Onkar Dabeer

On the limits of communication with low-precision analog-to-digital conversion at the receiver

As communication systems scale up in speed and bandwidth, the cost and power consumption of high-precision (e.g., 8-12 bits) analog-to-digital conversion (ADC) becomes the limiting factor in modern transceiver architectures based on digital signal processing. In this work, we explore the impact of lowering the precision of the ADC on the performance of the communication link. Specifically, we evaluate the communication limits imposed by low-precision ADC (e.g., 1-3 bits) for transmission over the 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 input distribution need not have any more than K+1 mass points to achieve the channel capacity. For 2-bin (1-bit) symmetric quantization, this result is tightened to show that binary antipodal signaling is optimum for any signal-to- noise ratio (SNR). For multi-bit quantization, a dual formulation of the channel capacity problem is used to obtain tight upper bounds on the capacity. The cutting-plane algorithm is employed to compute the capacity numerically, and the results obtained are used to make the following encouraging observations : (a) up to a moderately high SNR of 20 dB, 2-3 bit quantization results in only 10-20% reduction of spectral efficiency compared to unquantized observations, (b) standard equiprobable pulse amplitude modulated input with quantizer thresholds set to implement maximum likelihood hard decisions is asymptotically optimum at high SNR, and works well at low to moderate SNRs as well.

Detection of hiding in the least significant bit

In this paper, we apply the theory of hypothesis testing to the steganalysis, or detection of hidden data, in the least significant bit (LSB) of a host image. The hiding rate (if data is hidden) and host probability mass function (PMF) are unknown. Our main results are as follows. a) Two types of tests are derived: a universal (over choices of host PMF) method that has certain asymptotic optimality properties and methods that are based on knowledge or estimation of the host PMF and, hence, an appropriate likelihood ratio (LR). b) For a known host PMF, it is shown that the composite hypothesis testing problem corresponding to an unknown hiding rate reduces to a worst-case simple hypothesis testing problem. c) Using the results for a known host PMF, practical tests based on the estimation of the host PMF are obtained. These are shown to be superior to the state of the art in terms of receiver operating characteristics as well as self-calibration across different host images. Estimators for the hiding rate are also developed.

Signal Parameter Estimation Using 1-Bit Dithered Quantization

Motivated by the estimation of spatio-temporal events with cheap, simple sensors, we consider the problem of estimation of a parameter thetas of a signal s(x;thetas) 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 deterministic 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

Channel Estimation with Low-Precision Analog-to-Digital Conversion

We consider the problem of estimating the impulse response of a dispersive channel when the channel output is sampled using a low-precision analog-to-digital converter (ADC). While traditional channel estimation techniques require about 6 bits of ADC precision to approach full-precision performance, we are motivated by applications to multiGigabit communication, where we may be forced to use much lower precision (e.g., 1-3 bits) due to considerations of cost, power, and technological feasibility. We show that, even with such low ADC precision, it is possible to attain near full-precision performance using closed-loop estimation, where the ADC input is dithered and scaled. The dither signal is obtained using linear feedback based on the Minimum Mean Squared Error (MMSE) criterion. The dither feedback coefficients and the scaling gains are computed offline using Monte Carlo simulations based on a statistical model for the channel taps, and are found to work well over wide range of channel variations.

Convergence analysis of the constant modulus algorithm

We study the global convergence of the stochastic gradient constant modulus algorithm (CMA) in the absence of channel noise as well as in the presence of channel noise. The case of fractionally spaced equalizer and/or multiple antenna at the receiver is considered. For the noiseless case, we show that with proper initialization, and with small step size, the algorithm converges to a zero-forcing filter with probability close to one. In the presence of channel noise such as additive Gaussian noise, we prove that the algorithm diverges almost surely on the infinite-time horizon. However, under suitable conditions, the algorithm visits a small neighborhood of the Wiener filters a large number of times before ultimately diverging.

Capacity of the discrete-time AWGN channel under output quantization

We investigate the limits of communication over the discrete-time additive white Gaussian noise (AWGN) channel, when the channel output is quantized using a small number of bits. We first provide a proof of our recent conjecture on the optimality of a discrete input distribution in this scenario. Specifically, we show that for any given output quantizer choice with K quantization bins (i.e., a precision of log2 K bits), the input distribution, under an average power constraint, need not have any more than K + 1 mass points to achieve the channel capacity. The cutting-plane algorithm is employed to compute this capacity and to generate optimum input distributions. Numerical optimization over the choice of the quantizer is then performed (for 2-bit and 3-bit symmetric quantization), and the results we obtain show that the loss due to low-precision output quantization, which is small at low signal-to-noise ratio (SNR) as expected, can be quite acceptable even for moderate to high SNR values. For example, at SNRs up to 20 dB, 2-3 bit quantization achieves 80-90% of the capacity achievable using infinite-precision quantization.

Communication Limits with Low Precision Analog-to-Digital Conversion at the Receiver

We examine the Shannon limits of communication systems when the precision of the analog-to-digital conversion (ADC) at the receiver is constrained. ADC is costly and power- hungry at high speeds, hence ADC precision is expected to be a limiting factor in the performance of receivers which are heavily based on digital signal processing. In this paper, we consider transmission over an ideal discrete-time real baseband additive white Gaussian noise (AWGN) channel, and provide capacity results when the receiver ADC employs a small number of bits, generalizing our prior work on one bit ADC. We show that dithered ADC does not increase capacity, and hence restrict attention to deterministic quantizers. To compute the capacity, we use a dual formulation of the channel capacity problem, which is attractive due to the discrete nature of the output alphabet. The numerical results we obtain strongly support our conjecture that the optimal input distribution is discrete, and has at most one mass point in each quantizer interval. This implies, for example, that it is not possible to support large alphabets such as 64-QAM using 2-bit quantization on the I and Q channels, at least with symbol rate sampling.

Multivariate Signal Parameter Estimation Under Dependent Noise From 1-Bit Dithered Quantized Data

Motivated by applications in sensor networks and communications, we consider multivariate signal parameter estimation when only dithered 1-bit quantized samples are available. The observation noise is taken to be a stationary, strongly mixing process, which covers a wide range of processes including autoregressive moving average (ARMA) models. The noise is allowed to be Gaussian or to have a heavy-tail (with possibly infinite variance). An estimate of the signal parameters is proposed and is shown to be weakly consistent. Joint asymptotic normality of the parameters estimate is also established and the asymptotic mean and covariance matrices are identified.

On the Limits of Communication Performance with One-Bit Analog-To-Digital Conversion

As communication systems scale up in speed and bandwidth, the power consumption and cost of high-precision (e.g., 6-12 bits) analog-to-digital conversion (ADC) becomes the limiting factor in modern receiver architectures based on digital signal processing. In this paper, we consider the effects of lowering the precision of the ADC on the performance of the communication link. We focus on the most extreme scenario, in which the receiver employs one-bit ADC at baseband. While this constraint impacts every aspect of design ranging from synchronization and channel estimation to demodulation and decoding, in this initial exposition, we focus on the Shannon-theoretic limits imposed by one-bit ADC for an ideal discrete-time real baseband additive white Gaussian noise (AWGN) channel. Results are obtained both for non-spread and spread spectrum systems. While binary phase shift keying (BPSK) is optimal in a non-spread system, we obtain less predictable results for spread spectrum system, including the sub-optimality of BPSK and non-monotonicity of mutual information with signal-to-noise ratio for a fixed constellation

Analysis of an Adaptive Sampler Based on Weber's Law

Webers law suggests a logarithmic relationship between perceptual stimuli and human perception. The Weber sampler is an adaptive, nonuniform sampling mechanism that exploits Webers law to sample the signal at a minimum rate without significant perceptual degradation. In this paper, we introduce and analyze a regularized Weber sampler for smooth deterministic signals as well as smooth random processes. While analysis for a fixed Weber constant δ is analytically intractable except for some special cases, we exploit the fact that the Weber constant is usually small. Under suitable assumptions, for deterministic signals, we provide analytical approximations to the number of samples in a unit time interval and the intersample times. For a random signal, we give analytical approximations to the expectation and probability distribution function of the respective quantities. Our class of deterministic as well as random signals is quite wide, and in particular covers bandlimited signals and signals satisfying second order ordinary differential equations. We also present a number of simulations to demonstrate that our approximations are good up to a Weber constant of 0.2, which is the regime of practical interest. Our results provide a source model for the Weber sampler, which can be used in the study of transmission of perceptual stimuli signals over communication links.