Fast Fourier Transform (FFT) Algorithms

Transcription

1 Fast Fourier Trasform FFT Algorithms

2 Relatio to the z-trasform elsewhere, ozero, z x z X x [ ] 2 ~ elsewhere,, ~ e j x X x x π j e z z X X π 2 ~ The DFS X represets evely spaced samples of the z- trasform Xz aroud the uit circle.

3 Relatio to the DTFT w jw jw jw jw e X X e x e x e X 2π ~ ~ 2, 2 jw jw e X e X X w w ad w Let π π The DFS is obtaied by evely samplig the DTFT at w itervals. The iterval w is the samplig iterval i the frequecy domai. It is called frequecy resolutio because it tells us how close are the frequecy samples.

4 Samplig ad costructio i the z-domai + + ± ± m m m m j e z m W x m e x z X X j 2,,,, ~ 2 2 π π L r m r m r m m m m r x r m m x r m m x W m x W m W x W X x ~ ~ δ δ DFS & z- trasform IDFS

5 Commets Whe we sample Xz o the uit circle, we obtai a periodic sequece i the time domai. This sequece is a liear combiatio of the origial x ad its ifiite replicas, each shifted by multiples of or. If x for < ad >, the there will be o overlap or aliasig i the time domai.

6 Commets else x R x x for x x ~ ~ ~ R is called a rectagular widow of legth. THEOREM: Frequecy Samplig If x is time-limited fiite duratio to [,-], the samples of Xz o the uit circle determie Xz for all z.

7 Recostructio Formula Let x be time-limited to [,-]. The from Theorem we should be able to recover the z-trasform Xz usig its samples X~. ~ ~ ~ ~ ~ z W z W X z W X z W X z W X z x z x z X W - ~ z W X z z X

8 Examples Commets Zero-paddig is a operatio i which more zeros are appeded to the origial sequece. The resultig loger DFT provides closely spaced samples of the discretetimes Fourier trasform of the origial sequece. The zero-paddig gives us a high-desity spectrum ad provides a better displayed versio for plottig. But it does ot give us a high-resolutio spectrum because o ew iformatio is added to the sigal; oly additioal zeros are added i the data. To get high-resolutio spectrum, oe has to obtai more data from the experimet or observatios.

9 Traditioal cache solutio: Blocig Size 8 DFT p 2 radix 2 Size 4 DFT Size 4 DFT Size 2 DFT Size 2 DFT Size 2 DFT Size 2 DFT breadth-first, but with blocs of size cache requires program specialized for cache size

10 Recursive Divide & Coquer is Good depth-first traversal [Sigleto, 967] Size 8 DFT p 2 radix 2 Size 4 DFT Size 4 DFT Size 2 DFT Size 2 DFT Size 2 DFT Size 2 DFT evetually small eough to fit i cache o matter what size the cache is

11 Goal of a Efficiet computatio The total umber of computatios should be liear rather tha quadratic with respect to. Most of the computatios ca be elimiated usig the symmetry ad periodicity properties W W W + / 2 + W W + C log 2 If 2^, C will reduce to / times. Decimatio-i-time: DIT-FFT, decimatio-i-frequecy: DIF-FFT

12 Efficiet Computatio of Discrete Fourier Trasform Electrical scieces is full of sigal processig Digital computers paved way for reliable sigal processig DFT plays a importat role ad eeds efficiet procedure for computatio of X,

13 Direct Computatio of the DFT To idicate the importace of efficiet computatio schemes, it is istructive to cosider the direct evaluatio of the DFT equatio, Eq.2.. Sice x may be complex, we ca write

14 From Eq.2.2 it is clear that for each value of, the direct computatio of X requires 4 real multiplicatios ad 4-2 real additios. Total 4 2 real multiplicatios ad 4-2 real additios. The amout of time required for computatio becomes large.

15 Most approaches of improvig the efficiecy of the computatio of the DFT exploit oe or both of the followig special properties of the quatity

16 Cooley ad Tuey published a algorithm for computatio of DFT that is applicable whe is a composite umber. umber of such computatioal algorithms are ow as fast Fourier trasform, or simply FFT algorithms.

17 Radix-2 FFT Algorithms Let 2 v ; the we choose M2 ad L/2 ad divide x ito two /2-poit sequece. This procedure ca be repeated agai ad agai. At each stage the sequeces are decimated ad the smaller DFTs combied. This decimatio ads after v stages whe we have oe-poit sequeces, which are also oe-poit DFTs. The resultig procedure is called the decimatio-i-time FFT DIF-FFT algorithm, for which the total umber of complex multiplicatios is: C v *log 2 ; usig additioal symmetries: C v /2*log 2 Sigal flowgraph i Figure 5.9

18 Decimatio-i-frequecy FFT I a alterate approach we choose L2, M/2 ad follow the steps i We ca get the decimatio-frequecy FFT DIF- FFT algorithm. Its sigal flowgraph is a trasposed structure of the DIT-FFT structure. Its computatioal complexity is also equal to C v /2*log 2

19 Radix-2 FFT Algorithms To achieve the dramatic icrease i efficiecy, it is ecessary to decompose the DFT computatio ito successively smaller DFT computatios. I this process we exploit both the symmetry ad the periodicity of the complex expoetial.

20 Decimatio-i-time FFT Algorithm The priciple of decimatio-i-time is most coveietly illustrated by cosiderig the special case of a iteger power of 2; i.e. Sice is a eve iteger, we ca cosider computig X by separatig x ito two /2-poit sequeces cosistig of the eveumbered poits i x ad the oddumbered poits i x. With X give by

21 Separatig x ito eve-umbered ad odd-umbered poits, we obtai or with the substitutio of variables 2r for a eve ad 2r+ for odd,

22 But Cosequetly, Eq. 2.4 ca be writte as Each of the sums i Eq.2.5 is recogized as a /2- poit DFT. Each of sums eed oly be computed for betwee ad /2. Sice G ad H are each periodic i with period /2.

23 . The computatioal flow or the sigal flow i computig X accordig to Eq. 2.5 for a 8-poit sequece, i.e. 8 is show i Figure below.

24 Equatio 2.5 correspods to breaig the origial -poit DFT computatio ito two /2-poit DFT computatios. Each of the /2-poit DFT computatios ca be further broe ito two /4-poit DFTs. Thus G ad H i Eq.2.5 would be computed as idicated ext.

25 Similarly, where grx2r ad hrh2r+.

26 If the 4-poit DFTs i Figure 2. are computed accordig to Eqs. 2.6 ad 2.7, the that computatio would be carried out as idicated i Figure 2.2.

27 Isertig the computatio idicated i Figure 2.2 ito the flow graph of Figure 2., we obtai the complete flow graph of Figure 2.3. We have used the fact that /2 W 2.

28 I-place Computatios I view of Figure 2.4, the Figure 2.3 gives the complete computatioal flow graph for the - poit computatio of DFT of -poit sequece, for 8. A iterestig by-product of this derivatio is that, this flow graph, i additio to describig efficiet procedure for computig the DFT, also suggests a useful way of storig the origial data ad storig the results of the computatio i the itermediate arrays.

29 For 8, /4-poit DFT becomes 2-poit DFT. The 2- poit DFT of, for example, x ad x4 is depicted i Figure 2.4.

30 We shall deote the sequece of complex umbers resultig from the m th stage of computatio as X m l, where l,,..,-, ad m,2,.., formig a iput to the m+ st stage ad producig a output X m+ l as the output from the m+ st stage of computatios, it ca be see that the basic computatio i flow graph of Figure 2.3

31 The equatios represeted by this flow graph are Because of the appearace of the flow graph of Figure 2.5, this computatio is referred as a butterfly computatio. Equatios 2.8 suggest a meas of reducig the umber of complex multiplicatios by a factor of 2. To see this we ote that

32 So the equatios 2.8 become Equatios 2.9 are represeted i the flow graph of Figure 2.6.

33 Combiig the observatios i Figures 2.6, 2.5, 2.4 ad 2.3, the efficiet FFT algorithm i the computatioal flow graph represetatio for 8 is obtaied as show i Figure 2.7. The algorithm requires /2log 2 complex multiplicatios ad log 2 complex additios.

34 Decimatio-i-Frequecy FFT Algorithm The decimatio-i-time FFT algorithms were all based upo the decompositio of the DFT computatio by formig smaller ad smaller subsequeces. Alteratively decimatio-i-frequecy FFT algorithms are all based upo decompositio of the DFT computatio over X. For, a power of 2 i.e. we divide the iput sequece ito first half ad the last half of poits so that

35 Separatig -eve ad -odd, i.e. 2r ad 2r+, represetig the eve-umbered poits ad the odd-umbered poits, respectively, so that

36 Thus o the basis of Equatios 2. ad 2.2 with ad The DFT ca be computed by first formig the sequeces g ad h, the computig hw, ad fially computig the /2- poit DFTs of these two sequeces to obtai the eve-umbered output poits ad odd-umbered output poits, respectively.

37 The procedure suggested by Eqs. 2. ad 2.2 is illustrated through sigal flow graph for the case of 8- poit DFT i Figure 2.8.

38 Proceedig i a maer similar to that followed i derivig the decimatio-i-time algorithm, the fial sigal flow graph for computatio is show i Figure 2.9.

39 By coutig the arithmetic operatios i Figure 2.9, ad geeralizig, we see that the computatio of Figure 2.9 requires /2log 2 complex multiplicatios ad log 2 complex additios. Thus the total computatio is the same for decimatio-i-frequecy ad decimatio-i-time algorithms. Similar to decimatio-i-time algorithm the computatioal flow graph show i Figure 2.9 will idicate the i-place computatio capability of decimatio-i-frequecy algorithm. Figure 2.9 is the traspose of Figure 2.7.

40 Decimatio-I-Time FFT Algorithms Maes use of both symmetry ad periodicity Cosider special case of a iteger power of 2 Separate x[] ito two sequece of legth /2 Eve idexed samples i the first sequece Odd idexed samples i the other sequece j 2π / j 2π / j 2π [ ] x[]e x[]e + x[]e X eve odd Substitute variables 2r for eve ad 2r+ for odd / 2r 2r+ [ ] x[2r]w + x[2r + ]W X /2 r /2 r [ ] + W H [ ] G x[2r]w r /2 / 2 r x[2r + ]W G[] ad H[] are the /2-poit DFT s of each subsequece + W /2 r r /2

41 Decimatio I Time 8-poit DFT example usig decimatio-i-time Two /2-poit DFTs 2/2 2 complex multiplicatios 2/2 2 complex additios Combiig the DFT outputs complex multiplicatios complex additios Total complexity 2 /2+ complex multiplicatios 2 /2+ complex additios More efficiet tha direct DFT Repeat same process Divide /2-poit DFTs ito Two /4-poit DFTs Combie outputs

42 Decimatio I Time Cot d After two steps of decimatio i time Repeat util we re left with two-poit DFT s

43 Decimatio-I-Time FFT Algorithm Fial flow graph for 8-poit decimatio i time Complexity: log 2 complex multiplicatios ad additios

44 Butterfly Computatio Flow graph costitutes of butterflies We ca implemet each butterfly with oe multiplicatio Fial complexity for decimatio-i-time FFT /2log 2 complex multiplicatios ad additios

45 I-Place Computatio Decimatio-i-time flow graphs require two sets of registers Iput ad output for each stage ote the arragemet of the iput idices Bit reversed idexig X X X X X X X X [ ] x[ ] X [ ] [ ] x [ ] x[ 4] X [ ] [ ] x [ 2] x[ 2] X [ ] [ ] x [ 3] x[ 6] X [ ] [ ] x [ 4] x [ ] X [ ] x[ ] [ 5] x5 [ ] X [ ] x [ ] [ 6] x[ 3] X [ ] x[ ] [ 7] x[ 7] X [ ] x [ ]

46 Decimatio-I-Frequecy FFT Algorithm The DFT equatio [ ] X x[]w Split the DFT equatio ito eve ad odd frequecy idexes Substitute variables to get / 2 / 2 2r 2r [ ] x[]w x[]w + X 2r / 2 x[] W 2r + /2 2r [ ] x[]w + x[ + /2]W x[] + x[ + /2] X 2r / 2 / 2 2r W r /2 Similarly for odd-umbered frequecies /2 2r+ [ + ] x[] x[ + X 2r /2]W / 2

47 Decimatio-I-Frequecy FFT Algorithm Fial flow graph for 8-poit decimatio i frequecy

48 FFT vs. DFT The FFT is simply a algorithm for efficietly calculatig the DFT Computatioal efficiecy of a -Poit FFT: DFT: 2 Complex Multiplicatios FFT: /2 log 2 Complex Multiplicatios DFT Multiplicatios FFT Multiplicatios FFT Efficiecy ,536,24 64 : ,44 2,34 4 :,24,48,576 5,2 25 : 2,48 4,94,34, : 4,96 6,777,26 24, :

49 Bit Reversal The bit reversal algorithm used to perform the re-orderig of sigals. The decimal idex,, is coverted to its biary equivalet. The biary bits are the placed i reverse order, ad coverted bac to a decimal umber. Bit reversig is ofte performed i DSP hardware i the data address geerator DAG.

50 Iput sigal must be properly re-ordered usig a bit reversal algorithm DIT FFT I-place computatio umber of stages: log 2 Stage : all the twiddle factors are Last Stage: the twiddle factors are i sequetial order Stage Stage 2 Stage 3 Stage Log 2 umber of Groups /2 /4 /8 Butterflies per Group 2 4 /2 Dual-ode Spacig 2 4 /2 Twiddle Factor Expoets /2, /4,, /8,,, 2,3, to /2

51 Output sigal must be properly re-ordered usig a bit reversal algorithm DIF FFT I-place computatio umber of stages: log 2 Stage : the twiddle factors are i sequetial order Last Stage: all the twiddle factors are Stage Stage 2 Stage 3 Stage Log 2 umber of Groups 2 4 /2 Butterflies per Group /2 /4 /8 Dual-ode Spacig /2 /4 /8 Twiddle Factor Expoets, to /2-2, to /4-4, to /8 - /2,

52 Radix-4 Decimatio-I-Time FFT Algorithm A radix-4 FFT combies two stages of a radix-2 FFT ito oe, so that half as may stages are required. The radix-4 butterfly is cosequetly larger ad more complicated tha a radix-2 butterfly. /4 butterflies are used i each of log 2 /2 stages, which is oe quarter the umber of butterflies i a radix-2 FFT. Addressig of data ad twiddle factors is more complex, a radix-4 FFT requires fewer calculatios tha a radix-2 FFT. It ca compute a radix-4 FFT sigificatly faster tha a radix-2 FFT

53 Iverse Discrete Fourier Trasform IDFT The iverse discrete Fourier trasform IDFT is give by which is structurally similar to DFT, The chage we otice is i the multiplicatio factor /} ad replacemet of W by W -, ad the iterchage of sigals x ad X i the expressios ad the idex for summatio.

54 Thus i Figure 2.7 ad 2.9, if we exercise the above chages, the chaged sigal flow graphs will become algorithms for IDFT ad referred as IFFT algorithms.

55 Example Usig decimatio-i-time FFT algorithm compute DFT of the sequece {- } Solutio: Twiddle factors are

56 Solutio ad sigal flow graph of the example