A Multistage Approach to the Design of Prototype Filters for Modulated Filter Banks
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1 Proceeding of the World Congre on Engineering Vol II WCE, June 3 - July,, London, U.K. A ultitage Aroach to the Deign of Prototye Filter for odulated Filter Bank Neela R. Rayavarau(ember IEEE) and Neelam Ru Prakah(ember IEEE) Abtract: When filter bank are ued in multicarrier modulation the tranition bandwidth of the filter i uually very narrow. Thi i becaue the bandwidth of the filter i determined by the number of ubchannel and the greater the number of channel leer will be the tranition bandwidth. The order of the filter being inverely roortional to the tranition bandwidth i generally very high. Hence comutational comlexity i alo very high. We rooe a method that will hel reduce thi comlexity by deigning the decimator/interolator in the ubchannel uing a multitage aroach. We comare the aving obtained in comarion to direct deign aroache. We will alo comare the erformance of a coine modulated filter bank deigned in thi way with direct deign method. Keyword: decimator, interolator, multicarrier modulation, multitage, tranition bandwidth,. Introduction Digital Subcriber Line (xdsl) alication ue a multicarrier modulation technique called a Dicrete ultitone (DT) for the uroe of high bit rate tranmiion over the commonly ued twited air coer wire. Thi eliminate bottleneck in the data acce network between the central office and the end uer. Thi ha been made oible by the ue of, amongt other thing, advanced ignal roceing technology. One uch area of ignal roceing, filter bank, ha found vat alication i everal area of digital communication, uch a high eed DSL ervice for internet [][]. In Fig. i hown the block diagram of a multicarrier modulation ytem that ue multirate filter bank. The modulation and demodulation i erformed uing filter and fat tranform uch a anucrit ubmitted on February 5,.. Neela Rayavarau i with the Deartment of Electronic and Communication Engineering, Chitkara Intitute of Engineering and Technology, Rajura, Punjab, India. ( neela.rayavarau@chitkara.edu.in) FFT and DCT. The multirate filter bank i comoed of the ynthei and the analyi filter bank that are ued for erforming modulation and demodulation. Each channel of the ynthei filter bank conit of an uamler cacaded with a filter, together called a the interolator, and the analyi filter bank conit of a filter followed by a downamler together termed a a decimator. i/ Decoder,P/S convertor S/P convertor and encoder equalizer o/ odulat or Demodulat or P/S Convertor The u amling and the down amling factor will be determined by the number of ubchannel of the ytem and greater the number of channel greater will be uamling/downamling factor and maller will be the tranition bandwidth. Thi will obviouly lead to longer length filter being ued. Thi will in turn increae the comutational comlexity involved in the imlementation of the decimator and the interolator. In order to minimize comutation a multitage aroach to the decimator and interolator hown in Fig. and Fig 3. i ued. Thi i baed on the IFIR aroach uggeted by Neuvo et al [3]. We will briefly exlain thi method and ee how it may ued to obtain multitage imlementation. Channel S/P Convertor Fig. ulticarrier modulation Sytem uing multirate filter bank Dr. Neelam Ru Prakah i with the Deartment of Electronic and Communication Engineering, Punjab Engineering College, Chandigarh. India. ISBN: ISSN: (Print); ISSN: (Online) WCE
2 Proceeding of the World Congre on Engineering Vol II WCE, June 3 - July,, London, U.K. H(z) Reone of I(z) Fig.. Decimation Decimator Filter ω ω π- ω π H(z) (d) Fig 4. Stage of the IFIR filter. Fig. 3. U Samler Interolation Filter. IFIR filter Deired Narrow Band Reone H(z) ω ω π (a) Say the deired filter deigned according to certain et ecification ha order N and a tranition bandwidth of Δf. If thi filter i tretched fold then the tranition band will be Δf and the order of the tretched filter G(z) a hown in Fig.4(b) will be N/. The frequency reone of G(z ) i a hown in (c). It ha two a band, one the deired aband and the other i it image centered on π/l where L i the tretch factor, in thi cae. Thi image i to be ureed by cacading the filter G(z ) with an image ureor filter I(z) a hown in (d), to obtain the deired reone. The order of I(z) i very mall a it ha a very large tranition band. The order of G(z) (model filter) will be a little more than N/. Therefore ince the order of the two filter i low the amount of comutational comlexity involved in imlementing the filter i greatly reduced. Stretched filter G(z) I(z) G(z L ) ω ω π (b) Fig. 5 IFIR filter for tretch factor L Reone of G(z ) 3. ultitage Deign Deired undeired ω ω π Conider the analyi filter bank in the figure hown below. For number of ubchannel that i large the analyi filter H(z) will be narrowband and may deigned uing the IFIR aroach. (c) ISBN: ISSN: (Print); ISSN: (Online) WCE
3 Proceeding of the World Congre on Engineering Vol II WCE, June 3 - July,, London, U.K. Let u take the tretch factor to be a factor of the decimation factor. Then we may rereent the decimator tructure in each channel a a cacade of the image ureor filter then down amling by, the model filter G(z) and then down amling by a hown in Fig. 5 below. The comutational comlexity involved in the imlementation of the filter i deendent on the order of the filter and the order of the odel filter G(z), N g and the Image ureor filter I(z), N i can be obtained from the equation below. log.5 3 N g = 4.6( ) / * log.5 3 N i = 4.6[ ( ) ] / H (z) H (z) H - (z) () () The number of multilication er unit time(pu) i N g / + N i / (3) In the table hown below the direct and multitage deign are comared on the bai of the filter order and the number of addition and multilication involved in the imlementation of the filter for different value of and. Note that will remain fixed and can take different value for given Fig. 6 Analyi Filter Bank I(z) G(z ) I(z) G(z) Fig. 7 -Stage decimator x (n) x (n) x - (n) Table I: Variation of filter order with. Filter order Deign IFIR ethod 8 = =8 5 3 = = = = = =8 From Table. the following inference can be drawn. A the value of i increaed the order of the conventional filter aroximately double and the order of the model filter G(z) and of I(z) are ignificantly le. Alo the order of G(z) and I(z) i deendent on the value of the factor. A reduce the order of G(z) increae and that of I(z) reduce. Table. will how the effect of the multitage aroach on the comutation involved. The ecification we have choen are δ =., and δ =.. The aband and to band edge have been choen to be π/ and π/ reectively. Table II. Comarion on the bai of PU Deign IFIR ethod 8 = = = = = = = =8 Table III. Comarion on the bai of APU Deign IFIR ethod 8 = = = = = = = =8 ISBN: ISSN: (Print); ISSN: (Online) WCE
4 Proceeding of the World Congre on Engineering Vol II WCE, June 3 - July,, London, U.K. What can be inferred from the Table and 3 i the following. The cot of imlementation of the filter uing direct deign method i almot indeendent of the number of ubchannel. But the cot uing multitage deign varie for different. Alo for a given the comutational cot i different for different value of. Thi mean therefore that there mut be ome value of, where the comutational cot i minimum. We will now roceed to determine that value of where the cot of imlementation in term of PU i minimum for a fixed. Subtituting () and () in (3), differentiating the reult with reect to and equating to zero we will get the value of that will obtain minimum PU for a given value of. Thi i given by m agnitude in db magnitude reone of the 3 channel filter bank (otimal) = ( ) ( ) ( ) ( ) ( ) (4) Since the value of we obtain from (4) may not exactly be a factor of we mut chooe an aroriate value that i cloe to it. On teting the value for different and comaring, the value that we have tabulated in Table (3) match very cloely. So we may conclude that by factoring uitably we can obtain maximum aving in comutation cot. We have hown here, how the decimator of the analyi filter bank can be deigned uing the multitage aroach. The interolator in the cae of the ynthei filter bank may alo be deigned uing the multitage aroach. Next we will tudy that the characteritic of a rototye filter for a coine modulated filter bank deigned by thi method and ee how it comare with that of a filter obtained by direct deign method normalied frequency /i m agnitude in db 5 x -3 amlitude ditortion function 5-5 magnitude in db magnitude reone of G(z) and I(z) G(z) I(z) normalied frequency /i normalied frequency /i Comarion are obtained in the cae of a 3 channel filter bank, uing the method rooed by creuere and itra in [5] where the order of the filter by direct deign i 5 where a for the ame method uing the multitage imlementation the order of G(z) ia 69 and I(z) i 73 all ecification of the filter being the ame for both method. We have choen to be 8 i i the factor cloet to the otimal value calculated uing (4) In the cae of [5] the eak to eak amlitude ditortion wa marginally lower at. In concluion we may ay that the multitage imlementation allow u to deign the filter bank uing filter of much le order and it would be highly deirable to exlore thi method for imlementation of filter bank. ISBN: ISSN: (Print); ISSN: (Online) WCE
5 Proceeding of the World Congre on Engineering Vol II WCE, June 3 - July,, London, U.K. Reference [] Cherubini, G., Elfetheriou,E., Olcer,S., and Cioffi,J.., Filter bank modulation technique Very high eed digital ubcriber line, IEEE Communication agazine,.98-4, ay. [] Nowrouzian,B., Wang,L., Agha, W., An overview of dicrete multitone modulation/demodulation ytem in xdsl alication IEEE, [3] Neuvo,Y., Dong,C.Y. and itra,s.k., Interolated finite imule reone filter, IEEE Tran. on Acoutic, Seech and Signal Proceing, vol. ASSP-3, , June 984 [4] P.P Vaidyathan, ultirate Sytem and Filter Bank. Prentice Hall, Englewood Cliff, New Jerey, 997. [5] Creuere, C.D. and itra. S.K., A imle method for deigning high quality rototye filter for -channel Peudo QF bank, IEEE Tran. on Signal Proceing, vol. 43, no.4,.5-7, Aril 995. ISBN: ISSN: (Print); ISSN: (Online) WCE
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