On the performance of vector quantizers empirically designed from dependent sources

Studies the asymptotic characteristics of uniform scalar quantizers that are optimal with respect to mean-squared error (MSE). When a symmetric source density with infinite support is sufficiently well behaved, the optimal step size /spl Delta//sub N/ for symmetric uniform scalar quantization decreases as 2/spl sigma/N/sup -1/V~/sup -1/(1/6N/sup 2/), where N is the number of quantization levels, /spl sigma//sup 2/ is the source variance and V~/sup -1/(/spl middot/) is the inverse of V~(y)=y/sup -1/ /spl int//sub y//sup /spl infin// P(/spl sigma//sup -1/X>x) dx. Equivalently, the optimal support length N/spl Delta//sub N/ increases as 2/spl sigma/V~/sup -1/(1/6N/sup 2/). Granular distortion is asymptotically well approximated by /spl Delta//sub N//sup 2//12, and the ratio of overload to granular distortion converges to a function of the limit /spl tau//spl equiv/lim/sub y/spl rarr//spl infin//y/sup -1/E[X|X>y], provided, as usually happens, that /spl tau/ exists. When it does, its value is related to the number of finite moments of the source density, an asymptotic formula for the overall distortion D/sub N/ is obtained, and /spl tau/=1 is both necessary and sufficient for the overall distortion to be asymptotically well approximated by /spl Delta//sub N//sup 2//12. Applying these results to the class of two-sided densities of the form b|x|/sup /spl beta//e(-/spl alpha/|x|/sup /spl alpha//), which includes Gaussian, Laplacian, Gamma, and generalized Gaussian, it is found that /spl tau/=1, that /spl Delta//sub N/ decreases as (ln N)/sup 1//spl alpha///N, that D/sub N/ is asymptotically well approximated by /spl Delta//sub N//sup 2//12 and decreases as (ln N)/sup 2//spl alpha///N/sup 2/, and that more accurate approximations to /spl Delta//sub N/ are possible. The results also apply to densities with one-sided infinite support, such as Rayleigh and Weibull, and to densities whose tails are asymptotically similar to those previously mentioned.

In this paper, we create a framework for training-based channel estimation under different channel and interference statistics. The minimum mean square error (MMSE) estimator for channel matrix estimation in Rician fading multi-antenna systems is analyzed, and especially the design of mean square error (MSE) minimizing training sequences. By considering Kronecker-structured systems with a combination of noise and interference and arbitrary training sequence length, we collect and generalize several previous results in the framework. We clarify the conditions for achieving the optimal training sequence structure and show when the spatial training power allocation can be solved explicitly. We also prove that spatial correlation improves the estimation performance and establish how it determines the optimal training sequence length. The analytic results for Kronecker-structured systems are used to derive a heuristic training sequence under general unstructured statistics. The MMSE estimator of the squared Frobenius norm of the channel matrix is also derived and shown to provide far better gain estimates than other approaches. It is shown under which conditions training sequences that minimize the non-convex MSE can be derived explicitly or with low complexity. Numerical examples are used to evaluate the performance of the two estimators for different training sequences and system statistics. We also illustrate how the optimal length of the training sequence often can be shorter than the number of transmit antennas.

The Lagrangian formulation of variable-rate vector quantization is known to yield useful necessary conditions for quantizer optimality and generalized Lloyd algorithms for quantizer design. The Lagrangian formulation is demonstrated to provide a convenient framework for analyzing the empirical design of variable-rate vector quantizers. In particular, the consistency of empirical design based on minimizing the Lagrangian performance over a stationary and ergodic training sequence is shown for sources with finite second moment. The finite sample performance is also studied for independent training data and sources with bounded support.

The authors examined vector quantizer performance as a function of training sequence size for tree-structured and full-search vector quantizers. The performance was measured by the mean-squared error between the input image and the quantizer output at a given bit rate. The training sequence size was measured either by the number of training images, or by the number of training vectors. When the training vectors were counted, they were selected randomly from among the training images. For every training sequence size, vector quantizers were developed from several different training sequences, and the distortion was calculated for different test sequences in a cross validation procedure. Preliminary results suggest that plots of distortion vs. number of training images follow an algebraic decay, as expected from analogous results of learning theory. >

An antenna array on the transmit side provides the system with an extra spatial dimension that can be utilized for coding both in the spatial as well as the temporal domain. The recent development of such space time codes shows that there are ways of exploiting multiple transmit antennas while completely avoiding traditional beam-forming techniques need of accurate channel state information. In this project, we generate a framework for training based channel estimation under different channel and interference statistics. The minimum mean square error (MMSE) estimator for channel matrix estimation in Rician fading multi-antenna systems is analyzed, and exclusively the proposal of mean square error (MSE) minimizing training structures. By considering Kronecker-structured systems with a grouping of noise and interference and random training sequence length, we gather and simplify numerous earlier results in the framework. We simplify the circumstances for attaining the optimal training sequence structure and show when the spatial training power distribution can be explained unambiguously. We also prove that spatial correlation improves the estimation performance and establish how it determines the optimal training sequence length. The analytic results for Kronecker-structured systems are used to derive a heuristic training sequence under general unstructured statistics. The MMSE estimator of the squared Frobenius norm of the channel matrix is also derived and shown to provide far better gain estimates than other approaches. It is shown under which conditions training sequences that minimize the non-convex MSE can be derived explicitly or with low complexity. Numerical examples are used to evaluate the performance of the two estimators for different training classifications and system statistics. We also elucidate how the finest length of the training sequence often can be shorter than the number of transmit antennas. Keyword: mse, mmse estimator, kronecker-structured systems.

An antenna array on the transmit side provides the system with an extra spatial dimension that can be utilized for coding both in the spatial as well as the temporal domain.The recent development of such space time codes shows that there are ways of exploiting multiple transmit antennas while completely avoiding traditional beam-forming techniques need of accurate channel state information.In this project, we generate a framework for training based channel estimation under different channel and interference statistics.The minimum mean square error (MMSE) estimator for channel matrix estimation in Rician fading multi-antenna systems is analyzed, and exclusively the proposal of mean square error (MSE) minimizing training structures.By considering Kronecker-structured systems with a grouping of noise and interference and random training sequence length, we gather and simplify numerous earlier results in the framework.We simplify the circumstances for attaining the optimal training sequence structure and show when the spatial training power distribution can be explained unambiguously.We also prove that spatial correlation improves the estimation performance and establish how it determines the optimal training sequence length.The analytic results for Kronecker-structured systems are used to derive a heuristic training sequence under general unstructured statistics.The MMSE estimator of the squared Frobenius norm of the channel matrix is also derived and shown to provide far better gain estimates than other approaches.It is shown under which conditions training sequences that minimize the non-convex MSE can be derived explicitly or with low complexity.Numerical examples are used to evaluate the performance of the two estimators for different training classifications and system statistics.We also elucidate how the finest length of the training sequence often can be shorter than the number of transmit antennas.

The vector quantization method described by Linde et al. [IEEE Trans. Commun. 28, 84–95 (1980)] was used to optimally quantize speech spectral principal components. The spectral principal components, obtained from linear combinations of spectral band energies, are an optimal uncorrelated parameter set containing a very large amount of spectral variance with a relatively small number of dimensions. For a given percentage of quantization distortion, the vector quantizer required fewer bits per frame than individual quantization of principal components, thus indicating a clustering of the speech data in the uncorrelated parameter space. Using the principal components parameters to measure both gain and spectral shape and using eight bits per frame, the mean square vector quantization error was about 2.5% in a four-dimensional space, and about 5% in a seven-dimensional space. Speech synthesized from the vector quantized principal components using nine bits per frame was somewhat inferior in quality to speech synthesized with full precision principal components, but quite intelligible. [Work supported by NSF.]

This paper studies the asymptotic characteristics of optimum uniform scalar quantizers as N, the number of levels, becomes large. It is shown that the length of the support region increases as (ln N)/sup 1//spl alpha//, when applied to a random variable with a generalized Gaussian density of the form p(x)=ae(-b|x|/sup /spl alpha//). Moreover, the mean-squared error is asymptotically well approximated by /spl Delta//sup 2//12, where /spl Delta/ is the step size, and decreases as (ln N)/sup 2//spl alpha///N/sup 2/. >

As higher symbol rates are utilized in the intensity modulation and direct detection (IM/DD) scheme to meet the unrelenting growth of data traffic, overcoming the inter-symbol interference (ISI) induced by the limited bandwidth has become increasingly crucial. Channel equalization based on digital signal processing (DSP) is an effective solution, where least-mean squares (LMS) algorithm is adopted to adjust tap coefficients. However, the LMS algorithm usually has slow rate of convergence and requires lots of training symbols. This work proposes a novel training sequence to accelerate the LMS-based equalization. A first-order Markov chain (MC) is employed for sequence generation, which introduces correlation between samples and shapes signal spectrum. Compared with the conventional training sequence that consists of independent and identically distributed (i.i.d.) samples and has a white spectrum, the MC sequence enables faster convergence of tap coefficients and mean-squared error (MSE). Moreover, an experimental demonstration of a 43 Gbaud PAM-4 signal shows that the proposed sequence can achieve a lower pre-forward-error-correction (pre-FEC) bit error rate (BER) than that of the i.i.d. sequence with the same length. When the PAM-4 signal is transmitted over a 5-km standard single mode fiber (SSMF) with 6-dB system bandwidth of 10 GHz, more than 70% training sequence length reduction can be attained. When the fiber length is increased to 10 km and the signal suffers from severe power fading, more than 48% reduction can be achieved.

We analyze the behavior of a single-loop /spl Sigma//spl Delta/ modulator using a vector quantization (VQ) approach. We extract the encoder-decoder structure typical of VQ, existing in a /spl Sigma//spl Delta/ modulator. The study of the encoding part gives the intrinsic behavior of the modulator. Bounds to the intrinsic performance of the /spl Sigma//spl Delta/ modulator can be derived through this approach in the oversampling situation, assuming periodicity of the bandlimited input signals. >

For distributed multi-user and multi-relay cooperative networks, the received signal may be affected by multiple timing offsets (MTOs) and multiple channels that need to be jointly estimated for successful decoding at the receiver. This paper addresses the design of optimal training sequences for efficient estimation of MTOs and multiple channel parameters. A new hybrid Cramer-Rao lower bound (HCRB) for joint estimation of MTOs and channels is derived. Subsequently, by minimizing the derived HCRB as a function of training sequences, three training sequence design guidelines are derived and according to these guidelines, two training sequences are proposed. In order to show that the proposed design guidelines also improve estimation accuracy, the conditional Cramer-Rao lower bound (ECRB), which is a tighter lower bound on the estimation accuracy compared to the HCRB, is also derived. Numerical results show that the proposed training sequence design guidelines not only lower the HCRB, but they also lower the ECRB and the mean-square error of the proposed maximum a posteriori estimator. Moreover, extensive simulations demonstrate that application of the proposed training sequences significantly lowers the bit-error rate performance of multi-relay cooperative networks when compared to training sequences that violate these design guidelines.

This paper addresses the problem of optimal training for joint carrier frequency offset (CFO) and channel estimation in multiuser MIMO OFDM systems, where each user can utilize all available subcarriers. To choose the training sequence with the goal of providing the smallest estimation mean square error (MSE), the Cramer-Rao bound (CRB) is derived in close-form. However, little insight on the sequence selection can be obtained from the CRB directly due to its complicated dependence on training sequence and channel parameters. To proceed, asymptotic CRB, which has a much simpler dependence on the training, is derived. It is found that the optimal training sequences should be individually white and uncorrelated with each other. Simulation results illustrate the merits of the proposed training design.

Optimal scalar quantizer design for transmission over a finite-state channel is considered. The objective Is to minimize the mean-squared error when the channel is in the normal mode of operation, while guaranteeing a minimum fidelity when the channel is in the bad state. An optimal quantizer design algorithm for the general case where noisy state information is available both at the receiver and at the transmitter is derived. It is shown that using mixed strategies is necessary in order to achieve the optimal performance. Finally, the case where the observation is noisy is considered and it is shown that the optimal scheme in this case is to apply the algorithm for the no observation noise to the mean-squared estimate of the desired random variable from the noisy data.

To achieve reasonable bit error rate (BER) perfor- mance, accurate channel state information (CSI) is required for linear minimum mean squared error turbo equalizer(LMMSE- TE), which is hard to be obtained under the circumstance that long training sequence is not available or channel is time-varying. Soft and hard iterative channel estimation (HICE) algorithms were proposed as fine channel estimation schemes after coarse estimation. In this paper, a new iterative channel correction (ICC) algorithm that belongs to the HICE based on the least mean square (LMS) algorithm is proposed. A cost function using hard decision variables on the output of TE is formed. We show that the gradient descent (GD) algorithm is effective to iteratively minimize the cost function. Furthermore, the iterative step-size to guarantee the convergence for the ICC is analyzed, the mean squared error (MSE) lower bound of the estimated channels for the ICC is derived and validated by the simulations. Our simulations show that, given an imperfect CSI with MSE below the upper bound, the LMMSE-TE incorporated with the ICC yields small performance degradation compared to that with perfect CSI, while the traditional LMMSE-TE suffers from severe error floor even with more iterations, thus showing the robustness of our proposed algorithm to the imperfect channel estimations.

The central theme is the study of error propagation effects in decision feedback equalizers (DFEs). The thesis contains: a stochastic analysis of error propagation in a tuned DFE; an analysis of the effects of error propagation in a blindly adapted DFE; a deterministic analysis of error propagation through input-output stability ideas; and testing procedures for establishing correct tap convergence in blind adaptation. To a lesser extent, the decision directed equalizer (DDE) is also treated. Characterizing error propagation using finite state Markov process (FSMP) techniques is first considered. We classify how the channel and DFE parameters affect the FSMP model and establish tight bounds on the error probability and mean error recovery time of a tuned DFE. These bounds are shown to be too conservative for practical use and highlight the need for imposing stronger hypotheses on the class of channels for which a DFE may be effectively used. In blind DFE adaptation we show the effect of decision errors is to distort the adaptation relative to the use of a training sequence. The mean square error surface in a LMS type setting is shown to be a concatenation of quadratic functions exposing the possibility of false tap convergence to undesirable DFE parameter settings. Averaging analysis and simulation are used to verify this behaviour on some examples. Error propagation in a tuned DFE is also examined in a deterministic setting. A finite error recovery time problem is set up as an input-output stability problem. Passivity theory is invoked to prove that a DFE can be effectively used on a channel satisfying a simple frequency domain condition. These results give performance bounds which relate well with practice. Testing for false tap convergence in blind adaptation concludes our study. Simple statistic output tests are shown to be capable of discerning correct operation of a DDE. Similar tests are conjectured for the DFE, supported by proofs for the low dimensional cases.