0.5 Programs for circular convolution

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Programs for circular convolution

To write a program that computes the circular convolution of $h$ and $x$ using the bilinear form Equation 24 in Bilinear Forms for Circular Convolution we need subprograms that carry out the action of $P$ , ${P}^{t}$ , $R$ , ${R}^{t}$ , $A$ and ${B}^{t}$ . We are assuming, as is usually done, that $h$ is fixed and known so that $u={C}^{t}{R}^{-t}PJh$ can be pre-computed and stored. To compute these multiplicative constants $u$ we need additional subprograms to carry out the action of ${C}^{t}$ and ${R}^{-t}$ but the efficiency with which we compute $u$ is unimportant since this is done beforehand and $u$ is stored.

In Prime Factor Permutations we discussed the permutation $P$ and a program for it pfp() appears in the appendix. The reduction operations $R$ , ${R}^{t}$ and ${R}^{-t}$ 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 $A$ and ${B}^{t}$ we need to be able to carry out the action of ${A}_{{d}_{1}}\otimes \cdots \otimes {A}_{{d}_{k}}$ and this was discussed in Implementing Kronecker Products Efficiently . Note that since $A$ and ${B}^{t}$ 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 $x={P}_{9,5}x$ and $x={R}_{9,5}x$ . These are the first two commands in the program.

We noted above that bilinear forms for linear convolution, $\left({D}_{d},{E}_{d},{F}_{d}\right)$ , can be used for these cyclotomic convolutions.Specifically we can take ${A}_{{p}^{i}}={D}_{\phi \left({p}^{i}\right)}$ , ${B}_{{p}^{i}}={E}_{\phi \left({p}^{i}\right)}$ and ${C}_{{p}^{i}}={G}_{{p}^{i}}{F}_{\phi \left({p}^{i}\right)}$ . In this case Equation 20 in Bilinear Forms for Circular Convolution becomes

$A=1\oplus {D}_{2}\oplus {D}_{6}\oplus {D}_{4}\oplus \left({D}_{2}\otimes {D}_{4}\right)\oplus \left({D}_{6}\otimes {D}_{4}\right).$

In our approach this is what we have done. When we use the bilinear forms for convolution givenin the appendix, for which ${D}_{4}={D}_{2}\otimes {D}_{2}$ and ${D}_{6}={D}_{2}\otimes {D}_{3}$ , we get

$A=1\oplus {D}_{2}\oplus \left({D}_{2}\otimes {D}_{3}\right)\oplus \left({D}_{2}\otimes {D}_{2}\right)\oplus \left({D}_{2}\otimes {D}_{2}\otimes {D}_{2}\right)\oplus \left({D}_{2}\otimes {D}_{3}\otimes {D}_{2}\otimes {D}_{2}\right)$

and since ${E}_{d}={D}_{d}$ for the linear convolution algorithms listed in the appendix,we get

$B=1\oplus {D}_{2}^{t}\oplus \left({D}_{2}^{t}\otimes {D}_{3}^{t}\right)\oplus \left({D}_{2}^{t}\otimes {D}_{2}^{t}\right)\oplus \left({D}_{2}^{t}\otimes {D}_{2}^{t}\otimes {D}_{2}^{t}\right)\oplus \left({D}_{2}^{t}\otimes {D}_{3}^{t}\otimes {D}_{2}^{t}\otimes {D}_{2}^{t}\right).$

From the discussion above, we found that the Kronecker products like ${D}_{2}\otimes {D}_{2}\otimes {D}_{2}$ appearing in these expressions are best carried out by factoring the product in to factorsof the form ${I}_{a}\otimes {D}_{2}\otimes {I}_{b}$ . Therefore we need a program to carry out $\left({I}_{a}\otimes {D}_{2}\otimes {I}_{b}\right)x$ and $\left({I}_{a}\otimes {D}_{3}\otimes {I}_{b}\right)x$ . These function are called ID2I(a,b,x) and ID3I(a,b,x) and are listed in the appendix. The transposed form, $\left({I}_{a}\otimes {D}_{2}^{t}\otimes {I}_{b}\right)x$ , is called ID2tI(a,b,x) .

To compute the multiplicative constants we need ${C}^{t}$ . Using ${C}_{{p}^{i}}={G}_{{p}^{i}}{F}_{\phi \left({p}^{i}\right)}$ we get

$\begin{array}{ccc}\hfill {C}^{t}& =& 1\oplus {F}_{2}^{t}{G}_{3}^{t}\oplus {F}_{6}^{t}{G}_{9}^{t}\oplus {F}_{4}^{t}{G}_{5}^{t}\oplus \left({F}_{2}^{t}{G}_{3}^{t}\otimes {F}_{4}^{t}{G}_{5}^{t}\right)\oplus \left({F}_{6}^{t}{G}_{9}^{t}\otimes {F}_{4}^{t}{G}_{5}^{t}\right)\hfill \\ & =& 1\oplus {F}_{2}^{t}{G}_{3}^{t}\oplus {F}_{6}^{t}{G}_{9}^{t}\oplus {F}_{4}^{t}{G}_{5}^{t}\oplus \left({F}_{2}^{t}\otimes {F}_{4}^{t}\right)\left({G}_{3}^{t}\otimes {G}_{5}^{t}\right)\oplus \left({F}_{6}^{t}\otimes {F}_{4}^{t}\right)\left({G}_{9}^{t}\otimes {G}_{5}^{t}\right).\hfill \end{array}$

The Matlab function KFt carries out the operation ${F}_{{d}_{1}}\otimes \cdots {F}_{{d}_{K}}$ . The Matlab function Kcrot implements the operation ${G}_{{p}_{1}^{{e}_{1}}}\otimes \cdots {G}_{{p}_{K}^{{e}_{K}}}$ . 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 possibleto write a set of programs that take advantage of this. The programs we have generated call functions from a relativessmall 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.

• The Matlab function pfp implements this permutation of Prime Factor Permutations . Its transpose is implemented by pfpt .
• 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.
• ID2I and ID3I are Matlab functions for the operations $I\otimes {D}_{2}\otimes I$ and $I\otimes {D}_{3}\otimes I$ . These linear convolution operations are also described in the appendixBilinear Forms for Linear Convolution.' ID2tI and ID3tI` implement the transposes, $I\otimes {D}_{2}^{t}\otimes I$ and $I\otimes {D}_{3}^{t}\otimes I$ .

Operation counts

[link] 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 $x$ in the convolution of $x$ and $h$ .

The table in [link] 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 [link] require fewer additions than do ours. This is because the algorithms whose operation counts aretabulated in the table in [link] use a special ${\Phi }_{9}\left(s\right)$ convolution algorithm. It should be noted, however, that the efficient ${\Phi }_{9}\left(s\right)$ convolution algorithm of [link] 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 andyields a very compact program with function calls to a small set of functions.

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

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