Reducing Memory References for FFT Calculation

Transcription

1 Reducing Memoy Refeences fo FFT Calculation Mokhta A. Aboleaze Dept. of Compute Science and Engineeing Yok Univesity Toonto, ON. Canada Ayman I. Elnagga Dept. of Electical and Compute Engineeing Sultan Qaboos Univesity Muscat, Oman Abstact Fast Fouie Tansfom (FFT) is one of the most widely used algoithms in digital signal pocessing. It is used in many signal pocessing and communication applications. many of the FFT opeations ae pefomed in embedded systems. Since Embedded systems is vey small pocessos used in almost all type of appliances fom micowave ovens to cas, and many embedded systems ae potable and depend on small batteies fo powe; low enegy design is extemely impotant in embedded systems design. One of the majo enegy consumption souces in any pocesso is memoy access. Memoy access equies moe enegy than almost any opeation in a DSP (Digital Signal Pocesso), o embedded pocesso, educing memoy access plays a vey impotant ole in educing enegy consumption. In this pape we concentate on the enegy consumption in memoy in calculating FFT. we compae between thee diffeent techniques in calculating FFT with efeence to enegy consumption in memoy access. We also investigate the effect of the numbe of egistes in the CPU on educing enegy consumption in memoy access. 1. Intoduction Fast Fouie Tansfom (FFT) is pobably one of the most used signal pocessing algoithms in the wold. FFT is used in digital communication and in geneal in digital signal pocessing and is widely used as a mathematical tool in diffeent aeas. In the next few paagaphs, we biefly mention some of the applications of FFT. FFT is used to educe the computational time equied fo solving the poblem of electomagnetic scatteing fom wie antennas and conducting ectangula plates [18]. It is also used in solving a system of Toeplitz nomal equation [19]. It is also used in optimal fequency acquisition and measuements in Seach and Rescue Satellite Aidded Tacking (SARSAT) [1]. In intepolation techniques fo esampling of coelation pulse signals depating fom a small numbe of Nyquist samples [3]. FFT is also used in spectal estimation in [7] and [9], in single tone detection and fequency estimation in [], and in inceasing object detection in ada systems [17]. Also FFT is used in Othogonal Fequency Division Multiplexing (OFDM) which is the basis fo Multi Caie Code Division Multiple access (MC-CDMA), and its diect spead spectum vesion MC-DS-CDMA [6] and [8]. Windowed FFT is used in electic powe quality assessment in [11]. While in [5] the authos intoduced the genealized sliding FFT to efficiently implement the hopping FFT. Then they showed how to use it to implement the block LMS adaptive filte. Finally the FFT was used in 3- D induction well logging pobelm whee it could be used in chaacteization of oil esevois. Powe consumption is a vey impotant facto in the design of special pupose as well as geneal pupose pocessos [14]. Fo wieless devices, powe plays a cucial ole since the device opeates on a battey with a limited powe supply capability. Obviously that equie hadwae to use as little powe as it possibly could to pefom the job at hand. Fo geneal pupose pocessos, inceasing powe consumption leads to sophisticated cooling techniques which both incease the pice and educe the eliability of the pocesso. One of that main souces of enegy consumption in any pocesso chip is the memoy (o the cache if thee is a cache). It was epoted in [16] that instuction cache efeence amounts to 43% of the total powe consumption of the chip. Fo small embedded pocessos without cache, and since the main memoy consumes moe enegy than cache, that figue could be highe in small pocessos without cache. Minimizing the numbe of times the pocesso goes to the memoy, helps to educe the enegy consumption in the chip. While thee is nothing we can do to educe the memoy access to get instuctions (assuming a genal pupose pocesso achitectue); data access can be educed in two diffeent ways. The fist is using algoithms that equie less memoy access (eusing the accessed element as many times was possible befoe discading it, o eodeing memoy access). The second

2 is by using a lage set of egistes, thus accessed elemenst can be tempaily stoed in egistes fo futhe pocessing without memoy access. In this pape, we pesent a compaative study fo the memoy access in calculating in place FFT. We conside access to both data points and coefficients. We investigate thee diffeent techniques to calculate the FFT and show the numbe of memoy efeences of the diffeent techniques which is an indication of the powe dissipated in the memoy. The oganization of this pape is as follows. In Section, we pesent a bief oveview of FFT techniques. Section 3 discusses elated wok. In section 4, we pesent ou compaative study fo the numbe of memoy efeences in calculating in-place FFT using a vaiable numbe of egistes. Section 5 is a conclusion and futue wok.. FFT Algoithm The Discete Fouie Tansfom (DFT) of an N data samples is defined by N 1 ik X k = x i W N i = Whee W, ae the N th N = e j( π N) oot of unity, x is the oiginal sequence, and X is the FFT of x, k=,1,,...,n In geneal, both X, x and W ae complex numbes. This can be calculated using matix-vecto multiplication, whee ik W N ae aanged as N-by-N matix. That method equies O(N )complex multiplications. A faste way to calculate the DFT is the Fast Fouie Tansfom (FFT). FFT algoithm woks by dividing the input points into sets (k sets in geneal), calculate the FFT of the smalle sets (ecusively), and combining them togethe to get the Fouie tansfom of the oiginal set. That will educe the numbe of multiplications fom O(N ) to O(N log N). Figue 1. shows the 8-points decimation in time FFT. The 8 points ae divided into sets of 4 points each, then caculating the FFT fo these sets of 4 points each, then combining these sequences to poduce the 8 points FFT. The 4- points FFT is ecusively done in the same way. The points FFT is known as the buttefly opeation, and is shown at the bottom of Figue 1. As we can see fom Figue 1, Both data and coefficients ae accessed fom the memoy. Intemediate calculations ae done, then they ae stoed in the same memoy location they wee accessed fom (in place FFT calculation). By the end of the calculations (the end of the log N stages), the memoy contains the FFT sequence of the input (1) sequence. As we mentioned ealie, memoy access caies with it a heavy pice fom the enegy consumption point of view. Minimizing the numbe of memoy access goes a long way in educing the enegy consumed in calculating the FFT. Fo example, in the fist stage, we access both x(), and x(4). That esults in calculating new values fo x(), and x(4). Then we poceed to access the est of the data points. Then at the beginning of the second stage, again, we access x(), and x(). If the pevious values calculated fo x(), and x() ae stoed in a pai of egistes, then we don t need to go to the memoy in ode to access them again. If these values ae not stoed in egistes, then we have to go to the memoy to fetch them. the same could be said about the twiddle facto access. x() x(4) x() x(6) x(1) x(5) x(3) x(7) a b W N Figue 1. 8-points decimation in time FFT Whethe the equied data will be in the memoy o in egistes depends on the numbe of egistes in the CPU, the achitectue of the CPU, the instuction set of the CPU, and the compile used in geneating the machine code. Hee we concentate on the numbe of available egistes in the CPU, we also will make some assumptions egading the compile. 1 3 A = a + W N b B = a W N b X() X(1) X() X(3) X(4) X(5) X(6) X(7)

3 In this pape, we calculate the numbe of memoy access in calculating FFT fo diffeent algoithms, and fo diffeent values of N. Some of these algoithms ae designed specifically in ode to educe the numbe of memoy access (by eaanging the calculations), while othe ae designed to minimize the numbe of calculations, o to simplify the code. In the next section, we eview some of the pevious wok in minimizing powe consumption fo FFT calculations. The powe eduction falls in thee main categoies, educing memoy access fo data efeence, educing memoy access fo addess geneation, and educing the numbe of aithmetic calculations in ode to save enegy. 3. Related Wok Low powe design fo digital signal pocessing in geneal, and FFT in paticula has attacted a lot of attention. Low powe could be implemented on many levels. The algoithm level, the achitectue level, and the hadwae level. At the algoithm level, many attempts have been done to minimize powe consumption, highe adix FFT, mixed adix FFT [], Othe attempts ae made on both the achitectue level and the cicuit level. In [1], the authos consideed the poblem of coefficients addess geneation fo special pupose FFT pocessos. They designed the hadwae equied fo access of the Fouie coefficients fom the coefficients memoy. They showed that thei scheme can esult in powe saving of 7-8% compaed to Cohen s scheme (note that saving in the enegy consumption of the addess geneation unit only, not the entie pocesso powe). In [15], the authos poposed an addess geneation scheme fo FFT pocessos. Thei poposed hadwae complexity is 5% less than Cohen s scheme. Also thei scheme activate only half the memoy (they used memoy banks) thus saving moe enegy than peviously known techniques. A low-powe dynamic econfiguable FFT fabic was poposed in []. The pocesso can dynamically be configued fo 16-points to 14-points FFT calculation. The ovehead fo dynamic configuation is minimal while the powe eduction compaed to geneal pupose econfiguable achitectue is in the ange 3-9%. In [13] the authos poposed a novel FFT algoithm to educe the numbe of multiplications as well as the numbe of memoy accesses. Thei algoithm depends on e-aanging the computation in the diffeent stages of the Cooley-Tukey algoithm in ode to minimize the Twiddle facto access. Thei algoithm clustes togethe all the buttefly opeations that use a cetain Twiddle facto, access that twiddle facto to pefom the opeations, and neve access it again. That esult in 3% eduction in twiddle facto access compaed to the conventional DIF FFT. They also implemented thei algoithm on TI TMS3C6x digital signal pocesso. Thei implementation shows a significant eduction in the memoy access on the TI TMS3C6x digital signal pocesso. 4. Compaison The numbe of memoy access depends on many factos. The fist is the poduced machine code. The machine code depends of couse on both the high level souce code (if witten on high language souce code; othewise the assembly code), and the compile used. It also depends on the instuction set of the taget pocesso and the numbe of available egistes in the pocesso. The compile plays a vey impotant ole in the speed of the executed code. An efficient use of the esouces (including the available egistes) can speedup the execution. In this pape, we simulated thee diffeent algoithms fo FFT. We did not poduce the assembly code, but athe got the data access patten fom the C code. As mentioned befoe thee is no diect coespondence between the data access patten and the memoy access patten. data access could be diected to memoy if the data ae in the memoy, o a egiste if the equested data eside in a egiste. Anothe impotant facto is witing to the memoy. If a data element is changed, we can poceed to wite the changes to the memoy, o we can keep it in a egiste to be used in the computations and witten to the memoy eithe at the end of the code, o when we need to use the egiste to stoe anothe data item. In this pape, we assumed that the compile uses the available egistes to access and stoe the data. Once a data element is stoed in a egiste, it stays thee until the compile has to use the egiste to stoe new data. In that case the egistes ae eleased in a FIFO fashion. We also assumed a egiste-egiste achitectue, in which the opeands ae assumed to be in a egiste which is typical of a RISC achitectue. That puts an uppe bound on the pefomance, and depends to a geat extent on the compile to poduce an optimal code. The fist algoithm is the egula adix DIF FFT, the pseudo code in Figue. can be found in [1]. note that the last stage is sepaated fom the est of the stages since we do not need to multiply by the twiddle factos in that

4 stage, by sepaating the last stage fom the est of the stages, we save on memoy access as well as on multiplications. Radix-4 FFT algoithm is shown in Figue 3. Also taken fom [1]. Note that in both figues, that is a pseudo code whee the elements of x ae complex numbes. In eality it will take moe than a single addition/multiplications to epesent the addition o multiplications of a complex numbe. The thid algoithm we consideed is the one shown in [13], that minimizes the Twiddle facto memoy access. fft_dif(x[],m) n=^m; fo(i=m; i>1; i--) { m1=^i; mh=m1/; fo(j=;j<mh;j++) { e=exp(*pi*i*j/m1); fo(=; i<n; i=i+m1) { u=x[+j]; v=x[+j+mh]; a[+j]=u+v; a[+j+mh]=(u-v)*e; fo(=;<n;=+) { x[]=x[]+x[+1]; d a[+1]=a[]-a[+1]; Figue. DIF FFT Algoithm Table 1. shows the numbe of memoy access fo the thee above mentioned algoithms fo diffeent numbe of egistes anging fom 4 to. We also consideed diffent sizes fo FFT sequences with length N = n points. One can see that fo achitectues with a small numbe of egistes, the egula Radix- FFT pefoms best ove a wide ange of FFT sizes (fom 16 to 14). Fo achitectues with lage numbe of egistes, Radix-4 FFT pefoms the best ove the same ange of sizes. Fo achitectue with a modeate numbe of egistes (8,1,16) Radix- FFT and the educed memoy access method pefom equally good with a slight edge fo the simple adix-. In [13], the authos compaed between thei algoithm and the Radix- FFT unning on TI TMS3C6x and show that thei algoithm equie less memoy access than Radix-. We believe that this may be because of the VLIW achitectue of the pocesso, the numbe of egistes available, o the compile used. Howeve, we ou wok shows that with a vey good compile, adix- and 4 FFT can outpefom the educed memoy access algoithm. Table 1. The numbe of memoy accesses fo diffeent numbe of egistes, and FFT size = n # of Reg n Rad Reduced Rad Reduced Rad Reduced Rad Reduced Rad Reduced Rad Reduced Rad Reduced fft_ad4 n=^ldn; fo(ldm=ldn; ldm>1;ldm=ldm-) { m=^ldm; m=m/4; fo(j=;j<m;j++) { fo(=;<n;=+m) { u=a[+j]; u1=a[+j+m]; u=a[+j+*m]; u3=a[+j+3*m]; t=u+u+u1+u3; t1=u+u-u1-u3; t=u-u+(u1-u3)$ t3=u-u-(u1-u3)$ a[+j]=t; a[+j+m]=t*w; a[+j+*m]=t1*w a[+j+3*m]=t3*w Figue 3. Radix 4 DIF FFT n=4 n=5 n=6 n=7 n=8 n=9 n=1

5 5. Conclusions In this pape, we compaed between thee diffeent algoithms in the numbe of memoy access equied fo FFT calculations. We geneated data tace fom the C code and we use appoximate measues to estimate the numbe of memoy access equied to complete the computations assuming a vaiable numbe of egistes available to the compile. We did not assume a specific compile, just the data tace and a FIFO egiste set to be used with the compile. Ou wok shows that although the esults depends on the numbe of egistes, and the poblem size, howeve the simple Radix- FFT pefoms well compaed to the othe. 6. Refeences [1] Andt, J. Algoithms fo Pogammes, could be found at August 5. [] Chan, Y.T.; Ma, Q.; Inkol, R.; Evaluation of vaious FFT methods fo single tone detection and fequency estimation. IEEE Canadian Confeence on Electical and Compute Engineeing. Vol May 1997 pp [3] Coenen, A.; De Vos, A. FFT-baed intepolation fo multipath detection in GPS/GLONASS eceives. Electonic Lettes, Vol. 9, No. 19. Sept. 199 pp [4] Cohen, D. Simplified contol of FFT hadwae, IEEE Tansactions on Accousitc, Speech, and Signal Pocessing. Vol. 4, No. 6, Dec pp [5] Fahanq-Booujeny, B.; and Gazo, S. Genealized Sliding FFT and its application to implementation of block LMS adaptive filtes. IEEE Tansactions on Signal Pocessing. Vol. 4, Issue 3. Mach pp [6] Fedeiksen, F. B.; Pasad, R. An Oveview of OFDM and elated techniques towads development of futue wieless multimedia communications. IEEE Radio and Wieless Confeence RAWCON. 114 Aug.. pp 19- [7] Gan, R. Eman, K.; Wu S. An extended FFT algoithm fo ARMA spectal estimation, IEEE Tansactions on Accoustic Speech, and Signal Pocessing, Vol. 3, No. 1 Feb pp [8] Ghoashi, S. A.; Allen, B.; Ghavami, M.; Aghvami, A. H.; An oveview of MB-UWB OFDM. IEE Semina on Ulta Wide Band Communications Technologies and System Design. * July 4. pp 171. [9] Gough, P. T. A fast spectal estimation algoithm based on the FFT, IEEE Tansactions on Accoustic Speech, and Signal Pocessing, Vol. 44. No. 8. August 1996 pp [1] Hasan, M.; Aslan, T. Coefficients memoy addessing scheme fo high pefomance pocessos Electonic lettes. Vol. 37, No.. Oct. 1 pp1334 [11] Heydt, G. T.; Field, P. S.;, Piece, D.; Tu, L.; and Hensley, G. Applications of the windowed FFT to electic powe quality assessment IEEE Tansactions on Powe Delivey. Vol. 14, Issue 4. Oct pp [1] Holm, S. Optimum FFT-based fequency acquisition with application to COSPAS-SARSAT IEEE Tansactions on Aeospace and Electonic Systems. Vol. 9, No. Apil 1993 pp [13] Jiang, Y.; Zhou, T.; Tang, Y.; Wang Y. Twiddle-factobased FFT algoithm with educed memoy access Poc. of the 19th IEEE Intenational Paallel and Distibuted Pocessing Symposium IPDPS Apil. pp [14] Lahii, K.; Raghunathan, A.; Dey S.; Panigahi, D.; Battey-diven system desig: A new fontie in low powe design. poceeding of the 7th Asia and South Passific Design Automation Confeence ASP-DAC.71 jan., pp [15] Ma, Y.; Wanhamma, L. A Hadwae efficeint contol of memoy addessing fo high-pefomance FFT pocessos IEEE Tansactions on Signal Pocessing Vol. 48, No. 3 Mach pp [16] Montanao, J. et al A 16-MHz, 3-b.5-W CMOS RISC micopocesso Poc. IEEE Intl. Solid-State Cicuit Confeence., Feb pp [17] Pasad, N.; Shameem, V.; Desai, U. B.; Mechant S. N.; Impeovement in taget detection pefomance of pulse codded Dopple ada based on multicaie modulation with fast Fouie Tansfom (FFT). IEE Poceedings on Rada, Sona, and Navigation. Vol No. 1 Feb. 4 pp117. [18] Saka T.; Avas, E.; Rao, S. Application of FFT and conjugate gadient method fo the solution of electomagnetic adiation fom electically lage and small conduction bodies, in IEEE Tansactions on Antennas and Popagation. Vol. AP-34, No. 5, May 1986 pp [19] Yalagadda, R.; Babu, B. A note on the application of FFT to the solution of a system of Toeplitz nomal equation, IEEE Tansactions on Cicuits and Systems, Vol. 7, No. Feb. 198 pp [] Yeh, W.-C.; Jen, C.-W. High-speed and low-powe split adix FFT IEEE Tansactions on Signal Pocessing. Vol. 51, No. 3 Mach 3 pp [1] Zhang, Z. Q.; and Liu, Q.; H. Applications of the BCGS- FFT method to 3-D induction well logging poblems. IEEE Tansactions on Geoscience and Remote Sensing. Vol. 41, Issue 5. May 3. pp 9984 [] Zhao, Y.; Edogan A.; Aslan T. A low-powe and domain-specific econfiguable FFT fabic fo system-onchip applications Poc. of the 19th IEEE Intenational Paallel and Distibuted Pocessing Symposium IPDPS5 Apil 5 pp 169a7a