Dimension Of Signal Space Research Articles

Large System Spectral Analysis of Covariance Matrix Estimation

Eigendecomposition of estimated covariance matrices is a basic signal processing technique arising in a number of applications, including direction-of-arrival estimation, power allocation in multiple-input/multiple-output (MIMO) transmission systems, and adaptive multiuser detection. This paper uses the theory of non-crossing partitions to develop explicit asymptotic expressions for the moments of the eigenvalues of estimated covariance matrices, in the large system asymptote as the vector dimension and the dimension of signal space both increase without bound, while their ratio remains finite and nonzero. The asymptotic eigenvalue distribution is also obtained from these eigenvalue moments and the Stieltjes transform, and is extended to first-order approximation in the large sample-size limit. Numerical simulations are used to demonstrate that these asymptotic results provide good approximations for finite systems of moderate size.

Knowledge modelling in signal parameter evaluation

The aim of this work is to describe the available knowledge about the investigated signal to enable evaluation of uncertainty of the estimated parameter at each stage of the measurement process. For this purpose, signal is represented as a point in a finite-dimensional number space. Then, each information item described by a set of appropriate relationships is transformed into a set of possible signals. Thus, set-theory operations are applicable to data fusion modelling. As examples, sets of possible signals are presented for cases when the signal mean power is known, when the signal harmonics content is available or when signal maximum and minimum values are known. Finally, the necessary condition is formulated for a given information item usability in a defined parameter estimation. A bunch of subsets of signal space dimensions is defined and utilised for this purpose.

Capacity of sparse wideband channels with partial channel feedback

AbstractThis paper studies the ergodic capacity of wideband multipath channels with partial/limited channel state feedback. Our work builds on recent results that show a significant capacity gain in the wideband/low‐SNR regime when there is perfect channel state information (CSI) at the transmitter and the receiver, relative to the case of perfect CSI at the receiver only. Furthermore, this benchmark capacity gain can be achieved with just one bit of feedback per channel coefficient. However, the input signals used in these works are peaky; that is, they have large peak‐to‐average power ratios. Signal peakiness requirement is related to channel coherence and many recent measurement campaigns show that, in contrast to previous assumptions, wideband channels exhibit a sparse multipath structure that naturally leads to coherence in time and frequency. In this work, we show that multipath sparsity significantly relaxes the requirement of peaky signalling in attaining the capacity gains with channel state feedback. First, we show that the benchmark capacity gain, with perfect CSI at the transmitter and the receiver, is achievable even under an instantaneous power constraint. In the more realistic non‐coherent setting, we study the performance of a training‐based signalling scheme with one bit of feedback per channel coefficient. We show that multipath sparsity can be leveraged to achieve the benchmark capacity gain under both average as well as instantaneous power constraints as long as the channel coherence scales at a sufficiently fast rate with the signal space dimension (time‐bandwidth product). We also present guidelines for choosing signalling parameters as a function of the channel sparsity parameters to maximally exploit channel state feedback for capacity gains. Copyright © 2008 John Wiley & Sons, Ltd.

Maximizing MIMO Capacity in Sparse Multipath With Reconfigurable Antenna Arrays

Emerging advances in reconfigurable radio-frequency (RF) front-ends and antenna arrays are enabling new physical modes for accessing the radio spectrum that extend and complement the notion of waveform diversity in wireless communication systems. However, theory and methods for exploiting the potential of reconfigurable RF front-ends are not fully developed. In this paper, we study the impact of reconfigurable antenna arrays on maximizing the capacity of multiple input multiple output (MIMO) wireless communication links in sparse multipath environments. There is growing experimental evidence that physical wireless channels exhibit a sparse multipath structure, even at relatively low antenna dimensions. We propose a model for sparse multipath channels and show that sparse channels afford a new dimension over which capacity can be optimized: the distribution or configuration of the sparse statistically independent degrees of freedom (DoF) in the available spatial signal space dimensions. Our results show that the configuration of the sparse DoF has a profound impact on capacity and also characterize the optimal capacity-maximizing channel configuration at any operating SNR. We then develop a framework for realizing the optimal channel configuration at any SNR by systematically adapting the antenna spacings at the transmitter and the receiver to the level of sparsity in the physical multipath environment. Surprisingly, three canonical array configurations are sufficient for near-optimum performance over the entire SNR range. In a sparse scattering environment with randomly distributed paths, the capacity gain due to the optimal configuration is directly proportional to the number of antennas. Numerical results based on a realistic physical model are presented to illustrate the implications of our framework

Capacity of Sparse Multipath Channels in the Ultra-Wideband Regime

This paper studies the ergodic capacity of time-and frequency-selective multipath fading channels in the ultrawide-band (UWB) regime when training signals are used for channel estimation at the receiver. Motivated by recent measurement results on UWB channels, we propose a model for sparse multipath channels. A key implication of sparsity is that the independent degrees of freedom in the channel scale sublinearly with the signal space dimension (product of signaling duration and bandwidth). Sparsity is captured by the number of resolvable paths in delay and Doppler. Our analysis is based on a training and communication scheme that employs signaling over orthogonal short-time Fourier (STF) basis functions. STF signaling naturally relates sparsity in delay-Doppler to coherence in time and frequency. We study the impact of multipath sparsity on two fundamental metrics of spectral efficiency in the wideband/low-SNR limit introduced by Verdu: first- and second-order optimality conditions. Recent results by Zheng et al. have underscored the large gap in spectral efficiency between coherent and noncoherent extremes and the importance of channel learning in bridging the gap. Building on these results, our results lead to the following implications of multipath sparsity: (1) the coherence requirements are shared in both time and frequency, thereby significantly relaxing the required scaling in coherence time with SNR; (2) sparse multipath channels are asymptotically coherent - for a given but large bandwidth, the channel can be learned perfectly and the coherence requirements for first- and second-order optimality met through sufficiently large signaling duration; and (3) the requirement of peaky signals in attaining capacity is eliminated or relaxed in sparse environments.

Signal Sets From Functions With Optimum Nonlinearity

Signal sets with the best correlation property are desirable in code-division multiple-access (CDMA) systems. In this paper, the construction of Wootters and Fields for mutually unbiased bases is extended into a generic construction of signal sets using planar functions. Then, specific classes of planar functions and almost bent functions are employed to obtain (q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> +q,q) signal sets. The signal sets derived from planar functions are optimal with respect to the Levenstein bound, and those obtained from almost bent functions nearly meet the Levenstein bound. The signal sets constructed in this paper could have a very small alphabet size, and have applications in synchronous DS-CDMA systems, where the number of users is greater than the signal space dimension or the spreading factor

On the Limitation of Linear MMSE Detection

This correspondence highlights the performance limitation of linear minimum mean-squared error (mmse) detection in underdetermined vector Gaussian channels (as in overloaded code-division multiple-access (CDMA) systems) where the number of symbols (users) exceeds the signal space dimension (spread factor). It is shown that for such a simple receiver it is not possible to construct signal sets (or spreading codes) to even satisfy the basic requirement that every user's symbol error probability decays exponentially as noise power vanishes. This result holds for arbitrary received energies, modulation schemes, and any strictly underdetermined system with a finite signal space dimension and a finite number of users

A Single-Lead ECG Enhancement Algorithm Using a Regularized Data-Driven Filter

We presented a novel way of deriving a subspace filter for enhancing a noisy electrocardiogram (ECG) signal contaminated by electromyogram (EMG). The new subspace filter was based on a multiple cycle prediction (MCP) modeling of a single-lead ECG. The adoption of an MCP model resulted in a data matrix more suitable for separating noise and signal subspaces than the linear prediction (LP) model that is implicitly assumed in many existing subspace filters. Alignment of ECG cycles of different length is required for MCP modeling and was handled by a dynamic time warping (DTW) algorithm. A run-time procedure was designed for automatically determining the signal space dimension adaptively. To validate the new filter in a quantitative way, 12 clean realistic ECG segments with different degrees of heart rate variability generated using the ECGSyn program were mixed with different realizations of EMG noise in the MIT-BIH Noise Stress Test Database and locally acquired EMG at a typical 10-dB signal-to-noise ratio. The performance of the proposed method was compared to three existing ECG enhancement algorithms and achieved encouraging results. In addition, various ECG recordings from MIT-Arrythmia database were also mixed with EMG noise and subjected to the same four filters resulting in a qualitative comparison of them.

Optical intensity-modulated direct detection channels: signal space and lattice codes

Traditional approaches to constructing constellations for electrical channels cannot be applied directly to the optical intensity channel. This work presents a structured signal space model for optical intensity channels where the nonnegativity and average amplitude constraints are represented geometrically. Lattice codes satisfying channel constraints are defined and coding and shaping gain relative to a baseline are computed. An effective signal space dimension is defined to represent the precise impact of coding and shaping on bandwidth. Average optical power minimizing shaping regions are derived in some special cases. Example lattice codes are constructed and their performance on an idealized point-to-point wireless optical link is computed. Bandwidth-efficient schemes are shown to have promise for high data-rate applications, but require greater average optical power.

Codes over Finite Fields for Multidimensional Signals

In this paper, linear block codes over finite fields of the algebraic integer ring of the cyclotomic field Q(e2πi/n) modulo irreducible elements are presented, where n∈{3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84}. These codes can be used for coding over ϕ(n) dimensional signal space and can correct one error taken from the group of all roots of unity in the algebraic integer ring of Q(e2πi/n). These codes provide an algebraic approach in an area which is currently mainly dominated by nonalgebraic convolutional codes.

Noncoherent decision-feedback multiuser detection

The problem of noncoherent multiuser detection for multipulse modulation over the Gaussian multiuser channel is studied. It is assumed that neither the energies nor the carrier phases of the signals of any of the users are available at the receiver. Previously, a post-decorrelative generalized likelihood ratio test (GLRT) based detector was proposed as a solution to this problem. In this paper, we introduce the concept of noncoherent decision feedback (DF) that forms the basis for an improved solution to the same problem. The users are detected sequentially in some fixed order, and the decision for a particular user takes advantage of the reduction of uncertainty associated with the signal space resulting from decisions already made for previous users. In contrast to coherent DF, therefore, the feedforward and feedback transformations in the noncoherent case are themselves functions of the matched filter outputs. An efficient implementation of the noncoherent DF detector for M-ary modulation and /spl rho/-dimensional signal space requires O(M/sub /spl rho//) computational complexity per user. Upper and lower bounds on its symbol error probability are obtained. It is shown that significant improvements over the post-decorrelative GLRT-based detector are often possible. Sufficient conditions are obtained which guarantee, without ignoring error-propagation effects, that the high signal-to-noise performance of the DF strategy can be as good as its genie-aided version in which perfect past users' decisions are used.

A new multidimensional FFT based on one-dimensional decompositions

This work proposes an original multidimensional fast Fourier transform (FFT) algorithm where the computation is first organized into multiplier-free butterflies and then completed by 1-D FFTs. The properties of well-known 1-D FFT algorithms blend in quite nicely with those of the proposed multidimensional FFT scheme, extending their computational and structural characteristics to it. Strong points of the proposed method are that its total computational cost decreases as the signal space dimensions increase and that its efficiency is superior to that of any other multidimensional FFT algorithm.

Aspects on single symbol signaling on the frequency flat Rayleigh fading channel

The optimal single symbol detector on the Rayleigh fading channel computes a functional quadratic form in the time continuous received signal. A drawback is that closed-form solutions of integral equations based on the channel statistics are required. This makes simplified discrete receivers attractive. A class of suboptimal receivers that transforms the received random process to a set of discrete observables is derived. The set of observables constitutes a random vector in a finite dimensional receiver signal space. Given this vector, the maximum likelihood detector computes a quadratic form in the received vector. The discretization implies a loss of information, therefore such a detector is not, in general, optimal given the received time continuous signal. The purpose is, however, to achieve close to optimal performance when the number of observables becomes large. This class of detectors is analyzed using exact error probability calculations, which reveal several interesting properties. The length of the observation interval and the number of discrete observables have significant influences on the error probability when the time variations of the fading process are rapid compared with the symbol duration. By increasing the number of observables, the error floor is lowered, and the implicit diversity order is increased. This implicit diversity arises as soon as more than one observable per symbol interval is used and is a consequence of the information bearing signal being a random process. Matched filter receivers use few discrete observables per symbol interval, and thus suffer from high error floors and low implicit diversity orders on fast fading channels. The error probability is highly dependent on the shapes and durations of the modulator waveforms. For instance, pulses of long duration give lower error probabilities than shorter pulses, and for a certain type of orthogonal waveforms there is no error floor.

A reduced high dimensional FDTS for magneto-optical recording

FDTS (Fixed Delay Tree Search) sequence detection using metric calculation is demonstrated. FDTS with a long tree depth is performed using a High Dimensional Signal Space Detector. We propose a new HD (High Dimensional)-FDTS employing a reduction algorithm under conditions of a certain minimum RLL (Run Length Limitation) of code and with ISI (intersymbol interference). We made sure that the HD-FDTS minimum distance taking account of the RLL code was close to that of a Viterbi decoder. The decoder circuit is small and high speed and has a reasonable noise margin for an ideal equalized wave with AWGN (Additive White Gaussian Noise). The HD-FDTS may be applied to PLL (Phase Locked Loop) and AGC (Automatic Gain Control) because of the short decoding delay.

A high dimensional signal space implementation of FDTS/DF

A procedure for designing FDTS/DF as a high dimensional signal space detector is presented. The procedure is applied to the three dimensional case to illustrate the resulting detector structure. An equalization and code constraint reduce the number of boundaries and eliminate the multipliers from the general case. Simulation results show this new channel outperforms EPRML for user densities of 2.25 and higher.

Optimal simultaneous detection and estimation under a false alarm constraint

This paper addresses the problem of finite sample simultaneous detection and estimation which arises when estimation of signal parameters is desired but signal presence is uncertain. In general, a joint detection and estimation algorithm cannot simultaneously achieve optimal detection and optimal estimation performance. We develop a multihypothesis testing framework for studying the tradeoffs between detection and parameter estimation (classification) for a finite discrete parameter set. Our multihypothesis testing problem is based on the worst case detection and worst case classification error probabilities of the class of joint detection and classification algorithms which are subject to a false alarm constraint. This framework leads to the evaluation of greatest lower bounds on the worst case decision error probabilities and a construction of decision rules which achieve these lower bounds. For illustration, we apply these methods to signal detection, order selection, and signal classification for a multicomponent signal in noise model. For two or fewer signals, an SNR of 3 dB and signal space dimension of N=10 numerical results are obtained which establish the existence of fundamental tradeoffs between three performance criteria: probability of signal detection, probability of correct order selection, and probability of correct classification. Furthermore, based on numerical performance comparisons between our optimal decision rule and other suboptimal penalty function methods, we observe that Rissanen's (1978) order selection penalty method is nearly min-max optimal in some nonasymptotic regimes. >

Optimum detection of fading signals in impulsive noise

This paper deals with the synthesis and the analysis of optimum receivers to detect one out of M equally likely, equi-energy, fading signals in impulsive noise, modelled as a compound Gaussian, possibly correlated process. We show that the conventional coherent and incoherent detectors are still optimum, independent of the noise as well as the fading probability density functions. The performance analysis has been carried on with reference to the general case of arbitrarily distributed disturbance: in order to simplify the analysis, asymptotical expressions have been developed for high signal-to-noise ratios as well as high signal space dimensionality. Interestingly enough, this allows separating the effect of the noise spikyness from that of the fading law. Results indicate that, for deep fading, the noise marginal distribution does not dramatically affect the error probability, nor is it influential on the limit operating characteristics corresponding to infinite signal space dimensions. For non-fluctuating signals, instead, the noise distribution plays a primary role: spiky noise usually produces performance impairment; moreover, the limit performance in impulsive disturbance may exhibit marked deviations from the well-known stepwise shape which is typical of Gaussian channels. >

Quadrature-quadrature phase-shift keying

Quadrature-quadrature phase-shift keying (Q/sup 2/PSK) is a spectrally efficient modulation scheme which utilizes available signal space dimensions in a more efficient way than two-dimensional schemes such as QPSK and MSK (minimum-shift keying). It uses two date shaping pulses and two carriers, which are pairwise quadrature in phase, to create a four-dimensional signal space and increases the transmission rate by a factor of two over QPSK and MSK. However, the bit error rate performance depends on the choice of pulse pair. With simple sinusoidal and cosinusoidal data pulses, the E/sub b//N/sub 0/ requirement for P/sub b/(E)=10/sup -5/ is approximately 1.6 dB higher than that of MSK. Without additional constraints, Q/sup 2/PSK does not maintain a constant envelope, however, a simple block coding can provide a constant envelope. This coded signal substantially outperforms MSK and TFM (time-frequency multiplexing) in bandwidth efficiency. Like MSK, Q/sup 2/PSK also has self-clocking and self-synchronizing ability. An optimum class of pulse shapes for use in Q/sup 2/PSK format is presented. One suboptimum realization achieves the Nyquist rate of 2 b/s/Hz using binary detection.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A generalized sampling theorem

AbstractUsing the theory of pseudo‐biorthogonal base and the notion of reproducing kernel, we derive a very broad generalized sampling theorem with real pulse. In addition, it includes the traditional sampling theorem with ideal pulse as its special case. It also includes all the sampling theorems in the cases of bandpass‐type band‐limited signal space for nonuniformly spaced sampling points, for many variables, and for the case in which the notion of frequency is extended from Fourier transform to general integral transform. This generalized sampling theorem is effective also for so‐called undersampling whereby there are too few sampling points compared with the dimension of signal space, and for so‐called oversampling due to too many sampling points. Moreover, it is effective in the case where both occur; that is, in the case where it is undersampling from the standpoint of signal space while it is oversampling in a space of restored signal. For undersampling, the generalized sampling theorem provides the best approximation for each original signal.

Asymptotic quantization error of continuous signals and the quantization dimension

Extensions of the limiting qnanfizafion error formula of Bennet are proved. These are of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_{s,k}(N,F)=N^{-\beta}B</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is the number of output levels, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_{s,k}(N,F)</tex> is the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> th moment of the metric distance between quantizer input and output, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta,B&gt;0,k=s/\beta</tex> is the signal space dimension, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</tex> is the signal distribution. If a suitably well-behaved <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -dimensional signal density <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x)</tex> exists, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B=b_{s,k}[\int f^{\rho}(x)dx]^{1/ \rho},\rho=k/(s+k)</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b_{s,k}</tex> does not depend on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> . For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k=1,s=2</tex> this reduces to Bennett's formula. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</tex> is the Cantor distribution on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[0,1],0&lt;k=s/ \beta=\log 2/ \log 3&lt;1</tex> and this <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> equals the fractal dimension of the Cantor set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[12,13]</tex> . Random quantization, optimal quantization in the presence of an output information constraint, and quantization noise in high dimensional spaces are also investigated.