The reconstruction of a continuous-time signal s(t) from the sign of its (deliberately) contaminated samples is considered. Sequential, generally nonlinear estimates of s(t) are established and their performance is studied; error bounds and convergence rates are derived. The signal s(t) need not be band-limited. The convergence rates obtained here are faster than those obtained previously for nonsequential estimates. The degradation in the reconstruction of the signal, due to transmission over a noisy channel, is also investigated, and bounds on the additional error are obtained.

The reconstruction of analog signals from the sign of their noisy samples

Modern communication receiver architectures center around digital signal processing (DSP), with the bulk of the receiver processing being performed on digital signals obtained after analog-to-digital conversion (ADC). In this paper, we explore Shannon-theoretic performance limits when ADC precision is drastically reduced, from typical values of 8-12 bits used in current communication transceivers, to 1-3 bits. The goal is to obtain insight on whether DSP-centric transceiver architectures are feasible as communication bandwidths scale up, recognizing that high-precision ADC at high sampling rates is either unavailable, or too costly or power-hungry. Specifically, we evaluate the communication limits imposed by low-precision ADC for the ideal real discrete-time Additive White Gaussian Noise (AWGN) channel, under an average power constraint on the input. For an ADC with K quantization bins (i.e., a precision of log2 K bits), we show that the Shannon capacity is achievable by a discrete input distribution with at most K + 1 mass points. For 2-bin (1-bit) symmetric ADC, this result is tightened to show that binary antipodal signaling is optimum for any signal-to-noise ratio (SNR). For multi-bit ADC, the capacity is computed numerically, and the results obtained are used to make the following encouraging observations regarding system design with low-precision ADC : (a) even at moderately high SNR of up to 20 dB, 2-3 bit quantization results in only 10-20% reduction of spectral efficiency, which is acceptable for large communication bandwidths, (b) standard equiprobable pulse amplitude modulation with ADC thresholds set to implement maximum likelihood hard decisions is asymptotically optimum at high SNR, and works well at low to moderate SNRs as well.

Transceiver Design with Low-Precision Analog-to-Digital Conversion : An Information-Theoretic Perspective

Motivated by the estimation of spatio-temporal events with cheap, simple sensors, we consider the problem of estimation of a parameter of a signal corrupted by noise assuming that only 1-bit precision dithered quantized samples are available. An estimate that does not require the knowledge of the dither signal and the noise distribution is proposed, and it is analyzed in detail under variety of nonidealities. The consistency and asymptotic normality of the estimate is established for deter- ministic and random sampling, imprecise knowledge of sampling locations, Gaussian and non-Gaussian noise (with possibly infinite variance), a wide class of dither distributions, and under erroneous transmission of the binary observations via binary-symmetric channels (BSCs). It is also shown that if approximation to the log-likelihood equation in the full precision case yields a good estimate, then there is a corresponding good estimate based on 1-bit dithered samples. The proposed estimate requires no more computation than the maximum-likelihood estimate for the full precision case and suffers only a logarithmic rate loss compared to the full precision case when uniform dithering is used. It is shown that uniform dithering leads to the best rate among a broad class of dither distributions. A condition under which no dithering leads to a better estimate is also given.

Signal Parameter Estimation Using 1-Bit

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Signal Parameter Estimation Using 1-Bit