Contributor: BOB SCHOR { Ä Area: U-PASCAL |61 ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ Msg#: 5727 Date: 07-05-94 08:14 From: Bschor@vms.cis.pitt.edu Read: Yes Replied: No To: All Mark: Subj: FFT Algorithm in Pascal ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ From: bschor@vms.cis.pitt.edu Over the past several weeks, there have been questions about the Fast Fourier Transform, including requests for a version of the algorithm. The following is one such implementation, optimized for clarity (??) at the possible expense of a few percentage points in speed (it's pretty darn fast). It is written in "vanilla" Pascal, so it should work with all variants of the language. Note that buried in the comments is a reasonable reference for the algorithm. } PROGRAM fft (input, output); {****************************************} { } { Bob Schor } { Eye and Ear Institute } { 203 Lothrop Street } { Pittsburgh, PA 15213 } { } {****************************************} { test routine for FFT in Pascal -- includes real and complex } { Version 1.6 -- first incarnation } { Version 10.7 -- upgrade, allow in-place computation of coefficients } { Version 14.6 -- comments added for didactic purposes } CONST version = 'FFT Version 14.6'; CONST maxarraysize = 128; halfmaxsize = 64; maxfreqsize = 63; TYPE dataindextype = 1 .. maxarraysize; cmpxindextype = 1 .. halfmaxsize; freqindextype = 1 .. maxfreqsize; complex = RECORD realpart, imagpart : real END; dataarraytype = RECORD CASE (r, c) OF r : (rp : ARRAY [dataindextype] OF real); c : (cp : ARRAY [cmpxindextype] OF complex) END; cstermtype = RECORD cosineterm, sineterm : real END; fouriertype = RECORD dcterm : real; noiseterm : real; freqterms : ARRAY [freqindextype] OF cstermtype END; mixedtype = RECORD CASE (dtype, ctype) OF dtype : (dataslot : dataarraytype); ctype : (coefslot : fouriertype) END; CONST twopi = 6.2831853; VAR data : dataarraytype; didx : dataindextype; fidx : freqindextype; coefficients : fouriertype; mixed : mixedtype; { A note on declarations, above. Pascal does not have a base type of "complex", but it is fairly simple, given the strong typing in the language, to define such a type. One needs to write procedures (see below) that implement the common arithmetic operators. Functions would be even better, from a logical standpoint, but the language standard does not permit returning a record type from a function. . The FFT, strictly speaking, is a technique for transforming a complex array of points-in-time into a complex array of points-in- Fourier space (complex numbers that represent the gain and phase of the response at discrete frequencies). One typically has data, representing samples taken at some fixed sampling rate, for which one wants the Fourier transform, to compute a power spectrum, for example. Such data, of course, are "real" quantities. One could take these N points, make them the real part of a complex array of size N (setting the imaginary part to zero), and take the FFT. However, in the interest of speed (the first F of FFT means "fast", after all), one can also do a trick where the N "real" points are identified with the real, imaginary, real, imaginary, etc. points of a complex array of size N/2. The FFT now takes about half the time, and one needs to do some final twiddling to obtain the sine/cosine coefficients of the size N real array from the coefficients of the size N/2 complex array. . To clarify the dual interpretation of the data array as either N reals or N/2 complex points, the tagged type "dataarraytype" was defined. On input, it represents the complex data; on output from the complex FFT, it represents the complex Fourier coefficients. A final transformation on these complex coefficients can convert them into a series of real sine/cosine terms; for this purpose, the tagged type "mixed" was defined for the real FFT. . Finally, note that this, and most, FFT routines get their speed when the number of points is a power of 2. This is because the speed comes from a divide-and-conquer approach -- to do an FFT of N points, do two FFTs of N/2 points and combine the results. } PROCEDURE fftofreal (VAR mixed : mixedtype; realpoints : integer); { This routine performs a forward Fourier transform of an array "mixed", which on input is assumed to consist of "realpoints" data points and on output consists of a set of Fourier coefficients (a DC term, (N/2 - 1) sine and cosine terms, and a residual "noise" term). } CONST twopi = 6.2831853; VAR index, minusindex : freqindextype; temp1, temp2, temp3, w : complex; baseangle : real; { The following procedures implement complex arithmetic -- } PROCEDURE cadd (a, b : complex; VAR c : complex); { c := a + b } BEGIN { cadd } WITH c DO BEGIN realpart := a.realpart + b.realpart; imagpart := a.imagpart + b.imagpart END END; PROCEDURE csubtract (a, b : complex; VAR c : complex); { c := a - b } BEGIN { csubtract } WITH c DO BEGIN realpart := a.realpart - b.realpart; imagpart := a.imagpart - b.imagpart END END; PROCEDURE cmultiply (a, b : complex; VAR c : complex); { c := a * b } BEGIN { cmultiply } WITH c DO BEGIN realpart := a.realpart*b.realpart - a.imagpart*b.imagpart; imagpart := a.realpart*b.imagpart + b.realpart*a.imagpart END END; PROCEDURE conjugate (a : complex; VAR b : complex); { b := a* } BEGIN { conjugate } WITH b DO BEGIN realpart := a.realpart; imagpart := -a.imagpart END END; PROCEDURE forwardfft (VAR data : dataarraytype; complexpoints : integer); { The basic FFT is a recursive routine that basically works as follows: 1) The FFT is a linear operator, so the FFT of a sum is simply . the sum of the FFTs of each addend. 2) The FFT of a time series shifted in time is the FFT of the . unshifted series adjusted by a twiddle factor which looks . like a (complex) root of 1 (an nth root of unity). 3) Consider N points, equally spaced in time, for which you . want an FFT. Start by splitting the series into odd and . even samples, giving you two series with N/2 points, . equally spaced, but with the second series delayed in time . by one sample. Take the FFT of each series. Using property . 2), adjust the FFT of the second series for the time delay. . Now using property 1), since the original N points is simply . the sum of the two N/2 series, the FFT we want is simply the . sum of the FFTs of the two sub-series (with the adjustment . in the second for the time delay). 4) This is essentially a recursive definition. To do an N-point . FFT, do two N/2 point FFTs and combine the answers. All we . need to stop the recursion is to know how to do a 2-point . FFT: if a and b are the two (complex) input points, the . two-point FFT equations are A := a+b; B := a-b. 5) The FFT is rarely coded in its fully-recursive form. It . turns out to be fairly simple to "unroll" the recursion and . reorder it a bit, which simplifies the computation of the . roots-of-unity complex twiddle factors. The only drawback . is that the output array ends up scrambled -- if the array . indices are represented as going from 0 to M-1, then if one . represents the array index as a binary number, one needs to . bit-reverse the number to get the proper place in the array. . Thus, the next step is to swap values by bit-reversing the . indices. 6) There are numerous references on the FFT. A reasonable one . is "Numerical Recipes" by Press et al., Cambridge University . Press, which I believe exists in several language flavors. } CONST twopi = 6.2831853; PROCEDURE docomplextransform; VAR partitionsize, halfsize, offset, lowindex, highindex : dataindextype; baseangle, angle : real; bits : integer; w, temp : complex; BEGIN { docomplextransform } partitionsize := complexpoints; WITH data DO REPEAT halfsize := partitionsize DIV 2; baseangle := twopi/partitionsize; FOR offset := 1 TO halfsize DO BEGIN angle := baseangle * pred(offset); w.realpart := cos(angle); w.imagpart := -sin(angle); lowindex := offset; REPEAT highindex := lowindex + halfsize; csubtract (cp[lowindex], cp[highindex], temp); cadd (cp[lowindex], cp[highindex], cp[lowindex]); cmultiply (temp, w, cp[highindex]); lowindex := lowindex + partitionsize UNTIL lowindex >= complexpoints END; partitionsize := partitionsize DIV 2 UNTIL partitionsize = 1 END; PROCEDURE shufflecoefficients; VAR lowindex, highindex : dataindextype; bits : integer; FUNCTION log2 (index : integer) : integer; { Recursive routine, where "index" is assumed a power of 2. Note the routine will fail (by endless recursion) if "index" <= 0. } BEGIN { log2 } IF index = 1 THEN log2 := 0 ELSE log2 := succ(log2(index DIV 2)) END; FUNCTION bitreversal (index, bits : integer) : integer; { Takes an index, in the range 1 .. 2**bits, and computes a bit-reversed index in the same range. It first undoes the offset of 1, bit-reverses the "bits"-bit binary number, then redoes the offset. Thus if bits = 4, the range is 1 .. 16, and bitreversal (1, 4) = 9, bitreversal (16, 4) = 16, etc. } FUNCTION reverse (bits, stib, bitsleft : integer) : integer; { Recursive bit-reversing function, transforms "bits" into bit-reversed "stib. It's pretty easy to convert this to an iterative form, but I think the recursive form is easier to understand, and should entail a trivial penalty in speed (in the overall algorithm). } BEGIN { reverse } IF bitsleft = 0 THEN reverse := stib ELSE IF odd (bits) THEN reverse := reverse (bits DIV 2, succ (stib * 2), pred (bitsleft)) ELSE reverse := reverse (bits DIV 2, stib * 2, pred (bitsleft)) END; BEGIN { bitreversal } bitreversal := succ (reverse (pred(index), 0, bits)) END; PROCEDURE swap (VAR a, b : complex); VAR temp : complex; BEGIN { swap } temp := a; a := b; b := temp END; BEGIN { shufflecoefficients } bits := log2 (complexpoints); WITH data DO FOR lowindex := 1 TO complexpoints DO BEGIN highindex := bitreversal(lowindex, bits); IF highindex > lowindex THEN swap (cp[lowindex], cp[highindex]) END END; PROCEDURE dividebyn; { This procedure is needed to get FFT to scale correctly. } VAR index : dataindextype; BEGIN { dividebyn } WITH data DO FOR index := 1 TO complexpoints DO WITH cp[index] DO BEGIN realpart := realpart/complexpoints; imagpart := imagpart/complexpoints END END; BEGIN { forwardfft } docomplextransform; shufflecoefficients; dividebyn END; { Note that the data slots and coefficient slots in the mixed data type share storage. From the first complex coefficient, we can derive the DC and noise term; from pairs of the remaining coefficients, we can derive pairs of sine/cosine terms. } BEGIN { fftofreal } forwardfft (mixed.dataslot, realpoints DIV 2); temp1 := mixed.dataslot.cp[1]; WITH mixed.coefslot, temp1 DO BEGIN dcterm := (realpart + imagpart)/2; noiseterm := (realpart - imagpart)/2 END; baseangle := -twopi/realpoints; FOR index := 1 TO realpoints DIV 4 DO BEGIN minusindex := (realpoints DIV 2) - index; WITH mixed.dataslot DO BEGIN conjugate (cp[succ(minusindex)], temp2); cadd (cp[succ(index)], temp2, temp1); csubtract (cp[succ(index)], temp2, temp2) END; w.realpart := sin(index*baseangle); w.imagpart := -cos(index*baseangle); cmultiply (w, temp2, temp2); cadd (temp1, temp2, temp3); csubtract (temp1, temp2, temp2); conjugate (temp2, temp2); WITH mixed.coefslot.freqterms[index], temp3 DO BEGIN cosineterm := realpart/2; sineterm := -imagpart/2 END; WITH mixed.coefslot.freqterms[minusindex], temp2 DO BEGIN cosineterm := realpart/2; sineterm := imagpart/2 END END END; FUNCTION omegat (f : freqindextype; t : dataindextype) : real; { computes omega*t for particular harmonic, index } BEGIN { omegat } omegat := twopi * f * pred(t) / maxarraysize END; { main test routine starts here } BEGIN WITH mixed.dataslot DO FOR didx := 1 TO maxarraysize DO rp[didx] := (23 + 13 * sin(omegat (7, didx)) + 28 * cos(omegat (22, didx))); fftofreal (mixed, maxarraysize); WITH mixed.coefslot DO writeln ('DC = ', dcterm:10:2, ' ':5, 'Noise = ', noiseterm:10:2); FOR fidx := 1 TO maxfreqsize DO BEGIN WITH mixed.coefslot.freqterms[fidx] DO write (fidx:4, round(cosineterm):4, round(sineterm):4, ' ':4); IF fidx MOD 4 = 0 THEN writeln END; writeln; writeln ('The expected result should have been:'); writeln (' DC = 23, noise = 0, '); writeln (' sine 7th harmonic = 13, cosine 22nd harmonic = 28') END.