Capacity of the Discrete-Time AWGN Channel Under Output Quantization (original) (raw)
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Given a channel having binary input X = (x1, x2) having the probability distribution pX = (px 1 , px 2) that is corrupted by a continuous noise to produce a continuous output y ∈ Y = R. For a given conditional distribution p y|x 1 = φ1(y) and p y|x 2 = φ2(y), one wants to quantize the continuous output y back to the final discrete output Z = (z1, z2,. .. , zN) such that the mutual information between input and quantized-output I(X; Z) is maximized while the probability of the quantizedoutput pZ = (pz 1 , pz 2 ,. .. , pz N) has to satisfy a certain constraint. Consider a new variable ry = px 1 φ1(y) px 1 φ1(y) + px 2 φ2(y) , we show that the optimal quantizer has a structure of convex cells in the new variable ry. Based on the convex cells property, a fast algorithm is proposed to find the global optimal quantizer in a polynomial time complexity. In additional, if the quantizedoutput is binary (N = 2), we show a sufficient condition such that the single threshold quantizer is optimal.
IEEE Transactions on Wireless Communications, 2019
Low resolution analog-to-digital converter (ADC) has been considered as a promising solution to save power and cost in communication systems using high bandwidth and/or multiple RF chains. The goal of this work is to address the design of optimal signaling schemes and establish the capacity limit of Rician fading channels with low-resolution output quantization. This fading channel can be used to accurately model a wide range of wireless channels with line-of-sight (LOS) components, including emerging mmWave communications. The focus is on non-coherent fast fading channels where neither the transmitter nor the receiver knows the channel state information (CSI). By examining the continuity of the input-output mutual information, the existence of the optimal input signal is first validated. Then considering the case of 1-bit ADC, we show that the optimal input is π/2 circularly symmetric. A necessary and sufficient condition for an input signal to be optimal, which is referred to as the Kuhn-Tucker condition (KTC), and Lagrangian optimization problem are then established. By exploiting novel log-quadratic bounds on the Gaussian Q-function, it is then demonstrated that for a given mass point's amplitude, the corresponding rotated mass points through the phase of LOS component must form a square grid centered at zero. Furthermore, the amplitude of the mass points in the optimal distribution can take on only one value. As a result, the capacity-achieving input with 1-bit ADC is a rotated quadrature phase-shift keying (QPSK) constellation, and the rotation angle depends on the Rician factor. The characterization of the optimal input has also been extended to the case of multibit ADCs. Specifically, it is shown that for a K-bit ADC, the optimal input is discrete having at most 2 2K mass points. In both cases of 1-bit and K-bit ADCs, the channel capacities are established in closed-form.
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IEEE Transactions on Communications, 2009
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 quantization bins (i.e., a precision of log 2 bits), we show that the input distribution need not have any more than +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 signalto-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.