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Programs for Circular Convolution

Module by: Ivan Selesnick, C. Sidney Burrus. E-mail the authors

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Programs for Circular Convolution

To write a program that computes the circular convolution of hh and xx using the bilinear form Equation 24 in Bilinear Forms for Circular Convolution we need subprograms that carry out the action of PP, PtPt, RR, RtRt, AA and BtBt. We are assuming, as is usually done, that hh is fixed and known so that u=CtR-tPJhu=CtR-tPJh can be pre-computed and stored. To compute these multiplicative constants uu we need additional subprograms to carry out the action of CtCt and R-tR-t but the efficiency with which we compute uu is unimportant since this is done beforehand and uu is stored.

In Prime Factor Permutations we discussed the permutation PP and a program for it pfp() appears in the appendix. The reduction operations RR, RtRt and R-tR-t we have described in Reduction Operations and programs for these reduction operations KRED() etc, also appear in the appendix. To carry out the operation of AA and BtBt we need to be able to carry out the action of Ad1⊗⋯⊗AdkAd1⊗⋯⊗Adk and this was discussed in Implementing Kronecker Products Efficiently. Note that since AA and BtBt are block diagonal, each diagonal block can be done separately. However, since they are rectangular, it is necessary to be careful so that the correct indexing is used.

To facilitate the discussion of the programs we generate, it is useful to consider an example. Take as an example the 45 point circular convolution algorithm listed in the appendix. From Equation 19 from Bilinear Forms for Circular Convolution we find that we need to compute x=P9,5xx=P9,5x and x=R9,5xx=R9,5x. These are the first two commands in the program.

We noted above that bilinear forms for linear convolution, (Dd,Ed,Fd)(Dd,Ed,Fd), can be used for these cyclotomic convolutions. Specifically we can take Api=Dφ(pi)Api=Dφ(pi), Bpi=Eφ(pi)Bpi=Eφ(pi) and Cpi=GpiFφ(pi)Cpi=GpiFφ(pi). In this case Equation 20 in Bilinear Forms for Circular Convolution becomes

A = 1 ⊕ D 2 ⊕ D 6 ⊕ D 4 ⊕ ( D 2 ⊗ D 4 ) ⊕ ( D 6 ⊗ D 4 ) . A = 1 ⊕ D 2 ⊕ D 6 ⊕ D 4 ⊕ ( D 2 ⊗ D 4 ) ⊕ ( D 6 ⊗ D 4 ) . (1)

In our approach this is what we have done. When we use the bilinear forms for convolution given in the appendix, for which D4=D2⊗D2D4=D2⊗D2 and D6=D2⊗D3D6=D2⊗D3, we get

A = 1 ⊕ D 2 ⊕ ( D 2 ⊗ D 3 ) ⊕ ( D 2 ⊗ D 2 ) ⊕ ( D 2 ⊗ D 2 ⊗ D 2 ) ⊕ ( D 2 ⊗ D 3 ⊗ D 2 ⊗ D 2 ) A = 1 ⊕ D 2 ⊕ ( D 2 ⊗ D 3 ) ⊕ ( D 2 ⊗ D 2 ) ⊕ ( D 2 ⊗ D 2 ⊗ D 2 ) ⊕ ( D 2 ⊗ D 3 ⊗ D 2 ⊗ D 2 ) (2)

and since Ed=DdEd=Dd for the linear convolution algorithms listed in the appendix, we get

B = 1 ⊕ D 2 t ⊕ ( D 2 t ⊗ D 3 t ) ⊕ ( D 2 t ⊗ D 2 t ) ⊕ ( D 2 t ⊗ D 2 t ⊗ D 2 t ) ⊕ ( D 2 t ⊗ D 3 t ⊗ D 2 t ⊗ D 2 t ) . B = 1 ⊕ D 2 t ⊕ ( D 2 t ⊗ D 3 t ) ⊕ ( D 2 t ⊗ D 2 t ) ⊕ ( D 2 t ⊗ D 2 t ⊗ D 2 t ) ⊕ ( D 2 t ⊗ D 3 t ⊗ D 2 t ⊗ D 2 t ) . (3)

From the discussion above, we found that the Kronecker products like D2⊗D2⊗D2D2⊗D2⊗D2 appearing in these expressions are best carried out by factoring the product in to factors of the form Ia⊗D2⊗IbIa⊗D2⊗Ib. Therefore we need a program to carry out (Ia⊗D2⊗Ib)x(Ia⊗D2⊗Ib)x and (Ia⊗D3⊗Ib)x(Ia⊗D3⊗Ib)x. These function are called ID2I(a,b,x) and ID3I(a,b,x) and are listed in the appendix. The transposed form, (Ia⊗D2t⊗Ib)x(Ia⊗D2t⊗Ib)x, is called ID2tI(a,b,x) .

To compute the multiplicative constants we need CtCt. Using Cpi=GpiFφ(pi)Cpi=GpiFφ(pi) we get

C t = 1 ⊕ F 2 t G 3 t ⊕ F 6 t G 9 t ⊕ F 4 t G 5 t ⊕ ( F 2 t G 3 t ⊗ F 4 t G 5 t ) ⊕ ( F 6 t G 9 t ⊗ F 4 t G 5 t ) = 1 ⊕ F 2 t G 3 t ⊕ F 6 t G 9 t ⊕ F 4 t G 5 t ⊕ ( F 2 t ⊗ F 4 t ) ( G 3 t ⊗ G 5 t ) ⊕ ( F 6 t ⊗ F 4 t ) ( G 9 t ⊗ G 5 t ) . C t = 1 ⊕ F 2 t G 3 t ⊕ F 6 t G 9 t ⊕ F 4 t G 5 t ⊕ ( F 2 t G 3 t ⊗ F 4 t G 5 t ) ⊕ ( F 6 t G 9 t ⊗ F 4 t G 5 t ) = 1 ⊕ F 2 t G 3 t ⊕ F 6 t G 9 t ⊕ F 4 t G 5 t ⊕ ( F 2 t ⊗ F 4 t ) ( G 3 t ⊗ G 5 t ) ⊕ ( F 6 t ⊗ F 4 t ) ( G 9 t ⊗ G 5 t ) . (4)

The Matlab function KFt carries out the operation Fd1⊗⋯FdKFd1⊗⋯FdK. The Matlab function Kcrot implements the operation Gp1e1⊗⋯GpKeKGp1e1⊗⋯GpKeK. They are both listed in the appendix.

Common Functions

By recognizing that the convolution algorithms for different lengths share a lot of the same computations, it is possible to write a set of programs that take advantage of this. The programs we have generated call functions from a relatives small set. Each program calls these functions with different arguments, in differing orders, and a different number of times. By organizing the program structure in a modular way, we are able to generate relatively compact code for a wide variety of lengths.

In the appendix we have listed code for the following functions, from which we create circular convolution algorithms. In the next section we generate FFT programs using this same set of functions.

  • Prime Factor Permutations: The Matlab function pfp implements this permutation of Prime Factor Permutations. Its transpose is implemented by pfpt .
  • Reduction Operations: The Matlab function KRED implements the reduction operations of Reduction Operations. Its transpose is implemented by tKRED . Its inverse transpose is implemented by itKRED and this function is used only for computing the multiplicative constants.
  • Linear Convolution Operations: ID2I and ID3I are Matlab functions for the operations I⊗D2⊗II⊗D2⊗I and I⊗D3⊗II⊗D3⊗I. These linear convolution operations are also described in the appendix `Bilinear Forms for Linear Convolution.' ID2tI and ID3tI implement the transposes, I⊗D2t⊗II⊗D2t⊗I and I⊗D3t⊗II⊗D3t⊗I.

Operation Counts

Table 1 lists operation counts for some of the circular convolution algorithms we have generated. The operation counts do not include any arithmetic operations involved in the index variable or loops. They include only the arithmetic operations that involve the data sequence xx in the convolution of xx and hh.

The table in [2] for the split nesting algorithm gives very similar arithmetic operation counts. For all lengths not divisible by 9, the algorithms we have developed use the same number of multiplications and the same number or fewer additions. For lengths which are divisible by 9, the algorithms described in [2] require fewer additions than do ours. This is because the algorithms whose operation counts are tabulated in the table in [2] use a special Φ9(s)Φ9(s) convolution algorithm. It should be noted, however, that the efficient Φ9(s)Φ9(s) convolution algorithm of [2] is not constructed from smaller algorithms using the Kronecker product, as is ours. As we have discussed above, the use of the Kronecker product facilitates adaptation to special computer architectures and yields a very compact program with function calls to a small set of functions.

Table 1: Operation counts for split nesting circular convolution algorithms
N muls adds   N muls adds   N muls adds   N muls adds
2 2 4   24 56 244   80 410 1546   240 1640 6508
3 4 11   27 94 485   84 320 1712   252 1520 7920
4 5 15   28 80 416   90 380 1858   270 1880 9074
5 10 31   30 80 386   105 640 2881   280 2240 9516
6 8 34   35 160 707   108 470 2546   315 3040 13383
7 16 71   36 95 493   112 656 2756   336 2624 11132
8 14 46   40 140 568   120 560 2444   360 2660 11392
9 19 82   42 128 718   126 608 3378   378 3008 16438
10 20 82   45 190 839   135 940 4267   420 3200 14704
12 20 92   48 164 656   140 800 3728   432 3854 16430
14 32 170   54 188 1078   144 779 3277   504 4256 19740
15 40 163   56 224 1052   168 896 4276   540 4700 21508
16 41 135   60 200 952   180 950 4466   560 6560 25412
18 38 200   63 304 1563   189 1504 7841   630 6080 28026
20 50 214   70 320 1554   210 1280 6182   720 7790 30374
21 64 317   72 266 1250   216 1316 6328   756 7520 38144

It is possible to make further improvements to the operation counts given in Table 1[1], [2]. Specifically, algorithms for prime power cyclotomic convolution based on the polynomial transform, although more complicated, will give improvements for the longer lengths listed [1], [2]. These improvements can be easily included in the code generating program we have developed.

References

  1. Nussbaumer, H. J. (1980, April). Fast Polynomial Transform Algorithms for Digital Convolution. IEEE Trans. Acoust., Speech, Signal Proc., 28(2), 205-215.
  2. Nussbaumer, H. J. (1982). Fast Fourier Transform and Convolution Algorithms. Sringer-Verlag.

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