Download Fast Fourier Transform and Convolution Algorithms by Henri J. Nussbaumer PDF

By Henri J. Nussbaumer

Within the first variation of this booklet, we coated in bankruptcy 6 and seven the functions to multidimensional convolutions and DFT's of the transforms which we now have brought, again in 1977, and known as polynomial transforms. because the booklet of the 1st variation of this e-book, a number of very important new advancements about the polynomial transforms have taken position, and now we have integrated, during this variation, a dialogue of the connection among DFT and convolution polynomial rework algorithms. This fabric is roofed in Appendix A, besides a presentation of recent convolution polynomial remodel algorithms and with the appliance of polynomial transforms to the computation of multidimensional cosine transforms. we've got chanced on that the fast convolution and polynomial product algorithms of Chap. three were used commonly. This caused us to incorporate, during this variation, numerous new one-dimensional and two-dimensional polynomial product algorithms that are indexed in Appendix B. for the reason that our e-book is getting used as a part of a number of graduate-level classes taught at a variety of universities, we have now further, to this version, a collection of difficulties which hide Chaps. 2 to eight. a few of these difficulties serve additionally to demonstrate a little analysis paintings on DFT and convolution algorithms. i'm indebted to Mrs A. Schlageter who ready the manuscript of this moment version. Lausanne HENRI J. NUSSBAUMER April 1982 Preface to the 1st variation This ebook offers in a unified manner many of the quickly algorithms which are used for the implementation of electronic filters and the assessment of discrete Fourier transforms.

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Extra resources for Fast Fourier Transform and Convolution Algorithms

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A cyclotomic polynomial P(z) = (Z9 - I)J(z3 - 1) = Z6 + Z3 + 1 can be viewed, for example, as a polynomial Pz(ZI) = zI + Zl + 1 in which Z3 is substituted to Zl. In these cases, the multidimensional approach can be refined by calculating the polynomial product modulo P(z) as a two-dimensional polynomial product modulo (ZN, - Zl), Pz(z\). With this method, a polynomial multiplication modulo (Z4 + 1) is implemented with 9 multiplications and 15 additions (Sect. 2) by computing a polynomial product modulo (ZZ - Zl) on polynomials defined modulo (zr + I).

This completes the proof ofthe theorem. Note that this theorem provides not only a lower bound on the number of general multiplications, but also a practical algorithm for achieving this lower bound. However, the bound concerns only the number of general multiplications, that is to say, the multiplications where the two factors depend on the data. The bound does not include multiplications by constant factors which occur in the reductions modulo (z - a,) and in the Chinese remainder reconstruction.

63) The reductions and the Chinese remainder operations always have the same structure, regardless of the particular numerical value of N •. , N2 ... , but can be defined by a single program structure. (z) and (z - 1) are computed with N. 65) Thus, the only part of each convolution algorithm which needs to be individually stored is that corresponding to the polynomial products. With this method, the savings in storage can be quite significant. If we consider for example a simple convolution of 15 terms, the calculation can be performed by nesting a convolution of 3 terms with a convolution oflength 5.

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