Accurate Rotations Based on Coefficient Scaling

Abstract

Abstract-This brief presents a novel approach for improving the accuracy of rotations implemented by complex multipliers, based on scaling the complex coefficients that define these rotations. A method for obtaining the optimum coefficients that lead to the lowest error is proposed. This approach can be used to get more accurate rotations without increasing the coefficient wordlength as well as to reduce the wordlengh without increasing the rotation error.The paper analyzes two different situations where the optimization method can be applied: rotations that can be optimized independently and sets of rotations that require the same scaling. These cases appear in important signal processing algorithms such as the discrete cosine transform (DCT) and the fast Fourier transform (FFT). Experimental results show that the use of scaling for the coefficients clearly improves the accuracy of the algorithms. For instance, improvements of about 8 dB in the Frobenius norm of the FFT are achieved with respect to using non-scaled coefficients.

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

“…It approximates R cos α + jR sin α, which is the exact value of the rotation, scaled by the scaling factor R [2]. According to this, the normalized rotation error is According to (1), the angles of the FFT are given by…”

“…The values C i , S i and K i are obtained by the coefficient scaling method [2], optimized for the sets of angles {0, 2π N 3 M −i } radians, and modified to also calculate the number of adders for the kernels (where each kernel is a suggested set {K i , C i + jS i }). The calculated number of adders for a kernel is the greatest number used to implement any of the two angles of the kernel, and the other angle will then reuse the same adders by using multiplexers.…”

“…, N − 1} is the index of the rotation angle. One approach to implement the rotator is to store the twiddle factors in a ROM, and use a complex multiplier to carry out the rotation [2]. The complex multiplier can be implemented with three or four real multipliers [3].…”

“…Recently, a method based on scaling the coefficients in order to reduce the rotation error was presented in [2]. This approach uses complex multipliers, so it is not comparable with shiftand-add algorithms, but the described method is used in the proposed method to find efficient coefficients.…”

“…The base-3 structure uses, like the redundant CORDIC, the possibilities of a rotation by 0 • and, hence, it rotates either +α, 0, or −α. However, contrary to the redundant CORDIC, by using the coefficient scaling method [2], the scaling error can be considered in the selection of the coefficients. This assures a uniform scaling, i.e., the same gain for all angles.…”