GENERIC MODULE FourierTransform(R, RT, C, CT);
By Warren D. Smith, May 1985, March 1996. Gnu Copylefted.

Abstract: Fast Fourier Transforms (FFT's)

3/18/96 Warren Smith Initial version

3/23/96 Harry George Tweaked to fit m3na library


<* UNUSED *>
  Module = "FourierTransform.";
******************************************************** In place FFT routine with array length: N = a power of 2.

a{output}[k] = sum from m=0 to N-1 of a{input}[m]*exp(2 pi i direction m k/N)

where direction = +1 or -1. You have to do any scaling (by 1/N or 1/sqrt(N)) or zeroing of a[] by yourself.

Uses 2NlgN+2N+O(lgN) [real] multiplications, and 3NlgN+2N+O(lgN) [real] additions. (And hopefully the compiler optimizes the subscripting in the inner loop so that only 2NlgN total subscripting ops are needed.)

The test routine shown yields error after 1 forward and one backward use at least 3 decimal places better than slow FT has after only one use!

The basic FFT idea (for N a power of 2) is from J.Cooley&J.Tukey: An algorithm for the Machine Calculation of Complex Fourier Series, MOC 19 (1965) 297-301. (Knuth, Henrici, and Dahlquist/Bjorck also discuss the FFT.) The basic idea is that to calculate the DFT A[k] = sum(j=0..N-1)of a[j]*W^(k*j) for k=0..N-1 (where N = a power of 2, N>1, and W = a principal Nth root of unity, e.g. W = exp(2*i*PI/N) in the complex numbers) we may write A[k] = g[k] + h[k] * W^k where g[k] = sum(0<=2j<N)of a[2j] * (W^2)^(k*j) h[k] = sum(0<=2j<N)of a[2j+1] * (W^2)^(k*j) are FFTs of half the size (on the even and odd indexed a[]'s, respectively) note that g[k] and h[k] are periodic in k with period N/2, since W^(N/2) = -1, so that if T(N)=time to calculate FFT of size N, then T(N)=2T(N/2)+O(N) and so T(N)=O(NlgN). The inverse transform is a[k] = Ninverse * sum(j=0..N-1)of a[j]*Winverse^(k*j) for k=0..N-1 also an FFT and is calculable by the same method. (Ninverse*N=1. Winverse*W=1.) Cooley-Tukey also works for general highly composite N; a short and elegant implementation is: [Warren E.Ferguson: A simple derivation of glassman's general N FFT, Computers&Math. with Applics. 8 (1982) 401-411]. For an efficient implementation of the FFT, further refinements are desired. These include: 1.Removal of the recursion by reverse binary permuting the original data; 2.In place implementation with no extra storage requirement; 3.Calculation of the W^k's (k=0,1..) by efficient and stable recurrences (thus avoiding the need for transcendental functions); 4. Possibly use fast complex multiplication; 5.In specific but common applications, some further savings may be possible. Thus when doing a fast convolution of two arrays of N reals using an FFT, the two FFTs may be accomplished by ONE N point complex FFT, followed by linear time uncombining/termwise multiplication step, followed by a N/2 point reverse FFT. The reverse binary permuting stages may be avoided completely. The basic symmetry here is that the FFT of a real array is conjugate even, e.g. A[k] = CompConjugate(A[N-k]) if a[0..n-1] is real.

R.C.Singleton [e.g. see his algorithm for fast circular convolutions, CACM 12,3 (March 1969) 179; his FFT2 algorithm, CACM 11 (Nov 1968) 773-779; and his article CACM 10 (Oct 1967) 647-654] has suggested a second difference method [which keeps the real and imaginary parts of exp(ick) uncoupled] for evaluating exp(ick): exp(ic(k+1)) = exp(ick) + I[k+1]; I[k+1] = -4sin^2(c/2)exp(ikc) + I[k]; I[0] = 2isin(c/2) exp(-ic/2) which is both faster and experimentally far more stable (typically yielding 500 times smaller error for 128 point transform+inverse transform) than the straightforward multiply by exp(ic) method. (It's more stable because of the small multiplier -4sin^2(c/2), as opposed to 1.)

However a better idea which I recently thought of is to use this recurrence: exp(ic(k+1)) = exp(ic(k-1)) + 2isin(c) exp(ick) which requires only 2*,2+ per complex exponential (Singleton is 2*,4+; naive method is 4*,2+ and isn't stable) and is also stable since it involves the small multiplier 2isin(c). (In fact, this method runs faster than Singleton, is easier to program, and even yielded slightly better accuracy in the test program below, too!) Therefore this modification of Singleton is the method I've used.

Incidentally, Press et al. in their NR book use the Singleton algorithm but neglect to credit Singleton!

By a trivial modification of my code, one could generate the sines and cosines by repeated application of the bisection identities cos(t/2)=sqrt(0.5*(1.0+cos(t))), sin(t/2)=0.5*sin(t)/cos(t/2), starting from the values with t=pi and t=pi/2 as special cases, thus avoiding trig subroutine calls entirely. OK, I've now done this; now using precomputed table.

The time savings introduced by either of these is small, however. Finally, radix 8 transforms are experimentally the most efficient, typically 20% faster than radix 2 routines like this one, although far more complicated; and anyway I suspect the advantage is <20% in the modern cache-memory system world, since I suspect the radix 2 algorithm has better cache locality. However, that has not been tested.

Also you could take advantage of 1's and 0's to save a little time at the expense of considerably more space.

Another idea which I have chosen not to implement is the fact that two complex numbers may be multiplied in 3 real multiplications: thus E+iF = (a+bi)*(c+di) may be accomplished via the instructions bpa = b+a; bma = b-a; E = a*(c+d); F = bma*c+E; E -= bpa*d; and if bpa and bma are precomputed, this is a 3*,3+ method for a complex multiplication. This idea may be used to reduce (?) the box score from the present 2NlgN+2N+O(lgN) mults, 3NlgN+2N+O(lgN) adds to 1.5NlgN+2N+O(lgN) mults and 3.5NlgN+4N+O(lgN) adds. If a floating point multiplication is M times slower than a floating point addition, this idea pays iff lgN>4/(M-1). On PDP-11/44 C, however, rough timing has shown that M=1.08 (but with considerable standard deviation... it does about 5*10^4 additions/sec.) so this idea is not worth it unless N is enormous.

Some similar, but worse, ideas have been suggested by Buneman [If c=cos(m), s=sin(m), then precompute t1 = (1-c)/s = s/(1+c)= tan(m/2) and t2 = (1+s)/c = c/(1-s). Then X+iY = (a+bi)*(c+is) may be found in 3*,3+ by: if(|t1|<|t2|){ X = a-t1*b; Y = b+s*X; X -= t1*Y; } else{ X = b+t2*a; Y = c*X-a; X -= t2*Y; } tans may be updated by tan(x+y)-tan(x-y) = 2*tan(y)/(1-(tan(x)*tan(y))^2)), but the extra overhead seems not to be worth it] and also there is a symmetric 3*,5+ (4+ with precomputation) formula for E+iF = (a+bi)*(c+di): E = a*c-b*d; F = (a+b)*(c+d)-a*c-b*d.

Other FFT algorithms: Winograd has shown how to design FFTs with N prime (as opposed to the Cooley-Tukey approach which works for N highly composite) that run in O(NlgN) time and even with only O(N) multiplications; the latter figure is optimal. [S.Winograd: Math. of Comput. 32 (1978) 175-179; Advs in Math 32 (1979) 83-117]. Winograd's approach is based on a theorem that allows him (by a permutation of the original and transformed variables) to express FFTs for N prime in terms of a circular convolution of N-1 elements, plus some additions. He then shows how circular convolutions of k elements (for certain small k) may be computed in a small number of arithmetic operations, (for k=2..6, the number of multiplications Winograd uses is 2,4,5,10,8; for k prime, Winograd shows that a (2k-2)* scheme for CC(k) always exists) and further, how CC(n1) and CC(n2) algorithms may be composed to make a CC(n1*n2) algorithm, IF n1 and n2 are relatively prime, that uses mult(n1)*mult(n2) multiplications. Also, he shows how FFT(n1*n2) may be computed via FFT(n1) and FFT(n2) in mult(n1)*mult(n2) multiplications, IF n1,n2 relatively prime, and also gives methods for FFT(prime power). He gives two appendices containing optimized CC(2..6) and FFT(2..9) algorithms. Winograd's methods don't appear suitable for general N, but if N is specified in advance, they make it possible to do considerable fine tuning at the expense of large algorithm complexity.

Meanwhile, Nussbaumer [H.J.Nussbaumer: Fast Polynomial Transform algorithms for digital convolutions, IEEE Transactions on Audio, Speech, Signal Processing 28,2 (April 1980) 205-215 (this article has many references to other FFT schemes); see also Knuth 2: 503, 652-653] has found another way to do circular convolutions of arrays of (N=a power of 2) reals without any NTTs, FFTs, trig, or complex numbers. His approach is based on viewing circular convolutions as polynomial multiplications modulo certain simple polynomials, factoring the modular polynomials, divide and conquer, chinese remainder thm. His approach uses roughly NlgN *, NlgNlglgN +, is fairly complicated to program, and requires extra space.

[Nussbaumer&Quandalle: IBM JResDev 22 (1978) 134-144] show how some particularly efficient CC and FFT schemes for N in the range 10-3000 may be constructed; their approach is based on some novel ways to combine efficient small CC schemes that is rather like the NTT (Number theoretic transform) only in rings of polynomials rather than in the integers.

However even the best known arithmetic op count methods only improve on my method by perhaps 30%, and at the cost of considerable complexity. C.H.Papadimitriou [Optimality of the FFT, JACM 26 (1979) 95-102 and its refs] has shown that in some models of computation O(NlgN) is optimal for the FFT, while Patterson et al have shown a lower bound of O(NlgN/lglgN) for integer multiplication on multitape Turing machines, see Knuth 2.

FFTs in a finite field (if the ring ZmodK, called number theoretic transforms) are discussed in Aho,Hopcoft,Ullman: The Design and Analysis of Computer algorithms, Addison-Wesley 1974. They recommend using W=2, N=a power of 2, do all arithmetic in the ring of integers modulo 2^(N/2)+1 (in which W is an Nth root of unity, and in which the convolution theorem C[i]=A[i]B[i] <==> c[i] = sum(j=0..N-1)of a[j]*b[i-j mod N] [Which makes possible the calculation of discrete convolutions in NlgN time] still holds). [See also R.Agrawal&C.Burrus: NTTs to implement fast digital convolutions, ProcIEEE 63 (1975) 550; articles by H.Nussbaumer on Fermat and Mersenne transforms, IBMJR&D 21 (1976) 282 and 498.] These FFFFTs are of use in applications where it is desirable to completely eliminate roundoff error and floating point operations, e.g. all-integer convolutions. However, as you can see, NTTs have severe word length and transform length limitations; the need for high precision modular arithmetic can be a major stumbling block. On the other hand, multiplications by W=2 are easy, while modular arithmetic modulo a Fermat number is not that hard. Thus using N=16, modulo 65537 arithmetic, W=2 [left shift and modulo], Winverse=32769 [right shift; modular addition correction if inexact], and all numbers in 0..89 allows computation of CC(16) in 16*, many bit shifts and additions.

Rabiner,Schafer,Rader: The Chirp-Z transform and its Applications, BSTJ 48,3 (1969) 1249-1292 show how DFTs (for any N) may be calculated in NlgN time by using fast convolutions; the method also works for an extension of FFTs (to W=any complex number, not just the principal Nth root of unity): A[k] = sum(j=0..N-1)of a[j]*W^(k*j) for k=0..N-1 may be calculated in NlgN time by a fast convolution by the Chirp-Z transform identity A[k] = W^(k*k/2) * sum(j=0..N-1)of { W^(-((j-k)^2)/2) * W^(j*j/2)*b[j] }. Aho,Stieglitz,Ullman: Evaluating Polynomials at fixed sets of points, SIAMJComp 4,4 (Dec 1975) 533-539, demonstrate that a polynomial and all its derivatives at one point (or equivalently, an origin shift of an Nth degree polynomial) may be calculated in NlgN time by a fast convolution via the (binomial theorem) identity sum(j=0..N-1)of c[j]*(x+q)^j = sum(r=0..N-1)of x^r * d[r]/r! where d[r] = sum(j=r..N-1)of c[j]*j! * q^(j-r)/(j-r)! .

Some other applications of FFTs are:

Fast multiplication and division of N digit integers may be done in (roughly) NlgN time by using fast convolutions followed by a carry step. (See Aho-Hopcoft-Ullman: Design and Analysis of Computer algorithms, for further discussion.)

Base conversion of an N digit number may be done in N(lgN)^2 time by divide and conquer (convert the left and right half of the number recursively, then do a fast multiplication and addition to combine them).

Fast polynomial multiplication and division by fast convolutions in NlgN time are also discussed in AHU. (This may also be done for Chebyshev series...) Given the N roots of a polynomial, you can find its coefficients (as Chebyshev or as regular) in N(lgN)^2 time by fast polynomial multiplications on a binary tree. On the other hand, you can perform a root squaring transformation on a polynomial in (Chebyshev or power form) P(y) = -P(x)*P(-x) , y = x^2 in NlgN time by a fast multiplication, or alternatively can implement a Henrici-Gargantini or Korsak-Pease (or other) simultaneous all root iteration step, in N(lgN)^2 time by a fast multipoint evaluator, see below.

All shifted correlations of vectors (and/or autocorrelations) may be calculated in NlgN time by fast convolutions; this has application in signal processing, 1D pattern recognition, Electrical engineering.

Fast polynomial multiplication/division/remaindering and a divide and conquering of the Lagrange interpolation formula may be used to do fast Nth degree polynomial interpolation and N-point evaluation, as was shown by Borodin&Moenck [JCompSystSci 1974]. I have extended B&M's results to Chebyshev polynomials and less successfully to other polynomials.

Fast algorithms exist for power-series to continued fraction interconversion; these may also be generalized to Chebyshev series.

Fast polynomial evaluation/interpolation at special point sets (e.g. Z^k for some Z) may be accomplished in NlgN time by the Chirp-Z and FF transforms; this also carries over to Chebyshev. Thus fast Taylor and Chebyshev series calculations.

Fast composition of Taylor series - O((NlgN)^(3/2)) is also possible, as was discovered by Brent&Kung, via a block Horner approach. This may also be extended to Chebyshev.

Fast Elliptic linear PDE solvers (by finite differences or spectrally): there are many schemes based on FFTs that run in NlgN time, N=size of output.

A complete list of FFT applications is far too huge to discuss here... ***********************************************************************

CONST TrigTabSize = 40;

VAR                              (* CONST after initialization: *)
  (* cos( pi / 2^k ): *)
  PrecomputedCos: ARRAY [0 .. TrigTabSize - 1] OF R.T;

  (* sin( pi / 2^k ): *)
  PrecomputedSin: ARRAY [0 .. TrigTabSize - 1] OF R.T;
********************************************************** Uses repeated application of the bisection identities cos(t/2)=sqrt(0.5*(1.0+cos(t))), sin(t/2)=0.5*sin(t)/cos(t/2), starting from the values with t=pi and t=pi/2 as special cases. We see no reason to trust trig routines sin() and cos() although we will trust sqrt() and division. *************************************************************
PROCEDURE PreComputeTrigTables () =
    PrecomputedCos[0] := R.MinusOne;
    PrecomputedCos[1] := R.Zero;
    PrecomputedSin[0] := R.Zero;
    PrecomputedSin[1] := R.One;
    FOR k := 2 TO TrigTabSize - 1 DO
      PrecomputedCos[k] :=
        RT.SqRt(RT.Half * (R.One + PrecomputedCos[k - 1]));
      PrecomputedSin[k] :=
        RT.Half * PrecomputedSin[k - 1] / PrecomputedCos[k];
  END PreComputeTrigTables;
************************************************ Reorders array a[0..n-1] so that element[i] is swapped with element[reverse-bit-order[i]]. Two successive calls are identity. The algorithm below runs in O(n) time and is based on implementing a reverse-binary counter. The forward counter f increments 0..n/2-1 step 2 and if r, 0<=r<n/2 is the bit-reverse of f, we swap(f+1,n/2+r) with no test needed and we do swap(f,r) if r>f and also swap(n-1-f,n-1-r) if f,r both <n/2. **************************************************
  (* a[] overwritten by permutation. *)
  <* INLINE *>
  PROCEDURE Swap (x, y: CARDINAL; ) =
    VAR tmp: C.T;
      tmp := a[x];
      a[x] := a[y];
      a[y] := tmp;
    END Swap;
    r    : CARDINAL;
    j    : Word.T;
    n    : CARDINAL := NUMBER(a);
    nb2  : CARDINAL := n DIV 2;
    nb2m1: CARDINAL := nb2 - 1;
    nb4  : CARDINAL := n DIV 4;
    nm1  : CARDINAL := n - 1;
    <* ASSERT n > 0 *>
    <* ASSERT Word.And(n - 1, n) = 0 *>
    (** n must be a power of 2. **)
    r := 0;
    FOR f := 0 TO nb2m1 BY 2 DO
      <* ASSERT f + 1 < nb2 + r *>
      Swap(f + 1, nb2 + r);
      IF f < r THEN
        Swap(f, r);
        IF r < nb2 THEN Swap(nm1 - f, nm1 - r); END;
      (** increment the reverse binary counter r;
       * while loop executes O(1) times on average: *)
      j := nb4;
      WHILE Word.And(r, j) # 0 DO j := Word.RightShift(j, 1); END;
      r := Word.And(r + j + Word.LeftShift(j, 1), nb2m1);
  END ReOrder;
************************************** direction = +1 for inverse FFT, -1 for forward. (See FFT defn above.) a[] overwritten by transform. NOTE: You must call ReOrder(a) before calling this routine, because this routine assumes it has re-ordered a[]s as its input. To do an FFT of some data, therefore, we would call ReOrder(data); FFTwithWrongOrderedInput(data); I have separated the routines this way because I want to be able to avoid the ReOrder when computing convolutions and correlations. ***************************************
PROCEDURE FFTwithWrongOrderedInput
  (VAR a: ARRAY OF C.T; direction: [-1 .. 1]; ) =
    n                         : CARDINAL := NUMBER(a);
    nm1                       : CARDINAL := n - 1;
    ur, ui, wr, wi, tr, ti, zz: R.T;
    k, j, L2, L, ip           : CARDINAL;
    dir                                  := FLOAT(direction, R.T);
    <* ASSERT direction # 0 *>
    <* ASSERT n > 0 *>
    <* ASSERT Word.And(n - 1, n) = 0 *>
    (** n must be a power of 2 and n>=1. **)

    (* Now for FFT main loop *)
    L := 1;
    k := 0;
    WHILE L < n DO
      L2 := L + L;
      ur := R.One;
      ui := R.Zero;
      wr := PrecomputedCos[k];
      zz := PrecomputedSin[k] * dir;
      wi := -zz;
      zz := zz + zz;
      j := 0;
        FOR i := j TO nm1 BY L2 DO
          (** Press et al. refer to below as the "Danielson-Lanzcos
           * formula", since it was published by them in 1942 - much prior
           * to Cooley & Tukey. Typically, despite the fact that it was
           * known to D&L and even to Gauss (! see Goldstine: History of
           * Numerical Analysis), I'm told the FFT has been
           * patented. Of course, we pay no attention to this "patent". *)
          ip := i + L;
          tr := a[ip].re * ur - a[ip].im * ui;
          ti := a[ip].im * ur + a[ip].re * ui;
          a[ip].re := a[i].re - tr;
          a[ip].im := a[i].im - ti;
          a[i].re := a[i].re + tr;
          a[i].im := a[i].im + ti;
        IF j >= L THEN EXIT; END;
        (* And here is my superior recurrence to calculate the trig: *)
        tr := ur;
        ti := ui;
        ur := wr - zz * ti;
        ui := wi + zz * tr;
        wr := tr;
        wi := ti;
      L := L2;
  END FFTwithWrongOrderedInput;
*********************************************************** Given two real vectors x[0..N-1] and y[0..N-1], the scaled circular convolution z[0..] is defined by z[j] = scale * SUM(k=0..N-1)of x[j-k] * y[k]. If N is a power of 2, the below routine will compute the circular convolution of x[] and y[] where it is assumed that x[] had been copied into the real, and y[] into the imaginary, parts of complex array a[], on input. On output, z will occupy the real part of a[]. A non-circular convolution is got by use of the same routine, but with the large-indexed part of the x[] and y[] arrays zeroed so that the wraparound terms are all 0. To compute correlations, you can do a convolution with the y[] array in reverse order. *********************************** PROCEDURE CircularConvolution(a: ARRAY OF C.T; scale:R.T;) NOT IMPLEMENTED YET ********************************

************************************************** Slow FT routine, useful for debugging fast one. b[k] = sum from m=0 to N-1 of a[m]*exp(2 pi i direction m k/N) where direction = +1 or -1. ****************************************************

PROCEDURE SlowFT (READONLY a: ARRAY OF C.T; direction: [-1 .. 1]; ):
    n        := NUMBER(a);
    b        := NEW(REF ARRAY OF C.T, n);
    sum: C.T;
    dir      := FLOAT(direction, R.T);
    kn : R.T;
    <* ASSERT direction # 0 *>
    FOR k := 0 TO n - 1 DO
      sum := C.Zero;
      kn := dir * RT.TwoPi * FLOAT(k, R.T) / FLOAT(n, R.T);
      FOR m := 0 TO n - 1 DO
        sum := C.Add(sum, C.Mul(a[m], CT.ExpI(kn * FLOAT(m, R.T))));
      b[k] := sum;
    RETURN b;
  END SlowFT;
*** Test driver. ***
    a           := NEW(REF ARRAY OF C.T, 128);
    b           := NEW(REF ARRAY OF C.T, 128);
    n           := NUMBER(a^);
    x: CARDINAL;
    <* ASSERT NUMBER(b^) = n *>

    (* initialize a[] to psu-random complex numbers... *)
    x := 432531;
    FOR j := LAST(a^) TO FIRST(a^) BY -1 DO
      x := x * 57 MOD 1048583;   (* 57 is generator, mod this prime *)
      a[j].re := FLOAT(x, R.T);
      x := x * 57 MOD 1048583;   (* 57 is generator, mod this prime *)
      a[j].im := FLOAT(x, R.T);

    (* make copy b of a: *)
    b^ := a^;
    (* check reordering twice yields identity: *)
    FOR j := LAST(a^) TO FIRST(a^) BY -1 DO
      <* ASSERT CT.Norm1(C.Sub(a[j], b[j])) < FLOAT(0.000000001D0, R.T) *>

    (* forward transform of 'a' in place: *)
    FFTwithWrongOrderedInput(a^, 1);

    VAR c := SlowFT(b^, 1);

      (* check slow and fast give same result: *)
      FOR j := LAST(a^) TO FIRST(a^) BY -1 DO
        <* ASSERT CT.Norm1(C.Sub(a[j], c[j])) < FLOAT(0.0001D0, R.T) *>

    (* backward: *)
    FFTwithWrongOrderedInput(a^, -1);
    FOR j := LAST(a^) TO FIRST(a^) BY -1 DO
      a[j] := C.Scale(a[j], R.One / FLOAT(n, R.T));

    (* check get original back: *)
    FOR j := LAST(a^) TO FIRST(a^) BY -1 DO
      <* ASSERT CT.Norm1(C.Sub(a[j], b[j])) < FLOAT(0.0000001D0, R.T) *>
  END Test;

END FourierTransform.