Design of Reverse Converters for General RNS Moduli Sets $\{ 2^{k}, 2^{n}-1, 2^{n}+1, 2^{n+1}-1 \}$ and $\{ 2^{k}, 2^{n}-1, 2^{n}+1, 2^{n-1}-1 \}$ ($n$ even)

This paper presents the design methods of residue-to-binary (reverse) converters for two 4-moduli sets {2
^k
, 2
ⁿ
-1, 2
ⁿ
+1, 2
ⁿ⁺¹
-1} and {2
^k
, 2
ⁿ
-1, 2
ⁿ
+1, 2
^n-1
-1} for the pairs of positive integers n (n even) and arbitrary k > 0, which provide flexible dynamic ranges for the residue number system (RNS). These two moduli sets are generalizations of the 4-moduli sets {2
ⁿ
, 2
ⁿ
-1, 2
ⁿ
+1, 2
ⁿ⁺¹
-1} and {2
ⁿ
, 2
ⁿ
-1, 2
ⁿ
+1, 2
^n-1
-1} ( n even) with only a single parameter n. They facilitate selecting a moduli set which not only suits exactly the required dynamic range but also allows to balance delay disparity and hardware complexity between an even modulus and the odd moduli datapath channels. The new reverse converter for the moduli set {2
^k
, 2
ⁿ
-1, 2
ⁿ
+1, 2
ⁿ⁺¹
-1} not only enjoys slightly better performance than its known counterpart but also applies to arbitrary k. The reverse converter for the new moduli set {2
^k
, 2
ⁿ
-1, 2
ⁿ
+1, 2
^n-1
-1} is the first ever proposed. Synthesis results obtained for the 65 nm technology show that at least one of two versions of our converters is superior w.r.t. delay, power consumption, and area than its counterpart for the 4-moduli set {2
^k
, 2
ⁿ
-1, 2
ⁿ
+1, 2
ⁿ⁺¹
-1}, for all dynamic ranges from 14 to 64 bits.