On modulus replication for residue arithmetic computations of complex inner products

Abstract

Residue Number Systems require the selection of ring moduli whose product is greater than the predicted dynamic range of the computation being performed. The restriction that the moduli be relatively prime usually limits the set of available moduli and hence the maximum dynamic range. This is particularly the case when small moduli are to be considered for efficient hardware implementation. Severe restrictions occur when algebraic constraints, such as those posed by the necessity to implement quadratic residue rings, are a factor. This paper presents a technique for coding weighted magnitude components (e.g. bits) of numbers directly into polynomial residue rings, such that repeated use may be made of the same set of moduli to effectively increase the dynamic range of the computation. This effectively limits the requirement for large sets of relatively prime moduli. For practical computations over quadratic residue rings, at least 6-bit moduli have to be considered; we show, in this paper, that 5-bit moduli can be effectively used for large dynamic range computations.

Fault tolerant Fermat Number Transform domain adaptive filters based on modulus replication RNS architectures

“…Recently an approach to achieving fault tolerance in adaptive filters was proposed in [4], where the Fermat Number Transform (FNT) was combined with the transform domain block LMS algorithm (BLMS) to achieve computationally efficient fault tolerance. This paper extends the concept previously published in [4] by developing a hybrid combination of Fermat Number Transform block processing and Modular Replication RNS (MRRNS) coding [5] that results in adaptive filters that are immune to a wide range of transient errors. The proposed technique uses MRRNS arithmetic to implement the Fermat Number Transform (FNT) with improved dynamic range.…”

“…From (4) and (5) we see that both the filter output and the weight update steps involve convolution and correlation operations. As discussed earlier these operations can be efficiently implemented using the FNT.…”

Fault tolerant complex FIR filter architectures using a redundant MRRNS

“…After a multiplication, this will generate a 2nd order polynomial in y and so, in general, we will need a third order polynomial factor in y for g(x, y ) . If, however, we restrict the moduli to the form 4k + 1 , then we can use the QRNS mapping which maps the first-order polynomial multiplication back to a first-order polynomial (i.e., exactly what happens in normal complex multiplication) [4]. The MRRNS mapping can now be used to advantage since we can replicate a very small number of different moduli and select them for this property.…”

“…We use the same procedure as previ- 34 Q ( x ) = 2 + 2x + 33x2 + 9 x 3 + Ox4, (4) and the final result Q( 8) = 6738 .…”

“…2). The construction uses parallel MRRNS channels in which we embed a quadratic residue number system (QRNS) mapping [4]. We use channels computing over S(257 x 17) in order to allow the use of a Fermat ALU [8].…”