FFT, REALIZATION AND IMPLEMENTATION IN FPGA. Griffith University/Ericsson Microwave System AB 2000/2001. Magnus Nilsson


 Bertha Phyllis Rice
 1 years ago
 Views:
Transcription
1 EMWMS (Magus ilsso) EMW/FX/DC (Aders Waer) 1(5) FFT, REALIZATIO AD IMPLEMETATIO I FPGA Griffith Uiversity/Ericsso Microwave System AB /1 by Magus ilsso 1 cos(*pi*4.5/16*t)+i*si(*pi*4.5/16*t) Supervisor, EMW: Rue Olsso Supervisor, GU: Prof. Kuldip K. Paliwal Sigal Processig Laboratory, School of Microelectroic Egieerig, Griffith Uiversity Brisbae/Gotheburg /1 /proj/fmd/fft/report.fm
2 (5) Abstract Ericsso Microwave Systems develops radar systems for military ad civilia applicatios. I military eviromets high radar resolutio ad log rage are desired, thus high demads must be met by the geerated ad trasmitted radar sigal. I this report the desig of a parallel Radix4 Fast Fourier Trasform algorithm is described. A theoretical review regardig Fourier theory ad Fast Fourier Trasform (Radix ad Radix4) is doe. A complex parallel Radix4 algorithm is simulated, implemeted ad realized i hardware usig VHDL ad a Xilix VirtexE 1 FPGA circuit. The VHDL code was simulated ad sythesized i Ease ad Syplify eviromet. The desig was verified ad the output was idetical with the Matlab ad VHDL simulatios, provig speed improvemets due to a parallel approach.
3 3(5) Preface This thesis is a part of my educatio towards a Master degree i Computer ad Iformatio Egieerig at Griffith Uiversity, Brisbae, Australia. Project 1, ad 3. MEE797,MEE798 ad MEE799. The work has bee doe at Ericsso Microwave System AB i Möldal Swede, at the departmet FX/D I would like to thak the followig people who has bee of great help to me durig my work. My supervisor Rue Olsso, EMW. My maager Håka Olsso/Aders Waer, EMW. Prof. Kuldip K. Paliwal, supervisor GU. Daiel Wallström, EMW, for help with VHDL. Deis Eriksso, EMW, for help with Logical Aalyser/Patter geerator. ils Dagås ad Gabriel Gitye, EMW, for help with Matlab. I would also like to thak the remaiig staff at EMW/FX ad GU who have bee helpful to me.
4 4(5) Cotets Page 1 Itroductio Backgroud Task Techical fuctio JeaBaptisteJoseph Fourier The Fourier Trasform The Discrete Fourier Trasform Developmet of the Fast Fourier Trasform Theory of the Fast Fourier Trasform History of the Fast Fourier Trasform The Radix  Algorithm Fig.1. : FFTButterfly Fig.. : Radix DFT structure Fig.3. : Radix vs. Direct calculatio i flops Fig.4. : Radix algorithm comp. with MATLAB fuctio FFT. 4 7 The Radix4 Algorithm Fig.5. : Radix4 Butterfly, also referred to as Dragofly Fig.6. : Radix4 FFT algorithm compared with Matlab FFT Implemetatio ad Realizatio i hardware FPGA Fig.7. : CLB, Cofigurable logic block. Courtesy of Xilix Ic Complex FFT Fig.8. : Costructio cofiguratio Bitlegth Fig.9. : Radix4 FFT, 1bit legth of samples Fig.1. : Radix4 FFT, 14bit legth of samples Fig.11. : Radix4 FFT, 16bit legth of samples Radix4 FFT algorithm, = Fig.1. : Radix4 FFT, = Fig.13. : First FFT costructio vs. Matlab FFT Fig.14. : Timig diagram for Radix4 FFT, shared multiplier Radix4 FFT algorithm, = Fig.15. : Iput sigal X1 ad X Fig.16. : Iput sigal X3 ad X Fig.17. : Radix4 = Fig.18. : Timig diagram for Radix4 FFT legth 16, 16 bits.. 39 Fig.19. : Absolute value block
5 5(5) 9 Verificatio ad Results Test patter Matlab verificatio Fig.. : Output graph sigal X1, absolute = Fig.1. : Output graph sigal X, absolute = Fig.. : Output graph sigal X3, absolute = Fig.3. : Output graph sigal X4, absolute = Fig.4. : Output complex ad absolute, sigal 1 vs. Matlab...44 Fig.5. : Output complex ad absolute, sigal vs. Matlab...44 Fig.6. : Output complex ad absolute, sigal 3 vs. Matlab...45 Fig.7. : Output complex ad absolute, sigal 4 vs. Matlab Coclusio Ideas for further studies Refereces Appedix A Ease block structure of Radix4 FFT, = 64 AppedixA... Ease block structure of Radix4 FFT, = 64, shared mult Appedix A Ease block structure of Radix4 FFT, = 16 Appedix B Matlab code Appedix C Output listig
6 6(5) 1 ITRODUCTIO 1.1 BACKGROUD To implemet the DFT (FFT) i hardware (real time system) required expesive solutio ofte with ASIC (Applicatio Specific Itegrated Circuit). With the latest geeratio of FPGA (Field Programmable Gate Arrays) it is possible to implemet very large amouts of logic i a sigle itegrated circuit. A maufacturer of FPGA amed XILIX ow has a dropi module for their FPGAs which ca execute a 14poits FFT. It is iterestig to evaluate ad develop such a DFT or similar. 1. TASK To study, implemet ad evaluate the DFT (Discrete Fourier Trasform) i FPGA or similar. 1.3 TECHICAL FUCTIO The DFT shall collect data, execute a DFT or IDFT ad output the data. The implemetatio shall be optimized o executio time, size (area) ad cost.
7 7(5) JEABAPTISTEJOSEPH FOURIER The 1:st of March 1768, JeaBaptisteJoseph Fourier was bor. He was bor i poor circumstaces i the small village of Auxerre, Frace. JeaBaptisteJoseph Fourier itroduced the idea that a arbitrary fuctio, eve a fuctio defied by differet aalytic expressios i adjacet segmets of its rage (such as a staircase waveform) could evertheless be represeted by a sigle aalytic expressio. Fourier s ideas ecoutered resistace at the time but has prove to be a cetral theorem to may of the later developmets i mathematics, sciece ad also egieerig. As we all kow, it is at the heart of the electrical curriculum today. Fourier came across the idea i the coectio with the problem of flow of heat i solid bodies, icludig the heat from the earth. We have leared that Fourier was obsessed with heat, keepig his room really hot, ucomfortably hot for visitors, this whe eve wearig a heavy coat himself. Some has traced this obsessio back to Egypt where he wet 1798 with apoleo o a expeditio to civilize the coutry. By this time Fourier worked with his theories parallel to his official duties as a secretary of the Istitut d Egypte. At the time i Egypt, Fourier came i cotact with the Eglish Physisist Thomas Youg ( ), father of liearity, with whom he discussed his ideas ad worked together o, amog other thigs, the Rosetta Stoe. After returig back to Paris, Fourier had by 187, despite official duties, completed his theory of heat coductio, which depeded o the essetial idea of aalysig the temperature distributio ito spatially siusoidal compoets. He was very criticized for his theory amog the frech scietists, amog them where Biot ad Poisso. Eve though he was criticized for his theory he received a mathematic prize i 1811 for his heat theory. The publicatio of his writig report "Théorie aalytique de la chaleur" (The aalytical theory of heat) i 1815 was also met with some criticism ad this might be see as a idicatio of the deep ueasiess about Fourier aalysis that was felt by the great mathematicias of that day. JeaBaptisteJoseph Fourier died i Paris the 16:th of May 183. He ever got married.
8 8(5) 3 THE FOURIER TRASFORM Oe of todays pricipal aalysis tool i may of todays scietific challeges is the Fourier Trasform. Maybe the most kow applicatio of this mathematical techique is the aalysis of liear timeivariat system. As this might be the most well kow applicatio, the Fourier Trasform is essetially a uiversal problem solvig techique. Its importace is based o the fudametal property that oe ca examie a particular relatioship from a etirely differet viewpoit. Simultaeous visualizatio of a fuctio ad its Fourier Trasform is ofte the key to successful problem solvig. If we defie a sigal: lim yt () = t ± (EQ 1) 7 Trasiet sigal A trasiet sigals spectrum is characterised by the fact that it is cotiuous, this meas that it holds ifiite umbers of frequecy compoets, although usually they are i a fiite iterval.
9 9(5) Mathematically oe ca defie a sigal, that vary periodically with time, to be a sum of discrete frequecy compoets, where a simple relatioship exists betwee the frequecy compoets. This ca be defied as a formula: yt () = A { cos( ω t) + B si( ω t) } = 1 (EQ ) where ω = π T T = Periodicaltime If we look back o our trasiet sigal above, the mathematical cosequece will be that the coefficiets will be cotiues fuctios of the phase w. Equatio 3 becomes: yt () = A( ω) cos( ωt) + B( ω) si( ωt) (EQ 3)
10 1(5) If we compare this equatio with equatio we will see that the costat A / has disappeared, this though A / represets the time mea value of the sigal ad though it is a trasiet, the time mea value is zero. The Fourier coefficiets A(w) ad B(w) is defied by the Fourier itegrals: A( ω) = yt () cosωtdt (EQ 4) B( ω) = yt () siωtdt where (EQ 5) ω > Whe oe wats to calculate the Fourier coefficiets i the geeral case for the sigal f(t), oe should facilitate the calculatios by itroduce complex otatio.the startig poit for complex otatio of the Fourier Trasform is based o the formulas by Euler which gives a relatio betwee the complex umber j ad the trigoometrical fuctios sie ad cosie: cosα = e jα e jα (EQ 6) siα = e jα e jα j (EQ 7)
11 11(5) The equatios, 6 ad 7 will give us: yt () A( ω) e jωt + e jωt B( ω) e jωt e jωt = dω j yt () A e jωt + e jωt jb e jωt e jωt = dω yt () = 1  ( A jb)e jωt ( A+ jb)e j ωt dω (EQ 8) If we defie Y(w): Y ( ω) = 1  ( A( ω) jb( ω) ) Y ( ω) = 1  ( A( ω) + jb( ω) ) Equatio 9a ad 9b (EQ 9) The the equatio 8 ca be simplified by makig the itegratio over the real area: yt () = Y ( ω)e jωt (EQ 1)
12 1(5) This will the give us, by lookig at equatio 4 ad 5: Y ( ω) = yt () cosωtdt j y() t siωtdt Y ( ω) = yt ()( cosωt jsiωt) dt Y ( ω) = yt ()e jωt (EQ 11) We will the defie Y(w) as the Fourier Trasform of y(t) ad equatio 1 as the Iverse Fourier Trasform.
13 13(5) 4 THE DISCRETE FOURIER TRASFORM Whe samplig a arbitrary aalog sigal the sampled sigal ca be expressed as: yt () = y( )δ( ) + yt ( S )δ( t T S ) + y( T S )δ( t T S ) + + y( ( 1)T S )δ( t ( 1)T S ) Where (EQ 1) f T max S Accordig to the yquist theorem. The fuctio described above is a sum of time delayed delta fuctios, each of them with the height y(t S ). The Fourier Trasform for all of those fuctios equals the Fourier Trasform for the udelayed fuctio ie. F{ y( T S )δ( ) } = yt ( S )F{ δ( ) } multiplied with respectively time delay factor: Y ( ω) y( ) yt ( S )e jωt S = y( ( 1)T S )e jω 1 ( )T S 1 Y ( ω) yt ( S )e jωt S = = (EQ 13) Sice f = w/pi is a discrete variable whe we deal with a sampled sigal ad oly adopt the discrete values:
14 14(5) 1, , T , s T, s T s = k T s (EQ 14) Where k =,1,,...,1 Equatio 13 becomes: 1 j t πk S Y πk T = yt ( T s S )e s = = 1 = yt ( S ) e j π k (EQ 15) Oce agai k =,1,,...,1 If we simplify the equatio: W = e j π (EQ 16) We will get the fial expressio for the Fourier Trasform: 1 k Y( k) = y ( )W = (EQ 17) k =,1,,...,1 The factor W is called the Twiddle Factor.
15 15(5) 5 DEVELOPMET OF THE FAST FOURIER TRASFORM 5.1 THEORY OF THE FAST FOURIER TRASFORM If we cosider the equatio 17: 1 k Y( k) = y ( )W = k = 1,,,, 1 ad we cosider the amout of additios ad multiplicatios eeded for computig the algorithm. For istace, let us cosider the case whe = 4 Y ( ) = y W + y 1 W + y W + y 3 W Y ( 1) = y W + y 1 W 1 + y W + y 3 W 3 Y ( ) = y W + y 1 W + y W 4 + y 3 W 6 Y ( 3) = y W + y 1 W 3 + y W 6 + y 3 W 9 Or simplified i the compact form: Y( ) = W k yk ( ) (EQ 18) If we the cosider the twiddle factor ad y(k) we will i the worst case have two complex umbers. This fact will give us complex multiplicatios ad ()(1) complex additios. Suppose that we have a microprocessor that ca do a additio or a multiplicatio i 1 micro secod ad that this processor should compute a DFT o a 1 kbyte set of samples. If we have complex multiplicatios ad ()(1) complex additios ~ additios ad multiplicatios: x 14 x 1 micro secod =,1 secod. This without takig ito cosideratio the fact that the processor has to update poiters a so o. If we wat the aalyse to be made i real time we will have to have a distace betwee the samples that exceeds,1 secod:
16 16(5) =.5ms 14 Which gives us: = 488Hz.5 as the maximum samplig frequecy. By takig the yquist theorem i respect, we ca ot sample a sigal that holds a frequecy compoet that exceeds half the maximum samplig frequecy = 488Hz/ = 44Hz. There are two obvious ways to improve ad icrease the badwidth; a faster processor or optimizig the algorithm. 5. HISTORY OF THE FAST FOURIER TRASFORM I the begiig of the 196 s, durig a meetig of the Presidet s Scietific Advisory Commitee, Richard L. Garwi foud out that Joh W. Tukey was writig about the Fourier Trasform. Garwi was i his ow research i a desperate eed for a fast way to compute the Fourier Trasform. Whe questioed, Tukey outlied to Garwi essetially what has led to the famous CooleyTukey algorithm. To get some programmig techique, Tukey wet to IBM Research i Yorktow Heights ad meet there James W Cooley, who quickly worked out a computer program for this algorithm. After a while, request for copies ad a writeup bega accumulatig, ad also Cooley was asked to write a paper o the algorithm which i 1965 became the famous paper "A algorithm for the machie calculatio of complex Fourier series", that he published together with Tukey. Whe publishig this paper, reports of other people usig the same techique became kow, but the origial idea usually ascribe to Ruge ad Köig. The Cooley  Tukey algorithm is also called the Radix  algorithm, due to its sigal splittig.
17 17(5) 6 THE RADIX  ALGORITHM Oce agai cosider equatio 17: 1 k Y( k) = y ( )W = k = 1,,,, 1 ad we wat to aalyse the samples: { y ( )} = { y( ), y( 1), y( ),, y ( 1) } If we cosider the possibility to split the samples ito odd ad eve samples: { y( ) } = { y( ), y( ), y( 4), y ( ) } { y( + 1) } = { y( 1), y( 3), y( 5),, y ( 1) } Doig the DFT for those two sequeces will give: k Y( k) = y( )W + = = + 1 y( + 1)W ( )k (EQ 19) By extractig ad simplifyig the twiddle factor we are able to simplify eve further: k Y( k) = y( )W + = W k = k y( + 1)W (EQ )
18 18(5) W = e = e = e = j π j π j π W k Y( k) = y( )W + = W k = k y( + 1)W (EQ 1) (EQ ) By comparig this equatio with equatio 17 we will fid that this by defiitio are two DFT s with legth /. k Y( k) = Dk ( ) + W Ek ( ) k = 1,,,, 1 (EQ 3) Where D ad E represets the sums from equatio. The computatio gai by doig this will be: (as the multiplicatios i a ordiary DFT = ) = = This umber should be adjusted a bit though the twiddle factor should be multiplied with the odd sum, but this is of a first order of. If we study equatio 3, we will fid that k goes from to 1 but that D ad E represets DFT of /. Geerally for a DFT of legth is that it is periodical i k with. This leads to that D ad E i equatio 3 is periodical with /.
19 19(5) D k = Dk ( ) (EQ 4) E k = Ek ( ) (EQ 5) Calculatig the DFT: Y ( ) = D( ) + W 1 Y ( 1) = D( 1) + W Y ( ) = D( ) + W Y D = + W E( ) E( 1) E( ) E Y  D W E = = D( ) + W E( ) Y D W E = = D( 1) + W Y( 1) D ( 1) + W E( 1) D W = = + 1 E( 1) E (EQ 6) By symmetrically, the twiddle factor ca be expressed as:
20 (5) W k = W k (EQ 7) Which gives us: Y ( ) = D( ) + W 1 Y ( 1) = D( 1) + W Y D = + W E( ) E( 1) E Y  D ( = ) W E( ) Y D ( 1 1 = ) W Y( 1) = D W E( 1) E (EQ 8) By lookig ito equatio 8 we will fid oe elemetary buildigblock, the so called FFTButterfly. D(k) E(k) W K W K Y(k) Y(k+/) Fig.1. FFTButterfly
21 1(5) Which gives the equatios: k Y( k) = Dk ( ) + W Ek ( ) (EQ 9) Y( k) = Dk ( ) W k Ek ( ) (EQ 3) Sice dividig the sequeces ito smaller buildig blocks reduce the amout of multiplicatios, we will cotiue to divide the sequeces ito ew blocks. If we start with equatio ad divide the sum ito four ew sums: k Y( k) = y( 4)W + W k W k = = = k y( 4 + 1)W k y( 4)W = + k W k y( 4 + 1)W = = = y( 4 + )W y( 4 + 3)W k W ( + 1)k + ( + 1)k k + y( 4 + )W = = k y( 4 + 3)W (EQ 31)
22 (5) Ad sice W = W k Y( k) = y( 4)W 4 + W k = = k y( 4 + 1)W 4 k W + k W = = k y( 4 + )W 4 k + y( 4 + 3)W 4 (EQ 3) Ad by usig equatio 1 backwards k W = W k k Y( k) = y( 4)W 4 + W k = = k y( 4 + 1)W 4 W k + W k = = k y( 4 + )W 4 k + y( 4 + 3)W 4 (EQ 33) Ad if we cotiue to divide ito smaller sums util we oly have / poits DFT s we will get the structure described below, the costrai though is that the legth should be a power of two. The example below shows the structure for = 8.
23 3(5) x() Stage 1 Stage Stage 3 X() x(1) 1 X(1) x() W 81 X() x(3) 1 W 81 X(3) x(4) W 81 X(4) x(5) 1 W X(5) x(6) W 81 W 81 X(6) x(7) 1 W 81 W X(7) Fig.. Radix DFT structure Whe computig a DFT usig a Radix algorithm for the case whe = x the decimatio ito smaller sums ca be doe x=log times, ad this will give a total umber of complex multiplicatios = (/)log ad log complex additios. The gai whe comparig with a direct calculatio is eormous as show i the figure below: x 1 7 Radix vs. Direct Calculatio i flops 4.5 Radix Direct Calculatio umber of flops Fig.3. Radix vs. Direct calculatio i flops
24 4(5) Part of radix Matlab algorithm. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Iitialize variables. t = 1:1:14; x = si(*pi*.35*t)+si(*pi*.5*t); = legth(x); b = bidec(fliplr(decbi(:1:legth(x)1)))+1; MC = x(b); % Make i bit reversed order alfa = /; beta = 1; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculate Twiddle factor for = 1:/ W() = exp(j**pi*(1)/); W_r() = cos(*pi*(1)/); W_i() = si(*pi*(1)/); ed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculate FFT usig iplace orecursive DIT FFT, radix for h = 1:(log()/log()) b = ^(h1); a = 1; ao = 1; for d = 1:alfa c = 1; for e = 1:beta a+b; temp1 = W(c)*MC(a+b); temp = MC(a); MC(a) = MC(a) + temp1; MC(a+b) = temp  temp1; a = a + 1; c = c + alfa; ed a = ao + ^(h); ao = a; ed alfa = alfa/; beta = beta*; ed 6 5 y(x) = si(*pi*.35*t)+si(*pi*.5*t) Radix FFT algorithm y(x) = si(*pi*.35*t)+si(*pi*.5*t) MATLAB FFT algorithm Fig.4. Radix algorithm comp. with MATLAB fuctio FFT
25 5(5) 7 THE RADIX4 ALGORITHM By developig the Radix algorithm eve further ad usig the base 4 istead we will get the a more complex algorithm but with less computatio power. As we will uderstad, we will get the ew costrai where the umber of data poits i the DFT has to be the power of 4 (i.e.=4 x ). By doig i the same way as we did with the Radix algorithm we divide the data sequece ito four subsequece { y( 4) } = { y( ), y( 4), y( 8),, y ( 4) } { y( 4 + 1) } = { y( 1), y( 5), y( 9),, y ( 3) } { y( 4 + ) } = { y( ), y( 6), y( 1),, y ( ) } { y( 4 + 3) } = { y( 3), y( 7), y( 11),, y ( 1) } By usig the approach described i [8] ad by applyig: 3 lq lp X( p, q) = [ W Flq (, )]W 4 l = p = 13,,, where F(l,q) is give by: mq Flq (, ) = xlm (, )W m = l = 13,,, q = 1,,,, (EQ 34) Ad where: xlm (, ) = x( 4m+ l) X( p, q) = X p + q (EQ 35) (EQ 36)
26 6(5) Ad the four /4poit DFT s obtaied from equatio 35 are combied accordig to equatio 34 ad ca be combied to yield the poit DFT, as described i [8]: X(, q) X( 1, q) X(, q) X( 3, q) = j 1 j j 1 j W F(, q) q W F( 1, q) q W F(, q) 3q W F( 3, q) (EQ 37) We also have to ote that W = 1, which will give us three complex multiplicatios ad 1 complex additios per Radix4 butterfly. As the Radix 4 algorithm cosists of v steps (log()/log(4)) where each step ivolves /4 umber of butterflies we will get 3*v*/4 = (3/8)log umber of complex multiplicatios ad (3/)log complex additios. If compared with the computatioal power used by the Radix algorithm i chapter 5, we will fid that we have a computer gai of 5% regardig the complex multiplicatios, but that the umber of complex additios icreases by 5%. The matrix i equatio 37 is better described with a Radix4 butterfly: W i x A W q i1 x j 1 j B i W q x 11 C i3 W 3q x j 1 j D Fig.5. Radix4 Butterfly, also referred to as Dragofly
27 7(5) As we are iterested i a complex FFT we eed to derive the equatios for the complex radix4 algorithm. a = i 1 b = i1 W q c = i W q d = i3 W 3q A = a+ b + c + d B = a c jb ( d) C = a + c ( b + d) D = a c+ jb ( d) Which i the complex matter will give us, startig with the easiest oes: (r =real, i = imag) Ar = ar + br + cr + dr Ai = ai + bi + ci + di Cr = ar br + cr dr Ci = ai bi + ci di (EQ 38) Ad cotiuig with B gives: B = ar + ai cr ci jbr jbi + jdr + jdi B = ar + ai cr ci jbr + bi + jdr di (EQ 39) imag real (EQ 4) Divided ito real ad imagiary part: Br = ar cr + bi di Bi = ai ci br + dr (EQ 41)
28 8(5) Ad the last oe gives: D = ar + ai cr ci + jbr + jbi jdr jdi D = ar + ai cr ci + jbr bi jdr + di imag real (EQ 4) Divided ito real ad imagiary part: Dr = ar cr bi + di Di = ai ci + br dr (EQ 43) To get the iputs ar, ai, br, bi, cr, ci, dr ad di, we will have to multiply the iput ir, ii ad so o with the twiddle factor. This reder i: br = i1r cos( x) + i1i si( x) bi = i1i cos( x) i1r si( x) This is adequate for all iput sigals. X = dragofly specific value (twiddle factor) (EQ 44) As the goal of this project is to implemet a very fast fourier trasform i a realtime programmable logic system, we wat as few complex multiplicatios as possible, which yields lots of logic. With this i thoughts, to choose the Radix4 algorithm for implemetatio was obvious, as it has less complex multiplicatios tha the Radix algorithm.
29 9(5) Part of radix4 Matlab algorithm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Iitialize variables. t = 1:1:56; x = si(*pi*.35*t)+si(*pi*.38*t); x1 = x; = legth(x); t = log()/log(4); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Radix4 Algorithm for q = 1:t L = 4^q; r = /L; Lx = L/4; rx = 4*r; y = x; for j = :Lx1 for k = :r1 a = y(j*rx + k + 1); b = exp(i**pi*j/l)*y(j*rx + r + k + 1); c = exp(i**pi**j/l)*y(j*rx + *r + k + 1); d = exp(i**pi*3*j/l)*y(j*rx + 3*r + k + 1); t = a + c; t1 = a  c; t = b + d; t3 = b  d; x(j*r + k + 1) = t + t; x((j + Lx)*r + k + 1) = t1  i*t3; x((j + *Lx)*r + k + 1) = t  t; x((j + 3*Lx)*r + k + 1) = t1 + i*t3; ed ed ed 1 Radix 4 FFT algorithm Matlab FFT algorithm Fig.6. Radix4 FFT algorithm compared with Matlab FFT
30 3(5) 8 IMPLEMETATIO AD REALIZATIO I HARDWARE Classical implemetatio of the FFT algorithm, with a processor or i hardware usually requires a sequetial algorithm, i some cases recursive, this due to space ad memory requiremets. This slows dow the executio time. By utilizig moder programmable circuits, like a FPGA, a parallel approach to the realizatio of FFT is available. 8.1 FPGA The realtime FFT costructio was meat to be realized i a FPGA, a field programmable gate array, costructed ad maufactured by Xilix, Ic. The Xilix FPGA model VirtexE is a state of the art programmable gate array for high speed, high complex logical costructio. There is a great field of models, from small to large circuits. The logic iside a FPGA is costructed aroud a buildig block called CLB, Cofigurable logic block. Fig.7. CLB, Cofigurable logic block. Courtesy of Xilix Ic. Each of these blocks are divided ito two slices, where each slice cosists of two lookup tables ad some storage elemets. The slices are iterally coected i betwee ad are the basic highspeed logic i the circuit. Available for implemetatio of this project was a PCB with a Xilix VirtexE 1 mouted. This circuit holds a CLB array of 64 x 96 = 6144 CLB blocks. For more iformatio about Xilix VirtexE, refer to Xilix VertexE data book [13].
31 31(5) 8. COMPLEX FFT The Ericsso Microwave specificatio for the project was to simulate, realize ad implemet a complex FFT i a FPGA, a Xilix VirtexE 1. The specificatios for the FFT was: FFTlegth Miimum: 16 complex samples Maximum: 14 complex samples Typical: 64 or 56 (16) umber of bits for the iput sigal Miimum: 1 bits Maximum: 16 bits Typical: 1 The idea was to implemet the FFT as a buildigblock i a costructio, where the FFTblock will be placed after a quadrature divided A/D coverted sigal as described i the figure below: FPGA, Xilix VirtexE 1 f(x) A/D x() I/Q I Q FFT I Q Fig.8. Costructio cofiguratio
32 3(5) 8.3 BITLEGTH The first thig to cosider whe implemetig somethig discrete i hardware is to cosider the bit legth with which you wat to represet your sample. The best way to do this is to simulate differet types of bit legths ad compare the phase error ad amplitude error factor with the costrais for your costructio. 4 Radix 4 FFT, Bits = 1 si(*pi*f1*p/fs) 4 MATLAB FFT si(*pi*f1*p/fs) db db Amplitude error factor Radix 4 FFT/MATLAB FFT Phase error 15 db 1 Phase error Fig.9. Radix4 FFT, 1bit legth of samples 4 Radix 4 FFT, Bits = 14 si(*pi*f1*p/fs) 4 MATLAB FFT si(*pi*f1*p/fs) db db Amplitude error factor Radix 4 FFT/MATLAB FFT Phase error db Phase error Fig.1. Radix4 FFT, 14bit legth of samples 4 Radix 4 FFT, Bits = 16 si(*pi*f1*p/fs) 4 MATLAB FFT si(*pi*f1*p/fs) db db Amplitude error factor Radix 4 FFT/MATLAB FFT Phase error db Phase error Fig.11. Radix4 FFT, 16bit legth of samples
33 33(5) As the costrais for the realtime FFT costructio was to miimize the phase ad amplitude error as much as possible, but ot more tha that the costructio could be realizable. The simulatio results poited towards 16 bits, as this result had a small value of phase error. 8.4 RADIX4 FFT ALGORITHM, = 64 The first attempt of the implemetatio phase was to implemet a Radix4 FFT algorithm, with legth 64 complex samples. For a Radix4 FFT with legth = 64, there are 3 dragofly raks, with each rak comprisig 16 dragoflies. I the first revisio of the costructio, the bitlegth of the iput samples to the first dragofly rak was 1, this due to the precisio of the quadrature block i figure 8. Those iput samples were the multiplied with the phase factor for the correct block, also with a precisio of 1 bits. As the complex output of the multiplicatio will geerate 1 * 1 => 4 bits, the complex output of the multiplicatio is rouded of ad trucated to 14 bits. This results i a 14 bits iput to the secod rak of dragoflies, which will by usig the same model as for the first rak of dragoflies, geerate a complex output with the legth of 16 bits. As we also have a third rak of dragoflies, the complex output from our FFT costructio will have 18 bits. Radix4 FFT, = 64 I 1bits Q 1bits Dragofly rak I I I 14 bits 16 bits 18 bits Dragofly rak Dragofly rak Q Q Q 14 bits 16 bits 18 bits Fig.1. Radix4 FFT, = 64 The FFTblock was costructed usig the software EASE ad the programmig laguage VHDL, i.e. Very high speed itegrated circuit Hardware Descriptio Laguage. The software Ease is a block model descriptio laguage that lets you costruct the algorithm as blocks ad takes care of the itercoectio betwee the blocks ad the geerates the VHDL code for this itercoectio [1].
34 34(5) Part of the VHDL  code for oe of the dragoflies i the first of the raks is as follows: begi  process radix4 if clk'evet ad clk = '1' the ar_temp <= ir*cos_j; ai_temp <= ii*si_j; br_temp1 <= i1r*cos_1j; br_temp <= i1i*si_1j; bi_temp1 <= i1i*cos_1j; bi_temp <= i1r*si_1j; cr_temp1 <= ir*cos_j; cr_temp <= ii*si_j; ci_temp1 <= ii*cos_j; ci_temp <= ir*si_j; dr_temp1 <= i3r*cos_3j; dr_temp <= i3i*si_3j; di_temp1 <= i3i*cos_3j; di_temp <= i3r*si_3j; br_temp <= br_temp1 + br_temp; bi_temp <= bi_temp1  bi_temp; cr_temp <= cr_temp1 + cr_temp; ci_temp <= ci_temp1  ci_temp; dr_temp <= dr_temp1 + dr_temp; di_temp <= di_temp1  di_temp; ar_roud <= ar_temp((*1) dowto (3)) + roud; ai_roud <= ai_temp((*1) dowto (3)) + roud; br_roud <= br_temp((*1) dowto (3)) + roud; bi_roud <= bi_temp((*1) dowto (3)) + roud; cr_roud <= cr_temp((*1) dowto (3)) + roud; ci_roud <= ci_temp((*1) dowto (3)) + roud; dr_roud <= dr_temp((*1) dowto (3)) + roud; di_roud <= di_temp((*1) dowto (3)) + roud; temp1r <= ar_roud((+) dowto 1) + cr_roud((+) dowto 1); temp1i <= ai_roud((+) dowto 1) + ci_roud((+) dowto 1); tempr <= ar_roud((+) dowto 1)  cr_roud((+) dowto 1); tempi <= ai_roud((+) dowto 1)  ci_roud((+) dowto 1); ar_out <= temp1r + br_roud((+) dowto 1) + dr_roud((+) dowto 1); ai_out <= temp1i + bi_roud((+) dowto 1) + di_roud((+) dowto 1); br_out <= tempr + bi_roud((+) dowto 1)  di_roud((+) dowto 1); bi_out <= tempi  br_roud((+) dowto 1) + dr_roud((+) dowto 1); cr_out <= temp1r  br_roud((+) dowto 1)  dr_roud((+) dowto 1); ci_out <= temp1i  bi_roud((+) dowto 1)  di_roud((+) dowto 1); dr_out <= tempr  bi_roud((+) dowto 1) + di_roud((+) dowto 1); di_out <= tempi + br_roud((+) dowto 1)  dr_roud((+) dowto 1); ed if; ed process radix4; ed a ;  of Block1 The VHDL code above describes exactly the dragofly illustrated i figure 5. For this block the phase/twiddle factor is simple ad ca easy be realized as a right shift of the iput sigal, but as we get towards the last dragofly i the costructio, the phase factor costat gets more complex ad has to be realized as a high performace multiplier. The total block descriptio is described i appedix A1. The 64 complex iput sigals is shifted ito the FFTblock usig a shift register. This register is divided ito a real ad a imagiary part where the iput (complex) gets a ew sample every clock cycle. Whe the shift register gets full, it geerates a valid sigal that triggs the FFTblock that starts the FFT process. Whe the process is doe, a ew valid sigal is geerated, ad a output shift register is started. For every clock cycle, a ew processed value is delivered to the output. After all 64 values are delivered the valid sigal gets low ad shows that every sample has bee shifted out.
Soving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationSolving DivideandConquer Recurrences
Solvig DivideadCoquer Recurreces Victor Adamchik A divideadcoquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationDivide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015
CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationni.com/sdr Software Defined Radio
i.com/sdr Software Defied Radio Rapid Prototypig With Software Defied Radio The Natioal Istrumets software defied radio (SDR) platform provides a itegrated hardware ad software solutio for rapidly prototypig
More informationLearning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.
Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationDomain 1  Describe Cisco VoIP Implementations
Maual ONT (6428) 18004186789 Domai 1  Describe Cisco VoIP Implemetatios Advatages of VoIP Over Traditioal Switches Voice over IP etworks have may advatages over traditioal circuit switched voice etworks.
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationBasic Measurement Issues. Sampling Theory and AnalogtoDigital Conversion
Theory ad AalogtoDigital Coversio Itroductio/Defiitios Aalogtodigital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationTAYLOR SERIES, POWER SERIES
TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationStudy on the application of the software phaselocked loop in tracking and filtering of pulse signal
Advaced Sciece ad Techology Letters, pp.3135 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phaselocked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College
More informationTagore Engineering College Department of Electrical and Electronics Engineering EC 2314 Digital Signal Processing University Question Paper PartA
Tagore Egieerig College Departmet of Electrical ad Electroics Egieerig EC 34 Digital Sigal Processig Uiversity Questio Paper PartA UitI. Defie samplig theorem?. What is kow as Aliasig? 3. What is LTI
More informationAPPLICATION NOTE 30 DFT or FFT? A Comparison of Fourier Transform Techniques
APPLICATION NOTE 30 DFT or FFT? A Compariso of Fourier Trasform Techiques This applicatio ote ivestigates differeces i performace betwee the DFT (Discrete Fourier Trasform) ad the FFT(Fast Fourier Trasform)
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a stepbystep procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationSystems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 6985295 Email: bcm1@cec.wustl.edu Supervised
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70450) 18004186789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationEscola Federal de Engenharia de Itajubá
Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica PósGraduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationResearch Article Sign Data Derivative Recovery
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov
More informationA Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design
A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 168040030 haupt@ieee.org Abstract:
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationElementary Theory of Russian Roulette
Elemetary Theory of Russia Roulette iterestig patters of fractios Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationSoftware Engineering Guest Lecture, University of Toronto
Summary Beyod Software Egieerig Guest Lecture, Uiversity of Toroto Software egieerig is a ew ad fast growig field, which has grappled with its idetity: from usig the word egieerig to defiitio of the term,
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationConvention Paper 6764
Audio Egieerig Society Covetio Paper 6764 Preseted at the 10th Covetio 006 May 0 3 Paris, Frace This covetio paper has bee reproduced from the author's advace mauscript, without editig, correctios, or
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationDescriptive statistics deals with the description or simple analysis of population or sample data.
Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small
More informationNEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,
NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More informationEngineering Data Management
BaaERP 5.0c Maufacturig Egieerig Data Maagemet Module Procedure UP128A US Documetiformatio Documet Documet code : UP128A US Documet group : User Documetatio Documet title : Egieerig Data Maagemet Applicatio/Package
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationMultiplexers and Demultiplexers
I this lesso, you will lear about: Multiplexers ad Demultiplexers 1. Multiplexers 2. Combiatioal circuit implemetatio with multiplexers 3. Demultiplexers 4. Some examples Multiplexer A Multiplexer (see
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationFast Fourier Transform and MATLAB Implementation
Fast Fourier Trasform ad MATLAB Implemetatio by aju Huag for Dr. Duca L. MacFarlae Sigals I the fields of commuicatios, sigal processig, ad i electrical egieerig moregeerally, a sigalisay time varyig or
More informationEquation of a line. Line in coordinate geometry. Slopeintercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Pointslope form ( 點 斜 式 )
Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before
More informationChair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics
Chair for Network Architectures ad Services Istitute of Iformatics TU Müche Prof. Carle Network Security Chapter 2 Basics 2.4 Radom Number Geeratio for Cryptographic Protocols Motivatio It is crucial to
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationFOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More informationMTOMTS Production Systems in Supply Chains
NSF GRANT #0092854 NSF PROGRAM NAME: MES/OR MTOMTS Productio Systems i Supply Chais Philip M. Kamisky Uiversity of Califoria, Berkeley Our Kaya Uiversity of Califoria, Berkeley Abstract: Icreasig cost
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationTIEE Teaching Issues and Experiments in Ecology  Volume 1, January 2004
TIEE Teachig Issues ad Experimets i Ecology  Volume 1, Jauary 2004 EXPERIMENTS Evirometal Correlates of Leaf Stomata Desity Bruce W. Grat ad Itzick Vatick Biology, Wideer Uiversity, Chester PA, 19013
More informationODBC. Getting Started With Sage Timberline Office ODBC
ODBC Gettig Started With Sage Timberlie Office ODBC NOTICE This documet ad the Sage Timberlie Office software may be used oly i accordace with the accompayig Sage Timberlie Office Ed User Licese Agreemet.
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
More informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More information(VCP310) 18004186789
Maual VMware Lesso 1: Uderstadig the VMware Product Lie I this lesso, you will first lear what virtualizatio is. Next, you ll explore the products offered by VMware that provide virtualizatio services.
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More information3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average
5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a page formula sheet. Please tur over Mathematics/P DoE/November
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationNotes on exponential generating functions and structures.
Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a elemet set, (2) to fid for each the
More informationSolutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork
Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the
More informationiprox sensors iprox inductive sensors iprox programming tools ProxView programming software iprox the world s most versatile proximity sensor
iprox sesors iprox iductive sesors iprox programmig tools ProxView programmig software iprox the world s most versatile proximity sesor The world s most versatile proximity sesor Eato s iproxe is syoymous
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationChapter 10 Computer Design Basics
Logic ad Computer Desig Fudametals Chapter 10 Computer Desig Basics Part 1 Datapaths Charles Kime & Thomas Kamiski 2004 Pearso Educatio, Ic. Terms of Use (Hyperliks are active i View Show mode) Overview
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More information