Advanced Training Course on FPGA Design and VHDL for Hardware Simulation and Synthesis

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Peter Skinner
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1 Advaced Traiig Course o FPGA esig ad VHL for Hardware Simulatio ad Sythesis 26 October - 2 ovember, 29 igital Sigal Processig The iscrete Fourier Trasform Massimiliao olich EEI Facolta' di Igegeria Uiversita' degli Studi di Trieste via Valerio,, 427 Trieste Italy

2 The iscrete Fourier Trasform The discrete-time Fourier trasform (TFT) of a sequece is a cotiuous fuctio of!, ad repeats with period 2. I practice we usually wat to obtai the Fourier compoets usig digital computatio, ad ca oly evaluate them for a discrete set of frequecies. The discrete Fourier trasform (FT) provides a meas for achievig this. The FT is itself a sequece, ad it correspods roughly to samples, equally spaced i frequecy, of the Fourier trasform of the sigal. The discrete Fourier trasform of a legth sigal xœ, ; ; : : : ; is give by XŒk X xœe j.2= /k : This is the aalysis equatio. The correspodig sythesis equatio is xœ X k XŒke j.2= /k : Whe dealig with the FT, it is commo to defie the complex quatity W e j.2= / : With this otatio the FT aalysis-sythesis pair becomes XŒk xœ X X k xœw k XŒkW k : A importat property of the FT is that it is cyclic, with period, both i the

3 discrete-time ad discrete-frequecy domais. For example, for ay iteger r, XŒk C r X X xœw xœw k.kcr / XŒk; X xœw k.w /r sice W e j.2= / e j 2. Similarly, it is easy to show that xœ C r xœ, implyig periodicity of the sythesis equatio. This is importat eve though the FT oly depeds o samples i the iterval to, it is implicitly assumed that the sigals repeat with period i both the time ad frequecy domais. To this ed, it is sometimes useful to defie the periodic extesio of the sigal xœ to be QxŒ xœ mod xœ..// : Here mod ad..// are take to mea modulo, which has the value of the remaider after is divided by. Alteratively, if is writte i the form k C l for l <, the mod..// l: xœ QxŒ Similarly, the periodic extesio of XŒk is defied to be Q XŒk XŒk mod XŒ..k// : 2

4 It is sometimes better to reaso i terms of these periodic extesios whe dealig with the FT. Specifically, if XŒk is the FT of xœ, the the iverse FT of XŒk is QxŒ. The sigals xœ ad QxŒ are idetical over the iterval to, but may differ outside of this rage. Similar statemets ca be made regardig the trasform XŒk. Properties of the FT May of the properties of the FT are aalogous to those of the discrete-time Fourier trasform, with the otable exceptio that all shifts ivolved must be cosidered to be circular, or modulo. efiig the FT pairs xœ!xœk, x Œ the followig are properties of the FT: Symmetry:!X Œk, ad x 2 Œ!XŒk, XŒk X Œ.. k// RefXŒkg RefXŒ.. k// g ImfXŒkg ImfXŒ.. k// g jxœkj jxœ.. k// j ^XŒk ^XŒ.. k// Liearity: ax Œ C bx 2 Œ!aX Œk C bx 2 Œk. Circular time shift: xœ.. Circular covolutio: X m x Œmx 2 Œ.. m//!w km XŒk. m//!x ŒkX 2 Œk: Circular covolutio betwee two -poit sigals is sometimes deoted by x Œ xœ.

5 Modulatio: x Œx 2 Œ! X l X ŒlX 2 Œ..k l// : Some of these properties, such as liearity, are easy to prove. The properties ivolvig time shifts ca be quite cofusig otatioally, but are otherwise quite simple. For example, cosider the 4-poit FT XŒk X xœw k 4 of the legth 4 sigal xœ. This ca be writte as XŒk xœw4 k C xœw4 k C xœ2w4 2k C xœw4 k The product W4 k XŒk ca therefore be writte as W4 k k XŒk xœw4 C xœw4 2k C xœ2w4 k C xœw4 4k xœw4 k C xœw4 k C xœw4 2k C xœ2w4 k sice W4 4k W4 k. This ca be see to be the FT of the sequece xœ; xœ; xœ; xœ2, which is precisely the sequece xœ circularly shifted to the right by oe sample. This proves the time-shift property for a shift of legth. I geeral, multiplyig the FT of a sequece by W km -poit circular shift of the sequece by m samples. The covolutio properties ca be similarly demostrated. results i a It is useful to ote that the circularly shifted sigal xœ.. m// is the same as the liearly shifted sigal QxŒ m, where QxŒ is the -poit periodic extesio of xœ. 4

6 xœ QxŒ QxŒ m xœ.. m// O the iterval to, the circular covolutio x Œ x Œ x 2 Œ X m x Œmx 2 Œ.. m// ca therefore be calculated usig the liear covolutio product x Œ X m x Œm Qx 2 Œ m: Circular covolutio is really just periodic covolutio. Example: Circular covolutio with a delayed impulse sequece Give the sequeces x Œ x 2 Œ 5

7 the circular covolutio x Œ x Œ x 2 Œ is the sigal QxŒ delayed by two samples, evaluated over the rage to : x Œ Example: Circular covolutio of two rectagular pulses Let 8 < L x Œ x 2 Œ : otherwise: If L, the the -poit FTs are X Œk X 2 Œk X W k 8 < k : otherwise: Sice the product is 8 < 2 k X Œk X ŒkX 2 Œk : otherwise; it follows that the -poit circular covolutio of x Œ ad x 2 Œ is x Œ x Œ x 2 Œ ; : Suppose ow that x Œ ad x 2 Œ are cosidered to be legth 2L sequeces by augmetig with zeros. The 2L-poit circular covolutio is the see to be the same as the liear covolutio of the fiite-duratio sequeces x Œ ad x 2 Œ: 6

8 x Œ x 2 Œ L L x Œ x 2 Œ L 2 Liear covolutio usig the FT Usig the FT we ca compute the circular covolutio as follows Compute the -poit FTs X Œk ad X 2 Œk of the two sequeces x Œ ad x 2 Œ. Compute the product X Œk X ŒkX 2 Œk for k. Compute the sequece x Œ x Œ x 2 Œ as the iverse FT of X Œk. This is computatioally useful due to efficiet algorithms for calculatig the FT. The questio that ow arises is this: how do we get the liear covolutio (required i speech, radar, soar, image processig) from this procedure? 2. Liear covolutio of two fiite-legth sequeces Cosider a sequece x Œ with legth L poits, ad x 2 Œ with legth P poits. The liear covolutio of the sequeces, X x Œ x Œmx 2 Œ m m; is ozero over a maximum legth of L C P poits: 7

9 2 x [] L x 2 [] 8 P x [] L+P Therefore L C P covolutio. is the maximum legth of x Œ resultig from the liear The -poit circular covolutio of x Œ ad x 2 Œ is x Œ x 2 Œ X m x Œmx 2 Œ.. m// X m x Œm Qx 2 Œ m W It is easy to see that the circular covolutio product will be equal to the liear covolutio product o the iterval to as log as we choose L C P. The process of augmetig a sequece with zeros to make it of a required legth is called zero paddig. 2.2 Covolutio by sectioig Suppose that for computatioal efficiecy we wat to implemet a FIR system usig FTs. It caot i geeral be assumed that the iput sigal has a fiite duratio, so the methods described up to ow caot be applied directly: 8

10 h[] P x[] L 2L L The solutio is to use block covolutio, where the sigal to be filtered is segmeted ito sectios of legth L. The iput sigal xœ, here assumed to be causal, ca be decomposed ito blocks of legth L as follows: where xœ X x r Œ r rl; 8 < xœ C rl L x r Œ : otherwise: x [] x [] L x 2 [] L 2L 2L L 9

11 The covolutio product ca therefore be writte as X yœ xœ hœ y r Œ where y r Œ is the respose r rl; y r Œ x r Œ hœ: y [] y [] L+P y 2 [] L 2L Sice the sequeces x r Œ have oly L ozero poits ad hœ is of legth P, each respose term y r Œ has legth L C P. Thus liear covolutio ca be obtaied usig -poit FTs with L C P. Sice the fial result is obtaied by summig the overlappig output regios, this is called the overlap-add method. y[] L 2L L A alterative block covolutio procedure, called the overlap-save method, correspods to implemetig a L-poit circular covolutio of a P-poit

12 impulse respose hœ with a L-poit segmet x r Œ. The portio of the output that correspods to liear covolutio is the idetified (cosistig of L.P / poits), ad the resultig segmets patched together to form the output. Spectrum estimatio usig the FT Spectrum estimatio is the task of estimatig the TFT of a sigal xœ. The TFT of a discrete-time sigal xœ is X.e j! / X xœe j! : The sigal xœ is geerally of ifiite duratio, ad X.e j! / is a cotiuous fuctio of!. The TFT ca therefore ot be calculated usig a computer. Cosider ow that we trucate the sigal xœ by multiplyig with the rectagular widow w r Œ: w r [].5 The widowed sigal is the x w Œ xœw r Œ. The TFT of this widowed sigal is give by X w.e j! / X otig that the FT of x w Œ is X w Œk x w Œe j! X X x w Œe j 2k ; x w Œe j! :

13 it is evidet that X w Œk X w.e j! /ˇˇ!2k= : The values of the FT X w Œk of the sigal x w Œ are therefore periodic samples of the TFT X w.e j! /, where the spacig betwee the samples is 2=. Sice the relatioship betwee the discrete-time frequecy variable ad the cotiuous-time frequecy variable is! T, the FT frequecies correspod to cotiuous-time frequecies k 2k T : The FT ca therefore oly be used to fid poits o the TFT of the widowed sigal x w Œ of xœ. The operatio of widowig ivolves multiplicatio i the discrete time domai, which correspods to cotiuous-time periodic covolutio i the TFT frequecy domai. The TFT of the widowed sigal is therefore X w.e j! / 2 Z X.e j /W.e j.! / /d; where W.e j! / is the frequecy respose of the widow fuctio. For a simple rectagular widow, the frequecy respose is as follows: wrœ.5 8 jwr.e j! /j 8 The FT therefore effectively samples the TFT of the sigal covolved with the frequecy respose of the widow.! 2

14 Example: Spectrum aalysis of siusoidal sigals Suppose we have the siusoidal sigal combiatio xœ cos.=/ C :75 cos.2=/; < < : Sice the sigal is ifiite i duratio, the TFT caot be computed umerically. We therefore widow the sigal i order to make the duratio fiite: xœ 8 wrœ.5 8 xwœ 8 The operatio of widowig modifies the sigal. This is reflected i the discrete-time Fourier trasform domai by a spreadig of the frequecy compoets:

15 jx.e j! /j jwr.e j! /j jxw.e j! /j 5 2! 2 The operatio of widowig therefore limits the ability of the Fourier trasform to resolve closely-spaced frequecy compoets. Whe the FT is used for spectrum estimatio, it effectively samples the spectrum of this modified sigal at the locatios of the crosses idicated: 6 jxœkj ote that sice k correspods to!, there is a correspodig shift i the sampled values. I geeral, the elemets of the -poit FT of x w Œ cotai evely-spaced samples from the TFT X w.e j! /. These samples spa a etire period of the TFT, ad therefore correspod to frequecies at spacigs of 2=. We ca obtai samples with a closer spacig by performig more computatio. k 4

16 Suppose we form the zero-padded legth M sigal x M Œ as follows: 8 < xœ x M Œ : M : The M -poit FT of this sigal is X M Œk MX X x M Œe j 2 M k x w Œe j 2 M k X x w Œe j 2 M k The sample X p Œk ca therefore be see to correspod to the TFT of the the widowed sigal x w Œ at frequecy! k 2k=M. Sice M is chose to be larger tha, the trasform values correspod to regular samples of X w.e j! / with a closer spacig of 2=M. The followig figure shows the magitude of the FT trasform values for the 8-poit sigal show previously, but zero-padded to use a 2-poit FT: xm Œ jxœkj ote that this process icreases the desity of the samples, but has o effect o the resolutio of the spectrum. If W.e j! / is sharply peaked, ad approximates a irac delta fuctio at the k 5

17 origi, the X w.e j! / X.e j! /. The values of the FT the correspod quite accurately to samples of the TFT of xœ. For a rectagular widow, the approximatio improves as icreases: wrœ jwr.e j! /j jxw.e j! /j.5 2 2! The magitude of the FT of the widowed sigal is jxœkj which is clearly easier to iterpret tha for the case of the shorter sigal. As the widow legth teds to, the relatioship becomes exact. The rectagular widow iheret i the FT has the disadvatage that the peak sidelobe of W r.e j! / is high relative to the mailobe. This limits the ability of the FT to resolve frequecies. Alterative widows may be used which have preferred behaviour the oly requiremet is that i the time domai the k 6

18 widow fuctio is of fiite duratio. For example, the triagular widow wrœ.5 2 jxw.e j! /j jwr.e j! /j! leads to FT samples with magitude jxœkj Here the sidelobes have bee reduced at the cost of dimiished resolutio the mailobe has become wider. The method just described forms the basis for the periodogram spectrum estimate. It is ofte used i practice o accout of its perceived simplicity. However, it has a poor statistical properties model-based spectrum estimates geerally have higher resolutio ad more predictable performace. Fially, ote that the discrete samples of the spectrum are oly a complete k 7

19 represetatio if the samplig criterio is met. The samples therefore have to be sufficietly closely spaced. 4 Fast Fourier trasforms The widespread applicatio of the FT to covolutio ad spectrum aalysis is due to the existece of fast algorithms for its implemetatio. The class of methods are referred to as fast Fourier trasforms (FFTs). Cosider a direct implemetatio of a 8-poit FT: XŒk 7X xœw8 k ; k ; : : : ; 7: If the factors W8 k have bee calculated i advace (ad perhaps stored i a lookup table), the the calculatio of XŒk for each value of k requires 8 complex multiplicatios ad 7 complex additios. The 8-poit FT therefore requires 8 8 multiplicatios ad 8 7 additios. For a -poit FT these become 2 ad. / respectively. If 24, the approximately oe millio complex multiplicatios ad oe millio complex additios are required. The key to reducig the computatioal complexity lies i the observatio that the same values of xœw k 8 are effectively calculated may times as the computatio proceeds particularly if the trasform is log. The covetioal decompositio ivolves decimatio-i-time, where at each stage a -poit trasform is decomposed ito two =2-poit trasforms. That 8

20 is, XŒk ca be writte as XŒk =2 X r =2 X r xœ2rw 2rk X C =2 r xœ2r.w 2 /rk C W k otig that W 2 W =2 this becomes xœ2r C W.2rC/k =2 X r xœ2r C.W 2 /rk : XŒk =2 X r xœ2r.w =2 / rk C W k GŒk C W k HŒk: =2 X r xœ2r C.W =2 / rk The origial -poit FT ca therefore be expressed i terms of two =2-poit FTs. The =2-poit trasforms ca agai be decomposed, ad the process repeated util oly 2-poit trasforms remai. I geeral this requires log 2 stages of decompositio. Sice each stage requires approximately complex multiplicatios, the complexity of the resultig algorithm is of the order of log 2. The differece betwee 2 ad log 2 complex multiplicatios ca become cosiderable for large values of. For example, if 248 the 2 =. log 2 / 2. There are umerous variatios of FFT algorithms, ad all exploit the basic redudacy i the computatio of the FT. I almost all cases a off-the-shelf implemetatio of the FFT will be sufficiet there is seldom ay reaso to implemet a FFT yourself. 9

The Discrete Fourier Transform

The Discrete Fourier Transform The iscrete Fourier Trasform The discrete-time Fourier trasform (TFT) of a sequece is a cotiuous fuctio of!, ad repeats with period. I practice we usually wat to obtai the Fourier compoets usig digital