Improve4 Metliod of Generating Bit Reversed Numbers for Calculating Fast Fourier Transform

.Fast Foulier Transform (FFT ) is an important tool required for signal processing in defence applications. Thi~ paper reports an improved method for generating bit reversed numbers needecj in calculating FFf using radix-2. The refined algorithm takes advantage of some features of the bit reversed numbers, using intermediate arra~ lor storage and improved procedure for calculating base values required when generating bit reversed numbejrs. bit reversed numbers to place data at bit reversed positions, but only make use of efficient methods of swapping the data from the array for placing them at bit reversed positions. An improved method of generating bit reversed numbers is presented here, based on an earlier algorithm by SJresh4 (Hereafter referred to as basic algorithm. Methol;l is given in Appendix I). The modified algorithm generates a continuous stream of N bit reversed numbers for any given index n, i.e., N = 2n. I These bit reversed numbers can be us~d as indices of the data array for placing the data at bit reversed positions. 1 2. METHOD The features observed in the bit reversed numbers generated using basic algorithm are (Table I): I. (a) B~e values, which are the first values in the block of four numbers, are observed to be in bit reversed sequence. These are bitreversed numbers obtained with an index value of n -2. Thus there are 2n-2 base vaJues. 1. INTRODV:CTION , , Fast FourIer Transform (FfT) is an ubiquitous tool required for procefsing signals in defence applications, such as Iradars, doppler frequency measurements, moving tar~et inaicators, sonars, underwater communications, image reconstructions and restorations, digital fitters and ojther!. One of the noticeable features lobs'erved with 'in-place' F.FT calculation is that with irlput data placed in a natural I sequence, the output oqtained for each data point from the calculation is in bit reversed p(jsition. Thus, if input data are in natural order (x(O), x( 1 ), x(2), x(3 ), x( 4 ), . x(5), x(6), x(7», the output of the FFf calculation will have data at bit re~ersed positions (x(O), x(4), x(2), x(6), x(l), x(5), x(3), x(7».It is often found difficult to calculate FFT with input and output in natural sequence. I Thus, it is necess.ary to either load input data in natural order and then reorder the output, or place the input data at bit reverJed positions before the calculation to obtain I I the output in a natu,ral sequence. I I I It is essential to have a fast perIf1utation al.gorithm for reordering. Thi~ could be done~ either by placing each data directly at bit reversed p~sition in the array or hy rcor(lcrillg llIc 11:111\ I vl\ill\hlc i~, II or 111"1 1i('(III('II(.C to bit reversed posi'tions. The latter method is usually adoptedl-~ .These algorithms do not actually generate (b) The bit reversed numbers are divided into two halves. l;il~111111I tlll~ ("V(" II VIIIIIC~ wllilc 11,c Ilcxllllllfl:OIIIJlill1i odd values. 'These odd values are ~e incremental of the first half even values. Received 30 October 19951 revised 13 May 1996 I I 253 ,,~. ,.~. '; DEF SCI I, VOL 46, NO 4, OCfOBER 1996 'f1l1.1. 2. ) 1~."t.Ulloll Ip""d or II1I r..""r*-1 -llt lIlIlII 0111.111"" 011 I'C/4116 .2S M Ilz UIIIII C 1811K1I8&e; , TII<' f(!llowillg rc[ill~J 111~lhoJ is Ih~r~li)r~ "J()PI.~d laking inlo consider:llion Ihe above observalions. , Stcp I. With tlIC illilcx 11-2, calcullltc Ihc fir~t h:llf 2"-1 (If bil revcrscd numbcrs using basic algorithm. These are the first half of base values. Store lhem in an array. base (j), j = 0, ...2"-3-1 Step 2. Calculate 2"-1 bit reversed numbers using above base values as given in the basic algorithm. Step 3. Increment the base values in the array. base 0) = base 0) + I, for j = 0, ..2n-3-1 f )tep 4. Calculale next half2"-1 bit reversed numbers using the incremented base values from the array. Modified algorithms presented here have shown to perform better. Though this method makes Use of intermediate array of size 2n-3, its speed of calculation is found, to be ab-ove 25 per cent faster than the coml:norily used swapping methods for such p~rmutations in the calculation of FFr. Table I. Bit reversed num bers ror index n = 5 ACKNOWLEDGMEN1lS,The author is thankfulto ;Or. Ehrlich Desa,IDirector,NationalIn!;tituteof Oceanography,Goa, forI.,all the encouragement and Support. Thanks are also dueto Dr. Elgar Desa, for his ~uidance, discussions andvaluable suggestion. , REFERENCESRESULTSUsing this improved algorithm, the executionspeed has been found to be above six times faster thanthe basic algorithm. The performance of this modifiedalgorithm as compared to the algorithm given in3, isgiven in Table 2. These average execution speeds havebeen obtained with programs in C on PC 486/25 MHz.These are the average values obtained after 10,000execution cycles. 11 is seen that the speed has beenIimproved by using intermediate array and a modifiedmethod of generating base values. I2. 3.Ahmed, N. & Ra.o, K.R. Orthogonal'trnnsfonns fordigital signal processing. Springer-Verlag, New York,1986, p. 263.r ,,Rabiner, L.R. & Gold, B. Theory 3f1d application ofdigital signal processing. Prentice Hall, New Jersey,1975.'Willi~, H. Press, et'al. NuJle1cal recipes in fortran, Cambridge University Press, New York, 1992, p. 5or..,Suresh, T. Generating bit reyersed numbers forcakulating fast fourier transform. Co';'pute-r &Geosciences, 1995,21(2), T49-S2.j4.