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Efficient FPGA implementation of high-throughput mixed radix multipath delay commutator FFT processor for MIMO-OFDM.


Multiple Input Multiple Output-Orthogonal Frequency Division Multiplexing (MIMO-OFDM) is considered as one of the most promising solution to achieve high-throughput in today's wireless communication systems. It is based on combining Multiple Input Multiple Output (MIMO) signal processing techniques [1] with Orthogonal Frequency Division Multiplexing (OFDM) modulation [2]. MIMOOFDM significantly improves the reliability and throughput in wireless communication. Hence the technique is adopted with a wide range of current high-throughput communication standards, such as IEEE802.11, IEEE802.16, WiFi, WiMax, 3GPP and LTE [3-6]. The Fast Fourier Transform (FFT) and the Inverse Fast Fourier Transform (IFFT) processors are among the highest computationally complex and most critical modules within MIMO-OFDM based systems. They are required to process several independent high-throughput streams in parallel, while achieving real time requirements.

Literature indicates hardware design has migrated towards efficient and high performance FFT/IFFT architectures for MIMO-OFDM [7-14], and other applications, including signal and image processing [15-24]. Most of the existing architectures can be categorized into memory based designs, where only one butterfly is used, and pipeline designs, which uses a pipeline of butterfly stages. Pipeline architectures are suitable for MIMO-OFDM systems as they can achieve high-throughput and low-latency characteristics, which are required for continuous flow and real time applications [7], [11]. Pipeline architectures consist of two principal classes; Delay Feedback (DF) [15-18] and Delay Commutator [19-22]. The DF class includes Single-path Delay Feedback (SDF) and Multi-path Delay Feedback (MDF), where the DC class contains Single-path Delay Commutator (SDC) and Multipath Delay Commutator (MDC).

MIMO-OFDM based systems require FFT/IFFT of parallel independent data sequences to be calculated simultaneously. The traditional approach is to use several FFT/IFFT processors, where each processor deals with one sequence. However, as the number of independent data sequences increase, the hardware complexity of the system becomes very high. Hence, proposing efficient FFT/IFFT architectures using a single processor to process several streams is crucial for MIMO-OFDM designs. In this context, Lin and Lee [7], Fu and Ampadu [8] presented a 64/128 point FFT/IFFT processor based on mixed radix decomposition for up to four independent data sequences. Yang, Tsai and Chuang [9] proposed a MDC based architecture and a memory scheduling method to obtain a variable length FFT/IFFT processor. The circuit was fabricated using UMC 90 nm CMOS technology and can process up to four independent sequences. Tang, Liao and Chang [10] developed a Flexible-Radix-Configuration Multipath-Delay-Feedback (FRCMDF) architecture for variable length and multiple stream-based FFT processor. The architecture was fabricated using a TSMC 18 pm CMOS process. Tang, Tsai and Chang [11] introduced a mixed radix MDF based architecture for 2048 points FFT with up to eight parallel data streams. The processor was fabricated on 90 nm 1P9M CMOS and can obtain up to 2.4 GS/s throughput. Another mixed radix MDF based design for eight parallel data sequences was presented by Wang, Yan and Fu [12]. The design realizes a high-throughput and low-complexity 512-points FFT/IFFT processor. Additionally, Yan and Fu [12] proposed a single-Random Access Memory based group reorder that converts outputs into normal order. Tsai, Chen and Huang [13] proposed a systematic approach to automatically generate a variable size FFT/IFFT algorithm. Lin, Liu and Lee [14] proposed a Mixed Radix MDF architecture to implement a 128-point FFT/IFFT algorithm for Ultra Wide Band (UWB) communication systems.

FPGA FFT/IFFT algorithm implementations have been considered as a suitable solution to fulfil cost-efficient and high-performance implementations. Boopal, Garrido and Gustafsson [18] presented multi-streaming and variable-length Single-path Delay Feedback (SDF) based FFT architecture on Virtex-6 technology, which could process one stream of 2048-point, two streams of 1024-point or four streams of 512-point FFT operations. Ayinala, Brown and Parhi [20] proposed parallel MDC architectures using Radix-2, Radix-[2.sup.2] and Radix-[2.sup.3] independently to determine the FFT response of one real or complex sequence, which were synthesized on a Virtex-5 device. Garrido, Grajal, Sanchez and Gustafsson [21] presented a similar technique, which extended to include Radix-[2.sup.4] and any number of parallel paths which is a power of two, with implementation on Virtex-5. Further work from Garrido, Acevedo, Ehliar and Gustafsson [22] investigated FFT performance limitations carried out on Virtex-6. Wang, Liu, He, and Yu [23] proposed a Single-path Delay Commutator-Feedback (SDC-SDF) combination technique based on Radix-2 and synthesized on Virtex-5 devices. Uzun, Amira and Bouridane [24] presented a common framework for a memory-based FFT algorithm implemented on FPGA. This method is suitable for extension to include image processing applications.

Radix-[2.sup.k] MDC architecture has been more recently presented as an efficient solution for implementing FFT/IFFT algorithms [20-22]. Radix-[2.sup.k] MDC architecture can deal with any number of parallel data samples which are a power of two and can achieve high-throughput performance. In addition, the architecture requires fewer hardware operators compared to parallel feedback architecture MDF [21]. However, existing Radix-[2.sup.k] MDC designs are limited to processing single data streams in parallel and cannot process multiple independent data streams as required in MIMO-OFDM systems. Additionally, architectures based on single Radix-[2.sup.k] limits the design to processing sequences of data in correspondence with power of 2k sizes only.

This paper presents a high-frequency Mixed Radix-[2.sup.2] Radix-[2.sup.3] Multipath Delay Commutator ([MR2.sup.2]-[2.sup.3] MDC) FFT processor for MIMO-OFDM systems, with two and four data streams. The proposed design is suitable for implementing any power of two FFT size, where the design offers significant FPGA resource-space savings and significant utilization efficiency of embedded multipliers. Furthermore, the mixed radix approach allows FFT processor of size N to be constructed with various Radix-[2.sup.2] and Radix-[2.sup.3] module combinations. The remaining sections of this paper are organized as follows: Section II explains the Radix-[2.sup.2] and Radix-[2.sup.3] FFT algorithms. Section III introduces the proposed high-frequency [MR2.sup.2]-[2.sup.3] MDC. Section IV explains the design flow and implementation of the proposed architecture on FPGA, while Section V presents the experimental results of the implemented architecture on FPGA. Conclusions are provided in Section VI.


A. Radix [2.sup.2] Algorithm

The N-point Discrete Fourier Transform (DFT) is defined as:

X(k) = [N -1.summation over (n = 0)]x(n) x [W.sup.nk.sub.N] (1)

where x(n) and X(k) are complex samples in time and frequency domains, while the twiddle factor is defined as [W.sup.nk.sub.N] = exp (-j2[pi]nk/N)

Consider the following three-dimensional mapping of n and k [17]:

n = [< N/2 [n.sub.1] + N/4 [n.sub.2] + [n.sub.3] >.sub.N] (2)

k = [< [k.sub.1] + 2[k.sub.2] + 4[k.sub.3] >.sub.N] (3)

where the notation [< >.sub.N] represents the modulo-N operator.

Substituting equations (2) and (3) into equation (1) obtains:


Summing among [n.sub.1] and considering:


Equation (4) can be re-formulated as follows:


The critical concept of Radix-[2.sup.2] is to cascade the twiddle factor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the next step decomposition instead of multiplying with [B.sub.N/2] (N/4 [n.sub.2] + [n.sub.3], [k.sub.1]):


Substituting (7) into (6) and summing among [n.sub.2] determine:


where [H.sub.N/4]([n.sub.3], [k.sub.1], [k.sub.2]) can be expressed as follows:


Equation (5) represents the first butterfly structure containing only addition and subtraction operations. Equation (9) demonstrates the second butterfly structure containing addition, subtraction and multiplication with -j. The Radix-[2.sup.2] algorithm requires full multiplication every two stages as demonstrated in equation (8), opposed to multiplication occurring during every stage within Radix-2. Hence, Radix-[2.sup.2] demonstrates a 50% reduction of multiplication operations required in comparison to Radix-2. The complete Radix-[2.sup.2] algorithm can be obtained through decomposing equation (8) recursively to the remaining N/4 points DFTs.

The fundamental structure to obtain two consecutive stages in the Signal Flow Graph (SFG) for Radix-[2.sup.2] Decimation In Frequency (DIF) algorithm is illustrated in Fig. 1, where [x.sub.i](*) denotes the inputs of stage i and [x.sub.i+1](*) denotes the outputs of stage i + 1. When n is varied from 0 to Ni/4-l in the basic structure of Fig. 1 a group of [N.sub.i] inputs is formed, where [N.sub.i] is given by:


The group is repeated N/Ni times to obtain two consecutive stages in the complete SFG.

B. Radix [2.sup.3] Algorithm

Applying four dimensional linear index mapping to n and k can be expressed as follows:

n = [<N/2 [n.sub.1] + N/4 [n.sub.2] + N/8 [n.sub.3] + [n.sub.4]>.sub.N] (11)

k = [<[k.sub.1] + 2[k.sub.2] + 4[k.sub.2] + 8[k.sub.4]>.sub.N] (12)

The DFT function in equation (1) can be expressed in terms of four-dimensional linear index mapping, as follows:


Additionally, the twiddle factor [W.sup.nk.sub.N] with cascade decomposition can be expressed as:


Substitution of equation (14) into equation (13) and expending summation among [n.sub.1], [n.sub.2] and [n.sub.3] determine:


Eight DFT functions of length N/8 were expressed in equation (15), where a third butterfly structure [T.sub.N/8] ([n.sub.4], [k.sub.1], [k.sub.2], [k.sub.3]) is expressed as follows:


The third butterfly structure involves the twiddle factor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which can be realized with constant scalar operations. The initial radix-[2.sup.3] SFG stage contains only trivial multiplication with -j, the second stage multiplies with -j and constants [W.sup.1.sub.8] and [W.sup.3.sub.8], while the third stage contains multiplications with the general term. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In comparison, Radix-2 required multiplication during every stage, while Radix-[2.sup.2] only uses multiplication operations every second stage.

The fundamental structure to obtain three consecutive stages in the SFG of Radix-[2.sup.3] DIF algorithm is presented in Fig. 2. A group of [N.sub.i] inputs is obtained when n varies from 0 to [N.sub.i]/8-1. Three consecutive stages in the complete SFG are formed by AW, groups.


To the best of the authors' knowledge, current literature has not investigated improved high-frequency MDC architecture implementations on FPGA through added delay elements. Current architectures within literature have significantly limited maximum clock frequency responses when synthesized on FPGA without utilizing pipeline registers at element outputs. This poor maximum frequency performance occurs frequently when FFT size increases [1922]. Using an appropriate number of pipeline registers at the output element within the FFT architecture allows the length of critical paths within the processor circuit to be significantly reduced and achieve high-frequency response. The latency introduced with pipeline registers at each element output can be defined as the number of clock sample periods used for the block output delay. Simulation-based investigations indicate the appropriate number of pipeline registers to achieve maximum frequency is: four for complex multipliers, one for addition, subtraction and multiplexer and two for Block Random Access Memory (BRAM) operations. Pipeline register associated latencies cause synchronization issues within the architecture, affecting the FFT function response. Hence, it is critical to compensate the effect of pipeline register latency through inserting additional delay elements at specific points in the architecture design.

The general structure for the proposed MR-[2.sup.2]-[2.sup.3] MDc architecture is demonstrated in Fig. 3. The architecture is composed of an input memory-scheduling unit, m modules, where each module consists of either a Radix-[2.sup.2] or Radix-[2.sup.3] butterfly structures with a control unit. Let [m.sub.2] denote the number of Radix-[2.sup.2] modules, m 3 the number of Radix-[2.sup.3] modules and ns the number of stages in the Signal Flow Graph (SFG), ([n.sub.s] = [log.sub.2] (N)), then [n.sub.s], [m.sub.2], and [m.sub.3] are related as follows:

[n.sub.s] = 2[m.sub.2] + 3[m.sub.3] (17)

Equation (17) demonstrates that utilizing MR-[2.sup.2]-[2.sup.3] allows scaling FFT size to any power of two, which is not possible to achieve with single Radix-[2.sup.2] or Radix-[2.sup.3].

The Radix-[2.sup.2] function performs operations in two stages within the SFG as detailed in Fig. 1. In contrast, the Radix-[2.sup.3] function performs operations in three stages within the SFG as detailed in Fig. 2. For example, a FFT size N=32 demonstrates a SFG with five stages, which can be realized within architecture requiring one Radix-[2.sup.2] operation pipelined with one other Radix-[2.sup.3] operation.

A. Proposed Architecture for Two Data Streams

The proposed FFT system determines the DFT responses for two parallel data streams; A and B. The input memory scheduling system arranges the input sequences of data streams A and B as shown in Fig. 4. Data stream A = [x.sup.1] (0..N - 1) is separated into two sub-streams, obtaining upper and lower sub-streams [A.sub.1] = [x.sup.1](0..N/2 - 1) and [A.sub.2] = [x.sup.1] (N/2.. N - 1), where sub-stream length is N/2.

Similarly, data stream B is separated into upper sub-stream [B.sub.1] and lower sub-stream [B.sub.2]. The input data stream sequences are reshuffled to allow sub-streams [A.sub.1] and [A.sub.2] to be processed during the initial time index with Module 1 through m as shown in Fig. 3, followed by processing substeams [B.sub.1] and [B.sub.2]. Modules 1 through m can consist of Radix-[2.sup.2] or Radix-[2.sup.3] structures as shown in Fig. 5 and Fig. 6. Pipeline register latencies associated for operation functions have been presented within block diagrams. Delay elements are operated to maintain data synchronization with upper and lower sub-streams. Operations are applied to the upper [] and lower [] input data sub-streams to achieve the upper [U.sub.out] and lower [L.sub.out] output data sub-streams.

The Radix-[2.sup.2] MDC module includes two addition/substruction functions, one trivial multiplier, two standard multipliers, two Read Only Memories (ROM), two shuffling units, two multiplexers and three delay elements. The trivial multiplier performs real-imaginary swap with sign inversion operations.

Shuffling units are inserted between each consecutive stages, i.e. a shuffling unit is inserted between stage i and stage i + 1, as shown in Fig. 5. They are built using 2D delay elements, where D = N/[2.sup.(i+1)] and requires two multiplexers.

The delay elements are used for synchronizing the previous stage data outputs, i.e. stage i, where multiplexers allow routing data to the corresponding input of the Add/Sub unit of the stage i+1. The multiplexer control signal [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is directly obtained from a binary counter. It takes the value zero for D clock cycles followed by the value one for D others clock cycles, i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For D clock cycles, the upper data outputs of stage i U1 to UD are routed to the upper input of stage i+1, while simultaneously data outputs [U.sub.D+1] to [U.sub.2D] are routed to the lower input of stage i + 1. The process is similarly repeated with the lower data outputs of stage i during the next D clock cycles, i.e. lower data [L.sub.1] to [L.sub.D] are routed to the next stage upper input, while data [L.sub.D+1] to [L.sub.2D] are routed to the next stage lower input. Twiddle factors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with n = 0..[N.sub.i]/4 - 1 are stored in the upper path's ROM (ROM U) of size [N.sub.i]/4 while twiddle factors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are stored in the lower path's ROM (ROM L) of size [N.sub.i]/2. ROM U is addressed using bits [C.sub.0] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of a binary counter while ROM L is addressed using bits [C.sub.0] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the same counter.

The Radix-[2.sup.3] module utilizes three addition/substraction functions, one trivial multiplier, two standard multipliers, one Constant Complex Multiplication Unit (CCMU1), two ROMs, three shuffling units, one multiplexer and two delay elements. The CCMU1 operation applies rotation with -j, [W.sup.1.sub.8] and [W.sup.3.sub.8], where the associated structure is presented in Fig. 6. The CCMU1 contains one complex scaling operator, two trivial multipliers, three multiplexers in addition to delay elements. The same complex scaling operator used to rotate with [W.sup.1.sub.8], is cascaded with a trivial multiplier -J to accomplish rotation with [W.sup.3.sub.8]. Twiddle factors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with n = 0..[N.sub.i]/8 - 1 are stored in ROM U, while twiddle factors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are stored in ROM L. Both ROM U and ROM L are of size [N.sub.i]/2 and are addressed using bits [C.sub.0] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of a binary counter. It is important to note that all operations after the last addition unit are removed in both Radix-[2.sup.2] and Radix-[2.sup.3] functions, when they are implemented within the architecture end stage of module m.

B. Proposed Architecture for Four Data Streams

Input scheduling of four independent data streams A, B, C and D can be obtained using the input memory scheduling technique proposed by Yang, Tsai and Chuang [9], which rearranges the input data as shown in Fig. 7. Each of the parallel data streams A, B, C and D are separated into four equal length sub-streams adhering to the following format: Data stream A = [x.sup.1] (0..N - 1) is separated into [A.sub.1] = [x.sup.1](0..N/4 - 1), [A.sub.2] = [x.sup.1] (N/4..N/2 - 1), [A.sub.3] = [x.sup.1](N/2..3N/4 - 1), and [A.sub.4] = [x.sup.1] (3N/4..N - 1), where sub-stream lengths are defined as N/4. Streams B, C and D are split into four equal length substreams in a similar manner.

Once the input data streams have been rescheduled with the memory scheduling unit, then Modules 1 through m determine the DFT response of streams A, B, C and D consecutively.

The four parallel Radix-[2.sup.2] and Radix-[2.sup.3] MDC module process four samples in a continuous flow. Architecture of an initial stage four parallel Radix-[2.sup.2] module is shown in Fig. 8, while the architecture for intermediate and last modules are presented in Fig. 9.

Shuffling units keep the same structure shown in Fig. 5, with D delays element for each unit instead of 2D delays used within the two parallel data stream architecture. Twiddle factors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with n = 0..[N.sub.i]/4-1 are stored in ROML1, ROM U2 and ROML2 respectively. The three memories each of size [N.sub.i]/4 are addressed by bits C0 to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of a binary counter.

Architecture design for the initial four-parallel Radix-[2.sup.3] module is illustrated in Fig. 10, where the structure for intermediate and last modules are shown in Fig. 11. Twiddle factors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with n = 0..[N.sub.i]/8 - 1 are stored in memories ROM U1, ROM L1, ROM U2 and ROM L2 respectively. The four memories, each of size [N.sub.i]/4 are addressed by bits [C.sub.0] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of a binary counter.

It is important to note that both Radix-[2.sup.2] and Radix-[2.sup.3] initial and last modules (Module 1 and Module m) implement slightly modified architecture as presented in Fig. 8 to Fig. 11. Module 1 removes the shuffling operation at the output of the modules initial addition function, while Module m removes all operations after the last addition units.

C. Hardware Architecture Resource Requirements

The hardware architecture requirements for the proposed systems are presented and compared with other relevant work in Table I, where [phi] represents the number of independent data streams and p represents the number of parallel paths. The number of complex addition operators is dependent on the number of parallel data streams and FFT size N for any combination of Radix-[2.sup.2] and Radix-[2.sup.3] modules, where four data streams require double the amount of complex addition operators than a two data streams system. The memory architecture requirements presented in Table I are obtained with input scheduling unit, in addition to shuffling units inserted in Module 1 through m.

For the proposed architectures, the additional delay elements used for high-frequency design are also considered

In two data streams architecture, the input scheduling unit uses two memory operators each of size N/2, with four additional delays to maintain data synchronization, where the shuffling units contain delay elements of total size N-2. Radix-[2.sup.3] requires 11 additional delays regardless of module position within the architecture. Radix-[2.sup.2] implements six additional delays for module rank positions 1 through m-1 and one delay for module position rank m. Hence, the total memory size in the two parallel architecture is 2N + 2 + 6[m.sub.2] + 11[m.sub.3] - 5[k.sub.1], where [k.sub.1] = 1 if module m is a Radix-[2.sup.2], else [k.sub.1] = 0.

Systems with four parallel processing architectures involve an input scheduling unit with 12 memory banks, each of size N/4 in addition to six further delay elements for data synchronization, producing a total memory size of 3N+6. The shuffling units utilize delay elements of total size N-4. Each Radix-[2.sup.3] module obtains 21 additional delays independently of rank position. Additionally, Radix-[2.sup.2] requires seven additional delays for module rank position 1 through m-1, where module in rank position m uses three delays. Thus, the total memory size in the four parallel architecture is 4N + 2 + 21[m.sub.3] + 7[m.sub.2] - 4[k.sub.1].

Each module from rank position 1 through m-1 implemented within the two parallel data stream architecture systems contain two complex multipliers, determining the total multipliers used as 2(m-1). Furthermore, each Radix-[2.sup.3] module contains a single rotation by W8, carried out using the CCMU1 unit as shown in fig. 6. As a result, the total number of rotations by W8 within the two parallel architecture is equal to [m.sub.3].

For systems with four parallel data stream architecture, Radix-[2.sup.2] modules located in rank position 1 through m-1, contain three complex multipliers, while Radix-[2.sup.3] modules in comparison require four multipliers at the same rank position. Hence, the total number of complex multipliers required in the four parallel architecture is 3([m.sub.2] - [k.sub.1]) + 4([m.sub.3] + [k.sub.1]-1). Moreover, each Radix-[2.sup.3] module within the four parallel architecture contains two rotations by [W.sub.8] realized with CCMU2 and CCMU3 as illustrated in Fig. 10 and Fig. 11. Hence the total number of rotators by [W.sub.8] within the four parallel architecture is 2[m.sub.3].

The proposed hardware architecture requirements are presented and compared with other relevant work in Table I. The proposed architecture is suitable for parallel data stream processing with FFT size [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where other architectures evaluated have not taken these characteristics into design consideration.

Pipelined Radix-[2.sup.k] architectures presented in [21] and [22] are capable of processing one data stream with parallel data paths, requiring additional input modules to process parallel data streams. Table I illustrates that the proposed architecture involves the same number of required complex addition operators as architecture systems in [9, 21, 22] and fewer than [7, 8]. Furthermore, some Radix-[2.sup.2]-[2.sup.3] proposed combinations utilize fewer complex multipliers than equivalent architectures in [9, 21, 22]. For example, a FFT size N=256 would require 6 and 9 multipliers for two and four parallel data streams in [9, 21, 22], while the proposed [2.sup.3]-[2.sup.2]-[2.sup.3] combination uses 4 and 7 multipliers for two and four parallel data stream architecture. The proposed architecture combination obtains 33.3% and 22.2% reduction in required complex multipliers than architectures [9, 21, 22]. The FFT processor presented by Lin and Lee [7] requires two complex multipliers in addition to four modified complex multipliers which realize 0.62% of the standard multiplier area; this structure uses an equivalent of 4.48 multipliers in comparison to 6 multipliers used in this article presented method for the combination [2.sup.3]-[2.sup.2]-[2.sup.3]. However, work presented in [7] is limited to FFT size N = 64 and N=128, while the presented architecture is scalable to any power of two FFT size. Furthermore, architecture presented by Fu and Ampadu [8] requires fewer multipliers than the proposed system, however the architecture is also limited to FFT size N=64 and N=128. The additional memory usage is limited within the proposed architecture to achieve high-frequency in the implementation.


The proposed [MR2.sup.2]-[2.sup.3] MDC architectures for various FFT size N are designed using Xilinx System Generator (XSG). The design is based on fixed-point number representation, where scaling operations are introduced to avoid numerical saturation. Scaling is performed with bit shifting to the right of processed data within early stages of the FFT process. XSG integration with the Matlab-Simulink environment allowed FFT input generation, in addition to evaluating the obtained FFT output. For verification, the output from the proposed fixed-point FFT design is compared to the output from Matlab's floating-point FFT function. Designs from XSG are utilized for generating Register Transfer Level (RTL) files for migration to Xilinx ISE design suite environment for FPGA implementation. The architecture designs were synthesized and evaluated on a Virtex-5 FPGA XC5VSX240T-2FF1738 and Virtex-7 FPGA XC7VS485T-2FFG1661.


The proposed architecture was analyzed and evaluated with place and route synthesis. Mixed Radix-based methods allow FFT processors to be realized with various combinations of Radix-[2.sup.2] and Radix-[2.sup.3]. For example an FFT size N=256 can be built with four consecutive Radix-[2.sup.2] modules ([2.sup.2]-[2.sup.2]-[2.sup.2]-[2.sup.2]) or with a consecutive sequence of one Radix-[2.sup.3], one Radix-[2.sup.2] and one Radix-[2.sup.3] modules ([2.sup.3]-[2.sup.2]-[2.sup.3]), as it can be carried out with the sequence [2.sup.3]-[2.sup.3]-[2.sup.2].

The proposed architecture implements the FFT algorithm, for sizes in correspondence with N = 2"s ranging from 16 through 1024 with various combinations. The designs were evaluated in FPGA resource usage, performance and dynamic power consumption. FPGA Block Random Access Memory (BRAM) hardware was used in the scheduling unit and ROM operations. The Element delays implemented in shuffling and data synchronization operations were implemented with Shift Register Look-up tables (SRL). Mathematical DsP48E structures were used for standard multiplication, while all other design elements were synthesized with FPGA logic resources. Data Word Length (DWL) and Twiddle Factor Word Length (TFWL) were configured for 16 and 12 bits.

The FPGA utilized resources for the proposed high-frequency architecture are presented in Tables II and III in comparison with equivalent low-frequency architectures. Note low-frequency architectures refer to architectures without any pipeline registers or additional delay elements. A Resource Utilization Ratio (RUR) is utilized to provide a fair comparison, where the RUR compares the number of FPGA slices utilized in the proposed architecture [R.sub.p] against the number of slices involved in an equivalent low frequency design [R.sub.e]; RUR = [R.sub.p]/[R.sub.e].

Both high-frequency and low-frequency architecture designs obtained the same number of BRAM and DSP48E structures. Additionally, the RUR produced values did not exceed 1.74 with significantly high-level of throughput. Furthermore, building a FFT process with various Radix-[2.sup.2] and Radix-[2.sup.3] combinations offers significant advantages, including the reduction of FPGA resources and flexibility to FPGA platform resources. For example, a FFT size N=256 denoted with the architecture structure [2.sup.2]-[2.sup.2]-[2.sup.2]-[2.sup.2] requires 42 BRAM and 36 DSP48E structures, while in comparison a [2.sup.3]-[2.sup.2]-[2.sup.3] structure requires 38 BRAM and 28 DSP48E structures, while the [2.sup.3]-[2.sup.3]-[2.sup.2] combination requires 40

BRAM and 32 DSP48E structures.

It is important to note the FPGA's occupied area average, which is calculated with averaging the percentage of used BRAMs, slices and DsP48E blocks on the architecture implementation. The obtained FPGA occupied area average metrics range from 0.82% through 2.88 % within two parallel designs and from 1.89% through 4.96% within four parallel circuits. This demonstrates the proposed FFT process occupies a very small portion of the total area on a FPGA device, where the remaining area could be used to perform other MIMO-OFDM related functions.

The performances of the proposed high-frequency designs are compared with equivalent low-frequency designs in Table IV and Table V, demonstrating the system throughput in Mega-symbols per Second (Ms/S) and latency values. The proposed system comparison utilizes an Area to Throughput Ratio (ATR), calculated by dividing the number of occupied slices with the achieved throughput. The proposed architecture demonstrates significant throughput metric values ranging from 858 Ms/S to 916 Ms/S in two stream systems and from 1600 Ms/S to 1756 Ms/S in four stream systems. In comparison, the equivalent low-frequency architecture performance metrics range from 41.4 Ms/S to 125.6 Ms/S for two stream and from 86 Ms/S to 250.4 Ms/S in four stream designs.

The significant throughput improvement is not penalized with excessive resource usage and is efficient with device resource usage and achievable throughput as demonstrated with ATR values. Furthermore, the high-frequency architecture demonstrated significantly lower latency time than equivalent low-frequency architectures.

The dynamic power performance of the proposed high-frequency MR-[2.sup.2]-[2.sup.3] MDC architecture is compared with low-frequency equivalent architecture results and presented in Table VI and Table VII. High-frequency architecture demonstrates a significant increase of dynamic power consumption compared to low-frequency architecture. However, the Dynamic Power to Throughput Ratio (DPTR) for the proposed architecture is lower than their equivalent low-frequency architecture for FFT size above 64. This demonstrates there is no additional power consumption when using one high-frequency circuit opposed to several low-frequency circuits to achieve the same throughput. Additionally the four data stream high-frequency architecture obtained an average power increase of approximately 100% in comparison to the equivalent two data stream architectures. The dynamic power consumption demonstrates relatively little difference for large FFT size N with different Radix-[2.sup.2] and Radix-[2.sup.3] module combinations.

The proposed high-frequency architecture was carried out on a Virtex-7 XC7VS485T-2FFG1661 FPGA device to produce more recent device results obtained for FPGA area resources, maximum frequency and dynamic power consumption. The proposed designs FPGA area resource utilization is presented in Table VIII. The Virtex-7 implementation demonstrated significant savings in FPGA occupied slices, producing up to 28% and 23% slice reduction in comparison to Virtex-5 for two and four data stream implementations, which are presented in Tables II and Table III. Note the Virtex-7 FPGA's average occupied area is not exceeding 1.17% and 2.11% within two-parallel and four-parallel designs.

The maximum frequency for the proposed architectures synthesized on Virtex-7 and Virtex-5 platforms over increasing FFT size are presented in Fig. 12. Virtex-7 implemented architectures achieved higher frequency values in comparison to Virtex-5. Virtex-7 implementations initially achieved 477 MHz followed with very low-levels of decreasing frequency as FFT size increases. Virtex-5 implementations achieved lower frequency with 458 MHz and 439 MHz for two and four data stream architecture with relatively large frequency decrease as FFT size increases.

As shown in Fig. 13 the dynamic power consumption of the proposed architecture designs implemented on Virtex-7 demonstrated significant power savings in comparison to Virtex-5 implementations. For FFT size N=16 architecture at 400 MHz maximum frequency, two data stream demonstrated initial dynamic power consumption of 0.9 W and 0.4 W for Virtex-5 and Virtex-7 implementations, while four data streams obtained 2 W and 0.9 W for Virtex-5 and Virtex-7 implementations. It is important to note all designs demonstrated very low-levels of dynamic power consumption increase as FFT size increases.

The dynamic power consumption of high-frequency architecture for FFT size N=16 and 1024 with increasing frequency, synthesized for Virtex-5 and Virtex-7 platforms are presented in Fig. 14a and Fig. 14b. Virtex-7 implementations demonstrate significant dynamic power reductions in comparison to Virtex-5, where the dynamic power reduction magnitude increases as frequency increases.

The proposed high-frequency architecture is compared with recent and comparable techniques within literature as presented in Table IX. It is important to note Ayinala, Brown and Parhi [20] introduced a two-parallel architectures for one data stream without an input scheduling unit, while Wang, Liu, He and Yu [23] presented a single path FFT architecture to obtain a DFT response for a single stream. In order to provide a valid comparison the compared resources have been scaled to reflect the articles presented number of parallel paths. The proposed high-frequency design demonstrated the highest achieved frequency amongst the implementation techniques compared, where the proposed method obtains a significant advantage. The proposed architectures frequency improvement when compared against equivalent techniques increases as FFT size N increases.

The proposed realization involves fewer DSP mathematical hardware operators than techniques demonstrated in [23] and fewer or the same number of DSP operators than references [21, 22] for all FFT sizes investigated. Additionally, the presented architecture required fewer slices Look-Up Tables (LUT) than references [20] and [23] and fewer occupied slices than [20]. It is challenging to provide a valid occupied slice resource comparison with [21] and [22] as the compared techniques do not include an input shuffling unit function, which is necessary to process multiple streams in parallel.


This article presents an efficient FPGA implementation of high-frequency FFT architecture based on mixed Radix-[2.sup.2] Radix-[2.sup.3] MDC structures. The presented architecture design can process two or four independent data streams in parallel with efficient utilization of FPGA resources and demonstrates a significant high-throughput performance in comparison to other equivalent techniques. The designs were intended for MIMO-OFDM communication systems, where parallel data processing is a critical process, requiring efficient resource usage and high-throughput. The system is very simple to control with binary counter signals. The presented architecture is based on mixed Radix structures to realize scalable architectures to any FFT size [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Additionally, the mixed Radix approach offers advantages in the implementation flexibility, where a FFT processor for each size can be realized with several differing Radix-[2.sup.2] and Radix-[2.sup.3] module combinations. The designs were carried out and evaluated on Virtex-5 and Virtex-7 FPGA devices. Experimental results demonstrate the presented high-frequency architectures obtained throughput ratio improvements of 20.82 within two-parallel and 19.12 within four-parallel architectures in comparison to equivalent low-frequency designs. The proposed architecture obtained maximum frequency values of 458 MHz and 477 MHz for Virtex-5 and Virtex-7 devices, in addition to obtaining very low-levels of performance decrease as FFT size N increases. Additionally, Virtex-7 device implementations demonstrated significant dynamic power reduction while obtaining significantly higher throughput than Virtex-5.

Digital Object Identifier 10.4316/AECE.2017.01005


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Mohammed Dali (1) *, Abderezak Guessoum (2), Ryan M. Gibson (3), Abbes Amira (4,5), Naeem Ramzan (4)

(1) Research Laboratory in Electrical Engineering and Automatic, University of Medea,26000, Algeria

(2) Department of Electronics Engineering, University of Blida, 09000, Algeria

(3) School of Engineering and Built Environment, Glasgow Caledonian University, G4 0BA, UK

(4) School of Engineering and Computing, University of the West of Scotland, Paisley, PA1 2EB, UK

(5) Department of Computer Science and Engineering, Qatar University, Doha, 2713, Qatar

* dali.mohamed

Caption: Figure 1. SFG of the basic butterfly structure of Radix-[2.sup.2] DIF algorithm

Caption: Figure 2. SFG of the basic butterfly structure of Radix-[2.sup.3] DIF algorithm

Caption: Figure 3. General structure of the proposed [MR.sup.2]-R[2.sup.3] MDC architecture

Caption: Figure 4. Input reordering with Input Scheduling Unit for two parallel streams.

Caption: Figure 5. Architecture of the proposed high-frequency two parallel 22 module.

Caption: Figure 6. Architecture of the proposed high-frequency two parallel [2.sup.3] module

Caption: Figure 7. Input reordering by Scheduling Unit for four parallel data streams

Caption: Figure 8. Architecture of the proposed high-frequency four parallel Radix 22 module as first processing module (z=7)

Caption: Figure 9. Architecture of the proposed high-frequency four parallel Radix-[2.sup.2] module as intermediate module (whole diagram) or as last module (only components within dotted section)

Caption: Figure 10. Architecture of the proposed high-frequency four parallel Radix-[2.sup.3] module as first module (i=1)

Caption: Figure 11. Architecture of the proposed high-frequency four parallel Radix-[2.sup.3] module as intermediate module (whole diagram) or as last module (only diagram within dotted rectangle).

Caption: Figure 12. Maximum frequency versus FFT size N comparison between Virtex-7 and Virtex-5 implementation of proposed architectures

Caption: Figure 13. Dynamic power versus FFT size N comparison between Virtex-7 and Virtex-5 implementation of proposed architectures (with F=400 MHz)

Caption: Figure 14. Dynamic power versus frequency comparison between Virtex-7 and Virtex-5 implementation of proposed architectures. (a) Two parallel architecture. (b) Four parallel architecture

Designs           Type      Radix          N            [phi]   P

Proposed          MDC       MR-[2.sup.2]   2n           2       2

                            -[2.sup.3]                  4       4

Lin et al. [7]    SDF       R-2/R-8        64, 128      4       4
Fu et al. [8]     SDF/MDC   R2/R4          64, 128      4       4
Yang et al. [9]   MDC       R4/R8          128, 512,    4       4
                                           1024, 2048

Garrido et al.    MDC       R[2.sup.2]     4n           1       2
[21], [22]                                                      4
                            R[2.sup.3]     8n           1       2
Designs           Type      Complex         Memory size

Proposed          MDC       4[log.sub.4]N   2N + 2 + (11[m.sub.3] +
                                            6[m.sub.2] - 5[k.sub.1])*
                            8[log.sub.4]N   4N + 2 + (21[m.sub.3] +
                                            7[m.sub.2] - 4[k.sub.1])*

Lin et al. [7]    SDF       48              508
Fu et al. [8]     SDF/MDC   32              508
Yang et al. [9]   MDC       8[log.sub.4]N   3N + [[log.sub.4]N]-1.
                                            summation over (s = 1)]
Garrido et al.    MDC       4[log.sub.4]N   N
[21], [22]                  8[log.sub.4]N   N
                            4[log.sub.4]N   N
                            8[log.sub.4]N   N

Designs           Type      Complex            Rotation by
                            Multipliers        [W.sub.8]

Proposed          MDC       2(m-1)             [m.sub.3]

                            3[m.sub.2] +       2[m.sub.3]
                            4[m.sub.3] +
Lin et al. [7]    SDF       2 + (4 x 0.62)     0
Fu et al. [8]     SDF/MDC   4                  5
Yang et al. [9]   MDC       3([log.sub.4]N-1)  0

Garrido et al.    MDC       2([log.sub.4]N-1)  0
[21], [22]                  3([log.sub.4]N-1)  0
                            3(log8N-1)         [log.sub.8]N
                            4(log8N-1)         2[log.sub.8]N

()* presentitional delay elements for high throughput design

Table II. Virtex-5 FPGA XC5VSX240T-2FF1738 Resource Usage of the
Proposed MR22-R23 MDC for two Parallel Data Streams

N      Structure              Occupied Slices          BRAM   DSP48E
                              Low-     High-    RUR
                              Fre-     Fre-
                              quency   quency

16     [2.sup.2]-[2.sup.2]    279      347      1.24   8      8
32     [2.sup.3]-[2.sup.2]    400      501      1.25   8      8
       [2.sup.2]-[2.sup.3]    391      535      1.37   8      8
64     [2.sup.2]-[2.sup.2]    413      534      1.29   12     16
       [2.sup.3]-[2.sup.3]    553      674      1.22   8      8
128    [2.sup.2]-[2.sup.2]    537      684      1.27   12     16
       [2.sup.3]-[2.sup.2]    557      679      1.22   12     16
256    [2.sup.2]-[2.sup.2]    685      760      1.11   16     24
       [2.sup.3]-[2.sup.2]    709      906      1.28   12     16
       [2.sup.3]-[2.sup.3]    703      851      1.21   12     16
512    [2.sup.3]-[2.sup.3]    1012     1280     1.26   12     16
       [2.sup.3]-[2.sup.2]    937      1141     1.22   16     24
       [2.sup.2]-[2.sup.2]    781      1102     1.41   16     24
1024   [2.sup.2]-[2.sup.2]    1089     1372     1.26   20     32
       [2.sup.3]-[2.sup.3]    1171     1386     1.18   16     24
       [2.sup.3]-[2.sup.2]    1247     1655     1.33   16     24


N      Structure              Occupied Slices          BRAM   DSP48E
                              Low-     High-    RUR
                              Fre-     Fre-
                              quency   quency

16     [2.sup.2]-[2.sup.2]    508      619      1.22   30     12
32     [2.sup.3]-[2.sup.2]    693      991      1.43   32     16
       [2.sup.2]-[2.sup.3]    751      931      1.24   30     12
64     [2.sup.2]-[2.sup.2]    633      895      1.41   36     24
       [2.sup.3]-[2.sup.3]    924      1284     1.39   32     16
128    [2.sup.2]-[2.sup.2]    992      1123     1.13   36     24
       [2.sup.3]-[2.sup.2]    935      1262     1.35   38     28
256    [2.sup.2]-[2.sup.2]    629      1095     1.74   42     36
       [2.sup.3]-[2.sup.2]    1256     1744     1.39   38     28
       [2.sup.3]-[2.sup.3]    1222     1518     1.24   40     32
512    [2.sup.3]-[2.sup.3]    1656     2067     1.25   40     32
       [2.sup.3]-[2.sup.2]    1378     1596     1.16   44     40
       [2.sup.2]-[2.sup.2]    1607     1554     0.97   42     36
1024   [2.sup.2]-[2.sup.2]    1850     1659     0.90   48     48
       [2.sup.3]-[2.sup.3]    2063     2342     1.14   46     44
       [2.sup.3]-[2.sup.2]    1897     2433     1.28   44     40


N      Structure              Low-Frequency
                              Through-    ATR     Latency
                              put(Ms/S)           (nS)

16     [2.sup.2]-[2.sup.2]    125.6       2.22    127
32     [2.sup.3]-[2.sup.2]    91.0        4.40    352
       [2.sup.2]-[2.sup.3]    125.2       3.12    256
64     [2.sup.2]-[2.sup.2]    83.4        4.95    767
       [2.sup.3]-[2.sup.3]    87.0        6.36    736
128    [2.sup.2]-[2.sup.2]    80.0        6.71    1600
       [2.sup.3]-[2.sup.2]    60.6        9.19    2112
256    [2.sup.2]-[2.sup.2]    60.6        11.30   4224
       [2.sup.3]-[2.sup.2]    60.6        11.70   4224
       [2.sup.3]-[2.sup.3]    50.0        14.06   5120
512    [2.sup.3]-[2.sup.3]    50.0        20.24   10240
       [2.sup.3]-[2.sup.2]    48.8        19.20   10492
       [2.sup.2]-[2.sup.2]    58.8        13.28   8707
1024   [2.sup.2]-[2.sup.2]    47.6        22.88   21513
       [2.sup.3]-[2.sup.3]    41.4        28.29   24734
       [2.sup.3]-[2.sup.2]    47.8        26.09   21423

N      Structure              High-Frequency
                              Through-   ATR    Latency
                              put(Ms/S)         (nS)

16     [2.sup.2]-[2.sup.2]    916        0.38   48
32     [2.sup.3]-[2.sup.2]    910        0.55   81
       [2.sup.2]-[2.sup.3]    894        0.60   85
64     [2.sup.2]-[2.sup.2]    888        0.60   126
       [2.sup.3]-[2.sup.3]    878        0.77   139
128    [2.sup.2]-[2.sup.2]    884        0.77   217
       [2.sup.3]-[2.sup.2]    878        0.77   216
256    [2.sup.2]-[2.sup.2]    884        0.86   367
       [2.sup.3]-[2.sup.2]    872        1.04   383
       [2.sup.3]-[2.sup.3]    880        0.97   377
512    [2.sup.3]-[2.sup.3]    890        1.44   679
       [2.sup.3]-[2.sup.2]    880        1.30   675
       [2.sup.2]-[2.sup.2]    880        1.25   677
1024   [2.sup.2]-[2.sup.2]    858        1.60   1296
       [2.sup.3]-[2.sup.3]    862        1.61   1299
       [2.sup.3]-[2.sup.2]    866        1.91   1293


                              Low Frequency
N      Structure              Through-    ATR     Latency
                              put(Ms/S)           (nS)

16     [2.sup.2]-[2.sup.2]    250.4       2.03    64
32     [2.sup.3]-[2.sup.2]    186.4       3.72    172
       [2.sup.2]-[2.sup.3]    225.6       3.33    142
64     [2.sup.2]-[2.sup.2]    222.8       2.84    287
       [2.sup.3]-[2.sup.3]    174.4       5.30    367
128    [2.sup.2]-[2.sup.2]    200.4       4.95    639
       [2.sup.3]-[2.sup.2]    133.6       7.00    958
       [2.sup.2]-[2.sup.2]    195.6       3.22    1309
256    [2.sup.3]-[2.sup.2]    133.6       9.40    1916
       [2.sup.3]-[2.sup.3]    105.2       11.62   2433
       [2.sup.3]-[2.sup.3]    100.0       16.56   5120
512    [2.sup.3]-[2.sup.2]    117.6       11.72   4354
       [2.sup.2]-[2.sup.2]    190.8       8.42    2683
       [2.sup.2]-[2.sup.2]    170.4       10.86   6009
1024   [2.sup.3]-[2.sup.3]    86.0        23.99   11907
       [2.sup.3]-[2.sup.2]    118.0       16.08   8678

N      Structure              Through-    ATR    Latency
                              put(Ms/S)          (nS)

16     [2.sup.2]-[2.sup.2]    1756        0.35   36
32     [2.sup.3]-[2.sup.2]    1696        0.58   64
       [2.sup.2]-[2.sup.3]    1688        0.55   64
64     [2.sup.2]-[2.sup.2]    1684        0.53   88
       [2.sup.3]-[2.sup.3]    1668        0.77   101
128    [2.sup.2]-[2.sup.2]    1668        0.67   144
       [2.sup.3]-[2.sup.2]    1644        0.77   146
       [2.sup.2]-[2.sup.2]    1616        0.68   233
256    [2.sup.3]-[2.sup.2]    1652        1.06   240
       [2.sup.3]-[2.sup.3]    1668        0.91   237
       [2.sup.3]-[2.sup.3]    1636        1.26   416
512    [2.sup.3]-[2.sup.2]    1648        0.97   400
       [2.sup.2]-[2.sup.2]    1668        0.93   396
       [2.sup.2]-[2.sup.2]    1600        1.04   738
1024   [2.sup.3]-[2.sup.3]    1644        1.42   730
       [2.sup.3]-[2.sup.2]    1620        1.50   741


N      Structure              Low-Frequency      High-Frequency
                              Dynamic     DPTR   Dynamic     DPTR
                              Power (W)          Power (W)

16     [2.sup.2]-[2.sup.2]    0.128       1.02   1.079       1.18
32     [2.sup.3]-[2.sup.2]    0.110       1.21   1.095       1.20
       [2.sup.2]-[2.sup.3]    0.147       1.17   1.145       1.28
64     [2.sup.2]-[2.sup.2]    0.107       1.28   1.162       1.31
       [2.sup.3]-[2.sup.3]    0.111       1.28   1.176       1.34
128    [2.sup.2]-[2.sup.2]    0.106       1.33   1.180       1.33
       [2.sup.3]-[2.sup.2]    0.087       1.44   1.186       1.35
256    [2.sup.2]-[2.sup.2]    0.091       1.50   1.101       1.25
       [2.sup.3]-[2.sup.2]    0.092       1.52   1.044       1.20
       [2.sup.3]-[2.sup.3]    0.077       1.54   0.968       1.10
512    [2.sup.3]-[2.sup.3]    0.089       1.78   1.120       1.26
       [2.sup.3]-[2.sup.2]    0.087       1.78   1.190       1.35
       [2.sup.2]-[2.sup.2]    0.095       1.62   1.227       1.39
1024   [2.sup.2]-[2.sup.2]    0.092       1.93   1.345       1.57
       [2.sup.3]-[2.sup.3]    0.084       2.03   1.291       1.50
       [2.sup.3]-[2.sup.2]    0.088       1.84   1.364       1.58


N      Structure              Low Frequency      High-Frequency
                              Dynamic     DPTR   Dynamic    DPTR
                              Power (W)          Power (W)

16     [2.sup.2]-[2.sup.2]    0.295       1.18   2.16       1.23
32     [2.sup.3]-[2.sup.2]    0.236       1.27   2.251      1.33
       [2.sup.2]-[2.sup.3]    0.253       1.12   2.225      1.32
64     [2.sup.2]-[2.sup.2]    0.278       1.25   2.239      1.33
       [2.sup.3]-[2.sup.3]    0.225       1.29   2.346      1.41
128    [2.sup.2]-[2.sup.2]    0.264       1.32   2.033      1.22
       [2.sup.3]-[2.sup.2]    0.183       1.37   1.817      1.11
256    [2.sup.2]-[2.sup.2]    0.261       1.33   1.897      1.17
       [2.sup.3]-[2.sup.2]    0.195       1.46   2.069      1.25
       [2.sup.3]-[2.sup.3]    0.159       1.51   1.811      1.09
512    [2.sup.3]-[2.sup.3]    0.165       1.65   2.107      1.29
       [2.sup.3]-[2.sup.2]    0.182       1.55   2.071      1.26
       [2.sup.2]-[2.sup.2]    0.284       1.49   2.191      1.31
1024   [2.sup.2]-[2.sup.2]    0.270       1.58   2.132      1.33
       [2.sup.3]-[2.sup.3]    0.160       1.86   1.708      1.04
       [2.sup.3]-[2.sup.2]    0.201       1.70   2.113      1.30

PROPOSED [MR2.sup.2]-[2.sup.3]

N      Structure                   Occupied Slices

                              Two parallel   Four parallel

16     [2.sup.2]-[2.sup.2]    278            606
32     [2.sup.3]-[2.sup.2]    391            850
       [2.sup.2]-[2.sup.3]    413            882
64     [2.sup.2]-[2.sup.2]    388            814
       [2.sup.3]-[2.sup.3]    539            1141
128    [2.sup.2]-[2.sup.2]    556            1059
       [2.sup.3]-[2.sup.2]    529            1054
256    [2.sup.2]-[2.sup.2]    587            1033
       [2.sup.3]-[2.sup.2]    722            1352
       [2.sup.3]-[2.sup.3]    706            1367
512    [2.sup.3]-[2.sup.3]    958            1694
       [2.sup.3]-[2.sup.2]    828            1404
       [2.sup.2]-[2.sup.2]    858            1405
1024   [2.sup.2]-[2.sup.2]    1052           1569
       [2.sup.3]-[2.sup.3]    1182           1920
       [2.sup.3]-[2.sup.2]    1187           1872


N      Method                Device                Slice LUTs

                             XC5VSX240T-2FF1738    1355
       Proposed              XC7VS485T-2FFG1661    1575
                             XC6CSX475T-1 FF1156   1307
16                           XC5VLX20T             1136
       Garrido et al. [21]   XC5VSX240T-2FF1738    --
       Garrido et al. [22]   XC6CSX475T-1 FF1156   --
       Ayinala et al. [20]   XC5VLX20T             2348
       Wang et al. [23]      XC5VSX240T-2FF1738    2688
                             XC5VSX240T-2FF1738    2301 - 2327
32     Proposed              XC7VS485T-2FFG1661    2527 - 2534
                             XC5VLX20T             2109 - 2135
       Ayinala et al. [20]   XC5VLX20T             3088
                             XC5VSX240T-2FF1738    2059 - 3294
       Proposed              XC7VS485T-2FFG1661    2258 - 3466
                             XC6CSX475T-1 FF1156   2831 - 4028
64                           XC5VLX20T             2007 - 3026
       Garrido et al. [21]   XC5VSX240T-2FF1738    --
       Garrido et al. [22]   XC6CSX475T-1 FF1156   --
       Ayinala et al. [20]   XC5VLX20T             3832
       Wang et al. [23]      XC5VSX240T-2FF1738    4440
                             XC5VSX240T-2FF1738    2914 - 4152
       Proposed              XC7VS485T-2FFG1661    3117 - 4334
256                          XC6CSX475T-1 FF1156   2837-4046
       Garrido et al. [21]   XC5VSX240T-2FF1738    --
       Garrido et al. [22]   XC6CSX475T-1 FF1156   --
       Wang et al. [23]      XC5VSX240T-2FF1738    6932
                             XC5VSX240T-2FF1738    4932 - 5483
       Proposed              XC7VS485T-2FFG1661    5063 - 6307
1024                         XC6CSX475T-1 FF1156   4828-6011
       Garrido et al. [21]   XC5VSX240T-2FF1738    --
       Garrido et al. [22]   XC6CSX475T-1 FF1156   --
       Wang et al. [23]      XC5VSX240T-2FF1738    11216

N      Method                Slices        DSPs      Frequency

                             619           12        439
       Proposed              606           12        477
                             506           12        400
16                           534           12        384
       Garrido et al. [21]   386           12        458
       Garrido et al. [22]   567           12        335
       Ayinala et al. [20]   1046          --        370
       Wang et al. [23]                    16        322
                             931 - 991     16-Dec    422 - 424
32     Proposed              850 - 882     16-Dec    471 - 477
                             829 - 835     16-Dec    382 - 385
       Ayinala et al. [20]   1290          --        370
                             895 - 1284    16 - 24   417 - 421
       Proposed              814 - 1141    16 - 24   475 - 477
                             699 - 1006    16 - 24   394 - 400
64                           811 - 1151    16 - 24   375 - 394
       Garrido et al. [21]   695           24        384
       Garrido et al. [22]   782           24        335
       Ayinala et al. [20]   1560          --        370
       Wang et al. [23]      --            32        303
                             1095 - 1744   28 - 36   404 - 417
       Proposed              1033 - 1367   28 - 36   472 - 477
256                          928 - 1305    28 - 36   400
       Garrido et al. [21]   1024          36        389
       Garrido et al. [22]   924           36        240
       Wang et al. [23]      --            48        297
                             1659 - 2433   40 - 48   400 - 411
       Proposed              1569 - 1920   40 - 48   472 - 474
1024                         1434 - 1824             400
       Garrido et al. [21]   1425          48        270
       Garrido et al. [22]   1351          48        227
       Wang et al. [23]      --            64        298
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Author:Dali, Mohammed; Guessoum, Abderezak; Gibson, Ryan M.; Amira, Abbes; Ramzan, Naeem
Publication:Advances in Electrical and Computer Engineering
Article Type:Report
Date:Feb 1, 2017
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