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
$3.6\cdot 10^8$
2D DCTs of size 8×8 every second, with corresponding pixel rate of
$2.3\cdot 10^{10}$
pixels per second. The digital architecture is synthesized using 180 nm CMOS standard cells and shows a chip area of 7.41 mm
$^2$
. The CMOS design is predicted to operate at 893 MHz clock frequency, at a dynamic power consumption 13.22 mW/MHz
$\cdot$
V
$_{sup}^2$
.