A MULTIRATE APPROACH TO DDS ALGORITHM VERSUS TAYLOR-SERIES EXPANSION TECHNIQUE

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1 A ULTIRATE APPROACH TO DDS ALGORITH VERSUS TAYLOR-SERIES EXPANSION TECHNIQUE Ewa Hermanowiz 1 and iroslaw Rojewski Faulty of Eletronis, Teleommuniations and Informatis, Gdansk University of Tehnology, ul. Narutowiza 11/1, Gdansk, POLAND 1 hewa@eti.g.gda.l, ro@sound.g.gda.l ABSTRACT We show that a look-u table (LUT)-based diret digital synthesizer (DDS) an be treated as an multirate algorithm. Two aroahes to exloiting the LUT are onsidered and their erformane omared. First of them alies a frational delay filter in the flash-farrow struture. The seond aroah uses hase rotation based on the Taylor-series exansion method. Exeriments show that both aroahes used in a quadrature DDS are aable of yielding the surious free dynami range SFDR 80 db. However, the flash-farrow struture exhibits twie smaller numerial omlexity exressed in terms of the number of multiliations than the struture of a Taylor-series based rotator of the same order N=. DDS FDF FIR LUT QDDS QO RO SFDR R I ABBREVIATIONS AND NOTATION x 1/ I diret digital synthesizer frational delay filter finite imulse resonse look-u table quadrature DDS quadrature osillator read only memory surious free dynami range real numbers set integer numbers set int( x ) = x R is the integer art of x (rounds x to the nearest integer towards minus infinity) fra( x) = x int( x) / art of x.,1/ ) 1. INTRODUCTION x R is the frational Quadrature diret digital synthesizer (QDDS) lays an extremely imortant role eseially in modern digital ommuniations. It offers many advantages inluding reise and raid maniulation of its outut frequeny, the ability to tune with extremely fine frequeny resolution and to swith raidly between frequenies. ost QDDSs use a read only memory (RO) looku table (LUT) for funtion generation [1]. The onet of QDDS using a LUT of samles of one eriod of a omlex sinusoid (further alled the base waveform) having the normalized angular frequeny ω = π / [rad/sa], where Sa stands for a samle, resolves to one relationshi: s[ = ex( jϕ [ ) = ex( jϕ [ ) ; n = 0,1,,... (1) Here s[ stands for the desired omlex signal (sequene) to be generated by the QDDS for a given inut instantaneous frequeny ω [. The angular frequeny ω [ an be onstant: ω [ = ω with ω ( 0, π ) or varying in time, with ω[ n ] ( π, π ). The aumulate ϕ[ = φ 0 n k = 1 φ, n = 0 0, ϕ[ (, ) n ω[ k], n = 1,,3 L () stands for the theoretial, numerially intratable, unwraed hase at the outut of an hase aumulator (PA, see Fig.1) driven by ω [ at the QDDS inut, while ϕ [ = π fra( ϕ[ / π ), ϕ [ π, π ) n (3) stands for the instantaneous hase used in the omutation ratie, wraed to the interval ϕ [ π, π ) (1) an oth- (rinial hase wind). The sequene s[ erwise be written as s[ = ex( jω τ [ ) where (4) τ [ = ϕ [ ] / ω, τ [ [ /, / ) (5) n is a (virtual) hase-time in units determined by the time interval between samles of the base omlex sinusoid stored in the LUT. The delay τ [ an be deomosed into integer and frational arts in the following way τ [ = m [ (6) The integer art of τ [ for addressing the LUT is m [ = int( τ [ ) where m [ { /,,,0,1, L ( ) / } L. (7)

2 ω [ ϕ [ PA 1 ω τ [ int( ) m [ Selet from LUT ex( j m [ ) ω sˆ [ fra() Comute ex( jω d [ ) Fig.1. An idealized blok-sheme of the LUT-based QDDS. In ratie all waveforms are quantized whih leads to an finite reision algorithm. The frational art of (6) for the orretion of the result obtained from the LUT on the basis of (7), further alled the frational (subsamle) time-delay [15], is = fra( τ [ ), / Consequently the sequene, 1/ ) (8) s[ in (4) an be written as s[ = ex( jω ( m [ )) or equivalently in the fatorized form used in Fig.1 as s[ = ex( jω m [ ) ex( jω ) (9) (10) In many aliations, e.g. for ortable ommuniations, the ower onsumtion of frequeny synthesizer is very limited. One way to minimize the ower of the QDDS is to minimize the size of the LUT. For the QDDS it is suffiient to store in the LUT only /8 [1], [4], i.e. one otant, more effiient than the oular one quadrant [5], of the rinial eriod of the base waveform, due to sine/osine symmetry. For examle, assuming =64, only 8 omlex samles in the LUT are needed to generate the whole eriod of the base waveform, by roer addressing the stored samles.. CLASSIFICATION The imortane of mathematially equivalent formulas (9) and (10) is twofold. They lead to two different aroahes: A and B, to emloying the LUT in the QDDS algorithm. In the first aroah: A, based on (9), a hase hange of s [ by ω is erformed, where is the frational delay/advane relative to the integer hase address m [ for the LUT. Conetually this means a resamling of the base disrete-time omlex sinusoid having a onstant normalized angular frequeny ω into a omlex sinusoid having the instantaneous angular frequeny ω [. Suh a resamling is done generally, i.e. for an arbitrary bandlimited waveform, by means of a frational delay filter (FDF) [15]. When ω [ = ω is onstant, the samling rate onversion by a onstant fator of ω / ω is aomlished. In this ase the QDDS oerating as a quadrature osillator (QO), is in fat tworate. While the normalized angular frequeny of the omlex sinusoid stored in the LUT is ω, the normal- ized angular frequeny of a omlex sinusoid synthesized by the QDDS is ω. Ideally, the setra of these sinusoids, both having the initial hase equal to zero, are: πδ ( ω' ω ) and πδ ω' ' ω ), resetively. ( Here ω ', ω '' [ π, π ) are the setral frequenies defined relative to the samling intervals: for the LUT ontent ex( jω m), thus for disrete-time enumerated using m, and for the QDDS outut ex( jω n), thus for disrete-time enumerated using n. On the other hand, when ω[ is not a onstant, thus when the QDDS is oerating as an F-modulator, the angular frequeny ω[ beomes varying in time. Then we omly with a resamling with the timevariable ratio ω [ / ω. In the seond aroah: B, based on (10), we deal with a hase rotation of the samle ex( jω m [) from the LUT, by an angle (hase orretion) equal to ω. This an be ahieved, e.g., by using the Taylor-series exansion of the fator ex( jω ) [1], by diret linear interolation [13], by CORDIC [14], or any other angle-rotation algorithm. w[ m [ W ω b 1 b 3 b 1 1/ b b4 b5 b6 b7 b8 j 1/ Σ b 9 Σ Σ sˆ [ Fig.. The flash-farrow struture for the maximally flat (Lagrangian) FDF of order N= (15), (16). b b 8 b 3 b 4 b b5 6 1/ Fig.3. The struture of a rotator (17b) of order N= based on the Taylor-series exansion tehnique. Σ sˆ [ b 7

3 However, our hoie here is the Taylor-series exansion method as a referene for a omarison with the aroah A. In ratie of the aroah A, for a digital imlementation of the frational delay by aliation of an FDF of FIR tye and of length N = 3,5,7,K, an onurrent aess to N samles stored in the LUT is required. Well tailored for this aim is an algorithm, see Fig., for whih we have oined the name flash-farrow. In ontradition to that, the seond aroah, B, requires from the LUT only one samle: W m [ = ex( jω m [ ), where W = ex( jω ), for the Taylor-series based interolation. This samle has to be multilied by a omlex number (f. (10)) omuted on the sot ] W d [ n = ex( jω ) (11) Thus (10) an be rewritten shortly as follows m [ s[ = W W (1) In the aroah A we deal with a multirate task in general. In artiular, for the QDDS oerating as a QO, it resolves to a two-rate roblem with different inut and outut samling frequenies. This is the ase of most often met ratial multirates. 3. LUT-BASED DDS AS A ULTIRATE SYSTE A further larifiation is due to the reader. Text-book examles inulate us with a onvition that a tyial multirate/two-rate system has to have an inut and an outut like an interolator or deimator. But in the LUTbased DDS oerating as a QO the inut resides in the LUT and in fat there is only an outut. Then a question arises: how to disern that it is two-rate? Indeed, when the DDS is of LUT-less tye and omutes every outut samle ontinually on the basis of the same rule as, e.g., in [1], there is no lae for two rates. To the ontrary, when the DDS is LUT-based, then the LUT is a soure from whih the algorithm gathers the samles of the stored base sinusoid having the angular frequeny ω in order to onvert it into samles of the outut sinusoid having the angular frequeny ω ω. Then the algorithm we are dealing with is a samling frequeny onverter a tyial reresentative of digital signal roessing field known under the name of multi-rate DSP (f. hater 10 in [6]). The simlest algorithm for dereasing the samling frequeny, thus for inreasing the angular frequeny of the base sinusoid to be resamled, is an ordinary down-samler by a fator of k, ommonly denoted as k. The down-samler is used in the LUTbased DDS only when the angular frequeny ω of the target sinusoid is equal to the multile of the angular frequeny ω of the base sinusoid stored in the LUT: ω = kω (13) and when the initial hase is φ0 = m0ω, with k, m0 { /,...,,0,1,..., / } for even. A resamling algorithm for the generation of suh a sinusoid is simly a samling rate onverter whih yields at the DDS outut every k-th samle of the base omlex sinusoid from the LUT. In this ase = 0 n and m [, the address for the LUT, an be alulated aording to the following relationshi m [ = fra(( m0 kn) / ), n = 0,1,,... ω kω φ0 m0ω hene d [ 0, the ordinary down samling is insuffiient. Here the LUT-based DDS oerating as a QO has to be onsidered a two-rate algorithm onverting the angular frequeny ω of the base sinusoid stored in the LUT into the angular frequeny ω of the sinusoid at the QDDS outut (f. hater 10.5 in [6]). As an be seen from (9), in order to obtain the desired omlex sinusoid, e.g., s[ = ex( j ( ω n φ0)) with ω kω or φ0 m0ω, eah samle of the base sinusoid from the LUT having the address m [ has to be frationally delayed by the hase time:, and driven to the outut. The first ones who roosed a tehnique of samling frequeny onversion by using an FDF were R.Lagade and H.O. Kunz in 1981 [7]. Later on T. Ramstad [8], [9] alied this tehnique to samling frequeny onversion with an arbitrary ratio and F.. Gardner [10] to synhronization of samling instants in a digital modem. The first ones who alied to the DDS an interolation by means of a FDF in the Farrow struture [15] were A.. Eltavil and B. Daneshrad [11]. The FDF used in [11] was the filter from [3] invented by L. Eru, R.. Gardner and R.A. Harris. In [11] Fig. saled horizontally in both samling intervals shows samling instants orresonding to a sinusoid stored in the LUT and the samles and samling instants for the DDS outut. The authors also marked the frational time-delay in this figure. The Farrow struture of order N=3 was in [11] treated as an adjustable ieewise araboli interolator with a given base-oint with an address m k, whih orresonds to m [ here, and with the frational time-delay µ k, orresonding to here. Finding these two arameters is realized in every lok yle. Starting from this we an now modify the Farrow struture for the QDDS so as to make it flash. It is imortant to note that at every lok instant n two oerations are erformed. The value of for the Farrow struture is udated and the struture is rovided with three neighboring samles from the LUT with a new entral samle (base oint) having the address (14) Substitution of (14) for (7) means that for the generation of sinusoids having the angular frequeny (13) the hase aumulator is unneessary. In a more general ase, when 4. THE FARROW APPROACH TO QDDS or

4 m [. Both above mentioned oerations have to be realized simultaneously. A flash memory used for the LUT enables reading from different loations at the same time. Hene the unit time-delay bloks, z, in the Farrow struture an be removed. The result is further alled the flash-farrow struture. Fig. resents the flash-farrow struture for the maximally flat (Lagrangian) FDF [], [15] of order N=. We have used here the maximally flat FDF rather than that invented in [10] and used in [11] beause it is numerially more effiient and in the QDDS erforms equally well. The inut to the flash-farrow struture is The algorithm for this struture is given by m [ 1 m [ 1 m [ 1 = W ( W W (16) 1 m [ m [ 1 m [ 1 ( W W W )) where sˆ [ stands for an aroximate of s[. On the other hand, for the Taylor-series exansion tehnique we have used the following algorithm = W (1 jω (1 ω ω ω (17) j( )(1 j( ) L(1 j( )) K) 3 N = W (1 jω ) (17a) and for N=, shown in Fig.3 m [ ω sˆ [ = W (1 jω (1 j( ))) (17b) Wˆ w[ = { W m [ m [ m [ m [ 1, W, W In artiular from (17) we get for N=1 m [ 5. EXPERIENTS AND AIN RESULTS m [, FDF 1 d [ (1 osω ) j sin ω } (15) It was interesting to assess the otential of the aroah A, with an aliation ioneered in [11] based on a FDF, and to omare the results with the aroah B based on the Taylor-series exansion tehnique [1], known from roviding good results. Firstly we have it done analytially by resolving the aroriate algorithms: (16) and (17), to the form (1) with the following exressions for the estimates of (11) easy to handle: (18) for an FIR maximally flat FDF of order N=, thus of length 3 and m [ W, Taylor1 1 jω (19a) for N=1, thus for two-term Taylor-series aroximation and ˆ m [ 1 W, Taylor 1 ω d [ jω (19b) for N=, thus for three-term Taylor-series aroximation. The results obtained in floating oint arithmeti in ATLAB are resented in Fig.4. Fig.4a shows the relative omlex error magnitude as a funtion of Wˆ RCE( ) = 0 log (0) 10 W W and Fig.4b resents the hase error 180 (arg ˆ d PE( ) = W argw π a) [ b) ) (1) Fig.4. Errors: (0) (a) and (1) (b), for floating-oint aroximation of the fator (11) by using the maximally flat FDF and the Taylor-series exansion of order N=1,. As we an see from Fig.4 the relative omlex error magnitude for the maximally flat FDF-based aroximation of order N= omuted in floating oint arithmeti in ATLAB lies in between the Taylor-series exansion aroximants: of order N=1, meaning linear interolation [1], and of order N=. The hase error magnitude for the maximally flat FDF-based aroximation is greatest here. Note that this omarison is indeendent of the angular frequeny ω of the omlex sinusoid to be synthesized by the QDDS. Further on we have examined the erformane of the QDDS with quantization, as a QO of a omlex sinusoid having onstant angular frequeny ω[ = ω = πf, with f = 103 / 4096 ; a situation whih gives rise to surious frequenies [11]. We have used a LUT storing only /8=8 omlex samles of the base waveform whose rinial eriod inluded =64 samles. We

5 have simulated in ATLAB the QDDS for aroah A with the FDF from Fig. and the QDDS for aroah B, based on the rotator from Fig.3 with an algorithm using the Taylor-series exansion. Both were aimed at obtaining the surious free dynami range SFDR 80 db. The quantization reision for the FDF in the flash- Farrow struture from Fig. in bits was: b = b =11, 1 b 3 =10, b 4 =9, b5 =11, b6 =10, b 7 = b8 =11 and b9 =6 while for the rotator from Fig.3 using the Taylor-series exansion tehnique was: b =11, =8, b =9, =11, 1 b 3 b 5 =10, b 6 = b 7 =1 and b 8 =6. Hene our aim was ahieved by using the maximally flat FDF of 3 real-valued oeffiients (N=) and by the rotator of the same order N=. The result for the former (aroah A) is shown in Fig.5 where the surious eak level relative to the main setral line of the desired omlex sinusoid is 81. db. Thus the given requirement for the SFDR, defined as the surious eak level but with oosite sign, is fulfilled. The other aroah, B, with the rotator gave SFDR=80.8 db. Fig.5. Outut amlitude setrum with maximal value normalized to 0 db for QDDS with maximally flat FDF. Note that the number of arithmeti oerations erformed on real-valued numbers er one samle for aroah A, f. Fig., is: 4 multiliations, 10 summations and 4 saling oerations, and for aroah B, f. Fig.3, is: 8 multiliations, 4 summations and saling oerations. 6. CONCLUSIONS In this aer we have shown that a LUT-based QDDS an be onsidered a two-rate system. Two aroahes to exloiting the LUT have been onsidered. First of them alies a maximally flat FDF in the flash-farrow struture. The seond aroah uses hase rotation exloiting the Taylor-series exansion of the omlex fator ex( jω ). A omarison of these two aroahes has been resented. Exeriments have shown that QDDS algorithms for both these aroahes are aable of yielding the surious free dynami range SFDR 80 db. However, the flash-farrow struture exhibits twie smaller numerial omlexity exressed b 4 in terms of the number of multiliations er outut samle than the struture of the Taylor-series based rotator of the same order N=. In our exeriments we have used the π reresentation: π 355 / 113 (Zu Chongzi; AD [16]), aurate to 6 deimal laes. REFERENCES [1] A. Bellaouar,.S. O breht, A.. Fahim and.i. Elmasry, Low-ower diret digital frequeny synthesis for wireless ommuniations, IEEE Journal of Solid-State Ciruits, vol. 35, No. 3, , arh 000. [] E. Hermanowiz, Exliit formulas for weighting oeffiients of maximally flat tunable FIR delayers, Eletronis Letters, vol. 8, No 0, , 199. [3] L. Eru, R.. Gardner and R.A. Harris, Interolation in digital modems Part II, IEEE Transations on Communiations, vol. 41, 1993, [4] L.K. Tan and H. Samueli, A 00 Hz quadrature digital synthesizer/mixer in 0.8 µm COS, IEEE Journal of Solid-State Ciruits, vol. 30, No. 3, , [5] J..P Langlois and D. Al-Khalili, A low ower diret digital frequeny synthesizer with 60 db setral urity, The 1 th AC Great Lakes Sym. on VLSI, GLSVLSI, New York, Aril 18-0, , 00. [6] S.K. itra, Digital Signal Proessing A Comuter Based Aroah, nd ed., Graw-Hill, 001. [7] R. Lagade and H.O. Kunz, A new aroah to digital samling frequeny onversion, 68 th Convention of AES, Hamburg, arh 17-0, [8] T. Ramstad, Samle-rate onversion by arbitrary ratios, IEEE ICASSP, Paris, ay 198. [9] T. Ramstad, Digital methods for onversion between arbitrary samling frequenies, IEEE Trans. on ASSP, vol.3, No. 3, , June [10] R.. Gardner, Interolation in digital modems Part I: Fundamentals, IEEE Transations on Communiations, vol. 41, No. 3, , arh [11] A.. Eltavil and B. Daneshrad, Interolation based diret digital frequeny synthesis for wireless ommuniations, Pro. of the IEEE Wireless Communiations and Networking Conferene, vol. 1, , arh 00. [1] K.I. Palomaki and J. Niittylahti, A low-ower, memoryless diret digital frequeny synthesizer arhiteture, Proeedings of the IEEE Sym. on Ciruits and Systems, vol. II,.77-80, 003. [13] J..P. Langlois and D. Al-Khalili, Novel aroah to the design of diret digital frequeny synthesizers based on linear interolation, IEEE Transations on Ciruits and Systems-II: Analog and Digital Signal Proessing, vol. 50, No. 9, , 003. [14] A. adisetti, A.Y. Kwentus and A.N. Willson, A 100- Hz, 16-b, diret digital frequeny synthesizer with a 100-dB surious-free dynami range, IEEE Journal of Solid-State Ciruits, vol. 34, No. 8, [15] T. Laakso, V. Valimaki,. Karjalainen and U.K. Laine, Slitting the unit delay. Tools for frational delay filter design, IEEE Signal Proessing agazine, , January [16] A history of Pi, htt:/www-history.ms.st-andrews.a.uk /HistTois/Pi_through_the_ages.html.