Showing papers on "Split-radix FFT algorithm published in 2018"

A nonuniform fast Fourier transform based on low rank approximation

Abstract: By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast and quasi-optimal algorithm for computing the NUDFT based on the fast Fourier transform (FFT). Our key observation is that an NUDFT and DFT matrix divided entry by entry is often well approximated by a low rank matrix, allowing us to express a NUDFT matrix as a sum of diagonally scaled DFT matrices. Our algorithm is simple to implement, automatically adapts to any working precision, and is competitive with state-of-the-art algorithms. In the fully uniform case, our algorithm is essentially the FFT. We also describe quasi-optimal algorithms for the inverse NUDFT and two-dimensional NUDFTs.

Feedforward FFT Hardware Architectures Based on Rotator Allocation

Abstract: 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.

R-FFAST: A Robust Sub-Linear Time Algorithm for Computing a Sparse DFT

Abstract: The fast Fourier transform is the most efficiently known way to compute the discrete Fourier transform (DFT) of an arbitrary $ n$ -length signal, and has a computational complexity of $O( n\log n)$ . If the DFT $ \vec {X}$ of the signal $ \vec {x}$ has only $k$ non-zero coefficients (where $ k ), can we do better? We addressed this question and presented a novel fast Fourier aliasing-based sparse transform (FFAST) algorithm that cleverly induces sparse-graph alias codes in the DFT domain, via a Chinese-remainder-theorem-guided sub-sampling operation in the time-domain. The induced sparse-graph alias codes are then exploited to devise a fast and iterative onion-peeling style decoder that computes $ k$ -sparse DFT of an $ n$ -length signal using only $O( k)$ time-domain samples and $O( k\log k)$ computations. In this paper, we generalize the FFAST framework by Pawar and Ramchandran to the noisy setting where the time-domain samples are corrupted by white Gaussian noise. We show that the noise-robust R-FFAST algorithm computes a $ k$ -sparse DFT of an $ n$ -length signal using $O( k\log ^{3} n)$ noise-corrupted time-domain samples in $O( k\log ^{4} n)$ complexity, i.e., sub-linear sample and time complexity . In Section IX , we provide extensive simulation results validating the empirical performance of the R-FFAST algorithm, e.g., we show that the R-FFAST algorithm computes a 50-sparse DFT of an ≈ 10 million length signal using only 4800 noisy samples with an effective signal-to-noise ratio of 5 dB. We also provide comparison of the run-time performance of several existing sparse Fourier transform implementations with that of the R-FFAST and show that it is almost 20 times faster, for comparable settings, than the state-of-the-art algorithm, while simultaneously providing better support recovery guarantees. While our theoretical results are for signals with a uniformly random support of the non-zero DFT coefficients and additive white Gaussian noise, we provide simulation results, which demonstrate that the R-FFAST algorithm performs well even for signals like magnetic resonance images, that have an approximately sparse Fourier spectrum with a non-uniform support for the dominant DFT coefficients.

An all-optical soliton FFT computational arrangement in the 3NLSE-domain

Abstract: In this paper an all-optical soliton method for calculating the Fast Fourier Transform (FFT) algorithm is presented. The method comes as an extension of the calculation methods (soliton gates) as they become possible in the cubic non-linear Schrodinger equation (3NLSE) domain, and provides a further proof of the computational abilities of the scheme. The method involves collisions entirely between first order solitons in optical fibers whose propagation evolution is described by the 3NLSE. The main building block of the arrangement is the half-adder processor. Expanding around the half-adder processor, the "butterfly" calculation process is demonstrated using first order solitons, leading eventually to the realisation of an equivalent to a full Radix-2 FFT calculation algorithm.

Sparse fast Fourier transform for exactly sparse signals and signals with additive Gaussian noise

Abstract: In recent years, the Fourier domain representation of sparse signals has been very attractive. Sparse fast Fourier transform (or sparse FFT) is a new technique which computes the Fourier transform in a compressed way, using only a subset of the input data. Sparse FFT computes the desired transform in sublinear time, which means in an amount of time that is smaller than the size of data. In big data problems and medical imaging to reduce the time that patient spends in MRI machine, FFT algorithm is not ‘fast’ enough anymore; therefore, the concept of sparse FFT is very important. Similar to compressed sensing, sparse FFT algorithm computes just the important components in the frequency domain in sublinear time. In this work, sparse FFT algorithm is studied and implemented on MATLAB and its performance is compared with Ann Arbor FFT. A filter is used to hash the frequencies in the n dimensional frequency-sparse signal into B bins, where $$B=n/16$$ . The filter is used for analyzing an important fraction of the whole signal, and therefore, instead of computing n point FFT, B point FFT is computed, and this results in a faster Fourier transform. The probability of success of the implemented algorithm is investigated for noiseless and noisy signals. It is deduced that as the sparsity increases, the probability of perfect transform also increases. If the performances of the algorithm in both cases are compared, it is clearly seen that the performances degrade when there is noise. Therefore, it can be concluded that the algorithm should be improved especially for noisy considerations. The solvability boundary for a constant probability of error is deducted and added to give insight for future studies.

Canonic Composite Length Real-Valued FFT

Abstract: This paper presents a novel algorithm to compute real-valued fast Fourier transform (RFFT) that is canonic with respect to the number of signal values. A signal value can correspond to a purely real or purely imaginary value, while a complex signal consists of 2 signal values. For an N-point RFFT, each stage needs not compute more than N signal values, since the degrees of freedom of the input data are N. Any more than N signal values computed at any stage is inherently redundant. In order to reduce the redundant samples, a sample removal lemma, and two types of twiddle factor transformations are proposed: pushing and modulation. We consider 4 different cases for an N = P × Q point canonic RFFT: 1) P is odd, Q is odd; 2) P is odd, Q is even; 3) P is even, Q is odd; and 4) P is even, Q is even. No twiddle factor transformation is required when P is odd. It is shown that the number of twiddle factors can be reduced by performing modulation transformation when P is even and Q is odd, while 2 real twiddle factor operations are pushed to 1 complex twiddle factor operation when P and Q are both even. Canonic RFFT for any composite length can be computed by applying the proposed algorithm recursively. Performances of different RFFTs are also discussed in this paper. The major advantages of the canonic RFFTs are that they require the least number of butterfly operations, lead to more regular sub-blocks in the data-flow, and only involve real datapath when mapped to architectures. However, we also show that canonic RFFTs may not be desirable when taking the cost of twiddle factor operations as the major consideration.

Characteristic Analysis of 1024-Point Quantized Radix-2 FFT/IFFT Processor

Abstract: The precise analysis and accurate measurement of harmonic provides a reliable scientific industrial application However, the high-performance DSP processor is the important method of electrical harmonic analysis Hence, in this research work, the effort was taken to design a novel high-resolution single 1024-point fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) processors for improvement of the harmonic measurement techniques Meanwhile, the project is started with design and simulation to demonstrate the benefit that is achieved by the proposed 1024-point FFT/IFFT processor The pipelined structure is incorporated in order to enhance the system efficiency As such, a pipelined architecture was proposed to statically scale the resolution of the processor to suite adequate trade-off constraints The proposed FFT makes use of programmable fixed-point/floating-point to realize higher precision FFT

A novel word length optimization method for radix-2$^k$ fixed-point FFT

Abstract: Dear editor, Fast Fourier transform (FFT) is one of the most fundamental algorithms used in digital signal processing. Many applications such as orthogonal frequency division multiplexing (OFDM), long term evolution (LTE), and ultra-wideband (UWB) systems require an area efficient, high accuracy FFT processor. To design a high-precision and lowcomplexity FFT/IFFT processor architecture, the optimum bit sizing technique in each stage is usually adopted. Many fixed-point pipeline FFT processors are designed in previous studies [1–5]. However, most of the word length schemes in these studies are proposed based on long-time fixedpoint simulation. It is difficult to provide an accurate, fast word length scheme because of the diversity of FFT algorithms and the complexity of circuit structure. In this letter, we focus on the widely-used radix-2 decimation-in-frequency (DIF) fast Fourier transform (FFT) algorithm. Based on our previous research on fixed-point FFT signal-to-quantization-noise ratio (SQNR) assessment [6], the analytical expression of the word length in different stages is deduced. We further put forward a word length optimization method based on the analytical expression. In our previous work [6], we reached an SQNR analytical expression of radix-2 fixed-point FFT. We re-list the output SQNR expression as follows: