Nonuniform Frequency Sampling Research Articles

Inverse Scattering for Coupled Nonuniform Lossless Transmission Lines Based on a State-Space Method and Nonuniform Frequency Sampling

Inverse Scattering for Coupled Nonuniform Lossless Transmission Lines Based on a State-Space Method and Nonuniform Frequency Sampling

Dual-mode incoherent 3D imaging system with improved lateral resolution

In this paper, the imaging features of an incoherent holographic structure is analysed more thoroughly. The relation between the lateral resolution and frequency sampling step in an incoherent holographic system which is indeed a multi-view projection structure is formulated mathematically. Based on the presented mathematical relation the image resolution is improved through synthetically enlarging the lenses aperture. The non-uniform frequency sampling due to enlarging the lenses aperture is resolved through time-multiplexing. Through the proposed dual-mode, spatio-temporal, incoherent holographic structure, image quality and resolution is improved considerably. Experiments are performed to validate the claimed quality improvement.

Sparse Nonuniform Frequency Sampling for Fast SFCW Radar Imaging

Stepped frequency continuous wave (SFCW) radar achieves wide bandwidth by synthesizing series of monochromatic pulses in a consecutive manner. Uniform frequency sampling is often performed with a constant frequency step, which in turn limits the maximum unambiguous range achievable by the radar to reliably distinguish targets. Thus, a small frequency step must be selected when clutters exist at long distances even though the target of interests is located at much closer distances. This is costly since a large number of frequencies must be synthesized, which leads to slow acquisition speed. In this paper, a sparse nonuniform frequency sampling method is proposed to effectively reduce the number of frequencies while suppressing aliasing effects from clutters. The Poisson Sum Formula is utilized to derive a deterministic formula for choosing a discrete set of frequencies within a specified frequency band. A corresponding frequency weighting formula is added in order to maintain the same target impulse response in time-domain as the one achieved by dense uniform frequency sampling. Numerical and experimental results are presented to demonstrate the improved performances of the proposed sparse sampling method for SFCW radar imaging.

K-Space Aware Multi-Static Millimeter-Wave Imaging.

This paper focuses on an ef?cient approach of designing multi-static arrays for millimeter-wave imaging, based on the k-space or Fourier-spatial domain characteristic of imaging systems. Our goal is to decrease the redundancy of the data measured by each antenna, and to improve the resolution of the reconstructed image. The proposed technique is based on determining the role of each transmitter and receiver, in collecting the data from each voxel of the target in k-space domain and then rotating transmitters' beams to measure the desirable information. The effect of non-uniform redundant k-space domain frequency samples that act as an undesirable ?lter, is compensated using a modi?ed SAR back-projection algorithm. Experimental and simulation results are presented and compared with that of a sparse multi-static array with the same number of transmitters and receivers. Our simulations and measurements show signi?cant improvement in terms of overall quality and edge preservation in the reconstructed images. Also, the obtained results demonstrate that using the proposed structure and algorithm, the average improvement in peak-signalto-noise ratio (PSNR), structural similarity index measure (SSIM) and digital image correlation (DIC) metrics of 3.03 dB, 0.22 and 0.173, are achieved, respectively.

Some Notes on the Use of the Windowed Fourier Transform for Spectral Analysis of Discretely Sampled Data

The properties of the Gabor and Morlet transforms are examined with respect to the Fourier analysis of discretely sampled data. Forward and inverse transform pairs based on a fixed window with uniform sampling of the frequency axis can satisfy numerically the energy and reconstruction theorems; however, transform pairs based on a variable window or nonuniform frequency sampling in general do not. Instead of selecting the shape of the window as some function of the central frequency, we propose constructing a single window with unit energy from an arbitrary set of windows that is applied over the entire frequency axis. By virtue of using a fixed window with uniform frequency sampling, such a transform satisfies the energy and reconstruction theorems. The shape of the window can be tailored to meet the requirements of the investigator in terms of time/frequency resolution. The algorithm extends naturally to the case of nonuniform signal sampling without modification beyond identification of the Nyquist interval.

Comparative Study of Non-uniform Frequency Sampling Method in Equiripple Digital Filter Design

Comparative Study of Non-uniform Frequency Sampling Method in Equiripple Digital Filter Design

A Method for Designing FIR Filters with Arbitrary Magnitude Characteristic Used for Modeling Human Audiogram

Abstract —This paper presents an iterative method for designing FIR filters that implement arbitrary magnitude characteristics, defined by the user through a set of frequency-magnitude points (frequency samples). The proposed method is based on the non-uniform frequency sampling algorithm. For each iteration a new set of frequency samples is generated, by processing the set used in the previous run; this implies changing the samples location around the previous frequency values and adjusting their magnitude through interpolation. If necessary, additional samples can be introduced, as well. After each iteration the magnitude characteristic of the resulting filter is determined by using the non-uniform DFT and compared with the required one; if the errors are larger than the acceptable levels (set by the user) a new iteration is run; the length of the resulting filter and the values of its coefficients are also taken into consideration when deciding a re-run. To demonstrate the efficiency of the proposed method a tool for designing FIR filters that match human audiograms was implemented in LabVIEW. It was shown that the resulting filters have smaller coefficients than the standard one, and can also have lower order, while the errors remain relatively small. Index Terms—discrete Fourier transforms, error analysis, FIR filter, interpolation and non-uniform sampling.

Nonuniform frequency sampling with active learning: application to wide-band frequency-domain modeling and design

One must employ many frequency points to synthesize a wide-band time-domain signal scattered or radiated from a given linear device. If the structure is large relative to wavelengths of interest, the large number of required frequency-domain computations may require significant computational resources. An active learning framework is introduced to reduce the number of frequencies needed for frequency-domain numerical computations or measurements. This method is used to optimally select frequency points at which a computation of measurement should be performed, resulting in nonuniform frequency sampling and often a reduced number of total frequency points (vis-a/spl grave/-vis uniform sampling). In this paper we focus on wide-band frequency-domain numerical modeling, demonstrating a reduction in the required number of computations. The method consists of two basic steps, tied to fitting the frequency-domain data to a simple parametric model. One step involves estimation of model parameters using a least-square (LS) algorithm. This process is based on data from frequency points computed thus far by the rigorous numerical model. The next step is to optimally choose the next frequency point for analysis by the numerical model. This new frequency point is selected with the goal of reducing uncertainty in the simplified parametric model (quantified via the Fisher information matrix). Iterating these two steps, a sequence of numerical computations are performed, each at the most informative frequency for learning the parameters of the associated simpler parametric model. After demonstrating this technique with scattering problems, the idea is employed for design of an ultrawide-band antenna. In this design, a genetic algorithm (GA) is employed to optimize the geometry and the resistive loading of an antenna, with the number of required frequency-domain numerical computations reduced by the parametric model and active learning.

2D Zero-Phase FIR Filter Design with Nonuniform Frequency Sampling

In this paper, nonuniform frequency sampling techniques used in the design of two-dimensional zero-phase FIR filters are proposed and investigated. The design problem is treated as a bivariate interpolation problem with unevenly spaced data. The main idea is to select (find) sampling locations in the (ω1, ω2) frequency plane and corresponding sample values Hd(ω1k, ω2k) of the desired filter frequency response such that the designed filter performance is high. Although the filters designed with the proposed techniques are not optimal, the methods are conceptually simple and produce filters with high degree of shape regularity and approximation error comparable and sometimes even smaller than the conventional 2D FIR filter design methods.

2D ZERO-PHASE FIR FILTER DESIGN WITH NONUNIFORM FREQUENCY SAMPLING

In this paper, nonuniform frequency sampling techniques used in the design of two-dimensional zero-phase FIR filters are proposed and investigated. The design problem is treated as a bivariate interpolation problem with unevenly spaced data. The main idea is to select (find) sampling locations in the (ω1, ω2) frequency plane and corresponding sample values Hd(ω1k, ω2k) of the desired filter frequency response such that the designed filter performance is high. Although the filters designed with the proposed techniques are not optimal, the methods are conceptually simple and produce filters with high degree of shape regularity and approximation error comparable and sometimes even smaller than the "conventional" 2D FIR filter design methods.

Efficient robust DTMF decoding using the subband NDFT

Digital dual-tone multi-frequency (DTMF) decoding is often performed by applying a frequency transformation to the received DTMF signal to measure the energies present at the DTMF frequencies. A DTMF decoding algorithm is developed using the concepts of non-uniform frequency sampling and subband decomposition. Since DTMF signals occupy the low-frequency band of the telephone bandwidth, an efficient computation is achieved by retaining only the higher subband in a two-band decomposition of the received signal. Simulations based on the AT&T TouchTone standard are presented to compare the performances of DTMF decoding algorithms based on the subband non-uniform discrete Fourier transform (SB-NDFT), NDFT, SB-DFT and DFT. Results show that the SB-NDFT-based algorithm is more robust to the variation of DTMF frequencies within the specified tolerance bands. Besides, it also requires a significantly lower computational effort to attain a specified level of performance.

The nonuniform discrete Fourier transform and its applications in filter design. II. 2-D

For part I see ibid., vol. 43, no. 6, p. 422-33 (1996). The concept of the nonuniform discrete Fourier transform (NDFT) is extended to two dimensions to provide a basic framework for nonuniform sampling of 2-D sequences in the frequency domain. The 2-D NDFT of a sequence of size N/sub 1//spl times/N/sub 2/ is defined as samples of its 2-D z-transform evaluated at N/sub 1/N/sub 2/ distinct points located in the 4-D (z/sub 1/, z/sub 2/) space. These points are chosen appropriately so that the inverse transform exists. We discuss two special cases in which the choice of the sampling points is constrained so that the 2-D NDFT matrix is guaranteed to be nonsingular, and the number of operations required for computing its inverse is reduced, The 2-D NDFT is applied to nonuniform frequency sampling design of 2-D finite-impulse-response (FIR) filters. Nonseparable filters with good passband shapes and low peak ripples are obtained. This is illustrated by design examples, in which 2-D filters with various shapes are designed and compared with those obtained by other existing methods.

The nonuniform discrete Fourier transform and its applications in filter design. I. 1-D

For part I see ibid., vol. 43, no. 6, p. 422-33 (1996). The concept of the nonuniform discrete Fourier transform (NDFT) is extended to two dimensions to provide a basic framework for nonuniform sampling of 2-D sequences in the frequency domain. The 2-D NDFT of a sequence of size N/sub 1//spl times/N/sub 2/ is defined as samples of its 2-D z-transform evaluated at N/sub 1/N/sub 2/ distinct points located in the 4-D (z/sub 1/, z/sub 2/) space. These points are chosen appropriately so that the inverse transform exists. We discuss two special cases in which the choice of the sampling points is constrained so that the 2-D NDFT matrix is guaranteed to be nonsingular, and the number of operations required for computing its inverse is reduced, The 2-D NDFT is applied to nonuniform frequency sampling design of 2-D finite-impulse-response (FIR) filters. Nonseparable filters with good passband shapes and low peak ripples are obtained. This is illustrated by design examples, in which 2-D filters with various shapes are designed and compared with those obtained by other existing methods.

A fast algorithm for computing inverse cosine transforms for designing zero-phase FIR filters in frequency domain

A new algorithm for computing inverse cosine transforms or for designing zero-phase FIR filters from nonuniform frequency samples is presented. The algorithm is simple, fast, recursive and can be used in 1-D or 2-D applications. Based on the three-term recursive relation of the Chebyshev polynomials, the cosine matrix is decomposed into LU products using parallel computations. Two alternative approaches-a direct and a progressive-suitable for serial computations are also derived. Given N samples, the direct version requires 2.5N/sup 2/+O(N) flops for computing the inverse cosine transforms or for calculating the filter coefficients, whereas the progressive version needs only O(5N) flops when the next N+1th sample appears. The algorithm guarantees real results and produces accurate solutions even in cases of designing high-order 1-D or 2-D FIR filters or when the interpolation matrix is ill conditioned. It can be also used in LU-factorization and can be extended to m-D filter design. >

A novel method for designing FIR digital filters with nonuniform frequency samples

A new method of designing FIR digital filters using nonuniform frequency samples is presented. There is no restriction on the phase to be linear. The method is based on a Newton-type polynomial interpolating on the unit circle of the complex plane. Attractive features of the proposed method are the applicability to unequally spaced samples, the recursive and semipermanent computation of filter parameters, the capability of obtaining short transition bands or sharp cut-off frequency responses, and the design of efficient algorithms for real-time applications. In the serial case, when the next sample appears, the design parameters are evaluated only by updating the old ones with correction terms that could be used as indicators for convergence, approximation, or filter reduction. The method can be extended to m-D filter design, DFT calculation, design of parallel algorithms, etc. >

Two-dimensional polynomial interpolation from nonuniform samples

The authors derive a number of results on sufficient conditions under which the 2-D polynomial interpolation problem has a unique or nonunique solution. It is found that, if the sum of the degrees of the irreducible curves on which the interpolation points are chosen is small compared to the degree of the interpolating polynomial, then the problem becomes singular. Similarly, if there are too many points on any of the irreducible curves on which the interpolation points are chosen, then the interpolation problem runs into singularity. Examples of geometric distribution of interpolation points satisfying these conditions are shown. The examples include polynomial interpolation of polar samples, and samples on straight lines. The authors propose a recursive algorithm for computing 2-D polynomial coefficients for the nonsingular case where all the interpolation points are chosen on lines passing through the origin. The result is applied to the problem of nonuniform frequency sampling design for 2-D FIR filter design, and a few examples of such design are shown. >

Design of 2-D FIR filters by nonuniform frequency sampling

A method for the frequency-sampling design of two-dimensional FIR filters with nonuniformly spaced samples is presented. By imposing some mild constraints on sample location in the 2-D frequency plane, the method always provides a unique design solution. Important characteristics of the method are design flexibility through the use of nonuniform samples and computational efficiency. This method is compared with the uniform sampling, inverse discrete Fourier transform (DFT) approach and also with a general method for filter design called arbitrary sampling. The method presented is shown to require much less computation than the arbitrary sampling approach, which may lead to possible degenerate cases where there is no unique solution for the filter. The method proposed does not lead to such degeneracies and possesses more flexibility than the uniform sampling method. Examples are given in order to compare the new method with the uniform sampling method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

FIR design by non-uniform sampling in frequency domain

In this paper an expression for H(z) has been derived in terms of N independent samples {Hk }. It has then been used to examine effects of non-uniform frequency sampling on the resultant frequency response of FIR filters. This has been applied first to the non-uniform sampling in both the passband and stopband, of a desired filter, and then to non-uniform sampling in the stopband only. Several lowpass filter designs have been examined by these methods and results discussed.