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NASA NTRS2011

High-Throughput, Adaptive FFT Architecture for FPGA-Based Spaceborne Data Processors

Exponential growth in microelectronics technology such as field-programmable gate arrays (FPGAs) has enabled high-performance spaceborne instruments with increasing onboard data processing capabilities. As a commonly used digital signal processing (DSP) building block, fast Fourier transform (FFT) has been of great interest in onboard data processing applications, which needs to strike a reasonable balance between high-performance (throughput, block size, etc.) and low resource usage (power, silicon footprint, etc.). It is also desirable to be designed so that a single design can be reused and adapted into instruments with different requirements. The Multi-Pass Wide Kernel FFT (MPWK-FFT) architecture was developed, in which the high-throughput benefits of the parallel FFT structure and the low resource usage of Singleton s single butterfly method is exploited. The result is a wide-kernel, multipass, adaptive FFT architecture. The 32K-point MPWK-FFT architecture includes 32 radix-2 butterflies, 64 FIFOs to store the real inputs, 64 FIFOs to store the imaginary inputs, complex twiddle factor storage, and FIFO logic to route the outputs to the correct FIFO. The inputs are stored in sequential fashion into the FIFOs, and the outputs of each butterfly are sequentially written first into the even FIFO, then the odd FIFO. Because of the order of the outputs written into the FIFOs, the depth of the even FIFOs, which are 768 each, are 1.5 times larger than the odd FIFOs, which are 512 each. The total memory needed for data storage, assuming that each sample is 36 bits, is 2.95 Mbits. The twiddle factors are stored in internal ROM inside the FPGA for fast access time. The total memory size to store the twiddle factors is 589.9Kbits. This FFT structure combines the benefits of high throughput from the parallel FFT kernels and low resource usage from the multi-pass FFT kernels with desired adaptability. Space instrument missions that need onboard FFT capabilities such as the proposed DESDynl, SWOT (Surface Water Ocean Topography), and Europa sounding radar missions would greatly benefit from this technology with significant reductions in non-recurring cost and risk.

NguyenKobayashi, Kayla↗

NASA NTRS2010

A High-Throughput, Adaptive FFT Architecture for FPGA-Based Space-Borne Data Processors

Historically, computationally-intensive data processing for space-borne instruments has heavily relied on ground-based computing resources. But with recent advances in functional densities of Field-Programmable Gate-Arrays (FPGAs), there has been an increasing desire to shift more processing on-board; therefore relaxing the downlink data bandwidth requirements. Fast Fourier Transforms (FFTs) are commonly used building blocks for data processing applications, with a growing need to increase the FFT block size. Many existing FFT architectures have mainly emphasized on low power consumption or resource usage; but as the block size of the FFT grows, the throughput is often compromised first. In addition to power and resource constraints, space-borne digital systems are also limited to a small set of space-qualified memory elements, which typically lag behind the commercially available counterparts in capacity and bandwidth. The bandwidth limitation of the external memory creates a bottleneck for a large, high-throughput FFT design with large block size. In this paper, we present the Multi-Pass Wide Kernel FFT (MPWK-FFT) architecture for a moderately large block size (32K) with considerations to power consumption and resource usage, as well as throughput. We will also show that the architecture can be easily adapted for different FFT block sizes with different throughput and power requirements. The result is completely contained within an FPGA without relying on external memories. Implementation results are summarized.

Nguyen, Kayla↗

NASA NTRS1988

FFT applications to plane-polar near-field antenna measurements

The four-point bivariate Lagrange interpolation algorithm was applied to near-field antenna data measured in a plane-polar facility. The results were sufficiently accurate to permit the use of the FFT (fast Fourier transform) algorithm to calculate the far-field patterns of the antenna. Good agreement was obtained between the far-field patterns as calculated by the Jacobi-Bessel and the FFT algorithms. The significant advantage in using the FFT is in the calculation of the principal plane cuts, which may be made very quickly. Also, the application of the FFT algorithm directly to the near-field data was used to perform surface holographic diagnosis of a reflector antenna. The effects due to the focusing of the emergent beam from the reflector, as well as the effects of the information in the wide-angle regions, are shown. The use of the plane-polar near-field antenna test range has therfore been expanded to include these useful FFT applications.

Gatti, Mark S.↗

NASA NTRS1989

Performance of FFT methods in local gravity field modelling

Fast Fourier transform (FFT) methods provide a fast and efficient means of processing large amounts of gravity or geoid data in local gravity field modelling. The FFT methods, however, has a number of theoretical and practical limitations, especially the use of flat-earth approximation, and the requirements for gridded data. In spite of this the method often yields excellent results in practice when compared to other more rigorous (and computationally expensive) methods, such as least-squares collocation. The good performance of the FFT methods illustrate that the theoretical approximations are offset by the capability of taking into account more data in larger areas, especially important for geoid predictions. For best results good data gridding algorithms are essential. In practice truncated collocation approaches may be used. For large areas at high latitudes the gridding must be done using suitable map projections such as UTM, to avoid trivial errors caused by the meridian convergence. The FFT methods are compared to ground truth data in New Mexico (xi, eta from delta g), Scandinavia (N from delta g, the geoid fits to 15 cm over 2000 km), and areas of the Atlantic (delta g from satellite altimetry using Wiener filtering). In all cases the FFT methods yields results comparable or superior to other methods.

Forsberg, Rene↗

NASA NTRS1988

On the application of pseudo-spectral FFT technique to non-periodic problems

The reduction-to-periodicity method using the pseudo-spectral Fast Fourier Transform (FFT) technique is applied to the solution of nonperiodic problems including the two-dimensional Navier-Stokes equations. The accuracy of the method is demonstrated by calculating derivatives of given functions, one- and two-dimensional convective-diffusive problems, and by comparing the relative errors due to the FFT method with seocnd order Finite Difference Methods (FDM). Finally, the two-dimensional Navier-Stokes equations are solved by a fractional step procedure using both the FFT and the FDM methods for the driven cavity flow and the backward facing step problems. Comparisons of these solutions provide a realistic assessment of the FFT method indicating its range of applicability.

Biringen, S.↗

NASA NTRS1979

A comparison of WFTA and FFT programs

Bounds on the minimum number of data transfers (i.e., loads and stores) required by WFTA (Winograd Fourier Transform Algorithm) and FFT programs are presented. The analysis is applicable to those general-purpose computers with a small number of general processor registers (e.g., the IBM370, PDP-11, etc.). It is shown that the 1008-point WFTA requires about 21% more data transfers than the 1024-point radix-4 FFT; on the other hand, the 120-point WFTA has about 22% fewer data transfers than the 128-point radix-2 FFT. Finally, comparisons of the 'total' program execution times (multiplications, additions, and data transfers, but not indexing or permutations) are presented.

Nawab, H.↗

NASA NTRS1979

Bounds on the minimum number of data transfers in WFTA and FFT programs

Bounds on the minimum number of data transfers (i.e., loads, stores, copies) required by WFTA and FFT programs are presented. The analysis is applicable to those general-purpose computers with M general processor registers, where M is equal to or greater than 4 but much less than the transform length. It is shown that the 1008-point WFTA requires about 21 percent more data transfers than the 1024-point radix-4 FFT; on the other hand, the 120-point WFTA has about the same number of data transfers as the mixed radix (4 x 4 x 4 x 2) version of the 128-point FFT and 22 percent fewer than the radix-2 version. Finally, comparisons of the 'total' program execution times (multiplications, additions, and data transfers, but not indexing or permutations) are presented.

Nawab, H.↗

NASA NTRS1983

Numerical evaluation of the radiation from unbaffled, finite plates using the FFT

An iteration technique is described which numerically evaluates the acoustic pressure and velocity on and near unbaffled, finite, thin plates vibrating in air. The technique is based on Rayleigh's integral formula and its inverse. These formulas are written in their angular spectrum form so that the fast Fourier transform (FFT) algorithm may be used to evaluate them. As an example of the technique the pressure on the surface of a vibrating, unbaffled disk is computed and shown to be in excellent agreement with the exact solution using oblate spheroidal functions. Furthermore, the computed velocity field outside the disk shows the well-known singularity at the rim of the disk. The radiated fields from unbaffled flat sources of any geometry with prescribed surface velocity may be evaluated using this technique. The use of the FFT to perform the integrations in Rayleigh's formulas provides a great savings in computation time compared with standard integration algorithms, especially when an array processor can be used to implement the FFT.

Williams, E. G.↗

NASA NTRS1984

Pipelined digital SAR azimuth correlator using hybrid FFT-transversal filter

A synthetic aperture radar system (SAR) having a range correlator is provided with a hybrid azimuth correlator which utilizes a block-pipe-lined fast Fourier transform (FFT). The correlator has a predetermined FFT transform size with delay elements for delaying SAR range correlated data so as to embed in the Fourier transform operation a corner-turning function as the range correlated SAR data is converted from the time domain to a frequency domain. The azimuth correlator is comprised of a transversal filter to receive the SAR data in the frequency domain, a generator for range migration compensation and azimuth reference functions, and an azimuth reference multiplier for correlation of the SAR data. Following the transversal filter is a block-pipelined inverse FFT used to restore azimuth correlated data in the frequency domain to the time domain for imaging.

Wu, C.↗

NASA NTRS1991

A 640-MHz 32-megachannel real-time polyphase-FFT spectrum analyzer

A polyphase fast Fourier transform (FFT) spectrum analyzer being designed for NASA's Search for Extraterrestrial Intelligence (SETI) Sky Survey at the Jet Propulsion Laboratory is described. By replacing the time domain multiplicative window preprocessing with polyphase filter processing, much of the processing loss of windowed FFTs can be eliminated. Polyphase coefficient memory costs are minimized by effective use of run length compression. Finite word length effects are analyzed, producing a balanced system with 8 bit inputs, 16 bit fixed point polyphase arithmetic, and 24 bit fixed point FFT arithmetic. Fixed point renormalization midway through the computation is seen to be naturally accommodated by the matrix FFT algorithm proposed. Simulation results validate the finite word length arithmetic analysis and the renormalization technique.

Zimmerman, G. A.↗

NASA NTRS1991

Improved FFT-based numerical inversion of Laplace transforms via fast Hartley transform algorithm

The disadvantages of numerical inversion of the Laplace transform via the conventional fast Fourier transform (FFT) are identified and an improved method is presented to remedy them. The improved method is based on introducing a new integration step length Delta(omega) = pi/mT for trapezoidal-rule approximation of the Bromwich integral, in which a new parameter, m, is introduced for controlling the accuracy of the numerical integration. Naturally, this method leads to multiple sets of complex FFT computations. A new inversion formula is derived such that N equally spaced samples of the inverse Laplace transform function can be obtained by (m/2) + 1 sets of N-point complex FFT computations or by m sets of real fast Hartley transform (FHT) computations.

Hwang, Chyi↗

NASA NTRS1982

Numerical evaluation of the Rayleigh integral for planar radiators using the FFT

Rayleigh's integral formula is evaluated numerically for planar radiators of any shape, with any specified velocity in the source plane using the fast Fourier transfrom algorithm. The major advantage of this technique is its speed of computation - over 400 times faster than a straightforward two-dimensional numerical integration. The technique is developed for computation of the radiated pressure in the nearfield of the source and can be easily extended to provide, with little computation time, the vector intensity in the nearfield. Computations with the FFT of the nearfield pressure of baffled rectangular plates with clamped and free boundaries are compared with the 'exact' solution to illuminate any errors. The bias errors, introduced by the FFT, are investigated and a technique is developed to significantly reduce them.

Williams, E. G.↗

NASA NTRS1988

A high-performance FFT algorithm for vector supercomputers

Many traditional algorithms for computing the fast Fourier transform (FFT) on conventional computers are unacceptable for advanced vector and parallel computers because they involve nonunit, power-of-two memory strides. A practical technique for computing the FFT that avoids all such strides and appears to be near-optimal for a variety of current vector and parallel computers is presented. Performance results of a program based on this technique are given. Notable among these results is that a FORTRAN implementation of this algorithm on the CRAY-2 runs up to 77-percent faster than Cray's assembly-coded library routine.

Bailey, David H.↗

NASA NTRS1990

Interpolation And FFT Of Near-Field Antenna Measurements

Bivariate Lagrange interpolation applied to plane-polar measurement scans. Report discusses recent advances in application of fast-Fourier-transform (FFT) techniques to measurements of near radiation fields of antennas on plane-polar grid. Attention focused mainly on use of such measurements to calculate far radiation fields. Also discussion of use of FFT's in holographic diagnosis of distortions of antenna reflectors. Advantage of scheme, it speeds calculations because it requires fewer data and manipulations of data than other schemes used for this purpose.

Gatti, Mark S.↗

NASA NTRS1990

The use of the FFT for the efficient solution of the problem of electromagnetic scattering by a body of revolution

The enhancement of the computational efficiency of the body of revolution (BOR) scattering problem is discused with a view to making it practical for solving large-body problems. The problem of EM scattering by a perfectly conducting BOR is considered, although the methods can be extended to multilayered dielectric bodies as well. Typically, the generation of the elements of the moment method matrix consumes a major portion of the computational time. It is shown how this time can be significantly reduced by manipulating the expression for the matrix elements to permit efficient FFT computation. A technique for extracting the singularity of the Green function that appears within the integrands of the matrix diagonal is also presented, further enhancing the usefulness of the FFT. The computation time can thus be improved by at least an order of magnitude for large bodies in comparison to that for previous algorithms.

Gedney, Stephen D.↗

NASA NTRS1992

Efficient Two-Dimensional-FFT Program

Program computes 64 X 64-point fast Fourier transform in less than 17 microseconds. Optimized 64 X 64 Point Two-Dimensional Fast Fourier Transform combines performance of real- and complex-valued one-dimensional fast Fourier transforms (FFT's) to execute two-dimensional FFT and coefficients of power spectrum. Coefficients used in many applications, including analyzing spectra, convolution, digital filtering, processing images, and compressing data. Source code written in C, 8086 Assembly, and Texas Instruments TMS320C30 Assembly languages.

Miko, J.↗

NASA NTRS1991

The Filled Arm Fizeau Telescope (FFT)

Attention is given to the design of a Mills Cross imaging interferometer in which the arms are fully filled with mirror segments of a Ritchey-Chretien primary and which has sensitivity to 27th magnitude per pixel and resolution a factor of 10 greater than Hubble. The optical design, structural configuration, thermal disturbances, and vibration, material, control, and metrology issues, as well as scientific capabilities are discussed, and technology needs are identified. The technologies under consideration are similar to those required for the development of the other imaging interferometers that have been proposed over the past decade. A comparison of the imaging capabilities of a 30-m diameter FFT, an 8-m telescope with a collecting area equal to that of the FFT, and the HST is presented.

Synnott, S. P.↗

NASA NTRS1996

Error and Complexity Analysis for a Collocation-Grid-Projection Plus Precorrected-FFT Algorithm for Solving Potential Integral Equations with LaPlace or Helmholtz Kernels

In this paper we derive error bounds for a collocation-grid-projection scheme tuned for use in multilevel methods for solving boundary-element discretizations of potential integral equations. The grid-projection scheme is then combined with a precorrected FFT style multilevel method for solving potential integral equations with 1/r and e(sup ikr)/r kernels. A complexity analysis of this combined method is given to show that for homogeneous problems, the method is order n natural log n nearly independent of the kernel. In addition, it is shown analytically and experimentally that for an inhomogeneity generated by a very finely discretized surface, the combined method slows to order n(sup 4/3). Finally, examples are given to show that the collocation-based grid-projection plus precorrected-FFT scheme is competitive with fast-multipole algorithms when considering realistic problems and 1/r kernels, but can be used over a range of spatial frequencies with only a small performance penalty.

Phillips, J. R.↗