Manuel S. Stein

A Lower Bound for the Fisher Information Measure

The problem how to approximately determine the value of the Fisher information measure for a general parametric probabilistic system is considered. Having available the first and second moment of the system output in a parametric form, it is shown that the information measure can be bounded from below through a replacement of the original system by a Gaussian system with equivalent moments. The presented technique is applied to a system of practical importance and the potential quality of the bound is demonstrated.

Quantization-loss reduction for signal parameter estimation

Using coarse resolution analog-to-digital conversion (ADC) offers the possibility to reduce the complexity of digital receive systems but introduces a loss in effective signal-to-noise ratio (SNR) when comparing to ideal receivers with infinite resolution ADC. Therefore, here the problem of signal parameter estimation from a coarsely quantized receive signal is considered. In order to increase the system performance, we propose to adjust the analog radio front-end to the quantization device in order to reduce the quantization-loss. By optimizing the bandwidth of the analog filter with respect to a weighted form of the Cramér-Rao lower bound (CRLB), we show that for low SNR and a 1-bit hard-limiting device it is possible to significantly reduce the quantizationloss of initially -1.96 dB. As application, joint carrier-phase and time-delay estimation for satellite-based positioning and synchronization is discussed. Simulations of the maximum-likelihood estimator (MLE) show that the optimum estimator achieves the same quantization-loss reduction as predicted by the performance bound of the optimized system.

Overdemodulation for High-Performance Receivers with Low-Resolution ADC

The design of the analog demodulator for receivers with low-resolution analog-to-digital converters (ADCs) is investigated. For infinite ADC resolution, demodulation to baseband with M=2 orthogonal sinusoidal functions (quadrature demodulation) is an optimum design choice with respect to system performance. For receivers which are restricted to ADCs with low amplitude resolution, we show that this classical approach is suboptimal under an estimation and information theoretic perspective. To this end, we analyze the theoretical channel parameter estimation performance (Fisher information) under an ideal receive situation when forming M>2 analog demodulation channels prior to low-complexity 1-bit ADCs. To demonstrate the impact of overdemodulation for communication problems, we also provide a brief discussion on the achievable transmission rates (Shannon information) with 1-bit quantization of M>2 demodulation channels.

Performance analysis for pilot-based 1-bit channel estimation with unknown quantization threshold

Parameter estimation using quantized observations is of importance in many practical applications. Under a symmetric 1-bit setup, consisting of a zero-threshold hard-limiter, it is well known that the large sample performance loss for low signal-to-noise ratios (SNRs) is moderate (2/Π or -1.96dB). This makes low-complexity analog-to-digital converters (ADCs) with 1-bit resolution a promising solution for future wireless communications and signal processing devices. However, hardware imperfections and external effects introduce the quantizer with an unknown hard-limiting level different from zero. In this paper, the performance loss associated with pilot-based channel estimation, subject to an asymmetric hard limiter with unknown offset, is studied under two setups. The analysis is carried out via the Cramér-Rao lower bound (CRLB) and an expected CRLB for a setup with random parameter. Our findings show that the unknown threshold leads to an additional information loss, which vanishes for low SNR values or when the offset is close to zero.

Signal parameter estimation performance under a sampling rate constraint

Recently it has been found, that the receive filter design rule 2Br <; fs, known as the sampling theorem, does not necessarily lead to optimum signal parameter estimation performance. This is in particular the case if the transmit bandwidth Bt is higher than the sampling bandwidth of the receiver, i.e., 2Bt > fs. While this result has been obtained under the assumption of ideal low-pass filters, here we extend the analysis by taking into account the possibility of optimizing the filter transfer function. We formulate the problem of finding the best unconstrained analog receive filter with respect to the asymptotic estimation performance on the basis of the Fisher information. It is shown that the problem of optimizing the filter coefficients can be decomposed into N subproblems, each reducing to a binary decision rule for the spectral filter coefficients. Based on these findings we show examples of optimized ideal filters and compare their estimation performance with classical low-pass filters. The results give insights into favorable designs of realizable filters and allow to explore the theoretic performance limits of signal processing systems with sampling rate constraints.

Asymptotic Parameter Tracking Performance With Measurement Data of 1-Bit Resolution

The problem of signal parameter estimation and tracking with measurement data of low resolution is considered. In comparison to an ideal receiver with infinite receive resolution, the performance loss of a simplistic receiver with 1-bit resolution is investigated. For the case where the measurement data is preprocessed by a symmetric hard-limiting device with 1-bit output, it is well-understood that the performance for low SNR channel parameter estimation degrades moderately by 2/π(-1.96 dB). Here, we show that the 1-bit quantization loss can be significantly smaller if information about the temporal evolution of the channel parameters is taken into account in the form of a state-space model. By the analysis of a Bayesian bound for the achievable tracking performance, we attain the result that the quantization loss in dB is in general smaller by a factor of two if the channel evolution is slow. For the low SNR regime, this is equivalent to a reduced loss of √{2/π}(-0.98 dB). By simulating non-linear filtering algorithms for a satellite-based ranging application and a UWB channel estimation problem, both with low-complexity 1-bit analog-to-digital converter (ADC) at the receiver, we verify that the analytical characterization of the tracking error is accurate. This shows that the performance loss due to observations with low amplitude resolution can, in practice, be much less pronounced than indicated by classical results. Finally, we discuss the implication of the result for medium SNR applications like channel estimation in the context of mobile wireless communications.

Information-Preserving Transformations for Signal Parameter Estimation

The problem of parameter estimation from large noisy data is considered. If the observation size N is large, the calculation of efficient estimators is computationally expensive. Further, memory can be a limiting factor in technical systems where data is stored for later processing. Here we follow the idea of reducing the size of the observation by projecting the data onto a subspace of smaller dimension M ≪ N, but with the highest possible informative value regarding the estimation problem. Under the assumption that a prior distribution of the parameter is available and the output size is fixed to M, we derive a characterization of the Pareto-optimal set of linear transformations by using a weighted form of the Bayesian Cramér-Rao lower bound (BCRLB) which stands in relation to the expected value of the Fisher information measure. Satellite-based positioning is discussed as a possible application. Here N must be chosen large in order to compensate for low signal-to-noise ratios (SNR). For different values of M, we visualize the information-loss and show by simulation of the MAP estimator the potential accuracy when operating on the reduced data.

Optimum analog receive filters for detection and inference under a sampling rate constraint

The problem of optimum analog receive filtering for digital signal detection and parameter estimation is considered. Here the case of a signal source with bandwidth B_t and a receiver with fixed sampling rate f_s is discussed under the assumption that 2B_t > f_s. We investigate the impact of adjusting the receive bandwidth B_r of the analog pre-filter, which is applied prior to the sampler, with respect to the deflection coefficient or the Fisher information measure. This reveals that the design rule 2B_r <; f_s, known as the sampling theorem, does not necessarily lead to optimum system performance. Studying the two analytical information measures under a fix sampling rate f_s and an arbitrary choice of B_r, we provide an example where receive setups with 2B_r > f_s achieve higher detection and parameter estimation performance.

Multi-satellite time-delay estimation for reliable high-resolution GNSS receivers

Reliable estimation of position in time and space has become a key necessity in several technical applications like mobile navigation, precision farming or network synchronization. While the increasing amount of operating Global Navigation Satellite Systems (GNSS) offers diverse possibilities to receive GNSS signals worldwide and to determine position accurately, inter- and intrasystem interference has been identified as a problem of growing importance. The performance of receivers which track all in-view satellites individually degrades if mutual interference is not taken into account. Therefore, we consider the problem of joint signal parameter estimation. For scenarios where the signals of different satellites superimpose at the receiver a joint maximum likelihood estimator for all relevant signal parameters is derived. In order to keep the computation of the related likelihood function, and the determination of its maximum, feasible for a low-complexity receiver, an iterative Expectation-Maximization (EM) algorithm is applied. Simulations for different scenarios show that this approach is efficient in the estimation theoretic sense, and robust against interference that is caused by signals with known structure.

Measurement-driven quality assessment of nonlinear systems by exponential replacement

We discuss the problem how to determine the quality of a nonlinear system with respect to a measurement task. Due to amplification, filtering, quantization and internal noise sources physical measurement equipment in general exhibits a nonlinear and random input-to-output behaviour. This usually makes it impossible to accurately describe the underlying statistical system model. When the individual operations are all known and deterministic, one can resort to approximations of the input-to-output function. The problem becomes challenging when the processing chain is not exactly known or contains nonlinear random effects. Then one has to approximate the output distribution in an empirical way. Here we show that by measuring the first two sample moments of an arbitrary set of output transformations in a calibrated setup, the output distribution of the actual system can be approximated by an equivalent exponential family distribution. This method has the property that the resulting approximation of the statistical system model is guaranteed to be pessimistic in an estimation theoretic sense. We show this by proving that an equivalent exponential family distribution in general exhibits a lower Fisher information measure than the original system model. With various examples and a model matching step we demonstrate how this estimation theoretic aspect can be exploited in practice in order to obtain a conservative measurement-driven quality assessment method for nonlinear measurement systems.