Single Fast Fourier Transform Research Articles

Optical frequency shifted FMCW Lidar system for unambiguous measurement of distance and velocity

A new frequency-modulated continuous-wave (FMCW) Lidar system is proposed to measure distance and velocity unambiguously by using a frequency-shifted probe light. The optical frequency shift is induced by an acousto-optic modulator (AOM). The generated beat signal is up converted to high frequency, sparing sufficient RF bandwidth to avoid signal aliasing caused by high Doppler frequency shifts. The aliasing-free beat signal thus can be used to extract distance and velocity information unambiguously. Here, we experimentally demonstrated real-time measurement on a disk spinning up to 26.43 m/s at 1.267 m away. The precision of the measured distance and velocity are 16 mm and 0.037 m/s respectively. Furthermore, the proposed scheme is capable of discriminating the direction of rotation based on a single Fast Fourier Transform analysis. In addition, we apply a free space circulator to the system to weaken the mirror-scattered signals and thus to achieve continuous measurement of beats signal over wide frequency range. Finally, a three-dimensional image of the spinning disk is reconstructed.

Real-time realistic computer-generated hologram with accurate depth precision and a large depth range.

Holographic display is an ideal technology for near-eye display to realize virtual and augmented reality applications, because it can provide all depth perception cues. However, depth performance is sacrificed by exiting computer-generated hologram (CGH) methods for real-time calculation. In this paper, volume representation and improved ray tracing algorithm are proposed for real-time CGH generation with enhanced depth performance. Using the single fast Fourier transform (S-FFT) method, the volume representation enables a low calculation burden and is efficient for Graphics Processing Unit (GPU) to implement diffraction calculation. The improved ray tracing algorithm accounts for accurate depth cues in complex 3D scenes with reflection and refraction, which is represented by adding extra shapes in the volume. Numerical evaluation is used to verify the depth precision. And experiments show that the proposed method can provide a real-time interactive holographic display with accurate depth precision and a large depth range. CGH of a 3D scene with 256 depth values is calculated at 30fps, and the depth range can be hundreds of millimeters. Depth cues of reflection and refraction images can also be reconstructed correctly. The proposed method significantly outperforms existing fast methods by achieving a more realistic 3D holographic display with ideal depth performance and real-time calculation at the same time.

Machine-Learning-Aided Optical OFDM for Intensity Modulated Direct Detection

End-to-end learning systems are conceived for Orthogonal Frequency Division Multiplexing (OFDM)-aided optical Intensity Modulation paired with Direct Detection (IM/DD) communications relying on the Autoencoder (AE) architecture in deep learning. We first propose an AE-aided Layered ACO-OFDM (LACO-OFDM) scheme, termed as LACONet, for exploiting the increased bandwidth efficiency of LACO-OFDM. LACONet employs a Neural Network (NN) at the transmitter for bit-to-symbol mapping, and another NN at the receiver for recovering the data bits, which together form an AE and can be trained in an end-to-end manner for simultaneously minimizing both the BER and PAPR. Moreover, the detection architecture of LACONet is drastically simplified compared to classical LACO-OFDM, since the Fast Fourier Transform (FFT) operation is applied only once at the receiver. We further propose a generalized AE-aided optical OFDM scheme for IM/DD communications, termed as IMDD-OFDMNet, where the unipolarity of the Time Domain (TD) signal is no longer guaranteed by the Hermitian Symmetry, but rather by taking the absolute square value of the complex TD signal. As such, all available subcarriers of IMDD-OFDMNet are used for carrying useful information, hence it has a higher throughput than the LACO-based schemes. As a further benefit, its transceiver requires only a single Inverse FFT or FFT. Finally, simulation results are provided to show that our learning schemes achieve better BER and PAPR performance than their conventional counterparts.

Universal Filtered Multicarrier Receiver Complexity Reduction to Orthogonal Frequency Division Multiplexing Receiver

The Universal Filtered Multicarrier (UFMC) waveform technology is one of the promising waveforms for 5G and beyond 5G networks. Owing 2N-point Fast Fourier Transform (FFT) processor at the UFMC receiver, the computational and implementation complexity is two times more than the conventional Orthogonal Frequency Division Multiplexing (OFDM) receiver system. In this paper, we proposed a simplified UFMC receiver structure to reduce computational complexity as well as hardware requirements. The received UFMC symbol simplified exactly to its equivalent after performing 2N-point FFT and decimation operations. In which, the mathematical model of the frequency-domain UFMC signal is rederived after processing through 2N-point FFT and decimator, and the simplified signal is generated with an N-point FFT. Accordingly, the 2N-point FFT processor and decimator are replaced with a single N-point FFT processor. This approach reduces the 50% computational complexity at the FFT processor level hence the hardware and processing time. The computational complexity of the proposed receiver model is approximately equivalent to the OFDM receiver. Additionally, analyzed the mathematical model for the simplified UFMC receiver and the comparative performance of the UFMC system with the conventional model.

Fast calculation of computer generated hologram based on single Fourier transform for holographic three-dimensional display

We present an efficient method for the fast calculation of computer generated hologram (CGH). The 3D object is split into sub-layers according to its depth information. A 2D all-in-focus image is generated by sequential tiling all the layers in one plane. A Fourier hologram that contains all the information of 3D object is calculated from the fast Fourier transform (FFT) of the reassembled 2D image. By multiplying a pre-calculated multifocal off-axis digital phase mask (DPM) to the Fourier hologram, the content of each layer is axially relocated to different depth in the Fourier transform optical system to reconstruct the 3D object. The computation speed of the proposed method is greatly improved with only single FFT calculation process. Both of simulation and experimental results proves the validation of the proposed method.

Multi-channel bioimpedance spectroscopy based on orthogonal baseband shifting

Objective. Multi-channel bioimpedance spectroscopy (BIS) systems typically sample each channel’s impedance sequentially using multiplexers and a single analog-to-digital converter. These systems may lose their real-time capability with an increasing number of channels, especially for low excitation frequencies. We propose a new method, called orthogonal baseband shifting (OBS), for high-speed parallel BIS data acquisition at multiple excitation frequencies with low hardware and computational effort. Approach. Similar to orthogonal frequency-division multiplexing, used for digital data transmission, OBS systems use channel-specific orthogonal carrier frequencies to modulate the voltage response of the tissue. Given a suitable choice of carrier frequencies, the modulated signals of all channels sum up without loss of information and cross-talk. The fast Fourier transform (FFT) of the summed signal reveals a spectrum of non-overlapping, interleaved BIS data from which the corresponding BIS data of each channel can be calculated. Main results. In simulations, the system design requires a minimum signal-to-noise ratio of 30 dB to achieve amplitude errors below 1% and phase errors below 0.8°. The hardware realization, called ‘AixBIS’, has been evaluated for impedance measurements between 0.1 and 10 Ω with multi-frequency excitations between 45 and 180 kHz. The impedance values acquired had an averaged precision of 3.67 mΩ, which is only 0.65‰ in relation to the measured impedance. The phase had a mean precision of 0.46°. Moreover, in vitro measurements achieved 140 full spectrum acquisitions per second. The impedance change measured in a silicone heart phantom showed a high correlation of 0.83 with the ventricles volume change (flow). Significance. The proposed method enables very fast impedance acquisition of all channels. A complete measurement is performed in the time of a single FFT acquisition, which is equal to the resolution bandwidth of the FFT. In addition, portable and low-power multi-channel BIS devices profit from highly reduced hardware effort. The outstanding performance of OBS measurements with the AixBIS system have the potential for in vivo BIS measurements in real-time.

Analysis of the Single-FFT Receiver for Layered ACO-OFDM in Visible Light Communications

Layered asymmetrically clipped optical orthogonal frequency division multiplexing (LACO-OFDM) has a high spectral efficiency. However, an iterative detection with a pair of fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) is required for LACO-OFDM, which increases the complexity for the receiver of LACO-OFDM. In this paper, the performance of the receiver with a single FFT is analyzed for LACO-OFDM in visible light communications. In the single-FFT receiver, the different layers of ACO-OFDM signals are distinguished in time domain without the FFT/IFFT pair. Because only one FFT is employed in the proposed receiver, the complexity of O( N log2 N ) can be achieved, which is much lower than the complexity of the conventional LACO-OFDM receiver. Simulation results illustrate that the proposed scheme has a signal-to-noise ratio (SNR) penalty of 1dB ∼ 3dB compared with the conventional solution in additive white Gaussian noise (AWGN) channel. However, the proposed receiver outperforms conventional LACO-OFDM solution in light emitting diode (LED) nonlinearity channel.

PARALLEL PIPELINED MULTI RADIX VARIABLE LENGTH FAST FOURIER TRANSFORM ARCHITECTURE

There is an enormous demand for high speed data communication or high speed internet using Long Term Evolution (LTE) or LTE-Advanced communication methods. To achieve the high speed data rate in the receiver side, it is necessary to achieve high throughput in the Fast Fourier transform (FFT) architecture. Hence there is demand to improve the throughput of the FFT architectures used for high speed data communication. The FFT architecture is designed and optimized for LTE-A applications. However, the high throughput has been achieved by sacrificing hardware resources of the Field Programmable Gate Array (FPGA). Many fixed and variable length FFT processors are proposed by the researchers with improved performance focusing either on algorithmic modifications, novel architectural optimizations or radix selection. Among these FFT processors, the goal and main objectives of this paper are: 1. To design and implement a pipelined FFT architecture to give high throughput for LTE-A MIMO applications 2. To develop an intellectual property (IP) core for FFT computation with variable FFT size 3. To propose a parallel implementation to increase the performance of LTE-A baseband processing system. In an Orthogonal Frequency Division Multiplexing (OFDM) baseband communication system, FFT operation is one of the highest computationally serious tasks which directly influence the communication system performance factor. The baseband hardware has to be capable as well as efficient enough to calculate FFT in specific timing restrictions. Thus, the parallel pipelined multi radix Variable Length FFT architectures for LTE-A Multiple-Input-Multiple-Output (MIMO) applications have been designed. The proposed FFT architecture delivers a throughput of 550 MSPS at the maximum clock rate of 550 MHz. The proposed FFT support the FFT length of 64, 128, 256, 512, 1024 and 2048 for LTE-Advanced MIMO standard. The proposed FFT architecture has been implemented in Xilinx Artix-7 FPGA device and the performance metrics have been analysed. Even though the proposed FFT architecture consumes additional Block-RAMs (BRAM) and quite an amount of Xilinx-Xtreme Digital Signal Processor (DSP)/ DSP48 resources, the Power Delay Product (PDP) of the proposed FFT is excellent compared to the existing FFT architectures. The proposed multi radix mixed 2/3/5 parallel pipelined 2048 point FFT architectures exhibits very less latency of 11.382 µs at 550 MHz clock frequency compared to the existing system with the latency of 56.88 µs at 200 MHz clock frequency. Moreover, the designed FFT architecture utilizes 96 Xilinx Xtreme DSP blocks 25040 clock cycles to complete the FFT operation with an excellent throughput of 2200 MSPS with FFT computation time of 45.528 µs at the clock frequency of 550 MHz for the 4x4 MIMO- OFDM architecture. Therefore, the designed FFT provides high throughput rate in order to meet the modern wireless standard specifications. The proposed FFT architecture outperforms well in terms high throughput, low latency and better PDP with extra hardware as trade-off for both for Single FFT and for MIMO technology.

Hierarchical Particle Mesh: An FFT-accelerated Fast Multipole Method

Abstract I describe a modification to the original Fast Multipole Method (FMM) of Greengard & Rokhlin that approximates the gravitation field of an FMM cell as a small uniform grid (a “gridlet”) of effective masses. The effective masses on a gridlet are set from the requirement that the multipole moments of the FMM cells are reproduced exactly, hence preserving the accuracy of the gravitational field representation. The calculation of the gravitational field from a multipole expansion can then be computed for all multipole orders simultaneously, with a single Fast Fourier Transform, significantly reducing the computational cost at a given value of the required accuracy. The described approach belongs to the class of “kernel independent” variants of the FMM method and works with any Green function.

A fast algorithm for the inversion of Abel’s transform

We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier coefficients of a suitable periodic function associated with its Abel transform. This allows us to compute the Legendre coefficients of the inverse Abel transform in an easy, fast and accurate way by means of a single Fast Fourier Transform. The algorithm is thus appropriate also for the inversion of Abel integrals given in terms of samples representing noisy measurements. Rigorous stability estimates are proved and the accuracy of the algorithm is illustrated also by some numerical experiments.

Effect of data gaps: comparison of different spectral analysis methods

Abstract. In this paper we investigate quantitatively the effect of data gaps for four methods of estimating the amplitude spectrum of a time series: fast Fourier transform (FFT), discrete Fourier transform (DFT), Z transform (ZTR) and the Lomb–Scargle algorithm (LST). We devise two tests: the single-large-gap test, which can probe the effect of a single data gap of varying size and the multiple-small-gaps test, used to study the effect of numerous small gaps of variable size distributed within the time series. The tests are applied on two data sets: a synthetic data set composed of a superposition of four sinusoidal modes, and one component of the magnetic field measured by the Venus Express (VEX) spacecraft in orbit around the planet Venus. For single data gaps, FFT and DFT give an amplitude monotonically decreasing with gap size. However, the shape of their amplitude spectrum remains unmodified even for a large data gap. On the other hand, ZTR and LST preserve the absolute level of amplitude but lead to greatly increased spectral noise for increasing gap size. For multiple small data gaps, DFT, ZTR and LST can, unlike FFT, find the correct amplitude of sinusoidal modes even for large data gap percentage. However, for in-situ data collected in a turbulent plasma environment, these three methods overestimate the high frequency part of the amplitude spectrum above a threshold depending on the maximum gap size, while FFT slightly underestimates it.

Non-iterative phase hologram computation for low speckle holographic image projection.

Phase-only spatial light modulators (SLMs) are widely used in holographic display applications, including holographic image projection (HIP). Most phase computer generated hologram (CGH) calculation algorithms have an iterative structure with a high computational load, and also are prone to speckle noise, as a result of the random phase terms applied on the desired images to mitigate the encoding noise. In this paper, we present a non-iterative algorithm, where simple Discrete Fourier Transform (DFT) relations are exploited to compute phase CGHs that exactly control half of the desired image samples (those on even - or odd - indexed rows - or columns) via a single Fast Fourier Transform (FFT) and trivial arithmetic operations. The encoding noise appearing on the uncontrolled half of the image samples is reduced by the application of structured, non-random initial phase terms so that speckle noise is also kept low. High quality reconstructions are obtained under temporal averaging of several SLM frames. Interlaced video within half of the addressable image area is readily deliverable without frame rate division. Our algorithm provides about 6X and 20X reduction in computational cost compared to IFTA and FIDOC algorithms, respectively. Simulations and experiments verify that the algorithm constitutes a promising option for real-time computation of phase CGHs.

The expansion in Gegenbauer polynomials: A simple method for the fast computation of the Gegenbauer coefficients

We present a simple and fast algorithm for the computation of the Gegenbauer transform, which is known to be very useful in the development of spectral methods for the numerical solution of ordinary and partial differential equations of physical interest. We prove that the coefficients of the expansion of a function f(x) in Gegenbauer (also known as ultraspherical) polynomials coincide with the Fourier coefficients of a suitable integral transform of the function f(x). This allows to compute N Gegenbauer coefficients in O(Nlog2N) operations by means of a single Fast Fourier Transform of the integral transform of f(x). We also show that the inverse Gegenbauer transform is expressible as the Abel-type transform of a suitable Fourier series. This fact produces a novel algorithm for the fast evaluation of Gegenbauer expansions.

Large-scale fast Fourier transform on a heterogeneous multi-core system

As interest in hybrid computing systems increases, people are eager to find new ways to exploit the unique and efficient computational power of the heterogeneous multi-core systems. Although there has been much interest in implementing high-performance fast Fourier transform (FFT) libraries on this kind of system, most existing libraries focus on small-scale FFTs whose data can fit in the local storage of a single accelerator. Real-world FFT applications often require much larger scale FFTs, but it is extremely challenging for heterogeneous multi-core system with distributed architectures to make efficient large FFT implementations. In this paper, we introduce the first known FFT library for the heterogeneous multi-core system with distributed architecture that can solve one-dimensional FFTs larger than what fits in a single accelerator. Our implementation achieves 67% performance improvement of FFTW 3.2.2 (Fastest Fourier Transform in the West) and sustains over 36 single precision FFT GFLOPs ‘end-to-end’ Achieving such high performance requires novel schemes for large-scale FFT factorization, data permutation and all-to-all exchanges, and buffer designs to maximize use of the local storage while minimizing communication overhead. One important finding in this paper is that large-scale FFT on this kind of architecture behaves as data transfer bound which is quite different from other architectures. A significant contribution of this paper is that for each major component of our algorithm, we explore many possible design options and present quantitative performance comparisons. This provides value beyond specific architecture, as it illustrates the fundamental features associated with different communication paradigms and mechanisms for the heterogeneous multi-core system. Today’s computer systems are increasingly being designed to include general purpose accelerators. The techniques in this paper can also be applied to these architectures especially when they have limited local storage or steep cache hierarchies. We also provide insights on applying techniques in this paper to similar architectures.

Modelling bioclimate by means of Fourier analysis of NOAA‐AVHRR NDVI time series in Western Argentina

AbstractThis study assessed whether the relationship of climate with foliar phenology is sufficiently robust to use a measure of foliar phenology to interpolate climate statistics in areas where observations are sparse. The normalized difference vegetation index (NDVI) was used to represent vegetation activity. As a measure of foliar phenology, we used parameters obtained by modelling NDVI time series with a Fast Fourier Transform (FFT) applied to a 9‐year time series of monthly National Oceanographic and Atmospheric Administration (NOAA) advanced very high resolution radiometer (AVHRR) NDVI global area coverage (GAC) images. The FFT decomposes the series into an average signal and to sinusoidal components. The selected FFT parameters were mean NDVI, and amplitude and phase for a 1‐year period. Our specific objective was to relate the ratio of precipitation, P, over potential evapotranspiration, ETP, to the FFT parameters in two complementary ways. The first was to use them as attributes in a numerical classification to obtain a map of foliar isophenology, and then associate these classes with bioclimatic types, thus generating a bioclimatic map. The second was to fit a multiple linear regression model with P/ETP as predicted variable and the FFT parameters as predictive variables. The regression model was then applied to obtain a map of the ratio P/ETP. The latter gave a second bioclimatic map. Foliar isophenology classes show a north–south decrease in phase value and increase in amplitude and mean NDVI values, thus reflecting the transition in climate conditions from hotter and drier to wetter and cooler. The model explains 92% (p‐value < 10−12) of the spatial variation in the P/ETP ratio. When using a single FFT parameter, no significant relationship was obtained. The three parameters provide complementary information to understand phenological variability in response to climate variability. Modelling bioclimate by means of monthly NDVI series summarized by Fourier analysis is an adequate tool to extend climate data where they are sparse. Copyright © 2007 Royal Meteorological Society

A Synchronization Protocol for OFDM-Based Asynchronous Access Networks

To generate multiple orthogonal carriers in an OFDM (Orthogonal Frequency Division Multiplexing) symbol, an FFT (Fast Fourier Transform) and IFFT (Inverse FFT) pair is utilized at the transmitter and the receiver, respectively. Thus, it may be difficult for an MN (Mobile Node) to reconstruct signals received from several BSs (Base Stations), when single FFT module is used. Moreover, when a GPS (Global Position System) is not supported, ICI (Inter Carrier Interference) may occur over multi-cell environments. This paper introduces a synchronization protocol for OFDM-based asynchronous networks that do not use GPS. The network protocol is designed to perform the plug & access radio configuration for next generation networks.

Population Differentiation in a Complex Bird Sound: A Comparison of Three Bioacoustical Analysis Procedures

AbstractWe examined three bioacoustical analysis methods for comparing complex sounds among different populations. We chose the D‐syllable of the chick‐a‐dee call of the black‐capped chickadee (Poecile atricapilla) because it is a broadband sound representative of a class of vocalizations, common in many animals, that resists simple subjective classification for comparative studies. We examined the properties of the D‐syllable in field‐recorded samples from three different populations. The first method of data extraction sampled the amplitude values of a spectrum obtained in a single fast Fourier transform (SFFT) taken at the midpoint of each D‐syllable using multi‐speech software. The second method employed spectrogram cross‐correlation (SPCC) to obtain a matrix of similarity values between D‐syllables in the samples using canary software. The third method calculated similarity values obtained from the evaluation of four acoustic features of the D‐syllables derived from multi‐taper spectral analysis (MTSA) using sound analysis software. Following data extraction by these three techniques, we used multivariate statistical procedures to reduce the data for examination of differences among populations and to represent in scatter‐plots the patterns of clustering of the sounds. We found that the SFFT in the middle of the D‐syllable provided the poorest population discrimination following statistical processing, the SPCC method produced the next clearest population separation, and the MTSA method resulted in the most distinct separation of the three populations of D‐syllables. In carrying out these comparisons, we discovered that the characteristic environmental noise of a recording area can influence the signal properties of broadband sounds being compared by automated procedures, and could lead to faulty conclusions unless appropriate care is taken to mitigate the noise in which the signals of interest are embedded. Consequently we re‐analyzed our data following noise reduction and found less discrete population separation overall. However, the methods of SPCC and MTSA retained the ability to separate populations, with MTSA providing the sharpest discrimination among groups.

Real-Time Classification of Traffic Signs

A challenging real-time imaging problem is classifying video traffic signs in background clutter under rotation, scale, and translation invariant conditions. Normalized Gabor Wavelet Transform features from multi-resolution filters were originally biologically-based; however, optimized features proved more effective. Two whole image template matching techniques were unsuccessful. A statistical pattern recognition system recognized approximately 30% of the images for the original features and 50% for the optimized features; however, a multi-layer perceptron (mlp) detected over 70% of the images with the optimized features. The research demonstrated the possibility of a future automotive navigation aid which robustly collects sign images and classifies these images in real-time with a single Fast Fourier Transform (FFT), a bank of filters and a trained neural net.

Computation of the Complex Error Function

Rational expansions for computing the complex error function $w(z) = e^{ - z^2 } {\text{erfc}}( - iz)$ are presented. These expansions have the following attractive properties: (1) they can be evaluated using a polynomial evaluation routine such as Homer’s method, (2) the polynomial coefficients can be computed once and for all by a single Fast Fourier Transform (FFT), and (3) high accuracy is achieved uniformly in the complex plane with only a small number of terms. Comparisons reveal that in some parts of the complex plane certain competitors may be more efficient. However, the difference in efficiency is never great, and the new algorithms are simpler than existing ones: a complete program takes eight lines of Matlab code.

Half duplex integral vocoder modem system

Disclosed is a system whereby in a transmit mode analog speech is sampled d converted digitally to 12 bits of accuracy and then fed into a fast Fourier transform (FFT) processor which analyzes the speech into spectral and pitch parameters. These parameters are then quantized into a data stream which acts as an input to a differential phase shift keying modulator. The modulator constructs a multi-tone modem output signal from a 25 tone stack which is converted to an analog signal which is fed into a communications channel. In the receive mode a modem input signal from the communications channel is fed through the same analog to digital converter which was used for input speech. A data stream is now provided which is coupled back into the same digital FFT processor which now operates to provide pitch spectral coefficients which are then separated and used by a synthesizer to reconstruct a speech waveform. The speech waveform is applied to a digital analog converter which is the same converter used to produce the line signal when the processor operates in the transmit mode. The FFT processor implements a single FFT algorithm which is used for both vocoder and modem processing in both the transmit and receive modes.