Accurate Rotations Based on Coefficient Scaling

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 word length and to reduce the word length without increasing the rotation error. This brief 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 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.