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Based on the quadratic residue number system (QRNS), a novel optical matrix-vector multiplication is proposed. Complex number multiplication has higher parallelism than other algorithms. If the residue number is binary encoded, this multiplication can be easily implemented with a multichannel incoherent optical correlator. © 1996 John Wiley & Sons, Inc.

In this paper, we consider strictly proper digital filters with hypercomplex coefficients, and we also consider reduction of multiplications in hypercomplex algebra.As for the former problem, the first-order digital filter with hypercomplex coefficients can realize arbitrary-order strictly proper digital filters with real coefficients less than 4, whose order of numerator is smaller than that of denominator by 1.As for the latter problem, if we apply Quadratic Residue Number System to hypercomplex algebra, we can reduce the number of multiplications to 1/4 of the original one. And the most important thing of this result is that we only need 4 multiplications per multiplier and that these 4 multiplications can be done at the same time.

Algorithms and architectures are described for optical matrix–matrix and matrix–vector processing. Implementations are based on quadratic residue number system coding which has the remarkable property that computations in the real and imaginary channels are performed in parallel without any interactions. This approach significantly reduces the complexity of implementation. Residue-based processing provides increased dynamic range and a reduced number of computations with high speed and parallelism of the optics fully utilized. Several architectures of optical processors described in this paper include systolic and data flow arrays.

In this work, we revisit the implementation of polyphase filter banks in Quadratic Residue Number System (QRNS) for banks with a large number of channels by developing a new design methodology suitable for large systems required in the new generation of satellites. Furthermore, we compare the QRNS filter bank with an equivalent bank implemented in the traditional Complex Two's Complement System (CTCS) in terms of throughput, area and power dissipation. The results for large filter banks confirm the earnings in power consumption by using the QRNS.

Recursive and nonrecursive digital filters have been implemented using the proposed single-multiplexed dual-clock computational module (SDCM) in the bit-slice architecture. The amount of memory requirements has been reduced to 50% required for the quadratic-residue-number system (QRNS) or modified-QRNS filter architecture. These filter architectures are suitable for single-chip VLSI implementation of complex digital filters. Though the data rate is reduced by 50%, VLSI implementation of the filter architectures using the RNS principle is certainly possible in this approach. >

A new approach for processing complex numbers has been proposed, Basically, it is aimed at the implementation of radix-3 FFTs, but it can be used in any situation where the requirement to process Eisenstein integers arises, The comparison between the quadratic residue number system and ERNS provides information on available primes and dynamic ranges for the two number systems, This information can be used to decide on which number system is more appropriate for specific processing requirements.

The residue number system (RNS) is a useful tool for digital signal processing (DSP) since it can support parallel, carry-free, high-speed arithmetic. An RNS is defined by a set of relatively prime integers called the moduli set. The most important consideration when designing RNS systems is the choice of the moduli set. In order to maintain simple arithmetic, moduli sets with numbers of forms 2/sup n1/+1, 2/sup n2/-1 and 2/sup n3/ have been given serious consideration by RNS researchers in the past. The most popular such sets are the 3-moduli set {2/sup n/-1, 2/sup n/, 2/sup n/+1} and the 4-moduli set {2/sup n-1/-1, 2/sup n/-1, 2/sup n+1/-1, 2/sup n+1/} for even n. However, if very large dynamic ranges are desirable, the number of moduli should increase so that performance degradation is avoided. New multi-moduli RNS systems are presented that are based on moduli of forms 2/sup n1/+1, 2/sup n2/-1, 2/sup n3/. For a given dynamic range, these new systems exhibit performance of 2 to 3 times better than the performance of the known 3-moduli and 4-moduli systems.

The polynomial residue number system (PRNS) is a system in which the product of two polynomials can take place in parallel and with the minimum number of multiplications. The system is an extension of the quadratic residue number system (QRNS) which has been successfully used in complex digital signal processing. It is shown that an N-point complex linear convolution can be computed with 4N real multiplications when using the PRNS instead of 22 real multiplications when using the QRNS. The savings in the number of multiplications occur if some restrictions are placed on the modular ring used for performing the complex residue number system operations. >

A generic digital beamformer architecture that has been implemented to demonstrate the advantages of digital beamforming is described. The digital beamforming processor has been implemented using quadratic residue number system techniques to enhance performance. The custom processor chips are implemented in 1.2- mu m CMOS technology. A digital beamformer can provide significantly better sidelobe performance than an RF beamformer. This is shown for a 64-channel system having only 8-bit weights (I and Q) and 9-bit signals (I and Q). Sidelobe levels can be suppressed 45 dB or better, Practical system implementation considerations are addressed. >

The quadratic residue number system (QRNS) has been proven to be a system in which the complex multiplication can take place in a completely parallel fashion using only two real multiplications instead of four. The Letter shows how the QRNS can be used in computing skew cyclic convolutions.

Recently, the Quadratic Residue Number System ,(QRNS) has been introduced [5], [6], which allows the multiplication of complex integers with two real multiplications. The restriction is that this special residue number system has all moduli with prime factors of the form 4 k + 1 . If an increase in real multiplications from two to three can be tolerated, the restriction can be lifted to allow moduli of any form.