40 research outputs found

    Distributed Functional Scalar Quantization Simplified

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    Distributed functional scalar quantization (DFSQ) theory provides optimality conditions and predicts performance of data acquisition systems in which a computation on acquired data is desired. We address two limitations of previous works: prohibitively expensive decoder design and a restriction to sources with bounded distributions. We rigorously show that a much simpler decoder has equivalent asymptotic performance as the conditional expectation estimator previously explored, thus reducing decoder design complexity. The simpler decoder has the feature of decoupled communication and computation blocks. Moreover, we extend the DFSQ framework with the simpler decoder to acquire sources with infinite-support distributions such as Gaussian or exponential distributions. Finally, through simulation results we demonstrate that performance at moderate coding rates is well predicted by the asymptotic analysis, and we give new insight on the rate of convergence

    Linear Precoding with Low-Resolution DACs for Massive MU-MIMO-OFDM Downlink

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    We consider the downlink of a massive multiuser (MU) multiple-input multiple-output (MIMO) system in which the base station (BS) is equipped with low-resolution digital-to-analog converters (DACs). In contrast to most existing results, we assume that the system operates over a frequency-selective wideband channel and uses orthogonal frequency division multiplexing (OFDM) to simplify equalization at the user equipments (UEs). Furthermore, we consider the practically relevant case of oversampling DACs. We theoretically analyze the uncoded bit error rate (BER) performance with linear precoders (e.g., zero forcing) and quadrature phase-shift keying using Bussgang's theorem. We also develop a lower bound on the information-theoretic sum-rate throughput achievable with Gaussian inputs, which can be evaluated in closed form for the case of 1-bit DACs. For the case of multi-bit DACs, we derive approximate, yet accurate, expressions for the distortion caused by low-precision DACs, which can be used to establish lower bounds on the corresponding sum-rate throughput. Our results demonstrate that, for a massive MU-MIMO-OFDM system with a 128-antenna BS serving 16 UEs, only 3--4 DAC bits are required to achieve an uncoded BER of 10^-4 with a negligible performance loss compared to the infinite-resolution case at the cost of additional out-of-band emissions. Furthermore, our results highlight the importance of taking into account the inherent spatial and temporal correlations caused by low-precision DACs

    Multiple Description Quantization via Gram-Schmidt Orthogonalization

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    The multiple description (MD) problem has received considerable attention as a model of information transmission over unreliable channels. A general framework for designing efficient multiple description quantization schemes is proposed in this paper. We provide a systematic treatment of the El Gamal-Cover (EGC) achievable MD rate-distortion region, and show that any point in the EGC region can be achieved via a successive quantization scheme along with quantization splitting. For the quadratic Gaussian case, the proposed scheme has an intrinsic connection with the Gram-Schmidt orthogonalization, which implies that the whole Gaussian MD rate-distortion region is achievable with a sequential dithered lattice-based quantization scheme as the dimension of the (optimal) lattice quantizers becomes large. Moreover, this scheme is shown to be universal for all i.i.d. smooth sources with performance no worse than that for an i.i.d. Gaussian source with the same variance and asymptotically optimal at high resolution. A class of low-complexity MD scalar quantizers in the proposed general framework also is constructed and is illustrated geometrically; the performance is analyzed in the high resolution regime, which exhibits a noticeable improvement over the existing MD scalar quantization schemes.Comment: 48 pages; submitted to IEEE Transactions on Information Theor

    Distributed Scalar Quantization for Computing: High-Resolution Analysis and Extensions

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    Communication of quantized information is frequently followed by a computation. We consider situations of \emph{distributed functional scalar quantization}: distributed scalar quantization of (possibly correlated) sources followed by centralized computation of a function. Under smoothness conditions on the sources and function, companding scalar quantizer designs are developed to minimize mean-squared error (MSE) of the computed function as the quantizer resolution is allowed to grow. Striking improvements over quantizers designed without consideration of the function are possible and are larger in the entropy-constrained setting than in the fixed-rate setting. As extensions to the basic analysis, we characterize a large class of functions for which regular quantization suffices, consider certain functions for which asymptotic optimality is achieved without arbitrarily fine quantization, and allow limited collaboration between source encoders. In the entropy-constrained setting, a single bit per sample communicated between encoders can have an arbitrarily-large effect on functional distortion. In contrast, such communication has very little effect in the fixed-rate setting.Comment: 36 pages, 10 figure

    Message-Passing Estimation from Quantized Samples

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    Estimation of a vector from quantized linear measurements is a common problem for which simple linear techniques are suboptimal -- sometimes greatly so. This paper develops generalized approximate message passing (GAMP) algorithms for minimum mean-squared error estimation of a random vector from quantized linear measurements, notably allowing the linear expansion to be overcomplete or undercomplete and the scalar quantization to be regular or non-regular. GAMP is a recently-developed class of algorithms that uses Gaussian approximations in belief propagation and allows arbitrary separable input and output channels. Scalar quantization of measurements is incorporated into the output channel formalism, leading to the first tractable and effective method for high-dimensional estimation problems involving non-regular scalar quantization. Non-regular quantization is empirically demonstrated to greatly improve rate-distortion performance in some problems with oversampling or with undersampling combined with a sparsity-inducing prior. Under the assumption of a Gaussian measurement matrix with i.i.d. entries, the asymptotic error performance of GAMP can be accurately predicted and tracked through the state evolution formalism. We additionally use state evolution to design MSE-optimal scalar quantizers for GAMP signal reconstruction and empirically demonstrate the superior error performance of the resulting quantizers.Comment: 12 pages, 8 figure

    Adaptive Quantizers for Estimation

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    In this paper, adaptive estimation based on noisy quantized observations is studied. A low complexity adaptive algorithm using a quantizer with adjustable input gain and offset is presented. Three possible scalar models for the parameter to be estimated are considered: constant, Wiener process and Wiener process with deterministic drift. After showing that the algorithm is asymptotically unbiased for estimating a constant, it is shown, in the three cases, that the asymptotic mean squared error depends on the Fisher information for the quantized measurements. It is also shown that the loss of performance due to quantization depends approximately on the ratio of the Fisher information for quantized and continuous measurements. At the end of the paper the theoretical results are validated through simulation under two different classes of noise, generalized Gaussian noise and Student's-t noise

    Differential encoding techniques applied to speech signals

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    The increasing use of digital communication systems has produced a continuous search for efficient methods of speech encoding. This thesis describes investigations of novel differential encoding systems. Initially Linear First Order DPCM systems employing a simple delayed encoding algorithm are examined. The systems detect an overload condition in the encoder, and through a simple algorithm reduce the overload noise at the expense of some increase in the quantization (granular) noise. The signal-to-noise ratio (snr) performance of such d codec has 1 to 2 dB's advantage compared to the First Order Linear DPCM system. In order to obtain a large improvement in snr the high correlation between successive pitch periods as well as the correlation between successive samples in the voiced speech waveform is exploited. A system called "Pitch Synchronous First Order DPCM" (PSFOD) has been developed. Here the difference Sequence formed between the samples of the input sequence in the current pitch period and the samples of the stored decoded sequence from the previous pitch period are encoded. This difference sequence has a smaller dynamic range than the original input speech sequence enabling a quantizer with better resolution to be used for the same transmission bit rate. The snr is increased by 6 dB compared with the peak snr of a First Order DPCM codea. A development of the PSFOD system called a Pitch Synchronous Differential Predictive Encoding system (PSDPE) is next investigated. The principle of its operation is to predict the next sample in the voiced-speech waveform, and form the prediction error which is then subtracted from the corresponding decoded prediction error in the previous pitch period. The difference is then encoded and transmitted. The improvement in snr is approximately 8 dB compared to an ADPCM codea, when the PSDPE system uses an adaptive PCM encoder. The snr of the system increases further when the efficiency of the predictors used improve. However, the performance of a predictor in any differential system is closely related to the quantizer used. The better the quantization the more information is available to the predictor and the better the prediction of the incoming speech samples. This leads automatically to the investigation in techniques of efficient quantization. A novel adaptive quantization technique called Dynamic Ratio quantizer (DRQ) is then considered and its theory presented. The quantizer uses an adaptive non-linear element which transforms the input samples of any amplitude to samples within a defined amplitude range. A fixed uniform quantizer quantizes the transformed signal. The snr for this quantizer is almost constant over a range of input power limited in practice by the dynamia range of the adaptive non-linear element, and it is 2 to 3 dB's better than the snr of a One Word Memory adaptive quantizer. Digital computer simulation techniques have been used widely in the above investigations and provide the necessary experimental flexibility. Their use is described in the text

    Characterization of Information Channels for Asymptotic Mean Stationarity and Stochastic Stability of Non-stationary/Unstable Linear Systems

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    Stabilization of non-stationary linear systems over noisy communication channels is considered. Stochastically stable sources, and unstable but noise-free or bounded-noise systems have been extensively studied in information theory and control theory literature since 1970s, with a renewed interest in the past decade. There have also been studies on non-causal and causal coding of unstable/non-stationary linear Gaussian sources. In this paper, tight necessary and sufficient conditions for stochastic stabilizability of unstable (non-stationary) possibly multi-dimensional linear systems driven by Gaussian noise over discrete channels (possibly with memory and feedback) are presented. Stochastic stability notions include recurrence, asymptotic mean stationarity and sample path ergodicity, and the existence of finite second moments. Our constructive proof uses random-time state-dependent stochastic drift criteria for stabilization of Markov chains. For asymptotic mean stationarity (and thus sample path ergodicity), it is sufficient that the capacity of a channel is (strictly) greater than the sum of the logarithms of the unstable pole magnitudes for memoryless channels and a class of channels with memory. This condition is also necessary under a mild technical condition. Sufficient conditions for the existence of finite average second moments for such systems driven by unbounded noise are provided.Comment: To appear in IEEE Transactions on Information Theor
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