## Bilinear algorithms for discrete cosine transforms of prime lengths.(English)Zbl 1172.94342

Summary: This paper presents a strategy to design bilinear discrete cosine transform (DCT) algorithms of prime lengths. We show that by using multiplicative groups of integers, one can identify and arrange the computation as a pair of convolutions. When the DCT length $$p$$ is such that $$(p-1)/2$$ is odd, the computation uses two $$(p-1)/2$$ point cyclic convolutions. When $$(p-1)/2=2mq$$ with $$m>0$$ and $$q$$ odd, the computation requires one $$(p-1)/2$$ point cyclic convolution and a combination of a $$q$$ point cyclic convolution and a $$2m$$ point Hankel product. Using bilinear algorithms for convolutions and Hankel products, one gets a bilinear DCT algorithm. We also show that the additions required beyond the convolutions can be minimized by a small modification to the convolution algorithms. This minimization exploits the fact that efficient bilinear convolution algorithms are almost always based on Chinese Remainder Theorem.

### MSC:

 94A11 Application of orthogonal and other special functions 65T50 Numerical methods for discrete and fast Fourier transforms
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