlaed discrete ultitone odulation for igh Seed Coer Wire Communications," IEEE Journal on Selected Areas in Communications, Vol. 3, NO. 9, , December 995. []. A. Tzannes,. C. Tzannes, J. Proais, and P. N. eller, DT systems, DWT systems and digital filter bans, in Proc. Int. Conf. Communications, New Orleans, USA, ay 994, [3] A. N. Aansu and X. Lin, A comarative erformance evaluation of DT (OFD) and DWT (DSBT) based DSL communications systems for single and multitone interference, in Proc. IEEE Int. Conf. Acoustics, Seech, and Signal Processing, Seattle, USA, ay 998, vol. 6, [4] P. P. Vaidyanathan, ultirate Systems and Filter Bans. Prentice-all, Englewood Cliffs, NJ, 993. [5] F. Cruz-Roldán, P. Amo-Lóez, S. aldonado-bascón, and S. S. Lawson, An efficient and simle method for designing rototye filters for cosine-modulated seudo- QF bans, IEEE Signal Processing Letters, vol. 9, no.,. 9 3, Jan.. [6] P. artín, F. Cruz-Roldán, and T. Saramäi, A windowing aroach for designing critically samled nearly erfect-reconstruction cosine-modulated transmultilexers and filter bans, in Proc. Third Int. Sym. Image and Signal Processing and Analysis, Rome Italy, Set. 3, [7] C. D. Creusere and S. K. itra, A simle method for designing high-quality rototye filters for -band seudo-qf bans, IEEE Trans. Signal Processing, vol. 43, no. 4,. 5 7, Ar [8] Y.-P. Lin and P. P. Vaidyanathan, A Kaiser window aroach for the design of rototye filters of cosine modulated filter bans, IEEE Signal Processing Letters, vol. 5, no. 6,. 3 34, June 998. [9] F. Cruz-Roldán and. onteagudo, Efficient imlementation of nearly-erfect reconstruction cosinemodulated filterbans, IEEE Trans. Signal Processing, vol. 5, no. 9, , Set. 4. [] C. S. Burrus, A. W. Soewito, and R. A. Goinath, Least squared error FIR filter design with transition bands, IEEE Trans. Signal Processing, vol. 4, no. 6, , June 99.