Frequency Sampling Method Research Articles

Fourier Synthesis of Linear Arrays Based on the Generalized Scattering Matrix and Spherical Modes

This paper presents a novel, simple pattern synthesis procedure for linear equispaced arrays which can be characterized by a generalized scattering matrix (GSM) and whose radiated field can be expressed as a weighted sum of shifted spherical waves. It can be viewed as an extension of the classic design techniques of the Fourier series (FS) method or the Woodward-Lawson frequency sampling method, to the case in which the individual antenna elements' patterns and all interelement couplings are taken into account. The design procedure, which yields the excitations needed to achieve the desired pattern, is based on either the FS or the discrete Fourier transform (DFT) of the spherical mode expansion of the array radiated field, as well as on various properties associated to the FS or DFT coefficients. In this work, to compute the GSM of the array and the spherical mode expansion of the field, a validated hybrid full-wave methodology, based on the finite element method and rotation and translation properties of spherical waves, is used. Numerical results of different synthesized array patterns are presented for different arrays made up of dielectric resonator antennas and cavity-backed microstrip circular patches.

Correction of sampling errors due to laser tuning rate fluctuations in swept-wavelength interferometry

The frequency-sampling method is widely used to accommodate nonlinear laser tuning in swept-wavelength interferometric techniques such as optical frequency domain reflectometry (OFDR) and swept-wavelength optical coherence tomography (OCT). In this paper we analyze the frequency-sampling method and identify two sources of sampling errors. One source of error is the limit of an underlying approximation for long interferometer path mismatches and fast laser tuning rates. A second source of error is transmission delays in data acquisition hardware. We show that the measurement system can be configured such that the two error sources cancel to second order. We present experimental verification of sampling error correction using a general swept-wavelength interferometer with a significantly nonlinear laser sweep.

Design of IIR linear‐phase nonuniform‐division filter banks with signed powers‐of‐two coefficients

AbstractThis paper deals with the minimax design of two‐channel linear‐phase (LP) nonuniform‐division filter (NDF) banks using infinite impulse response (IIR) digital all‐pass filters (DAFs) with signed powers‐of‐two (SPT) coefficients. Based on the theory of two‐channel NDF banks using two IIR DAFs, the design problem is appropriately formulated to result in an appropriate Chebyshev approximation for the desired phase responses of the IIR DAFs. Through a frequency sampling and iterative approximation method, the optimization problem for finding the SPT coefficients for the IIR DAFs can be solved by utilizing a weighted least‐squares approach in conjunction with a coordinate rotational digital computer (CORDIC) algorithm. The resulting two‐channel SPT coefficient NDF banks can possess approximately LP response without magnitude distortion. Several simulation examples are presented for illustration and comparison. Copyright © 2008 John Wiley & Sons, Ltd.

First-order PMD compensation using optical FIR filter

We present a synthesis algorithm to design an optical finite impulse response (FIR) filter for compensating a first-order polarization mode dispersion (PMD) by minimizing the differential group delay (DGD). The desired frequency response was approximated using two widely used methods in designing digital FIR filters: the Fourier series expansion method and the frequency sampling method. A numerical simulation was performed for an eighth-order filter to demonstrate the difference between the two methods. The simulation results produced a sharper cutoff for the Fourier series expansion and higher stopband attenuation for the frequency sampling method. The Fourier series method produced better results in reducing the DGD.

Stoer‐Bulirsch adaptive frequency sampling method for efficient analysis of frequency selective surfaces

AbstractIn this article, a Stoer‐Bulirsch adaptive frequency sampling method is proposed in conjunction with the method of moments to analyze the broadband electromagnetic wave scattering from frequency selective surfaces. On comparing with other sampling methods, the Stoer‐Bulirsch algorithm is a recursive tabular method and requires no matrix inversion. Therefore, it can obtain a rational interpolation function without suffering from singularity problems. Furthermore, since final results are obtained by interpolating with the adaptive frequency sampling method, no more information is needed except the sampled frequency and its true value. Hence, it can be widely used in most of the electromagnetic simulations. Numerical results show that our proposed sampling strategy performs well in terms of both simulation time and computation accuracy. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 755–758, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23204

Designing Waveforms for Temporal Encoding Using a Frequency Sampling Method

In this paper a method for designing waveforms for temporal encoding in medical ultrasound imaging is described. The method is based on least squares optimization and is used to design nonlinear frequency modulated signals for synthetic transmit aperture imaging. By using the proposed design method, the amplitude spectrum of the transmitted waveform can be optimized, such that most of the energy is transmitted where the transducer has large amplification. To test the design method, a waveform was designed for a BK8804 linear array transducer. The resulting nonlinear frequency modulated waveform was compared to a linear frequency modulated signal with amplitude tapering, previously used in clinical studies for synthetic transmit aperture imaging. The latter had a relatively flat spectrum which implied that the waveform tried to excite all frequencies including ones with low amplification. The proposed waveform, on the other hand, was designed so that only frequencies where the transducer had a large amplification were excited. Hereby, unnecessary heating of the transducer could be avoided and the signal-to-noise ratio could be increased. The experimental ultrasound scanner RASMUS was used to evaluate the method experimentally. Due to the careful waveform design optimized for the transducer at hand, a theoretic gain in signal-to-noise ratio of 4.9 dB compared to the reference excitation was found, even though the energy of the nonlinear frequency modulated signal was 71% of the energy of the reference signal. This was supported by a signal-to-noise ratio measurement and comparison in penetration depth, where an increase of 1 cm was found in favor for the proposed waveform. Axial and lateral resolutions at full-width half-maximum were compared in a water phantom at depths of 42, 62, 82, and 102 mm. The axial resolutions of the nonlinear frequency modulated signal were 0.62, 0.69, 0.60, and 0.60 mm, respectively. The corresponding axial resolutions for the reference waveform were 0.58, 0.65, 0.62, and 0.60 mm, respectively. The compression properties of the matched filter (mismatched filter for the linear frequency modulated signal) were tested for both waveforms in simulation with respect to the Doppler frequency shift occurring when probing moving objects. It was concluded that the Doppler effect of moving targets does not significantly degrade the filtered output. Finally, in vivo measurements are shown for both methods, wherein the common carotid artery on a 27-year-old healthy male was scanned.

Introduction to digital signal processing and filter design

Preface. 1. Introduction. 1.1 Introduction. 1.2 Application of DSP. 1.3 Discrete-Time Signals. 1.4 History of Filter Design. 1.5 Analog and Digital Signal Processing. 1.6 Summary. Problems. References. 2. Time-Domain Analysis and z Transform. 2.1 A Linear, Time-Invariant System. 2.2 z Transform Theory. 2.3 Using z Transform to Solve Difference Equations. 2.4 Solving Difference Equations Using the Classical Method. 2.5 z Transform Method Revisited. 2.6 Convolution Revisited. 2.7 A Model from Other Models. 2.8 Stability. 2.9 Solution Using MATLAB Functions. 2.10 Summary. Problems. References. 3. Frequency-Domain Analysis. 3.1 Introduction. 3.2 Theory of Sampling. 3.3 DTFT and IDTFT. 3.4 DTFT of Unit Step Sequence. 3.5 Use of MATLAB to Compute DTFT. 3.6 DTFS and DFT. 3.7 Fast Fourier Transform. 3.8 Use of MATLAB to Compute DFT and IDFT. 3.9 Summary/ Problems. References. 4. Infinite Impulse Response Filters. 4.1 Introduction. 4.2 Magnitude Approximation of Analog Filters. 4.3 Analog Frequency Transformations. 4.4 Digital Filters. 4.5 Impulse-Invariant Transformation. 4.6 Bilinear Transformation. 4.7 Digital Spectral Transformation. 4.8 Allpass Filters. 4.9 IIR Filter Design Using MATLAB. 4.10 Yule-Walker Approximation. 4.11 Summary. Problems. References. 5. Finite Impulse Response Filters. 5.1 Introduction. 5.2 Linear Phase Fir Filters. 5.3 Fourier Series Method Modified by Windows. 5.4 Design of Windowed FIR Filter Using MATLAB. 5.5 Equiripple Linear Phase FIR Filters. 5.6 Design of Equiripple FIR Filters Using MATLAB. 5.7 Frequency Sampling Method. 5.8 Summary. Problems. References. 6. Filter Realizations. 6.1 Introduction. 6.2 FIR Filter Realizations. 6.3 IIR Filter Realizations. 6.4 Allpass Filters in Parallel. 6.5 Realization of FIR and IIR Filters Using MATLAB. 6.6 Summary. Problems. References. 7. Quantized Filter Analysis. 7.1 Introduction. 7.2 Filter Design-Analysis Tool. 7.3 Quantized Filter Analysis. 7.4 Binary Numbers and Arithmetic. 7.5 Quantization Analysis of IIR Filters. 7.6 Quantization Analyis of FIR Filters. 7.7 Summary. Problems. References. 8. Hardware Design Using DSP Chips. 8.1 Introduction. 8.2 Simulink and Real-Time Workshop. 8.3 Design Preliminaries. 8.4 Code Generation. 8.5 Code Composer Studio. 8.6 Simulator and Emulator. 8.7 Conclusion. References. 9. MATLAB Primer. 9.1 Introduction. 9.2 Signal Processing Toolbox. References. Index.

Design of narrowband frequency sampling FIR filters using Z-window

In this work a method for automatic design of narrowband frequency sampling low-pass digital FIR filters is presented. Spectral response of Zhong (Z)-window has been used to provide the filtering action. Z-windows are used as variable windows by varying the side-lobe dip position. The proposed design methodology is a hybridization of the concepts of frequency sampling method and window method of filter design. This proposed design technique produces low-pass FIR filters featuring sharp cut off, exactly marked pass-band and cut off frequencies along with tolerable pass-band ripple and variable stop-band attenuation. For the filter applications requiring brick-wall response, considerable saving in the filter order can be achieved with subsequent reduction in computational and storage requirements by this method.

Design of narrowband frequency sampling FIR filters using Z-window

In this work a method for automatic design of narrowband frequency sampling low-pass digital FIR filters is presented. Spectral response of Zhong (Z)-window has been used to provide the filtering action. Z-windows are used as variable windows by varying the side-lobe dip position. The proposed design methodology is a hybridization of the concepts of frequency sampling method and window method of filter design. This proposed design technique produces low-pass FIR filters featuring sharp cut off, exactly marked pass-band and cut off frequencies along with tolerable pass-band ripple and variable stop-band attenuation. For the filter applications requiring “brick–wall” response, considerable saving in the filter order can be achieved with subsequent reduction in computational and storage requirements by this method.

A New Sampling Frequency Selection Scheme in Undersampling Systems

In software defined radio systems, placing the analog-to-digital converter (ADC) near the antenna part in the block diagram of the receiver is desired to improve the flexibility of the system. The radio frequency (RF) sampling method, in which the received signal is sampled at the RF stage, realizes such structure. The undersampling is a potential method to sample the RF signal using the existing consumer ADCs because high speed ADCs are required in the traditional methods, such as Nyquist sampling or the oversampling of the RF signal. This paper presents a technique to determine the minimum sampling frequency to undersample the separated multiple wireless systems simultaneously. In addition, this paper proposes a frequency selecting scheme that enables selection of a lower sampling frequency by receiving at least the desired transmission channels in the wireless system signals. This paper also provides a result of performance analysis of the proposed scheme.

Optimal Design of Complex Two-Channel IIR QMF Banks with Equiripple Response

The optimal design of complex infinite impulse response (IIR) two-channel quadrature mirror filter (QMF) banks with equiripple frequency response is considered. The design problem is appropriately formulated to result in a simple optimization problem. Therefore, based on a variant of Karmarkar's algorithm, we can efficiently solve the optimization problem through a frequency sampling and iterative approximation method to find the complex coefficients for the IIR QMFs. The effectiveness of the proposed technique is to form an appropriate Chebyshev approximation of a desired response and then find its solution from a linear subspace in several iterations. Finally, simulation results are presented for illustration and comparison.

Minimax Design of Two-Channel Nonuniform-Division Filterbanks Using IIR Allpass Filters

The design of two-channel linear-phase nonuniform-division filter (NDF) banks constructed by infinite impulse response (IIR) digital allpass filters (DAFs) in the sense of L/sub /spl infin// error criteria is considered. First, the theory of two-channel NDF bank structures using two IIR DAFs is developed. Then, the design problem is appropriately formulated to result in a simple optimization problem. Utilizing a variant of Karmarkar's algorithm, we can efficiently solve the optimization problem through a frequency sampling and iterative approximation method to find the coefficients for the IIR DAFs. The resulting two-channel NDF banks can possess approximately linear-phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of a desired phase response and then to find its solution from a linear subspace in a few iterations. Several simulation examples are presented for illustration and comparison.

Design of two-channel linear-phase QMF banks based on real IIR all-pass filters

The design of two-channel linear-phase quadrature mirror filter (QMF) banks constructed by real infinite impulse response (IIR) digital all-pass filters is considered. The design problem is appropriately formulated to result in a simple optimisation problem. Using a variant of Karmarkar's algorithm, the optimisation problem can be efficiently solved through a frequency sampling and iterative approximation method to find the real coefficients for the IIR digital all-pass filters. The resulting two-channel QMF banks possess an approximately linear phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of the desired phase response and then finding its solution from a linear subspace in a few iterations. Finally, several simulation examples are presented for illustration and comparison.

A real-time programmable audio bandwidth filter incorporating a high-level design interface

A simple but highly integrated digital signal processing system is described for real time filtering of audio bandwidth signals. It includes the necessary processing and communications hardware, the processing code itself, and a high-level software interface that enables the user to design and download arbitrary finite impulse response filters to the run-time system. The filter coefficients are calculated using the Frequency Sampling Method. A unique feature of the design is the manner in which the software and hardware components have been organized as an intelligent system, obviating on the part of the user a detailed knowledge of filter design theory or any abilities in processor architecture and assembly code programming.

Reduced complexity beam space broadband frequencyinvariant beamforming

A frequency-invariant beamformer (FIB) that uses a reduced number of adaptive coefficients is described. A beam space extension of a previously proposed uniform linear array FIB is used in the design procedure that is analogous to the frequency-sampling method in filter design. Simulations show similar synthesis performance between the conventional and proposed FIB, with the latter offering reduced complexity in adaptive applications.

A real time programmable digital filter for biomedical signal enhancement incorporating a high-level design interface

A simple but highly integrated digital signal processing system is described for real time filtering of biomedical signals. It includes the necessary processing and communications hardware, the processing code itself and a high-level software interface that enables the user to design and download arbitrary finite impulse response filters to the run-time system. The filter coefficients are calculated using the frequency sampling method. Since the filters are realized using a finite impulse response, no phase distortion is introduced into the processed signals. A unique feature of the design is the manner in which the software and hardware components have been organized as an intelligent system, obviating on the part of the user a detailed knowledge of filter design theory or any abilities in processor architecture and assembly code programming.

B-spline design of maximally flat and prolate spheroidal-type FIR filters

A digital FIR filter is described that offers excellent passband and stopband characteristics for general applications. Design formulae include parameters that adjust the magnitude response from one having characteristics like the maximally flat designs of Hermann (1971) and Kaiser (1975, 1979) to one having characteristics like the minimum-sidelobe energy approximations of Kaiser and Saramaki (1989). The impulse response coefficients are more straightforward to obtain than these filter designs while offering preferable response characteristics in many instances. Unlike FIR filters designed by window- or frequency-sampling methods, the filter coefficients are determined from the inverse Fourier transform in closed form once B-splines have been used to replace sharp transition edges of the magnitude response. Although the filters are developed in the frequency domain, a convergence window is identified in the convolution series and compared with windows of popular FIR filters. By means of example, adjustment of the transitional parameter is shown to produce a filter response that rivals the stopband attenuation and transition width of prolate spheroidal designs. The design technique is extended to create additional transitional filters from prototype window functions, such as the transitional Hann window filter. The filters are particularly suitable for precision filtering and reconstruction of sampled physiologic and acoustic signals common to the health sciences but will also be useful in other applications requiring low passband and stopband errors.

The design of linear phase FIR filters using the IDFT

The paper presents a comprehensive set of equations for specifying the transfer function of linear phase finite-impulse response (LPFIR) filters. A procedure for designing the eight possible LPFIR filters (symmetric and anti-symmetric, even and odd length, type 1 and type 2 filters), using the inverse discrete Fourier transform (IDFT) is described. This technique can be equally applied to the window method and the frequency sampling method.

A recursive frequency-sampling method for designing zero-phase FIR filters by nonuniform samples

A novel frequency-sampling method for designing zero-phase FIR filters from nonuniform samples is presented. The method is fast, simple, recursive and can be used in the design of 1D or 2D zero-phase FIR filters by imposing some mild constraints on sample locations in the 2D frequency plane. Based on a novel Newton representation of the filter transfer function the proposed method guarantees real results, saves a number of operations and produces accurate solutions even in cases of designing high-order filters or when the interpolation matrix is ill-conditioned. In the progressive case when the next sample appears, the design parameters are evaluated by updating the old ones with correction terms that could be used as indicators for convergence, approximation, or filter reduction. The method can be used in mD filter design, in LU-factorization or in inversion of cosine matrices. >

Design of two-dimensional FIR filter using hybrid domain approximation

This paper proposes a zero-phase, two-dimensional (2-D) FIR filter design using a hybrid domain where a spatial variable and frequency variable are mixed. For a desired amplitude response given in 2-D frequency plane, the approximation method of 1-D FIR filter design is applied in two steps and the desired response is approximated. First, the approximation coefficients in the hybrid domain are obtained by min-max approximation for one of the frequency variables. Then filter coefficients are obtained by the frequency sampling method or inconsistent Čhebyshev approximation method for another of the variables. Moreover, the error distribution of the designed 2-D filter is evaluated; and, when the magnitude of the error is uneven in the frequency domain, then the method of redesigning is proposed to obtain good approximation. Since this method decomposes the calculation of the 2-D filter into the 1-D case, its design is very simple. Moreover, this method has the advantage of free setting the magnitude of error in the passband and stopband of filter.