Optimum Circuits for Bit-Dimension Permutations

Abstract

In this paper we presents a systematic approach to design hardware circuits for bit-dimension permutations. The proposed approach is based on decomposing any bit-dimension permutation into elementary bit-exchanges. Such decomposition is proven to achieve the theoretical minimum number of delays required for the permutation. This offers optimum solutions for multiple well-known problems in the literature that make use of bit-dimension permutations. This includes the design of permutation circuits for the fast Fourier transform (FFT), bit reversal, matrix transposition, stride permutations and Viterbi decoders.

Multiplexer and Memory-Efficient Circuits for Parallel Bit Reversal

“…The number of delays (D), the number of multiplexers (M ) and the latency (Lat) of circuits that calculate an elementary bit exchange are summarized in Table I. This table is brought from [2] and the description of the circuits can be found in [2], [13]. The circuits for elementary bit-exchange are classified into circuits that exchange two serial dimensions, i.e., serialserial (ss), two parallel dimensions, i.e., parallel-parallel (pp)…”

“…Regarding the storage elements, each ss permutation can be carried out either by a memory or by a circuit for serial-serial permutation [2], [4]. The latter consists of a buffer and two multiplexers.…”

“…The latter consists of a buffer and two multiplexers. Each pp permutation admits two possibilities: Either it only requires to interconnect each input to the corresponding output by a wire [2], which does not require any hardware, or uses a set of multiplexers in parallel to route the input data [5]- [9]. The former case occurs when all the values coming from the same input follow the same path, whereas the latter case occurs when they may follow different paths.…”

“…B IT REVERSAL [1] is a type of bit-dimension permutation [2], [3] that permutes a set of indexed data according to a reversing of the bits of the index. Its main use is to sort out the outputs of the fast Fourier transform (FFT), which are generally provided in the so called bit-reversed order.…”

“…REVIEW OF BIT-DIMENSION PERMUTATIONS This section reviews the main ideas related to bit-dimension permutations, which are required for understanding this paper. For a detailed description of bit-dimension permutations, the reader is encouraged to read [2].…”