The fast Fourier transform (FFT) algorithm was developed by Cooley and Tukey in 1965. It could reduce the computational complexity of discrete Fourier transform significantly from $$O(N^2)$$
 to $$O(N\log _2 {N})$$
 . The invention of FFT is considered as a landmark development in the field of digital signal processing (DSP), since it could expedite the DSP algorithms significantly such that real-time digital signal processing could be possible. During the past 50 years, many researchers have contributed to the advancements in the FFT algorithm to make it faster and more efficient in order to match with the requirements of various applications. In this article, we present a brief overview of the key developments in FFT algorithms along with some popular applications in speech and image processing, signal analysis, and communication systems.

50 Years of FFT Algorithms and Applications

In this paper, we present new feedforward FFT hardware architectures based on rotator allocation. The rotator allocation approach consists in distributing the rotations of the FFT in such a way that the number of edges in the FFT that need rotators and the complexity of the rotators are reduced. Radix-2 and radix-2 k feedforward architectures based on rotator allocation are presented in this paper. Experimental results show that the proposed architectures reduce the hardware cost significantly with respect to previous FFT architectures.

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Feedforward FFT Hardware Architectures Based on Rotator Allocation

In this paper, an efficient mapping of the pipeline single-path delay feedback (SDF) fast Fourier transform (FFT) architecture to field-programmable gate arrays (FPGAs) is proposed. By considering the architectural features of the target FPGA, significantly better implementation results are obtained. This is illustrated by mapping an R22SDF 1024-point FFT core toward both Xilinx Virtex-4 and Virtex-6 devices. The optimized FPGA mapping is explored in detail. Algorithmic transformations that allow a better mapping are proposed, resulting in implementation achievements that by far outperforms earlier published work. For Virtex-4, the results show a 350% increase in throughput per slice and 25% reduction in block RAM (BRAM) use, with the same amount of DSP48 resources, compared with the best earlier published result. The resulting Virtex-6 design sees even larger increases in throughput per slice compared with Xilinx FFT IP core, using half as many DSP48E1 blocks and less BRAM resources. The results clearly show that the FPGA mapping is crucial, not only the architecture and algorithm choices.

Efficient FPGA Mapping of Pipeline SDF FFT Cores

The field of pipelined FFT hardware architectures has been studied during the last 50 years. This paper is a survey that includes the main advances in the field related to architectures for complex input data and power-of-two FFT sizes. Furthermore, the paper is intended to be educational, so that the reader can learn how the architectures work. Finally, the paper divides the architectures into serial and parallel. This classification puts together those architectures that are conceived for a similar purpose and, therefore, are comparable.

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A Survey on Pipelined FFT Hardware Architectures

This paper presents the fastest fast Fourier transform (FFT) hardware architectures so far. The architectures are based on a fully parallel implementation of the FFT algorithm. In order to obtain the highest throughput while keeping the resource utilization low, we base our design on making use of advanced shift-and-add techniques to implement the rotators and on selecting the most suitable FFT algorithms for these architectures. Apart from high throughput and resource efficiency, we also guarantee high accuracy in the proposed architectures. For the implementation, we have developed an automatic tool that generates the architectures as a function of the FFT size, input word length and accuracy of the rotations. We provide experimental results covering various FFT sizes, FFT algorithms, and field-programmable gate array boards. These results show that it is possible to break the barrier of 100 GS/s for FFT calculation.

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World’s Fastest FFT Architectures: Breaking the Barrier of 100 GS/s

This brief presents a new type of fast Fourier transform (FFT) hardware architectures called serial commutator (SC) FFT. The SC FFT is characterized by the use of circuits for bit-dimension permutation of serial data. The proposed architectures are based on the observation that, in the radix-2 FFT algorithm, only half of the samples at each stage must be rotated. This fact, together with a proper data management, makes it possible to allocate rotations only every other clock cycle. This allows for simplifying the rotator, halving the complexity with respect to conventional serial FFT architectures. Likewise, the proposed approach halves the number of adders in the butterflies with respect to previous architectures. As a result, the proposed architectures use the minimum number of adders, rotators, and memory that are necessary for a pipelined FFT of serial data, with 100% utilization ratio.

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The Serial Commutator FFT

This paper presents a new radix-2^k multi-path FFT architecture, named MSC FFT, which is based on a single-path radix-2 serial commutator (SC) FFT architecture. The proposed multi-path architecture has a very high hardware utilization that results in a small chip area, while providing high throughput. In addition, the adoption of radix-2^k FFT algorithms allows for simplifying the rotators even further. It is achieved by optimizing the structure of the processing element (PE). The implemented architecture is a 128-point 4-parallel multi-path SC FFT using 90 nm process. Its area and power consumption at 250 MHz are only 0.167 mm2 and 14.81 mW, respectively. Compared with existing works, the proposed design reduces significantly the chip rea and the power consumption, while providing high throughput.

A 128-point Multi-Path SC FFT Architecture

This paper presents a new radix-2k multi-path FFT architecture, named MSC FFT, which is based on a single-path radix-2 serial commutator (SC) FFT architecture. The proposed multi-path architecture has a very high hardware utilization that results in a small chip area, while providing high throughput. In addition, the adoption of radix-2k FFT algorithms allows for simplifying the rotators even further. It is achieved by optimizing the structure of the processing element (PE). The implemented architecture is a 128-point 4-parallel multi-path SC FFT using 90 nm process. Its area and power consumption at 250 MHz are only 0.167 mm2 and 14.81 mW, respectively. Compared with existing works, the proposed design reduces significantly the chip area and the power consumption, while providing high throughput.

Shen-Jui Huang

Papers

The Serial Commutator FFT

Feedforward FFT Hardware Architectures Based on Rotator Allocation

A 128-point Multi-Path SC FFT Architecture

A 128-Point Multi-Path SC FFT Architecture