Computation of 2D 8×8 DCT Based on the Loeffler Factorization Using Algebraic Integer Encoding

This paper proposes a computational method for 2D 8×8 DCT based on algebraic integers. The proposed algorithm is based on the Loeffler 1D DCT algorithm, and it is shown to operate with exact computation—i.e., error-free arithmetic—up to the final reconstruction step (FRS). The proposed algebraic integer architecture maintains error-free computations until an entire block of DCT coefficients having size 8×8 is computed, unlike algorithms in the literature which claim to be error-free but in fact introduce arithmetic errors between the column- and row-wise 1D DCT stages in a 2D DCT operation. Fast algorithms are proposed for the final reconstruction step employing two approaches, namely, the expansion factor and dyadic approximation. A digital architecture is also proposed for a particular FRS algorithm, and is implemented on an FPGA platform for on-chip verification. The FPGA implementation operates at 360 MHz, and is capable of a real-time throughput of <inline-formula><tex-math notation="LaTeX"> $3.6\cdot 10^8$</tex-math><alternatives><inline-graphic xlink:href="cintra-ieq4-2837755.gif"/></alternatives> </inline-formula> 2D DCTs of size 8×8 every second, with corresponding pixel rate of <inline-formula> <tex-math notation="LaTeX">$2.3\cdot 10^{10}$</tex-math><alternatives> <inline-graphic xlink:href="cintra-ieq6-2837755.gif"/></alternatives></inline-formula> pixels per second. The digital architecture is synthesized using 180 nm CMOS standard cells and shows a chip area of 7.41 mm<inline-formula> <tex-math notation="LaTeX">$^2$</tex-math><alternatives><inline-graphic xlink:href="cintra-ieq7-2837755.gif"/> </alternatives></inline-formula>. The CMOS design is predicted to operate at 893 MHz clock frequency, at a dynamic power consumption 13.22 mW/MHz <inline-formula><tex-math notation="LaTeX">$\cdot$</tex-math><alternatives> <inline-graphic xlink:href="cintra-ieq8-2837755.gif"/></alternatives></inline-formula> V<inline-formula> <tex-math notation="LaTeX">$_{sup}^2$</tex-math><alternatives><inline-graphic xlink:href="cintra-ieq9-2837755.gif"/> </alternatives></inline-formula>.

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