IIR Flters Usng Stochastc Arthmetc Naman Saraf, Ka Bazargan, avd J Llja, and Marc Redel epartment of Electrcal and Computer Engneerng Unversty of Mnnesota, Twn Ctes Mnneapols, MN, USA {saraf012, ka, llja, mredel} @umnedu Abstract We consder the desgn of IIR flters operatng on oversampled sgmadelta modulated bt streams usng stochastc arthmetc Conventonal dgtal flters process multbt data at the Nyqust rate usng multbt multplers and adders Hgh resoluton ACs based on the sgmadelta modulaton generate random bts at an oversampled rate as ntermedate data We propose to flter the sgmadelta modulated bt streams drectly and present frst and second order low pass IIR flters based on the stochastc ntegrator Expermental results show a sgnfcant reducton n hardware area by usng stochastc flters Keywords Stochastc computng; Oversamplng; Sgmadelta modulaton; Stochastc ntegrator; IIR flters I INTROUCTION Flterng s a fundamental operaton n sgnal processng to modfy the spectral characterstcs of sgnals, wth numerous applcatons n mage and speech processng ue to the prolferaton of VLSI technology, bulk of the flterng today s performed n the dgtal doman gtal flterng outperforms analog sgnal processng by offerng better nose mmunty, repeatablty and a superor tolerance to process varatons Conventonal dgtal flterng s based on Nyqustrate data convertors and multbt dgtal flters as depcted n Fg1 The analog nput s converted to a multbt dgtal representaton by a Nyqust rate analogtodgtal convertor (AC), and processed by a multbt dgtal flter The fltered dgtal sgnal s then converted back to the analog doman by a dgtaltoanalog convertor (AC) The key attrbutes of a dgtal flterng archtecture are the data processng rate and the resoluton Conventonal dgtal flterng processes data at the Nyqust rate, whch s equal to twce the maxmum frequency n the analog nput sgnal (samplng theorem) [1] whle the resoluton s determned by the AC resoluton espte the pervasveness of conventonal dgtal flterng, the archtecture suffers from a number of lmtatons The modest resoluton of modern Nyqust rate ACs lmts the resoluton of the archtecture to bts [2], [3] In addton, the multbt dgtal flters requre multbt multplers and adders, leadng to a large area and power dsspaton A sgnfcant mprovement over the conventonal Nyqustrate dgtal flterng s possble by usng oversampled sgmadelta modulator () based data convertors [4] An operates at a processng rate much hgher than the Nyqust rate whle convertng the analog nput to a much lower resoluton one bt representaton The therefore trades ampltude resoluton n favor of resoluton n tme nput f N Nyqust rate AC Multbt gtal Flter AC output Fg1 Conventonal dgtal flterng usng Nyqust rate ACs/ACs nput AC R Multbt gtal Flter R AC output ecmaton Interpolaton Rf N f N Rf N Fg2 Conventonal dgtal flterng based on ACs/ACs based ACs offer a hgh resoluton (upto 4 bts) at a fracton of the power dsspated by Nyqust rate ACs [3] Moreover, based ACs are bult usng fewer analog components leadng to a lower mplementaton cost [3] and better ntegraton wth the standard CMOS technology The bt stream at the oversampled rate R (R ) s processed by a decmaton flter to obtan a multbt dgtal representaton of the analog nput at the Nyqust rate Therefore, Nyqustrate data convertors n the conventonal dgtal flterng archtecture may be replaced by based data convertors to mprove the resoluton However, the resultng archtecture (Fg2) stll requres multbt dgtal flters, and more mportantly addtonal decmaton and nterpolaton flters to nterface the at the oversampled rate We propose to flter the oversampled bt streams drectly to avod the nterface flters and smplfy the dgtal flters to process bt stream data The proposton of flterng analog sgnals encoded as bt streams by dgtal flters constructed usng one bt adders and multplers promses major savngs n area The dea of flterng oversampled bt streams drectly has been explored n lterature Numerous flter desgns have demonstrated the feasblty of the archtecture n Fg3 such as the based bt stream FIR flters [57], IIR flters [810] and LMS adaptve flters [11], [12] Tutoral papers explanng the approach wth potental applcatons have also appeared [13], [14] nput Rf N AC bt stream flter Rf N AC Rf N output Fg3 based oversampled bt stream flterng 9783981537024/ATE14/ 2014 EAA/
x (n) a 1 a N y( n 1) y( n N) Fg4 The conventonal IIR flter structure Exstng based bt stream IIR flters encode ether the nput sgnal x(n) or the coeffcents a n the bt stream format to smplfy the multplers gtal flters are classfed nto FIR or IIR flters based on the flter transfer functon IIR flters meet a gven set of flter specfcatons usng fewer memory and logc resources than FIR flters and are the focus of the current work Exstng based bt stream IIR flters [810] make lmted use of the oversampled bt streams, encodng ether the nput sgnal or the flter coeffcents n the bt stream format, whle representng the other n a conventonal multbt format leadng to smpler multplers Moreover, the number of s requred n the flters ncreases lnearly wth the flter order We propose an IIR flter structure based on the stochastc ntegrator [15] to process analog nputs encoded as oversampled bt streams Our approach dffers from the conventonal IIR flter structure (Fg4) by avodng explct multplers and adders We develop an orgnal transform analyss of the stochastc ntegrator and present a modfed stochastc ntegrator wth a gan parameter We dscuss the desgn of frst and second order low pass IIR flters based on the modfed stochastc ntegrator and present frequency doman smulaton results We compare the hardware costs of our proposed flters to the conventonal IIR flters and conclude wth a dscusson on the advantages and tradeoffs of based bt stream flterng In the followng sectons, we assume that the oversampled bt streams are avalable for processng and do not dscuss the desgn of s II STOCHASTIC INTEGRATOR Consder an bt bnary up/down counter wth states labelled as Let n (n) denote the bt bnary number stored n the counter at tme n The counter nputs and n update the counter state every clock cycle accordng to the rule n Table I Let the counter nputs be drven by ndependent Bernoull random bt streams such that, (n) ( n ) (n) TABLE I Counter state update rule 1 0 0 1 0 0 1 1 The next counter state n (n ) s a random varable wth a probablty dstrbuton functon shown n Table II TABLE II Probablty dstrbuton functon of the counter state ( ) ( ) The expected value of the next counter state s computed as, [ n (n )] ( n (n) )( (n) (n) (n)) ( n (n) )( (n) (n) (n)) n (n)( (n) (n) (n) (n)) whch smplfes to, [ n (n )] n (n) (n) (n) In addton, the counter state s compared wth an bt unformly dstrbuted random nteger usng a comparator The comparator output s logc 1 f the counter state exceeds the random nteger, else logc 0 Let (n) denote the probablty of the comparator output to be logc 1 when the counter state s n (n) Then, ( a a a n) (n) (n) n (n) (4) The probablty of the comparator output to be logc 1 n the next clock cycle, (n ) s computed from (n) and the nput probablty values (n) and (n) usng the law of total probablty as, (n ) ( (n) ) ( (n) (n) (n)) ( (n) ) ( (n) (n) (n)) (n)( (n) (n) (n) (n)) (n ) (n) (n) (n) We notce that and dffer only by a factor of In fact, a smpler method to arrve at s to compute the expected value of the next counter state as a functon of the present state and the nput probablty values, and dvdng the expresson by the number of states Intutvely, the output of the system s logc 1 wth a probablty that s equal to the expected value of the counter state scaled by We wll use the observaton later to derve equvalent expressons for the modfed stochastc ntegrator We nterpret the probablty varables n as real values n the nterval [ ] by applyng the bpolar transformaton, (n) (n) x (n) (n) Equaton s now expressed as,
T clk Unform Random Integer The ntegrator state s now updated accordng to the rule, n (n ) n (n) ( 4) Bnary Comparator The expected value of the next state s therefore gven by, x 1( n ) x ( n 2 ) up Bnary Counter down T clk C N Fg5 Stochastc ntegrator count (n) We have thus constructed a dynamcal system descrbed by the recurrence relaton n wth real valued nputs and outputs n the nterval [ ] We refer to the system as the stochastc ntegrator depcted n Fg5 To verfy the analogy wth a dscretetme ntegrator, we consder the frequency doman representaton of usng the ztransform Let the transform pars of the varables be, (n) x (n) x (n) Therefore, the transfer functons of the system become, The transfer functon of a dscretetme ntegrator s gven by, The system therefore performs dscrete tme ntegraton of the nput random bt streams x (n) and x (n) wth a gan of to produce the output random bt stream (n) The nput x (n) s connected to the nonnvertng nput of the stochastc ntegrator whle x (n) s connected to the nvertng nput A stochastc ntegrator s characterzed by the number of states and the state update rule Equaton descrbes a stochastc ntegrator where the state s updated every clock cycle and the next state takes on one of three possble values shown n Table I Let us now consder an ntegrator wth states where the state s updated every clock cycles The ntegrator observes random bts at each of the two nputs before transtonng to the next state We assume that the nput probablty values do not vary sgnfcantly over the clock cycles and reman equal to (n) and (n) Let the random bts at the two nputs be denoted by, wth expected values, [ ] (n) ( (n)) (n) [ ] (n) ( (n)) (n) [ n (n )] [ n (n)] [ ] (n ) (n) n (n) [ ] [ ] [ ] n (n) (n) (n) ( (n) (n)) Applyng the bpolar transformaton yelds, ( ) Equaton descrbes a stochastc ntegrator wth a gan of that operates at a rate tmes lower than the clock rate wth transfer functons gven by, A hardware realzaton of the parameterzed stochastc ntegrator s shown n Fg6 An up/down counter wth states, computes the dfference n the number of logc 1 bts observed at the two ntegrator nputs over the clock cycles The sgned bnary count n s added to the ntegrator state stored n at the end of the clock and the counter s reset Note that a stochastc ntegrator wth s dentcal to the structure n Fg5 In the followng sectons, we wll represent a stochastc ntegrator by the symbol shown n Fg7 wth the nvertng nput labeled as x x ( n) 2 1 n up down Bnary Counter C K T clk T clk Unform Random Integer Bnary Comparator State Regster KT clk Bt stream Multbt number C N count (n) Fg6 Modfed stochastc ntegrator wth parameter K,N Fg7 Stochastc ntegrator symbol wth parameter
III STOCHASTIC IIR FILTERS We now dscuss bt stream IIR flters based on the stochastc ntegrator A general order IIR flter s descrbed by the dfference equaton, x(n) K,N (n) a (n ) wth a frequency doman representaton, x(n ) a where x(n) and (n)are the flter nput and output values at tme n The constant coeffcents a and determne the zeros and poles of the flter The zeros and poles govern the flter frequency response and are obtaned from the flter transfer functon as roots of the numerator and the denomnator polynomal Poles near n the complex plane realze low pass dgtal flters Moreover, every pole of a stable flter must le nsde the unt crcle The nput and output values for a conventonal dgtal IIR flter are determnstc multbt numbers However, n the context of our stochastc ntegrator based bt stream IIR flters, x(n) and (n) represent the nstantaneous probabltes of the nput and output bt streams The nput bt streams to our stochastc IIR flters are always generated by based ACs Autocorrelaton studes have revealed that the bt stream generated by an s a Bernoull process [13] Moreover, bt streams generated by dstnct sources are uncorrelated Consder the stochastc ntegrator confguraton depcted n Fg8 where the nput bt stream x(n) s derved from an based AC The recurrence relaton of the flter s gven by, (n ) (n) (x(n) Thus, the transfer functon becomes, (n)) (n) x(n) The only pole of the flter s located at Snce, and the system s a stable frst order low pass IIR flter A second order low pass stochastc IIR flter s realzed by the system n Fg9 wth the recurrence relatons, (n ) (n) (x(n) (n)) x(n) Fg8 Frst order low pass stochastc IIR flter y 1 ( n) 1,N 1 K K,N 2 2 Fg9 Second order low pass stochastc IIR flter The transfer functon of the system s obtaned by solvng the followng par of equatons, whch yelds ( ) ( ) The poles of the second order stochastc IIR flter are gven by, whle the zeros are located at 4 4 The second order IIR flter may possess two real valued poles or a par of complex conjugate poles dependng on the stochastc ntegrator parameters Second order systems, thus, exhbt a broader range of behavor than frst order systems Snce, the system s a stable second order low pass IIR flter IV EXPERIMENTAL RESULTS In ths secton we present expermental results on stochastc flters based on the frst and second order IIR flter structures descrbed n the prevous secton Table III lsts the stochastc ntegrator parameters for the flters used n our experments The parameter values were selected to demonstrate a dverse set of flter responses The samplng frequency and the oversamplng rato R were set at and R Therefore, the hghest nput frequency processed by the stochastc IIR flters s gven by, (n ) (n) ( (n) (n)) ( 4) R
TABLE III Stochastc ntegrator parameters used n the experments Flter Order LPF_1A 1 1 64 LPF_1B 1 1 256 LPF_2A 2 1 64 1 256 LPF_2B 2 1 256 1 64 We measure the magntude responses of the stochastc flters n Table III by applyng a snusodal test nput of frequency to an based AC and flterng the resultng bt stream The multbt stochastc ntegrator state correspondng to the output bt stream s the flter output The flter response at frequency s determned from the FFT of the flter output The test nput frequency s swept from C to to generate the curves shown n Fg10 and Fg11 Each plot shows the smulated magntude response of the stochastc IIR flter wth the expected response of a conventonal IIR flter Fg10 depcts the magntude responses of the frst order stochastc IIR flters Recall that the pole of a frst order IIR flter s located at The pole of LPF_1B s much closer to than the pole of LPF_1A, leadng to a sharper rolloff and lower bandwdth The magntude responses of the second order stochastc IIR flters are shown n Fg11 Flter LPF_2B has two smple real poles whle LPF_2A has a par of complex conjugate poles, whch s evdent from the overshoot n the magntude response The output resoluton of a stochastc flter s determned by the number of ntegrator states A conventonal IIR flter wth an bt resoluton s equvalent to a stochastc IIR flter havng an ntegrator wth states However, the pole locaton of a stochastc flter s strongly affected by the value of, unlke conventonal flters where the resoluton has no mpact on the pole locaton Ths requres a careful selecton of values for and whle desgnng stochastc IIR flters to smultaneously meet the requrements on the output resoluton and the pole locatons BW 1A Fg11a Magntude response of LPF_2A Fg11b Magntude response of LPF_2B We compare the hardware costs of the stochastc IIR flters wth the conventonal IIR flters havng an equvalent output resoluton for a 45nm CMOS technology The results n Table IV clearly ndcate that stochastc flters occupy a smaller area than the conventonal flters The random number generator n the stochastc ntegrators was constructed usng an LFSR TABLE IV Comparson of the hardware area (lbrary unts) Flter Conventonal IIR Flter Stochastc IIR Flter LPF_1A 554 1184 LPF_1B 8972 1561 LPF_2A 10674 *** 2700 LPF_2B 12483 *** 2700 *** consderng dfferent resoluton n the two stages to make a comparson V A ESIGN EXAMPLE Frst order IIR flters are of consderable nterest n audo sgnal processng due to ther low mplementaton cost We consder the desgn of frst order low pass IIR flters for audo sgnals known as treblecut flters The specfcatons for an example treblecut audo flter are summarzed n Table V TABLE V Specfcatons for a treblecut audo flter Fg10a Magntude response of LPF_1A BW 1B Nyqust frequency (Hz) 44100 Oversamplng rato 64 Samplng frequency (Hz) 2822400 Computaton resoluton (bts) 9 cutoff frequency (Hz) 800 The desgn of a dgtal frst order low pass IIR flter begns wth an analog frst order low pass transfer functon gven by, Fg10b Magntude response of LPF_1B where s the analog cutoff frequency n rad/s
The analog transfer functon pole at s mapped to a plane pole usng the mpulse nvarance method [16] as, Based on the locaton of the plane pole n and the flter specfcatons, we select and as the parameters for a frst order stochastc treblecut audo flter We test the stochastc flter by applyng a mxture of snusodal sgnals at frequences, and The nput and output sgnal spectrums n Fg12 and Fg13 verfy the operaton of the stochastc treblecut audo flter The sgnal at frequency s at the flter cutoff frequency and s attenuated by 3dB The sgnal at s removed from the output whle the sgnal at s unaffected Table VI compares the area of the frst order stochastc treblecut audo flter wth a conventonal mplementaton We observe that the stochastc mplementaton requres a much smaller area than the conventonal IIR flter flow flow f med Fg12 Input sgnal spectrum f med Fg13 Fltered output sgnal spectrum fhgh fhgh TABLE VI Comparson of the hardware area for the audo flter (lbrary unts) Flter Conventonal IIR Flter Stochastc IIR Flter Treblecut flter 9182 1752 VI CONCLUSIONS AN FUTURE WORK We have presented an approach to flter analog sgnals encoded as bt streams by IIR flters constructed usng the stochastc ntegrator The proposed archtecture smplfes the arthmetc operatons n the conventonal IIR flters and offers sgnfcant savngs n area However, unlke conventonnal dgtal flters, stochastc flters operate on probablstc data and are susceptble to random nose Future work on stochastc IIR flters nvolves extendng the low pass structures to desgn band pass and hgh pass flters based on the stochastc ntegrator and dervng addtonal performance metrcs such as the power dsspaton evelopng an accurate nose model to predct the SNR at the output of the stochastc flters s a challengng task The fnal valdaton of the approach would requre desgnng stochastc flters to solve complex flterng problems wth an acceptable performance VII ACKNOWLEGMENT Ths work was funded n part by the Natonal Scence Foundaton under Grant CCF1241987 REFERENCES [1] CE Shannon, Communcaton n the presence of nose, Proceedngs of the IEEE, vol86, no2, pp447457, 1998 [2] RH Walden, todgtal converter survey and analyss, IEEE Journal on Selected Areas n Communcatons, vol17, no4, pp539550, 1999 [3] Bn Le, TW Rondeau, JH Reed, CW Bostan, todgtal converters, IEEE Sgnal Processng Magazne, vol22, no6, pp6977, 2005 [4] JM de la Rosa, Sgmaelta Modulators: Tutoral overvew, desgn gude, and stateoftheart survey, IEEE Transactons on Crcuts and Systems I: Regular Papers, vol58, no1, pp121, 2011 [5] PW Wong, RM Gray, FIR flters wth sgmadelta modulaton encodng, IEEE Transactons on Acoustcs, Speech and Sgnal Processng, vol38, no6, pp979990, 1990 [6] PW Wong, Fully sgmadelta modulaton encoded FIR flters, IEEE Transactons on Sgnal Processng, vol40, no6, pp16051610, 1992 [7] SM Kershaw, S Summerfeld, MB Sandler, M Anderson, Realsaton and mplementaton of a sgmadelta btstream FIR flter, IEE Proceedngs on Crcuts, evces and Systems, vol143, no5, pp267 273, 1996 [8] A Johns, M Lews, esgn and analyss of deltasgma based IIR flters, IEEE Transactons on Crcuts and Systems II: and gtal Sgnal Processng, vol40, no4, pp233240, 1999 [9] A Johns, M Lews, Cherepacha, Hghly selectve analog flters usng ΔΣ based IIR flterng, IEEE Internatonal Symposum on Crcuts and Systems, pp13021305, vol2, 1993 [10] BR Owen, A Johns, A snglecolumn structure for deltasgma based IIR flters, IEEE Internatonal Symposum on Crcuts and Systems, vol5, pp2413416, 1992 [11] E Pfann, RW Stewart, MW Hoffman, Oversampled sgmadelta LMS adaptve FIR flters, IEE Proceedngs on Vson, Image and Sgnal Processng, vol147, no5, pp385392, 2000 [12] Q Huang, GS Moschytz, multplerless LMS adaptve FIR flter structures, IEEE Transactons on Crcuts and Systems II: and gtal Sgnal Processng,vol40, no12, pp790794, 1993 [13] F Malobert, Nonconventonal sgnal processng by the use of sgma delta technque: a tutoral ntroducton, IEEE Internatonal Symposum on Crcuts and Systems, vol6,pp26452648 vol6, 1992 [14] V da Fonte as, Sgmadelta sgnal processng," IEEE Internatonal Symposum on Crcuts and Systems, vol5, pp421424, 1994 [15] BR Ganes, Stochastc Computng Systems, Advances n Informaton Systems Scence, JFTou, vol2, chapter 2, pp37172, New York: Plenum, 1969 [16] JG Proaks, gtal Sgnal Processng, 4 th Edton