Is Phase Shift Keying Optimal for Channels with Phase-Quantized Output?

The use of low-resolution analog-to-digital converters (ADCs) has recently gained significant research interest because it addresses practical problems and scalability issues in 5G core technologies such as massive data processing, high power consumption, and cost [1]. Most studies on low-resolution ADCs have been more focused on investigating the fundamental limits and practical detection strategies in the context of multiple-input multiple-output (MIMO) and millimeter wave (mmWave) systems [2, 3, 4, 5]. However, these studies did not properly address the structure of the capacity-achieving input and only analyzed performance via capacity bounds using simplified analytical models. Low-resolution receiver design requires a shift in signal/code construction since Gaussian signaling is no longer optimal in channels with quantized output [6].

Some research efforts have been invested in analyzing the capacity limits of channels with low-resolution quantization and finding the optimum signaling schemes for such channels. One of the first studies on this topic showed that binary antipodal signaling is optimal for real AWGN channel with 1-bit quantized output [7]. Extension of capacity analysis to other wireless channels with 1-bit in-phase and quadrature (I/Q) output quantization revealed that QPSK is the optimum signaling for coherent/noncoherent Rayleigh channel [8][9], noncoherent Rician channel [6], and zero-mean Gaussian mixture channel [10]. However, identifying the structure of capacity-achieving input analytically for static and fading channels with multi-bit I/Q quantization still remains an open problem [6].

Motivated by the above discussion, we aim to extend the capacity results of 1-bit I/Q quantization to multi-bit quantization. However, we shall investigate multi-bit phase quantization instead of the conventional I/Q quantization. Phase quantization ignores the amplitude component thus eliminating the necessity for automatic gain control [11]. Furthermore, phase quantizers can be easily implemented in practice using analog phase detectors and 1-bit comparators which consume negligible power (in the order of mW) [12]. Information rate of phase-quantized block noncoherent receiver has been studied before in [11] but the proponents of the study did not show the optimality of phase shift keying. Error rate analysis of low-resolution phase-modulated communication has been done for the single-input single-output (SISO) fading channel [13, 12], relay channel [14], and multiuser MIMO channel [15] but only investigated uncoded transmissions. In this work, we provide a rigourous proof that phase shift keying is indeed capacity-achieving for static channels with phase-quantized output. In fact, the analytical tractability of the 1-bit ADC case in [8, 7, 6, 9, 10] comes from the tractability of the more general multi-bit phase quantization. We then extend the analysis to phase-quantized Rayleigh fading channel with channel state information (CSI) known only at the receiver and give some insights about the outage exponent (or diversity order) of its outage probability. In particular, our numerical results reveal a threshold effect in the outage exponent when the required transmission rate $R$ of an $M$-PSK scheme exceeds a certain value. The proofs can be found in the supplement material [16].

We consider a discrete-time baseband model shown in Figure 1. The transmitter sends a signal $X$ which has an average power constraint $\mathbb{E}[|X|^{2}]\leq P$. $g_{\text{LoS}}$ is a complex constant representing the gain and direction of the line-of-sight (LoS) component and $N\sim\mathcal{CN}(0,\sigma^{2})$ is an additive noise. We can express the received signal prior to quantization as

The signal is then fed to a symmetric $b$-bit phase quantizer $\mathcal{Q}_{\phi}(\cdot)$ to produce an integer-valued output $Y\in[0,2^{b}-1]$. To be more precise, the output of the phase quantizer is $Y=y$ if $\angle Z\in\mathcal{R}^{\text{PH}}_{y}$, where $\mathcal{R}^{\text{PH}}_{y}$ is given by

Due to the circular structure of the phase quantizer, the addition operation $Y+k$ for some $k\in\mathbb{Z}$ constitutes a modulo $2^{b}$ addition. In this quantization model, only a coarse phase information of the received signal is retained and the goal of the receiver is to reliably recover the message encoded in $X$ using the phase quantizer output, $Y$. It should be noted that the discrete-time channel model we considered implicitly assumes that the phase quantizer is symmetric and that the channel output is sampled at the Nyquist rate. However, such quantization and sampling strategy may not be optimal in some cases as pointed out in [17, 18].

We identify the probability quantities essential to express the mutual information. Suppose we define $U=\frac{g_{\text{LoS}}}{\sigma}X$ and $Z^{\prime}=\frac{Z}{\sigma}$. The conditional PDF $p_{Z^{\prime}|U}(z^{\prime}|u)$ is given by

Note that in the phase-quantized receiver, we discard any information on the magnitude. Suppose we represent the random variables in polar form (i.e. use $Z^{\prime}=\frac{\sqrt{R}}{\sigma}e^{j\Phi}$ and $U=\sqrt{\alpha}e^{j\Theta}$). The probability of $\angle Z^{\prime}=\Phi$ given $U=\sqrt{\alpha}$ is transmitted can be written as

where the last equality is obtained from [19, equation (10)]. $Q(\cdot)$ is the tail probability of the standard normal distribution. The conditional PDF $p_{Y|U}(y|u)$ (or $p_{Y|A,\Theta}(y|\alpha,\theta)$), denoted as $W_{y}^{(b)}(u)$ (or $W_{y}^{(b)}(\alpha,\theta)$), is given by

Equation (II) has no closed-form expression. However, we can still use it to identify the optimal input distribution and numerically compute the capacity of the phase-quantized system. Now, consider an input distribution $F_{U}(u)$ with density function $f_{U}(u)$. With slight abuse of notation, we use $F_{U}$ and $f_{U}$ to denote $F_{U}(u)$ and $f_{U}(u)$, respectively. For a given $F_{U}$, the probability mass function (PMF) of $Y$ is therefore

We use the above notation to emphasize that the PMF of $Y$ is induced by the choice of the distribution $F_{U}$. Given the above probability quantities, we can now express the mutual information between $U$ and $Y$ as follows:

We use the notation $I(F_{U})$ since the mutual information is a result of choosing a specific input distribution $F_{U}$. All $\log(\cdot)$ functions in this paper are in base 2 unless stated otherwise. Let $P^{\prime}=\frac{|g_{\text{LoS}}|^{2}}{\sigma^{2}}P$. The capacity for a given power constraint is the supremum of mutual information between $U$ and $Y$ over the set of all input distributions $F_{U}$ satisfying the power constraint $\mathbb{E}[|U|^{2}]\leq P^{\prime}$. In other words,

where $\Omega$ is the set of all input distributions which have average power less than or equal to $P^{\prime}$. In the next section, we establish the properties of the capacity-achieving input distribution $F^{*}_{U}$ when $g_{\text{LoS}}$ is known at the transmitter.

The mutual information $I(F_{U})$ is concave with respect to $F_{U}$ [20, Theorem 2.7.4] and the power constraint ensures that $\Omega$ is convex and compact with respect to weak* topology¹¹1This is the coarsest topology in which all linear functionals of $dF_{U}$ of the form $\int f(u)dF_{U}$, where $f(u)$ is a continuous function, are continuous. [21]. The existence of $F_{U}^{*}$ is equivalent to showing that $I(F_{U})$ is continuous over $F_{U}$. The finite cardinality of phase quantizer output trivially ensures this and the proof follows closely to the method in [7, Appendix A] and [6, Lemma 1]. We first show that the optimal input distribution satisfies a certain phase symmetry. Then, we prove that $F_{U}^{*}$ should have a single amplitude level. Finally, we identify the structure of the optimal input by establishing its discreteness and locating its mass points.

We have shown that $2^{b}$-PSK is optimal for an AWGN channel with $b$-bit phase-quantized output and $h$ is known at the transmitter. We now ask if this continues to hold when channel information is unavailable at the transmitter. Ultimately, is $2^{b}$-PSK still the best choice in fading environment without channel state feedback? We now consider a quasi-static Rayleigh flat fading environment. The fixed channel gain $g_{\text{LoS}}$ in Figure 1 is replaced by a random fading gain $G\sim\mathcal{CN}(0,1)$. We further assume that the fading state $g$ is known only at the receiver. Without loss of generality, we assume $\sigma^{2}=1$. We define the function

as the maximum rate of reliable communication supported by a modulation scheme $\mathcal{X}_{M}$ and a fading realization $g$ at some SNR. Here, the symbols $x\in\mathcal{X}_{M}$ have the form $x=\sqrt{SNR}e^{j\beta}$ so that Propositions 1 and 2 are satisfied³³3We omit the proof that these necessary conditions for optimum $\mathcal{X}_{M}$ hold even when $g$ is unknown at the transmitter.. If the transmitter encodes the data at a rate $R$ bits/channel use, then an outage occurs when $I(\mathcal{X}_{M}|G=g)<R$ since the error rate cannot be made arbitrarily small whatever coding scheme is used. The function $r\left(\gamma,\angle g,\mathcal{X}_{M}\right)$ is a convex decreasing function of $\gamma$ (Lemmas 2 and 3). Thus, it follows that its inverse function with respect to $\gamma$ has one-to-one mapping and is also decreasing. The outage probability is expressed as

The third line follows by noting that $|g|^{2}$ is exponentially-distributed for Rayleigh fading and $\angle g$ is uniformly-distributed. It is difficult to analytically derive the outage probability so the expression for $P_{\text{out}}(SNR)$ is evaluated numerically to provide some more insight. In order to characterize the outage probability, we focus on the outage exponent (or diversity order) which is the asymptotic slope of the outage probability as a function of SNR. Mathematically, this is defined as

Figure 3 depicts the outage probability of Rayleigh fading channel with 3-bit phase-quantized output for different $R$ and $\mathcal{X}_{M}=\{\text{8-PSK, 16-PSK, $\infty$-PSK}\}$. One noteworthy observation is the sudden decrease of the outage exponent when $R$ is increased from $2.00$ to $2.05$ for 8-PSK. DVO drops from $1$ to $\frac{1}{2}$. This is also the case for 16-PSK when $R$ is increased from $2.50$ to $2.55$. This can be partially explained by rates of 8-PSK and 16-PSK for varying rotations (as seen in Figure 4). Since the transmitter cannot compensate the phase rotation induced by fading, choosing an $R$ that exceeds the worst-case rates of 8-PSK and 16-PSK in Figure 4 causes outage even with high SNR. Lastly, we note that $\infty$-PSK is invariant of the channel phase. As such, the choice of $R$ does not affect its outage exponent provided $R<b$. However, the input distribution that achieves the best outage performance for a particular SNR and quantizer resolution still needs to be addressed by further research.

In this work, we analyzed the capacity of channels with phase-quantized output. The first contribution of this work is a rigorous proof that a rotated $2^{b}$-phase shift keying is optimal for static channels with $b$-bit phase quantization. Using the capacity-achieving input, a channel capacity expression is established. Numerical examples were provided to demonstrate the optimality of the capacity-achieving input. For phase-quantized Rayleigh fading case, the outage performance was analyzed numerically for different $M$-PSK modulation schemes and $b=3$. Our numerical findings showed that transmitting at a rate $R$ that is above the information rate of $M$-PSK signaling with worst-case $\angle g+\beta$ would significantly impact the robustness of the system against outage. A threshold effect in the outage exponent was observed in 8-PSK and 16-PSK when $R$ exceeded these values. Further research needs to be conducted to be able to generalize these results to different types of fading channels. Ergodic capacity of phase-quantized fading channel is also considered for future work.