Low-complexity rotators for the FFT using base-3 signed stages

Abstract

Abstract-Rotations by angles that are fractions of the unit circle find applications in e.g. fast Fourier transform (FFT) architectures. In this work we propose a new rotator that consists of a series of stages. Each stage calculates a micro-rotation by an angle corresponding to a power-of-three fractional parts. Using a continuous powers-of-three range, it is possible to carry out all rotations required. In addition, the proposed rotators are compared to previous approaches, based of shift-and-add algorithms, showing improvements in accuracy and number of adders.

Low-Complexity Multiplierless Constant Rotators Based on Combined Coefficient Selection and Shift-and-Add Implementation (CCSSI)

“…The kernel achieves a precision of 22.69 correct bits by using 10 adders, as shown in Table VII. 3) Comparison: Figures 13 and 14 compare the proposed rotators from Table VII with other multiplierless rotators for W 16 and W 32 in the literature. The previous approaches include rotators based on MCM [5] 1 , Booth encoding [25], trigonometric identities [7], [24], base-3 rotators [27], MSR-CORDIC [23] 2 and non-redundant CORDIC [13]. The number of adders in the figures are for rotations in the range [0, π/4].…”

“…This is done in [27], where all the rotations are generated by combining a small set of FFT angles. This set fits the rotation angles of the FFT better than that of the CORDIC, which results in a reduction in the rotation error, number of adders and latency of the circuit.…”

“…Id. [24] Base−3 [27] CSD−based [4] MSR−CORDIC [23] CORDIC [13] Proposed (Table V R. Booth [25] Trig. Id.…”

“…Likewise, the CORDIC elementary angles have been used since the CORDIC algorithm was proposed half a century ago, without questioning if another selection of angles might provide better results. Now, there are results that demonstrate the existence of better angle sets than the CORDIC one for the FFT rotations [27].…”

“…CORDIC-based approaches for constant rotators [18]- [23] are based on selecting stages of the conventional CORDIC [18]- [20] or increasing the amount of micro-rotation angles [21]- [23]. Multiplier-based approaches for constant rotators [4]- [7], [24]- [27] base on techniques to optimize realvalued constant multiplications [4]- [6], trigonometric identities [7], [24], optimization of the coefficient encoding [25], [26], and angle generation by a base-3 system [27].…”