585 research outputs found
Linear Matrix Inequality Formulation of Spectral Mask Constraints With Applications to FIR Filter Design
Abstract-The design of a finite impulse response (FIR) filter often involves a spectral "mask" that the magnitude spectrum must satisfy. The mask specifies upper and lower bounds at each frequency and, hence, yields an infinite number of constraints. In current practice, spectral masks are often approximated by discretization, but in this paper, we will derive a result that allows us to precisely enforce piecewise constant and piecewise trigonometric polynomial masks in a finite and convex manner via linear matrix inequalities. While this result is theoretically satisfying in that it allows us to avoid the heuristic approximations involved in discretization techniques, it is also of practical interest because it generates competitive design algorithms (based on interior point methods) for a diverse class of FIR filtering and narrowband beamforming problems. The examples we provide include the design of standard linear and nonlinear phase FIR filters, robust "chip" waveforms for wireless communications, and narrowband beamformers for linear antenna arrays. Our main result also provides a contribution to system theory, as it is an extension of the wellknown Positive-Real and Bounded-Real Lemmas
Design of Waveform Set for Multiuser Ultra-Wideband Communications
The thesis investigates the design of analogue waveform sets for multiuser and UWB communications using suitably
chosen Hermite-Rodriguez basis functions. The non-linear non-convex optimization problem with time and frequency
domains constraints has been transformed into suitable forms and then solved using a standard optimization package.
The proposed approach is more flexible and efficient than existing approaches in the literature. Numerical results show
that orthogonal waveform sets with high spectral efficiency can be produced
Approximation of L\"owdin Orthogonalization to a Spectrally Efficient Orthogonal Overlapping PPM Design for UWB Impulse Radio
In this paper we consider the design of spectrally efficient time-limited
pulses for ultrawideband (UWB) systems using an overlapping pulse position
modulation scheme. For this we investigate an orthogonalization method, which
was developed in 1950 by Per-Olov L\"owdin. Our objective is to obtain a set of
N orthogonal (L\"owdin) pulses, which remain time-limited and spectrally
efficient for UWB systems, from a set of N equidistant translates of a
time-limited optimal spectral designed UWB pulse. We derive an approximate
L\"owdin orthogonalization (ALO) by using circulant approximations for the Gram
matrix to obtain a practical filter implementation. We show that the centered
ALO and L\"owdin pulses converge pointwise to the same Nyquist pulse as N tends
to infinity. The set of translates of the Nyquist pulse forms an orthonormal
basis or the shift-invariant space generated by the initial spectral optimal
pulse. The ALO transform provides a closed-form approximation of the L\"owdin
transform, which can be implemented in an analog fashion without the need of
analog to digital conversions. Furthermore, we investigate the interplay
between the optimization and the orthogonalization procedure by using methods
from the theory of shift-invariant spaces. Finally we develop a connection
between our results and wavelet and frame theory.Comment: 33 pages, 11 figures. Accepted for publication 9 Sep 201
Fast Fourier Optimization: Sparsity Matters
Many interesting and fundamentally practical optimization problems, ranging
from optics, to signal processing, to radar and acoustics, involve constraints
on the Fourier transform of a function. It is well-known that the {\em fast
Fourier transform} (fft) is a recursive algorithm that can dramatically improve
the efficiency for computing the discrete Fourier transform. However, because
it is recursive, it is difficult to embed into a linear optimization problem.
In this paper, we explain the main idea behind the fast Fourier transform and
show how to adapt it in such a manner as to make it encodable as constraints in
an optimization problem. We demonstrate a real-world problem from the field of
high-contrast imaging. On this problem, dramatic improvements are translated to
an ability to solve problems with a much finer grid of discretized points. As
we shall show, in general, the "fast Fourier" version of the optimization
constraints produces a larger but sparser constraint matrix and therefore one
can think of the fast Fourier transform as a method of sparsifying the
constraints in an optimization problem, which is usually a good thing.Comment: 16 pages, 8 figure
FIR Filter Design by Convex Optimization Using Directed Iterative Rank Refinement Algorithm
The advances in convex optimization techniques have offered new formulations of design with improved control over the performance of FIR filters. By using lifting techniques, the design of a length-L FIR filter can be formulated as a convex semidefinite program (SDP) in terms of an L× L matrix that must be rank-1. Although this formulation provides means for introducing highly flexible design constraints on the magnitude and phase responses of the filter, convex solvers implementing interior point methods almost never provide a rank-1 solution matrix. To obtain a rank-1 solution, we propose a novel Directed Iterative Rank Refinement (DIRR) algorithm, where at each iteration a matrix is obtained by solving a convex optimization problem. The semidefinite cost function of that convex optimization problem favors a solution matrix whose dominant singular vector is on a direction determined in the previous iterations. Analytically it is shown that the DIRR iterations provide monotonic improvement, and the global optimum is a fixed point of the iterations. Over a set of design examples it is illustrated that the DIRR requires only a few iterations to converge to an approximately rank-1 solution matrix. The effectiveness of the proposed method and its flexibility are also demonstrated for the cases where in addition to the magnitude constraints, the constraints on the phase and group delay of filter are placed on the designed filter. © 2015 IEEE
Optimal Waveforms Design for Ultra-Wideband Impulse Radio Sensors
Ultra-wideband impulse radio (UWB-IR) sensors should comply entirely with the regulatory spectral limits for elegant coexistence. Under this premise, it is desirable for UWB pulses to improve frequency utilization to guarantee the transmission reliability. Meanwhile, orthogonal waveform division multiple-access (WDMA) is significant to mitigate mutual interferences in UWB sensor networks. Motivated by the considerations, we suggest in this paper a low complexity pulse forming technique, and its efficient implementation on DSP is investigated. The UWB pulse is derived preliminarily with the objective of minimizing the mean square error (MSE) between designed power spectrum density (PSD) and the emission mask. Subsequently, this pulse is iteratively modified until its PSD completely conforms to spectral constraints. The orthogonal restriction is then analyzed and different algorithms have been presented. Simulation demonstrates that our technique can produce UWB waveforms with frequency utilization far surpassing the other existing signals under arbitrary spectral mask conditions. Compared to other orthogonality design schemes, the designed pulses can maintain mutual orthogonality without any penalty on frequency utilization, and hence, are much superior in a WDMA network, especially with synchronization deviations
Optimal Linear Precoding Strategies for Wideband Non-Cooperative Systems based on Game Theory-Part I: Nash Equilibria
In this two-parts paper we propose a decentralized strategy, based on a
game-theoretic formulation, to find out the optimal precoding/multiplexing
matrices for a multipoint-to-multipoint communication system composed of a set
of wideband links sharing the same physical resources, i.e., time and
bandwidth. We assume, as optimality criterion, the achievement of a Nash
equilibrium and consider two alternative optimization problems: 1) the
competitive maximization of mutual information on each link, given constraints
on the transmit power and on the spectral mask imposed by the radio spectrum
regulatory bodies; and 2) the competitive maximization of the transmission
rate, using finite order constellations, under the same constraints as above,
plus a constraint on the average error probability. In Part I of the paper, we
start by showing that the solution set of both noncooperative games is always
nonempty and contains only pure strategies. Then, we prove that the optimal
precoding/multiplexing scheme for both games leads to a channel diagonalizing
structure, so that both matrix-valued problems can be recast in a simpler
unified vector power control game, with no performance penalty. Thus, we study
this simpler game and derive sufficient conditions ensuring the uniqueness of
the Nash equilibrium. Interestingly, although derived under stronger
constraints, incorporating for example spectral mask constraints, our
uniqueness conditions have broader validity than previously known conditions.
Finally, we assess the goodness of the proposed decentralized strategy by
comparing its performance with the performance of a Pareto-optimal centralized
scheme. To reach the Nash equilibria of the game, in Part II, we propose
alternative distributed algorithms, along with their convergence conditions.Comment: Paper submitted to IEEE Transactions on Signal Processing, September
22, 2005. Revised March 14, 2007. Accepted June 5, 2007. To be published on
IEEE Transactions on Signal Processing, 2007. To appear on IEEE Transactions
on Signal Processing, 200
The design of digital all-pass filters using second-order cone programming (SOCP)
This brief proposes a new method for designing digital all-pass filters with a minimax design criterion using second-order cone programming (SOCP). Unlike other all-pass filter design methods, additional linear constraints can be readily incorporated. The overall design problem can be solved through a series of linear programming subproblems and the bisection search algorithm. The convergence of the algorithm is guaranteed. Nonlinear constraints such as the pole radius constraint of the filters can be formulated as additional SOCP constraints using Rouche's theorem. It was found that the pole radius constraint allows an additional tradeoff between the approximation error and the stability margin. The effectiveness of the proposed method is demonstrated by several design examples and comparison with conventional methods. © 2005 IEEE.published_or_final_versio
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