On the Comparison of Different Number Systems in the Implementation of Complex FIR Filters

International audienceIn modern electronic systems, complex arithmetic computation plays an important role in the implementation of different Digital Signal Processing (DSP) and scientific computation algorithms [1], [2]. Most of the interest in complex signal processing is related to the implementation of wireless communication systems based on new concepts and architectures [3]. A very interesting tutorial paper on complex signal processing and its applications has been presented recently [4]. In [4], the importance of the use of complex signal processing in wireless communications systems has been shown. Regarding communication systems, one of the most critical computation to be implemented in hardware is complex FIR filtering. In fact, FIR filters are generally characterized by a high order (number of taps) to obtain sharp transition bands that, in case of high speed real time computation, require many resources and have high power dissipation. In particular, for complex FIR filters, the hardware complexity is mostly determined by the number of complex multipliers (i.e. each complex multiplication is actually implemented with four scalar multiplications). Different solutions have been proposed to lower the hardware complexity of the complex multiplication either at algorithmic level (Golub Rule) [5], or by using different number systems such as the Quadratic Residue Number System (QRNS) [6], [2] and the Quater-Imaginary Number System (QINS) [7]