DESIGN AND IMPLEMENTATION OF SPLIT RADIX ALGORITHM FOR LENGTH - 6 M DFT USING VLSI AND FPGA

Transcription

1 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS Vol. Issue.7, Pg.: -45 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS ISS -745 DESIG AD IMPLEMETATIO OF SPLIT RADIX ALGORITHM FOR LEGTH - M DFT USIG VLSI AD FPGA G.aesh 1, S.Suthi 1 Assistant Pofesso Depatent Of E.C.E See Vidyaniethan Engineeing College, Tiupati, A.P ganeantinaesh@gail.co P.G. Student Schola, M.Tech (VLSI), Depatent of E.C.E See Vidyaniethan Engineeing College, Tiupati, A.P suthiseasapua@gail.co Abstact The Discete Fouie Tansfo (DFT) is a tansfo used fo Fouie Analysis of a sapled sequence of a signal. The Fast Fouie Tansfo (FFT) is a nueically efficient algoith used to copute the DFT. In this pape we popose to design a 1-point DFT and also find its coputation tie. The 1-point DFT can be calculated by adix- and adix- FFT with deciation in tie. It is a vaiant of split adix and can be flexibly ipleent a length of X DFT. ovel ode peutations of Sub-DFT s and eduction of the nube of aithetic opeations ipove the viability of the poposed algoith. It povides a wide choice of accessible FFT s lengths. The six point DFT and the twelve point DFT can be siulated in Modelsi using Syste Veilog language. The algoith can evaluate a non-powe-of-six DFT, as long as its length can be divisible by. In ode to educe the nube of opeations, all sub-dft s ae eodeed satisfactoily. The poposed algoith shows that its ipleentation equies fewe opeations as copaed with the ealie nown algoiths. Keywods: Discete Fouie Tansfos (DFT), Fast Fouie Tansfos (FFT), Invese Fast Fouie Tansfos (IFFT), Geneal Split Radix, Radix- 1. Intoduction The Discete Fouie Tansfo (DFT) is a tansfo used fo Fouie analysis of a finite sapled sequence of a signal. The DFT is widely eployed in signal pocessing and elated fields to analyze the fequencies contained in a signal. The Fast Fouie Tansfo (FFT) is a nueically efficient algoith used to copute the DFT. Fo a coplex -point Fouie tansfo, the FFT educes the nube of coplex ultiplications fo the ode of to the ode of log. Mixed adix-//4/5 FFTs can be used to ipleent the DFT algoith with educed coputation if the nube of DFT point = l 4 5n (, l,, and n ae positive integes). S. S u t h i e t a l Page

2 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS Vol. Issue.7, Pg.: -45 It is a Fouie tansfo fo a finite doain, discete tie peiodic function, which is suitable fo pocessing data stoed in coputes. Basically it convets discete tie data into discete fequency data and vice vesa. The need fo this convesion is that ou signals can be viewed in diffeent doain, inside which diffeent difficult pobles becoe siple to analyze. The inceasing application of digital equipent caused the coputation of discete Fouie tansfo to becoe an ipotant poble. In the past few yeas, a nube of algoiths have been poposed fo coputing the discete Fouie tansfo. To deteine the DFT oe quicly and with less coplexity, Fast Fouie tansfo algoiths have been developed which ae geneally nown as FFT. Most of these algoiths deal with powe-of- sequence lengths. The fist widely nown achieveent in this aea was the adix- FFT. The nube of aithetic opeations equied fo calculating the FFT is one of the ipotant factos in evaluating any FFT algoith. The adix- FFT algoith is in the long list of pactical DFT algoiths with educed aithetical coplexity fo data sizes =, being an intege. The inceased usage of FFTs ade us concentate on the coplexity, eoy usage, and powe consuption of the algoiths when used in digital signal pocessing applications. This lead to the ipoveent of FFT fo diffeent length sequences such as adix-, adix-, and adix-1 DFTs. These FFT algoiths ae developed fo adix- FFTs and they ae found to be bette than the existing algoiths. Siultaneously, the eseaches on the algoiths fo coputing length- n= DFT have esulted in the pesentation of the ethods fo = and =. Due to the poo efficiency, the algoiths fo ae of tivial pactical eanings when =. Howeve, thee exist any applications in which the sequence lengths ae and [1]. So an algoith fo sequence length-= have been developed which shows inceased pefoance than the existing algoiths.. FFT algoiths.1 Radix-/8 FFT Algoith fo Length qx DFTs A new adix-/ 8 fast Fouie tansfo (FFT) algoith have been poposed fo coputing the discete Fouie tansfo of an abitay length =qx, whee is an odd intege []. It educes substantially the opeations such as data tansfe, addess geneation, and twiddle facto evaluation o access to the looup table, which contibute significantly to the execution tie of FFT algoiths. It is shown that the aithetic coplexity (ultiplications, additions) of the poposed algoith is, in ost cases, the sae as that of the existing split-adix FFT algoith. The basic idea behind the poposed algoith is the use of a ixtue of adix- and adix-8 index aps. The algoith is expessed in a siple atix fo, theeby facilitating an easy ipleentation of the algoith, and allowing fo an extension to the ulti diensional case. Fo the stuctual coplexity, the ipotant popeties of the Cooley Tuey appoach such as the use of the buttefly schee and in-place coputation ae peseved by the poposed algoith. It is suitable only fo DFT of sequence length =qx.. Radix / Split- Radix FFT Algoith A adix-/1 deciation-in-fequency (DIF) fast Fouie tansfos (FFT) algoith and its highe adix vesion, naely adix-4/1 DIF FFT algoith, have been poposed by suitably ixing the adix-, adix-4 and adix-1 index aps, and cobing soe of the twiddle factos []. It is shown that the poposed algoiths and the existing adix-/4 and adix-/8 FFT algoiths equie exactly the sae nube of aithetic opeations (ultiplications, additions). By using this technique, it can be shown that all the possible split-adix FFT algoiths of the type adix- / s fo coputing a point DFT equie exactly the sae nube of aithetic opeations. This algoith is suitable only fo sequence of length =, is intege.. ew Radix- FFT algoith A new adix- FFT algoith suitable fo ultiply-add instuction have been poposed. The new adix- FFT algoith equies fewe floating-point instuctions than the conventional adix- FFT algoiths on pocessos that have a ultiply-add instuction. Techniques to obtain an algoith fo coputing adix- FFT with fewe floating-point instuctions than conventional adix- FFT algoiths have been poposed []. The S. S u t h i e t a l Page 7

3 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS Vol. Issue.7, Pg.: -45 nube of floating-point instuctions fo the new adix- FFT algoith is copaed with those of conventional adix- FFT algoiths on pocessos with ultiply-add instuction..4 The -Point Radix / Split Radix Algaith.4.1 The Radix- and Radix- FFT Appoach The poposed Radix / algoith is based on ixtue of Radix- and Radix- FFT algoiths. The definition of DFT is given by, 1 X ( x( n) e n j / 1 x = n (1) j / nk e cos j sin () In (1) and () is the nube of data, j= -1 and is the twiddle facto. is called the -point DFT of the sequence of x(n). Fo each value of, the value of X ( epesents the Fouie tansfo at the fequency. The Radix- DIT-FFT can be deived as, 1 X ( xn n = 1 1 ( 1) ) n x(n) x(n 1 n n 1 (n) x(n ) n 1 n x 1 ( n) n x(n 1) 1 n x(n ) P( Q( R( () Each of the sus, P(, Q(, and R(, in () is ecognized as an /-point DFT. The tansfo X( can be boen into thee pats as shown in (4). X ( P( Q( R( X ( ) P( Q( ( ) R( / 1/ P( Q( R( (4),1,... 1 In (4), the peiodicity popety is used to siplify and can be expessed as shown in (5) 1.The coplex nubes 4 1 S. S u t h i e t a l Page 8

4 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS Vol. Issue.7, Pg.: j / 1 e j j/ 1 e j (5) The Radix- DIT-FFT can be deived as, 1 X ( xn n = 1 1 ( ) x n / x(n 1) n n / 1 1 ( ) ( ) n n / x n x n n 1 n 1 n X (n 4) x(n 5) x(n ) / / P( 5 U( Q( V ( R( S( 4 T( ().4. Split Radix / FFT Appoach The Algoith decoposes a DFT of size-= into one length-/ and fou length-/ sub DFTs. The flexibility of the decoposition enables the algoith to be copetent at the ipleentation of a non-poweof-six DFT, while its length can exactly divided by. Appopiate peutations ae used fo sub DFT input sequences to educe the coputational capacity. The definition of DFT is X 1 x n nw (7) j / hee e, j 1, the length of sequence x(n) is assued as an intege, which is divisibly by six. Fo lengths of DFT, powes-of-six would be best fo the poposed algoith. Obviously, the DFT can be divided into thee length / sub-dfts. In ode to deive a best possible algoith, we continue to decopose the thee sub-dfts. Due to no scaling facto in font of it, the fist sub-dft should be let as it is and diectly go into the ecusive decoposition of the next stage. The othe two sub DFTs ae divided into fou sub-dfts of length-/. Actually, if the length of a DFT can be divided by, the DFT can be decoposed by the algoith. The genealized length- can be assued as = x, whee -1. The decoposition of a DFT of size = x is denoted by, 1 1 n n / n n / X ( x w x K 1 1 / n n n n / w x w x 1 w w x n n / (8) hee the fou length- / sub DFTs ae eodeed. To siplify the desciption, (8) can be expessed by, S. S u t h i e t a l Page 9

5 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS Vol. Issue.7, Pg.: -45 X ( A w w B C w E w w F (9) hee, A 1 x n / n 1 B x n n / 1 C x n n / 1 E x n n / 1 n n / (1) F x In (9), w B and w w F can be teated in pais, since w w F and w w F is a conjugatepai. In the siila way, and can be handled, with in pais. The diect ipleentation of (9) pefos any unnecessay opeations, since the coputations of X, X, X, X X X tun out to 4 5 ', shae any calculations each othe. In paticula, if we add to, the size-/ to DFT ae not changed (because they ae peiodic in, while the size-/ DFT is unchanged if we add to / to. So, the only things that change ae, and following six identities ae necessay, X A ( w w B w w F ) tes. In ode to educe the nube of the opeations, the ( w C w w w B w w F ) X A ( w w w B w w w F ) ( w w C w w E ) (11) (1) X A ( w w w B w w w F ) ( w w C w w E ) (1) X A ( w w w B w w w F ) ( w w C w w E ) 1 1 / 1 1 (14) / X A ( w w B w w w F ) ( w C w E ) (15) S. S u t h i e t a l Page 4

6 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS Vol. Issue.7, Pg.: -45 X A ( w w w B w w w F ) ( w w C w w E ) 5 / (1) A coplete output set {X } can be obtained if we let ange fo to / 1in the above six equations. e now suaize the schee of the poposed adix-/ FFT algoith. The initial input sequence of length- is decoposed into five sub-sequences. This pocess is epeated successively fo each of new sub-sequences, until the sizes of all sub DFTs ae indivisible by. Figs,, 4 illustate the flow gaph of, and 1 point adix / algoith (-points and 4-points FFT can be pefoed with SRFFT)..4. Pefoance Analysis of the Algoith In this section, we conside the pefoance of the poposed algoith by analysing the coputational coplexity and copaing it with existing algoiths. Let M and A be, espectively the nube of ultiplications and additions. e assue that a -point DFT equies 4 eal ultiplications and 1 eal additions (soe algoith assues that a -point DFT is calculated with eal ultiplication and 1 eal additions, since one need not ultiply ½ and the ultiplication by 1/ can be evaluated with bit shift). Figue 1: Bloc diaga of split adix /FFT Figue : Flow gaph of -point FFT Figue : Flow gaph of -point / FFT S. S u t h i e t a l Page 41

7 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS Vol. Issue.7, Pg.: -45 Figue 4: Flow gaph of 1-point / FFT The decoposition in the poposed algoith is conducted ecusively until the lengths of all sub DFTs cannot be exactly divided by. In geneal, thee ae only 1 the fist special buttefly (if 1and 1), 1 the second special case buttefly (if and 1), 1 the thid special case buttefly and 1 the fouth special case buttefly (if and 1). The total nube of the fifth and sixth type of butteflies is Thus, the aithetic coplexity of the poposed algoith can be given as follows, M / 4M / 8 1, 1 8 M 4M 8, 1 1 M / 4M / 8, 1 4 / / M (19) A/ 4A/ 1, 1 8 A 4A, 1 1 A/ 4A/, / / A (). IFFT ALGORITHMS.1 Pefoance Analysis of the Algoith Ipleentation of Radix / IFFT Design and veification using syste veilog will be done. Due to being an iegula intege fo the sequence lengths; it is difficult to gain a copletely accuate foula of coputational coplexity. The IFFT can be pefoed by fist swapping the eal and iaginay pats of the incoing data and then pefoing the fowad FFT on the and once again swapping the eal and iaginay pats of the data at the output. This ethods allows one to pefo the IFFT without changing any intenal coefficients and thus, esulting in oe efficient hadwae ipleentation. S. S u t h i e t a l Page 4

8 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS Vol. Issue.7, Pg.: -45 Figue 5: 1-Point Split Radix / IFFT 4. Siulation Results The 1 point DFT sequence has been ipleented in VLSI and siulated using odelsi based on adix / FFT algoith. Fig., 7 shows the siulation esults of 1 point DFT sequence. Fig.8, 9 shows the siulation esults of 1-Point Split Radix / IFFT. Figue : Input to 1-Point DFT Figue 7: Output of 1-Point DFT S. S u t h i e t a l Page 4

9 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS Vol. Issue.7, Pg.: -45 Figue 8: Input to 1-Point IFFT Figue 9: Output of 1-Point IFFT. 5. Conclusion In this pape, a adix / FFT algoith is pesented fo length- DFT. The poposed algoith is a ixtue of adix- and adix- algoith. It can evaluate a non-powe-of-six DFT, as long as its length- can be divided by. In ode to educe the nube of opeations, all sub DFTs ae eodeed favouably. The poposed algoith shows that its ipleentation equies less eal opeations as copaed with the published algoiths. Coputational coplexity is appoxiately 4log-+8. Due to being an iegula intege fo the sequence lengths; it is difficult to gain a copletely accuate foula of coputational coplexity. The device utilization suay shows that the aea occupied by the algoith is vey low. The poposed algoith can be used in OFDM systes, Signal and Iage Pocessing, Discete tie Signal pocessing, and DCT applications. In the futue, since the poposed algoith shows advantages fo coputing length-=q X DFT, we will do soe wos in this way. The algoiths pesented in can be ipoved by scaled DFT, and the copaison of its coputational coplexity with the poposed algoith needs to ae. Refeences [1] eihua Zheng and Kenli, Split Radix Algoith fo Length DFT,IEEE signal pocessing Lettes, VOL., O. 7, July 1. [] en-chang Yeh and Chein-ei Jen, High-Speed and Low-Powe Split-Radix FFT, IEEE Tansactions on Signal Pocessing, Vol. 51, o., Mach. [] Saad Bouguezel, M. Oai Ahad, and M..S. Sway, An Efficient Split-Radix FFT Algoith confeence poceedings of Intenational syposiu on cicuits and systes ISCAS,Vol.4, [4] M.Figo and S.Johnson, The design and ipleentation of ff, Poc. IEEE, vol. 9, o., pp. 1-1, 5. [5] J.Keine S. Kunis, and D. Potts, Using nfft-a softwae libay fo vaious non equispaced fast Fouie tansfos, ACM Tans. Math. Softw. (TOMS), vol., no.4, pp , 5. [] S. Bouguezel, M. Ahad, and M. Sway, A new adix-/8 FFT algoith fo length-dfts, IEEE Tans. Cicuits Syst. I, vol.51, no. 9, pp , Sep. 4. [7] I. Kaa and Y. Elcheif, Conjugate pai fast Fouie tansfo, Electon. Lett., vol. 5, no.5, pp. 4 5, Ap S. S u t h i e t a l Page 44

10 ITERATIOAL JOURAL OF RESEARCH I COMPUTER APPLICATIOS AD ROBOTICS Vol. Issue.7, Pg.: -45 [8] Y. Suzui, T. Sone, and K. Kido, A new FFT algoith of adix,, and 1, IEEE Tans. Acoust., Speech, Signal Pocess., vol. ASSP-4,no., pp. 8 8, Ap [9] S. Paash and V. Rao, A new adix- FFT algoith, IEEE Tans. Acoust., Speech, Signal Pocess., vol. ASSP-9, no. 4, pp ,Aug [1] E. Dubois and A. Venetsanopoulos, A new algoith fo the adix- FFT, IEEE Tans. Acoust., Speech, Signal Pocess,, vol. ASSP-, pp. 5, Jun AUTHORS PROFILE G. ARESH oing as Assistant Pofesso in See Vidyaniethan Engineeing College, Tiupati. He copleted M.Tech fo Sathyabhaa Univesity, Chennai and Copleted B.Tech fo SVCET, Chittoo. His aeas of inteest ae Low Powe VLSI and FPGA. Eail:ganeantinaesh@gail.co S. SRUTHI Copleted he B.Tech in Electical and Electonics Engineeing fo Annaachaya Institute of Technology and Sciences, Tiupati. He inteests ae Low Powe VLSI and Digital Syste Design. Pesently, she is pusuing M.tech in See Vidyaniethan Engineeing College, Tiupati, Andha Padesh. Eail:suthiseasapua@gail.co S. S u t h i e t a l Page 45