**7. Complexity comparison**

Table 1 presents the total hardware requirements and conversion delays of the reverse converters for the large dynamic range four-moduli sets in terms of logic gates and FAs. Note that *A*FA and *D*FA indicate the area and delay of one FA, respectively. It can be seen that the fastest converter is the converter for moduli set {2*n*–1, 2*n*+1, 2*2n*, 2*2n*+1}. Because, the dynamic range of this set is 6*n*-bit while the dynamic range of moduli set {2*n*–1, 2*n*, 2*n*+1, 2*2n*+1} is 5*n*-bit. Therefore, for providing the same dynamic range, the value of *n* for the first


Table 1. Hardware requirements and conversion delays of the reverse converters for the large dynamic range four-moduli sets

has been done to design efficient reverse converters. This chapter presents a study on the state-of-the-art reverse converters which have been designed for the recently introduced large dynamic range RNS four-moduli sets. We provide an overview about different reverse conversion algorithms, the recent four-moduli sets, and the reverse converter architectures.

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set is smaller than the second set. Furthermore, the reverse converter for the moduli set {2*n*– 1, 2*n*, 2*n*+1, 2*2n*+1–1} relies on less hardware requirements than others. From another side, the moduli sets {2*n*–1, 2*n*, 2*n*+1, 22*n*+1–1} and {2*n*–1, 2*n*+1, 2*2n*, 2*2n*+1–1} results in faster RNS arithmetic units than the moduli sets {2*n* –1,2*n*, 2*n* +1, 22*n* +1} and {2*n*–1, 2*n*+1, 22*n*, 22*n*+1}.

Fig. 4. The converter for moduli set {2*n*–1, 2*n*+1, 2*2n*, 2*2n*+1–1} (Molahosseini & Navi, 2010)

#### **8. Conclusion**

The Residue Number System has been recognized as one of the efficient alternative number systems which can be used to high-speed hardware implementation of Digital Signal Processing computation algorithms. However, forward and reverse converters are needed to act as interfaces between RNS and the conventional binary digital systems. The overhead of these converters can frustrate the speed efficiency of RNS, and due to this a lot of research has been done to design efficient reverse converters. This chapter presents a study on the state-of-the-art reverse converters which have been designed for the recently introduced large dynamic range RNS four-moduli sets. We provide an overview about different reverse conversion algorithms, the recent four-moduli sets, and the reverse converter architectures.
