We consider the classical problem of estimating the covariance matrix of a subgaussian distribution from i.i.d. samples in the novel context of coarse quantization, i.e., instead of having full knowledge of the samples, they are quantized to one or two bits per entry. This problem occurs naturally in signal processing applications. We introduce new estimators in two different quantization scenarios and derive non-asymptotic estimation error bounds in terms of the operator norm. In the first scenario we consider a simple, scale-invariant one-bit quantizer and derive an estimation result for the correlation matrix of a centered Gaussian distribution. In the second scenario, we add random dithering to the quantizer. In this case we can accurately estimate the full covariance matrix of a general subgaussian distribution by collecting two bits per entry of each sample. In both scenarios, our bounds apply to masked covariance estimation. We demonstrate the near-optimality of our error bounds by deriving corresponding (minimax) lower bounds and using numerical simulations.
In this self-contained chapter, we revisit a fundamental problem of multivariate statistics: estimating covariance matrices from finitely many independent samples. Based on massive Multiple-Input Multiple-Output (MIMO) systems we illustrate the necessity of leveraging structure and considering quantization of samples when estimating covariance matrices in practice. We then provide a selective survey of theoretical advances of the last decade focusing on the estimation of structured covariance matrices. This review is spiced up by some yet unpublished insights on how to benefit from combined st…
Fixed-Threshold One-Bit Toeplitz Covariance Estimation under Sparse-Ruler Sampling
School of Computer and Artificial Intelligence, Chaohu University（巢湖大学计算机与人工智能学院）；School of Electronic and Optical Engineering, Nanjing University of Science and Technology（南京理工大学电子与光学工程学院）
In this paper, we investigate the quantization and the feedback of downlink spatial covariance matrix for massive multiple-input multiple-output (MIMO) systems with cascaded precoding. Massive MIMO has gained a lot of attention recently because of its ability to significantly improve the network performance. To reduce the overhead of downlink channel estimation and uplink feedback in frequency-division duplex massive MIMO systems, cascaded precoding has been proposed, where the outer precoder is implemented using traditional limited feedback while the inner precoder is determined by the spatia…
Reliable one-bit quantization of bandlimited graph data via single-shot noise shaping
Ludwig-Maximilians-Universität（路德维希-马克西米利安大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Technical University of Munich（慕尼黑工业大学）；Munich Data Science Institute（慕尼黑数据科学研究所）
In this paper we investigate the achievable rate of LOS MIMO systems that use 1-bit quantization and spatial oversampling at the receiver. We propose that by using additional antennas at the receiver, the loss incurred due to the strong limitation of the quantization can be decreased. Mutual information results show that considerable rate gains can be achieved depending on the number and arrangement of the antennas. In some of the cases, even the full available rate from the transmitter can be attained. Furthermore, the results also reveal that two-dimensional antenna arrays can benefit more f…
To relax power consumption requirements in multigigabit/s communications systems low resolution quantization can be used. Information-theoretic results have shown that systems employing 1-bit quantization and oversampling are a viable option for this. This work investigates such a structure under the influence of additive Gaussian noise and two matched pulse shaping filters. It is described how a BCJR algorithm, based on a finite-state channel assumption, can be used to reconstruct symbols of higher order modulation schemes that allow the transmission of more than one bit per symbol. Furthermo…
Oversampling combined with low quantization resolutions has been shown to be a viable option when aiming for energy efficiency in multigigabit/s communications systems. This work considers the case of 1-bit quantization combined with oversampling and shows how the performance of such a system can be improved by using matched pulse shaping filters and faster than Nyquist signaling. The channel is considered with additive Gaussian noise and the performance of the system is evaluated in terms of achievable information rate under symbol-by-symbol detection.
We construct high order low-bit Sigma-Delta $(ΣΔ)$ quantizers for the vector-valued setting of fusion frames. We prove that these $ΣΔ$ quantizers can be stably implemented to quantize fusion frame measurements on subspaces $W_n$ using $\log_2( {\rm dim}(W_n)+1)$ bits per measurement. Signal reconstruction is performed using a version of Sobolev duals for fusion frames, and numerical experiments are given to validate the overall performance.
Quantized channel state information (CSI) plays a critical role in precoding design which helps reap the merits of multiple-input multiple-output (MIMO) technology. In order to reduce the overhead of CSI feedback, we propose a deep learning based CSI quantization method by developing a joint convolutional residual network (JCResNet) which benefits MIMO channel feature extraction and recovery from the perspective of bit-level quantization performance. Experiments show that our proposed method substantially improves the performance.
Millimeter wave (mmWave) spectrum has drawn attention due to its tremendous available bandwidth. The high propagation losses in the mmWave bands necessitate beamforming with a large number of antennas. Traditionally each antenna is paired with a high-speed analog-to-digital converter (ADC), which results in high power consumption. A hybrid beamforming architecture and one-bit resolution ADCs have been proposed to reduce power consumption. However, analog beamforming and one-bit quantization make channel estimation more challenging. In this paper, we propose a narrowband channel estimation algo…
A channel with continuous phase modulation and 1-bit ADC with oversampling is considered. Due to oversampling, higher-order modulations yield a higher achievable rate and this work presents a method to approach this with sophisticated coding. A tailored bit mapping is proposed where two bits can be detected reliably at high SNR and only the third bit suffers from an error floor. With this, a convolutional code is only applied to the third bit. Our experiments with a turbo receiver show that the proposed method corresponds to a lower BER in comparison to the case of conventional channel coding.
We develop a direction of arrival (DoA) and channel estimation algorithm for the one-bit quantized millimeter-wave (mmWave) massive multiple-input multiple-output (MIMO) system. By formulating the estimation problem as a noisy one-bit compressed sensing problem, we propose a computationally efficient gridless solution based on the expectation-maximization generalized approximate message passing (EMGAMP) approach. The proposed algorithm does not need the prior knowledge about the number of DoAs and outperforms the existing methods in distinguishing extremely close DoAs for the case of one-bit q…
We propose a novel linear minimum-mean-squared-error (MMSE) precoder design for a downlink (DL) massive multiple-input-multiple-output (MIMO) scenario. For economical and computational efficiency reasons low resolution 1-bit digital-to-analog (DAC) and analog-to-digital (ADC) converters are used. This comes at the cost of performance gain that can be recovered by the large number of antennas deployed at the base station (BS) and an appropiate precoder design to mitigate the distortions due to the coarse quantization. The proposed precoder takes the quantization non-linearities into account and…
Communication systems with low-resolution analog-to-digital-converters (ADCs) can exploit channel state information at the transmitter (CSIT) and receiver. This paper presents initial results on codebook design and performance analysis for limited feedback systems with one-bit ADCs. Different from the high-resolution case, the absolute phase at the receiver is important to align the phase of the received signals when the received signal is sliced by one-bit ADCs. A new codebook design for the beamforming case is proposed that separately quantizes the channel direction and the residual phase.
This paper studies the optimal bit allocation for shape-gain vector quantization of wireless channels in multiuser (MU) multiple-input multiple-output (MIMO) downlink systems based on linear precoding. Our design minimizes the mean squared-error between the original and quantized channels through optimal bit allocation across shape (direction) and gain (magnitude) for a fixed feedback overhead per user. This is shown to significantly reduce the quantization error, which in turn, decreases the MU interference. This paper makes three main contributions: first, we focus on channel gain quantizati…
We consider the problem of channel estimation for uplink multiuser massive MIMO systems, where, in order to significantly reduce the hardware cost and power consumption, one-bit analog-to-digital converters (ADCs) are used at the base station (BS) to quantize the received signal. Channel estimation for one-bit massive MIMO systems is challenging due to the severe distortion caused by the coarse quantization. It was shown in previous studies that an extremely long training sequence is required to attain an acceptable performance. In this paper, we study the problem of optimal one-bit quantizati…
We consider the one-bit quantizer that minimizes the mean squared error for a source living in a real Hilbert space. The optimal quantizer is a projection followed by a thresholding operation, and we provide methods for identifying the optimal direction along which to project. As an application of our methods, we characterize the optimal one-bit quantizer for a continuous-time random process that exhibits low-dimensional structure. We numerically show that this optimal quantizer is found by a neural-network-based compressor trained via stochastic gradient descent.
For improving coarsely quantized decoding of LDPC codes, we propose a check node aware design of the variable node update. In contrast to previous works, we optimize the variable node to explicitly maximize the mutual information preserved in the check-to-variable instead of the variable-to-check node messages. The extended optimization leads to a significantly different solution for the compression operation at the variable node. Simulation results for regular LDPC codes confirm that the check node aware design, especially for very coarse quantization with 2- or 3-bit messages, achieves perfo…
Low-resolution devices are promising for systems that demand low energy consumption and low complexity as required in IoT systems. In this study, we propose a novel waveform for bandlimited channels with 1-bit quantization and oversampling at the receivers. The proposed method implies that the information is conveyed within the time instances of zero-crossings which is then utilized in combination with a Gray-coding scheme. Unlike the existing method, the proposed method does not require optimization and transmission of a dynamic codebook. The proposed approach outperforms the state-of-the-art…
It is known that the estimated energy consumption of digital-to analog converters (DACs) is around 30\% of the energy consumed by analog-to-digital converters (ADCs) keeping fixed the sampling rate and bit resolution. Assuming that similarly to ADC, DAC dissipation doubles with every extra bit of resolution, a decrease in two resolution bits, for instance from 4 to 2 bits, represents a 75$\% $ lower dissipation. The current limitations in sum-rates of 1-bit quantization have motivated researchers to consider extra bits in resolution to obtain higher levels of sum-rates. Following this, we devi…
Over-the-air federated edge learning (Air-FEEL) is a communication-efficient framework for distributed machine learning using training data distributed at edge devices. This framework enables all edge devices to transmit model updates simultaneously over the entire available bandwidth, allowing for over-the-air aggregation. A one-bit digital over-the-air aggregation (OBDA) scheme has been recently proposed, featuring one-bit gradient quantization at edge devices and majority-voting based decoding at the edge server. However, the low-resolution one-bit gradient quantization slows down the model…
Stanford University（斯坦福大学）；University of New South Wales（新南威尔士大学）
In this paper, we compress convolutional neural network (CNN) weights post-training via transform quantization. Previous CNN quantization techniques tend to ignore the joint statistics of weights and activations, producing sub-optimal CNN performance at a given quantization bit-rate, or consider their joint statistics during training only and do not facilitate efficient compression of already trained CNN models. We optimally transform (decorrelate) and quantize the weights post-training using a rate-distortion framework to improve compression at any given quantization bit-rate. Transform quant…
Today's communication systems typically use high resolution analog-to-digital converters (ADCs). However, considering future communication systems with data rates in the order of 100Gbit/s the ADC power consumption becomes a major factor due to the high sampling rates. A promising alternative are receivers based on 1-bit quantization and oversampling w.r.t. the signal bandwidth. Such an approach requires a redesign of modulation, receiver synchronization, and demapping. A zero crossing modulation is a natural choice as the information needs to be carried in the zero crossing time instants. The…
Motivated by recent high bandwidth communication systems, Inter-Symbol Interference (ISI) channels with 1-bit quantized output are considered under an average-power-constrained continuous input. While the exact capacity is difficult to characterize, an approximation that matches with the exact channel output up to a probability of error is provided. The approximation does not have additive noise, but constrains the channel output (without noise) to be above a threshold in absolute value. The capacity under the approximation is computed using methods involving standard Gibbs distributions. Mark…
Deep learning has been recently applied to physical layer processing in digital communication systems in order to improve end-to-end performance. In this work, we introduce a novel deep learning solution for soft bit quantization across wideband channels. Our method is trained end-to-end with quantization- and entropy-aware augmentations to the loss function and is used at inference in conjunction with source coding to achieve near-optimal compression gains over wideband channels. To efficiently train our method, we prove and verify that a fixed feature space quantization scheme is sufficient …
