Low-Resolution Quantization in Phase Modulated Systems: Optimum Detectors and Error Rate Analysis

This paper studies optimum detectors and error rate analysis for wireless systems with low-resolution quantizers in the presence of fading and noise. A <italic>universal</italic> lower bound on the average symbol error probability (<inline-formula> <tex-math notation="LaTeX">$\mathsf {SEP}$ </tex-math></inline-formula>), correct for all <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary modulation schemes, is obtained when the number of quantization bits is not enough to resolve <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> signal points. In the special case of <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary phase shift keying (<inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-PSK), the maximum likelihood detector is derived. Utilizing the structure of the derived detector, a general average <inline-formula> <tex-math notation="LaTeX">$\mathsf {SEP}$ </tex-math></inline-formula> expression for <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-PSK modulation with <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-bit quantization is obtained when the wireless channel is subject to fading with a circularly-symmetric distribution. For the Nakagami-<inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> fading, it is shown that a transceiver architecture with <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-bit quantization is <italic>asymptotically</italic> optimum in terms of communication reliability if <inline-formula> <tex-math notation="LaTeX">$n \geq \log _{2}M +1$ </tex-math></inline-formula>. That is, the decay exponent for the average <inline-formula> <tex-math notation="LaTeX">$\mathsf {SEP}$ </tex-math></inline-formula> is the same and equal to <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> with infinite-bit and <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-bit quantizers for <inline-formula> <tex-math notation="LaTeX">$n\geq \log _{2}M+1$ </tex-math></inline-formula>. On the other hand, it is only equal to <inline-formula> <tex-math notation="LaTeX">$\frac {1}2$ </tex-math></inline-formula> and 0 for <inline-formula> <tex-math notation="LaTeX">$n = \log _{2}M$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n < \log _{2}M$ </tex-math></inline-formula>, respectively. An extensive simulation study is performed to illustrate the accuracy of the derived results, energy efficiency gains obtained by means of low-resolution quantizers, performance comparison of phase modulated systems with independent in-phase and quadrature channel quantization and robustness of the derived results under channel estimation errors.


A. Background and Motivation
Analog-to-digital converters (ADCs) are known to consume most of the power dissipated at a base station [1].It is shown that the power consumed by ADCs grows exponentially with their resolution level and linearly with their sampling rate [2], [3].Thus, using highresolution quantization with high sampling rates can significantly degrade the energy efficiency of a communication system.With the introduction of massive multiple-input-multiple-output (MIMO) and millimeter wave (mmWave) technology, this is even more prominent in next generation wireless systems.Because, massive MIMO systems use hundreds of antennas where each antenna is connected to a dedicated radio frequency (RF) chain equipped with highresolution ADCs.MmWave systems, on the other hand, use much larger bandwidths that require higher sampling rates.In fact, the typical power consumption of a high speed (≥ 20 GSamples/s) and high-resolution (8-12 bits) ADC is around 500 [mWatts].Therefore, a future mmWave massive MIMO system with 256 RF chains and 512 ADCs will require around 256 [Watts] of power [4], which is potentially unaffordable.Consequently, the idea of replacing power hungry high-resolution ADCs with low-resolution ADCs could provide a viable solution to the power consumption concerns in future wireless systems.Indeed, low-resolution ADCs have long been known to provide significant energy savings in digital transceiver implementations [5]- [7].Their other benefits include simplification in design (especially with 1-bit ADCs) and reduction in transceiver form-factor [4], [8], [9].Furthermore, the future long-term evolution (LTE) networks are also expected to support a wide range of Internet-of-Things (IoT) applications through protocols such as LTE-M, NB-IoT and EC-GSM, where devices are usually battery power-limited [10].In these future application scenarios, lowresolution ADC based digital transceivers have the potential to prolong the battery lifetime of remote IoT devices as well, and thereby lessening the operating costs and the need for frequent human intervention.This paper investigates the performance of a wireless communication system with low resolution ADCs, in a symbol error probability (SEP) perspective.In our analysis, we consider the optimum maximum likelihood (ML) detector, and to provide a thorough discussion, we focus on single-input single-output (SISO) channels.Most of the previous work on low-resolution ADCs have focused on the abstract case of 1-bit quantization [5], [9], [11]- [22], where a simple comparator forwards the sign of the signal to the digital domain and discards all the information about the analog signal amplitude.Such comparators consume negligible power and does not require an automatic gain control circuit.Thus, they lead to cost and power effective implementation of RF chains [14].In this paper, we take a different approach in which we allow the number of bits in the quantizer to vary until the transceiver architecture becomes asymptotically optimum in terms of communication reliability.Focusing on a special phase quantizer, we derive analytical expressions of the average SEP when the channels are subject to Nakagami-m fading and the transmitted bits are modulated using M-PSK modulation.More importantly, our asymptotic results reveal a fundamental ternary behaviour in the error probability performance of a wireless communication system with low-resolution ADCs, providing an important insight to system designers when choosing the required number of quantization levels.
Using low-resolution ADCs in wireless communication systems has been investigated in various aspects.The performance of communication systems with low-resolution ADCs is lower than that of the idealized systems without quantization or traditional systems with high-resolution ADCs.Therefore, performance analysis of low-resolution quantization is a key research area.It was shown in [18] that the capacity of point-to-point MIMO channel with 1-bit ADCs is lower bounded by the rank of the channel in the high signal-to-noise ratio (SNR) regime.Results in [20] and [21] show that the channel capacity reduces by a factor of 2/π (1.96 dB) in the low-SNR regime for a MIMO system with 1-bit ADCs, when compared to a conventional highresolution system.Further, the results in [22] establish the fact that the performance loss due to employing 1-bit ADCs can be overcome by having approximately 2.5 times more antennas at the base station.[23] focuses on the information rate of a quantized block non-coherent channel with 1-bit ADCs.The results in this paper show that around 80 − 85% of the mutual information attained with unquantized observations can also be attained with 3-bit quantization for QPSK modulation and SNR greater than 2-3 dB.In [24], Liang et al. presented a mixed-ADC architecture for MIMO systems in which some of the high-resolution ADCs were replaced with 1-bit ADCs.Their results show that the proposed architecture can achieve a near-similar performance as conventional architecture while reducing the energy consumption considerably.
Signal detection rules developed for receivers with high-resolution ADCs often become suboptimal for receivers with low-resolution ADCs [4].In [25], the authors propose a linear minimum mean square error (LMMSE) receiver when in-phase and quadrature components of the received signal are independently quantized by using a low-resolution ADC.They provide an approximation for the mean squared error between the transmitted symbol and the received one, and derive an optimized linear receiver which performs better than the conventional Weiner filter.
Results in [25] were further extended to an iterative decision feedback receiver with quantized observations in [26].For the same quantizer structure of independent quantization of in-phase and quadrature signal components, an ML detector was obtained in [11] by using only 1-bit ADCs.The complexity of the ML detector proposed in [26] grows exponentially with high signal constellations, number of transmit antennas and network size, which is not practical for real-world deployments.To overcome this difficulty, a near-optimum ML detector was proposed in [12] by using the convex optimization techniques.Although the SEP performance of the proposed near-optimum ML detector is better than the performance of linear detectors, it has been numerically observed that the proposed near-optimum ML detector still suffers from an error floor as SNR increases [4], [12].Complementing this critical observation, in our work, we show the existence of a universal error floor below which the average SEP cannot be pushed down for any M-ary modulation scheme and quantizer structure if the number of quantization bits is less than log 2 M.

B. Main Contributions
In this paper, we consider a point-to-point wireless communication system, where data transmission is corrupted by fading and noise.Motivated by the capacity achieving property of circularly symmetric input distributions for low-resolution ADCs [27], we assume that the transmitted symbols are modulated using M-ary phase shift keying (M-PSK).For such a system, we design a low-resolution ADC that quantizes the phase of the received signal in such a way that only the information about the quantization region in which the received signal landed is sent to the detector.The use of phase quantization in our model is further motivated by the following two factors.First, considering channel impairments as phase rotations in transmitted signals, quantization and decision regions for M-PSK modulation are conveniently modelled as convex cones in the complex plane [28], and without requiring the use of automatic gain control.Second, phase quantizers can be implemented using 1-bit ADCs that consist of simple comparators, and they consume negligible power (in the order of mWatts).Our main contributions are summarized as follows.
• For any M-ary modulation scheme and quantizer structure, we show the existence of an error floor below which the average symbol error probability (SEP) cannot be pushed if the number of quantization bits n is less than log 2 M.
• For M-PSK modulation with M ≥ 2, we derive the optimum ML detection rule for signal detection with low-resolution ADCs.
• We obtain analytical expressions for the average SEP attained by the derived ML rule with n-bit quantization when the wireless channel is subjected to Nakagami-m fading.
• We establish a fundamental ternary average SEP behaviour with low-resolution ADCs and M-PSK modulation under the Nakagami-m fading model.In particular, we show that the decay exponent of the average SEP is the same with that of an infinite-bit quantization, which is equal to m, when n is larger than or equal to log 2 M + 1.We also show that it is equal to 1 2 and 0 for n = log 2 M and n < log 2 M, respectively.• We perform a detailed numerical analysis in the high-SNR regime to corroborate the derived analytical results and to illustrate the energy gains obtained by low-resolution ADCs.
From a system design point of view, our results show that using one additional bit on top of log 2 M of them can achieve optimum communication robustness in the high-SNR regime.
In particular, for fading environments with a large value of m, using an extra quantization bit improves communication reliability significantly.On the other hand, it may be more beneficial to use log 2 M bits for small values of m, without sacrificing from communications robustness too much but doubling system energy efficiency.

C. Notation
We use uppercase letters to represent random variables and calligraphic letters to represent sets.We use R, R 2 and N to denote the real line, 2-dimensional Euclidean space and natural numbers, respectively.For a pair of integers i ≤ j, we use [i : j] to denote the discrete interval {i, i + 1, . . ., j}.For two functions f and g, we will say | for some c > 0 when x is sufficiently close to x 0 .Similarly, we will say f (x) = Ω (g(x)) The set of complex numbers C is R 2 equipped with the usual complex addition and complex multiplication.We write z = z re + z im to represent a complex number z ∈ C, where  = √ −1 is the imaginary unit of C, and z re and z im are called, respectively, real and imaginary parts of z [29].Every z ∈ C has also a polar representation z = |z| e θ = |z| (cos (θ) +  sin (θ)), where |z| z2 re + z 2 im is the magnitude of z and θ = Arg (z) ∈ [−π, π) is called the (principle) argument of z. 1 As is common in the communications and signal processing literature, Arg (z) will also be called the phase of z (modulo 2π).For a complex random variable Z = Z re + Z im , we define its mean and variance as respectively.We say that Z is circularly-symmetric if Z and e θ Z induce the same probability distribution over C for all θ ∈ R [30], [31].For x > 0, log x and log 2 x will denote natural logarithm of x and logarithm of x in base 2, respectively.

A. Channel Model and Signal Modulation
We consider the classical point-to-point wireless channel model with flat-fading.For this channel, the received discrete-time baseband equivalent signal Y can be expressed by where X ∈ C ⊂ C is the transmitted signal, C is the constellation set of information signals in C, SNR is the ratio of the transmitted signal energy to the additive white Gaussian noise (AWGN) spectral density, H ∈ C is the unit power channel gain between the transmitter and the receiver, and W is the circularly-symmetric zero-mean unit-variance AWGN, i.e., W ∼ CN (0, 1).In order to formalize the receiver architecture and the optimum signal detection problem below, we will assume that C = e π( 2k+1 in the remainder of the paper, which is the classical M-ary phase shift keying (M-PSK) signal constellation 2 and for ease of exposition, we only consider the case in which M is an integer power of 23 .

B. Receiver Architecture
The receiver architecture is based on a low-resolution ADC.As illustrated in Fig. 1, the received signal Y is first sent through a low-resolution quantizer, and then the resulting quantized signal information is used to determine the transmitted symbol X.More specifically, if n bits are used to quantize Y , the quantizer Q divides the complex domain C into 2 n quantization regions and outputs the index of the region in which Y lies as an input to the detector.As such,

Estimate of the Transmitted Symbol
Since information is encoded in the phase of X with the above choice of constellation points, we choose R k as the convex cone given by We also assume that full channel state information is available at the receiver.The assumption on the availability of full channel state information at the receiver is justified by the previous work on channel estimation with low-resolution ADCs [8], [32]- [34].In particular, it was shown in [8] that it is possible to attain a near full-precision channel estimation performance with the use of low-resolution ADCs by increasing the number of training symbols in the closed-loop estimation process.Further, mixed-ADC architectures can also be employed to achieve high channel estimation accuracy [24].

III. OPTIMUM SIGNAL DETECTION
The aim of the detector is to minimize the SEP by using the knowledge of Q(Y ) and channel state information, which can be represented as selecting a signal point x (k, h) satisfying The main performance figure of merit for the optimum detector is the average SEP given by It is important to note that p (SNR) depends on SNR as well as the number of quantization bits.
Our first result indicates that there is an SNR-independent error floor such that the average SEP values below which cannot be attained for n < log 2 M. The following theorem establishes this result formally.
Theorem 1: Let p min be the probability of the least probable transmitted symbol.If n < log 2 M, then for any choice of modulation scheme and quantizer structure for all SNR ≥ 0.
Proof: See Appendix A.
Firstly, we note that the error floor in ( 4) is always a valid lower bound since P min ≤ 1 M .Secondly, it does not depend on the fading model.The average SEP values below M −2 n 2 n p min cannot be achieved due to the inherent inability of low-resolution ADC receivers to resolve different signal points when n < log 2 M. We also note that the Fano's inequality can also be used to obtain similar, perhaps tighter, lower bounds on p (SNR) [35].However, this will require the calculation of equivocation between X and Q(Y ) for each choice of modulation scheme and quantizer structure.Hence, it is not clear how the minimization is carried out over the modulation and quantizer selections in this approach.
Next, we will assume that all signal points in C are equiprobable, with probability 1 M , and hence the optimum detector in (2) is equivalent to the ML detector given by SNRhx and variance Var (Y ) = 1, we can write the probability in (5) as where the integral in ( 6) is with respect to the standard Borel measure in C [36].The next theorem describes the operation of the ML detector for the above signal detection problem.
Theorem 2: Assume H has a continuous probability density function (pdf).Then, x (k, h) is unique with probability one, i.e., the set of h values for which arg max is singleton has probability one, and the ML detection rule for the low-resolution ADC based receiver architecture can be given as where which is defined as dist (z, A) inf s∈A |z − s|, and We note that the half-hyperplane H k in Theorem 2 bisects the kth quantization region R k into two symmetric regions.Hence, Theorem 2 indicates that the most probability mass is accumulated in the region R k when the unit-variance proper complex Gaussian distribution with mean closest to H k is integrated over R k .Next we use the structure of the ML detection rule to derive integral expressions for p (SNR) for M ≥ 2 in Section IV.Further, in order to characterize the communication robustness with low-resolution ADCs in the high SNR regime, we also provide a detailed analysis on the asymptotic decay exponent of p (SNR) in Section V.

A. Symbol Error Probability for n ≥ log 2 M
Let us first obtain a key lemma that simplifies the calculations for deriving p (SNR) when the number of quantization bits is at least log 2 M. Note that this lemma holds for general circularlysymmetric fading processes without assuming any specific functional form.
Lemma 1: Let H = Re Λ be a circularly-symmetric fading coefficient with R and Λ denoting the magnitude and the phase of H, respectively.Let the joint pdf of R and Λ be given by where Using Lemma 1, next we obtain integral expressions for p (SNR) when H is circularlysymmetric with the generalized Nakagami-m fading magnitude.We note that the Nakagami-m fading model characterizes a broad range of fading phenomena ranging from severe to moderate and no fading conditions as m varies over [0.5, ∞) [37], [38] and it reduces to Rayleigh fading for m = 1.
Considering these advantages, we will focus on the Nakagami-m fading model for H to derive integral expressions for p (SNR) in the remainder of the paper.This will be done so for all parameter combinations of M ≥ 2 (as an integer power of 2), n ≥ log 2 M and m ≥ 0.5.
It will be seen that the derived integral expressions are easy to calculate numerically and they reduce to simple closed-form expressions in some special cases.Further, we will also show that using log 2 M + 1 bits is enough to achieve the maximum communication robustness achieved by using infinite number of quantization bits.
Theorem 3: Assume H is a unit-power fading coefficient distributed according to a circularlysymmetric distribution with Nakagami-m fading magnitude.Let Q (•) be the complementary distribution function of the standard normal random variable and Γ (•) be the gamma function [39].Then, for n ≥ log 2 M and M ≥ 2, p (SNR) is given according to , where Proof: In the following we provide the proof for M ≥ 4. Please note that the proof for M = 2 is similar and simpler.With a slight abuse of notation, we define where the set E is defined as in Lemma 1.The probability in ( 14) can be calculated by conditioning on the real part of W , which is denoted by W re .By using Fig. 2 as a visual guide, we can write p (SNR, h) after conditioning on W re as for w ≥ − √ SNRr cos θ.Similarly, for w < − √ SNRr cos θ, we get Integrating (15) and ( 16) with respect to the pdf of W re , which is given by f Wre (w) = 1 √ π e −w 2 , we obtain p (SNR, h) as For Nakagami-m fading distribution with shape parameter m ≥ 0.5 and spread parameter Ω > 0 [40], we can write the pdf of the fading magnitude as We set Ω = 1 in our calculations to make sure that H has unit-power.We average p (SNR, h) over the fading distribution and solve the resulting integral based on Lemma 1, and the fact that θ lies between 0 and 2π M , to obtain p (SNR) in Theorem 3.

B. Centering Property: Impact of Quantization Bits on the Average SEP
In this subsection, we will present an intuitive explanation as to why p (SNR) improves with increasing number of quantization bits.In particular, we will observe that one extra bit, on top of log 2 M of them, provides a desirable centering property that steers the received signal away from the error-prone decision boundaries.This intuition will help to understand the underlying dynamics leading to the ternary behaviour for the decay exponent of p (SNR) that we establish in the high SNR regime in Section V.
be the ith signal point in the constellation set C and where This means that all the received signal points in E i,k will be detected as x i , and hence E i,k can be considered as the region of attraction of x i .This also means that if the received signal lands in E i,k when x i is transmitted, then there will not be any detection errors.Let us consider an example for QPSK modulation with 2-bit and 3-bit quantization.Without loss of generality, we will assume that x 3 = e  π M is the transmitted signal.Our analysis will be for two cases of λ = π 18 and λ = 4π 18 , where λ = Arg (h).Table I summarizes these two cases, and Fig. 3 illustrates them.In this figure, we show both the original signal points (indicated by '⋄') and the rotated ones (indicated by '•') after multiplying with √ SNR and h.
For both 2-bit and 3-bit quantization, we observe that h ∈ D 2 and h ∈ D 4 for λ = π 18 and λ = 4π 18 , respectively.Therefore, for 2-bit quantization, the region of attraction for x 3 will be E 3 for both cases.Here, we can see that the rotated constellation point √ SNRhx 3 is very close to the decision boundary when λ = 4π 18 .Hence, there is a high probability that the received signal √ SNRhx 3 + w lands in the adjacent quantization region for λ = 4π 18 .In this instance, we will have a detection error.However, with the addition of one bit to the quantizer (i.e. with 3-bit quantization), the region of attraction of x 3 will be e  π 4 E 3 , and hence the ML detector can correctly decode the transmitted signal even if the received one lands in the adjacent quantization region.
Table I: Centering property for QPSK modulation with 2-bit and 3-bit quantization.E i,k is the region of attraction of the symbol x i when the quantizer output  This is illustrated in Fig. 3(d).Therefore, the addition of one extra bit to the quantizer, steers the received signal away from the error-prone decision boundaries to improve p (SNR).Similarly, when the number of bits in the quantizer continues to increase, the quantization regions will become thinner, and hence the regions of attraction will be better centered around the received signal points.This is the fundamental phenomenon that explains why the average SEP improves with a larger number of quantization bits.

V. THE DECAY EXPONENT FOR THE AVERAGE SYMBOL ERROR PROBABILITY
In this section, we will analyze the communication robustness that can be achieved with low-resolution ADCs by focusing on the decay exponent for p (SNR), which is given by4 Following the convention in the field, we will call DVO the diversity order, although there is only a single diversity branch in our system.It should be noted that Nakagami-m amplitude distribution can be obtained as the envelope distribution of m independent Rayleigh faded signals for integer values of m [37].Hence, visualizing a Nakagami-m wireless channel as a predetection analog square-law diversity combiner will put the results of this section into context.
We devote the rest of the current section to the proof of this important finding.We will first start with a definition that will simplify the notation below.
Proof: See Appendix F.
The following lemma establishes lower and upper bounds on SEP in (9).We note that the bounds in Lemma 5 hold for all circularly-symmetric fading processes, including Nakagami-m magnitude pdf as a special case.
Proof: See Appendix G.

Theorem 4:
The DVO of a low-resolution ADC based receiver architecture with M-PSK modulation and Nakagami-m fading is given by Proof: The proof for M ≥ 4 directly follows from Lemmas 2, 3, 4 and 5.For BPSK modulation (i.e., M = 2) and n = 1, we have By using the change of variables θ = θ − π 2 in the second integral term of ( 23), we have This expression is equivalent to p 1 (SNR) + p 2 (SNR) for M = 4 and n = 2. Hence, by using Lemma 2, we can conclude that for BPSK modulation with 1-bit quantization and m ≥ 1 2 .For BPSK modulation with n > log 2 (M), we have Therefore . Define the function g SNR (θ, β) as SNR increases, we can use the monotone convergence theorem [41] to write We note that the last integral is finite since g ∞ (θ, β) is continuous and finite over the range of integration.Therefore, for BPSK modulation with n > log 2 (M), we get The DVO analysis above helps to discover the first-order effects of the low-resolution ADC based receivers on the SEP system performance.In particular, we observe that it is enough to use log 2 M + 1 bits for quantizing the received signal to extract full diversity, which is equal to m for Nakagami-m faded wireless channels.Considering the fact that energy consumption increases exponentially with the number of quantization bits [42], this finding indicates that a significant energy saving is possible by means of low-resolution ADC based receivers without any (first order) loss in communication robustness.
We also observed that the DVO is only equal to 1 2 when n = log 2 M. Together with the universal bound obtained in Theorem 1, the discovered ternary behaviour has significant implications in terms of how to choose the number of quantization bits for low-resolution ADC based receivers.In particular, for fading environments with m close to 1  2 , a system designer may decide to trade off reliability for energy consumption, without having too much degradation in average SEP by using log 2 M bits.On the other hand, for fading environments with large m, it is more beneficial to use one extra bit to have a major improvement in average SEP.

VI. PERFORMANCE ANALYSIS FOR QPSK MODULATION
In this section, we conduct a performance analysis for QPSK modulation with low-resolution ADCs by using our results in previous sections.We first present a simplified version of the average SEP expression in (9) for QPSK modulation, and then we analyze the effect of lowresolution quantization under Rayleigh fading.

A. Symbol Error Probability for QPSK modulation
Nakagami-m Fading: In the special case of QPSK modulation (i.e., M = 4), the average SEP expression in (9) can be further simplified to produce because tan 2π M = ∞ for M = 4.By using hypergeometric function 2 F 1 [•] [39], we can simplify (28) for 2-bit quantization (i.e., M = 4 and n = 2) as Rayleigh Fading: For special case of Rayleigh fading, which is obtained by setting m = 1, the expression in (28) can be re-expressed as where ϑ = . Furthermore, for 2-bit quantization with Rayleigh fading (i.e., M = 4, n = 2 and m = 1), we can obtain p (SNR) in closed form as This closed-form analytical expression is very easy to compute without resorting to any numerical integration.

B. Analysis of Quantization Penalty for QPSK Modulation
By using the Taylor series expansion for high SNR values, we can re-express the average SEP expressions for QPSK modulation under Rayleigh fading given in (29) as While phase quantization with less number of quantization bits is desirable, due to less processing complexity at the receiver, it deteriorates the average SEP performance of the system.In the following, we quantify the increase in the average SEP as a quantization penalty defined as where p A (SNR, ∞) is the average SEP with infinite number of quantization bits.Based on (30), we can derive p A (SNR, ∞) as where we have used the small-angle approximation tan(x) = x as x → 0. Substituting (30) and ( 32) into (31) and doing some mathematical manipulations, we can derive the quantization penalty in terms of average SEP with n-bit quantization as In Section VII, we use Ψ (SNR, n) to quantify the increase in average SEP as we change from infinite-bit to n-bit quantization.
We further notice that, in order to achieve the same average SEP as with n-bit quantization, we need to transmit the signal using a higher power if we use only (n − 1)-bit quantization.In  the following, we quantify the increase in the transmit power as another quantization penalty defined by where SNR n and SNR n−1 are the SNR values required to achieve a certain average SEP with n and n−1 quantization bits, respectively.Substituting (30) into (34) and doing some mathematical manipulations, we can derive the quantization penalty with n-bit quantization as In Section VII, we use Φ (SEP, n) to quantify the required transmit power increase as we change from n-bit to (n − 1)-bit quantization.

VII. NUMERICAL RESULTS
In this section, we present analytical and simulated SEP results for M-PSK modulation with n-bit quantization.Channel fading is unit-power and circularly-symmetric with Nakagami-m distributed magnitude, and additive noise is complex Gaussian with zero mean and unit variance.results are generated using Monte Carlo simulation, while the analytical results are generated using our expression in (9).As the plot illustrates, the analytical results accurately follow the simulated results for all cases.We observe a noteworthy improvement in the average SEP when n changes from 2 to 3-bit quantization for QPSK modulation in both m = 1 and 2. This jump in the average SEP performance is expected in the light of Theorem 4, which states that using one extra bit, on top of log 2 M bits, improves the DVO from 1 2 to m.We also observe that the average SEP reduces as we increase n, but the amount by which it reduces also gets smaller as we increase n.This can be clearly observed from the zoomed-in section in Fig. 4. As expected, DVO = m for all n ≥ 3. Furthermore, DVO = 1 2 for any m, when n = 2. Fig. 5 plots the average SEP as a function of SNR for QPSK, 8-PSK and 16-PSK modulations schemes while keeping the Nakagami-m shape parameter fixed at m = 1, which is the classical Rayleigh fading scenario.We plot the average SEP for each modulation scheme by using n = log 2 M, log 2 M + 1 and log 2 M + 2 bits.From the plots, we can clearly observe that QPSK with 2-bit, 8-PSK with 3-bit and 16-PSK with 4-bit quantization have a DVO of 1  2 .Further, we can observe that QPSK with 3 or more bits, 8-PSK with 4 or more bits, 16-PSK with 5 or more bits quantizations have a DVO of 1, which is equal to m in this case.To further emphasize this point, the zoomed-in section in Fig. 5  bit quantization.Similarly, the average SEP for 16-PSK has a lower bound of 0.75 with 2-bit quantization and a lower bound of 0.5 with 3-bit quantization.It should be noted that the error floor given in Theorem 1 is more conservative than those observed in Fig. 7.This is because it is a universal lower bound that holds for all modulation schemes, quantizer types and fading environments, not only for very specific ones used to plot average SEP curves in Fig. 7.
Finally, Fig. 8 illustrates the quantization penalty and plots the asymptotic average SEP curves as a function of SNR for QPSK modulation with n = 2, 3, 4 and ∞ under Rayleigh fading.
The asymptotic plots are generated by using the expressions in (30).We observe that a DVO of half is achieved with 2-bit quantization, and the full DVO of one is achieved with n > 2. When the SNR is fixed at 18 dB, we observe a quantization penalty of Ψ ( 18

VIII. CONCLUSIONS AND FUTURE GENERALIZATIONS
In this paper, we performed a theoretical analysis of a low-resolution based ADC communication system and obtained fundamental performance limits, optimum ML detectors and a general analytical expression for the average SEP for M-PSK modulation with n-bit quantization.These results were further investigated for Nakagami-m fading model in detail.We conducted an asymptotic analysis to show that the decay exponent for the average SEP is the same and equal to m with infinite-bit and n-bit quantizers for n ≥ log 2 M + 1.We also performed an extensive numerical study to illustrate the accuracy of the derived analytical expressions.
In most parts of the paper, we have focused on phase modulated communications.Phase modulation has an important and practical layering feature enabling the quantizer and detector design separation in low-resolution ADC communications.For a given number of bits, the quantizer needs to be designed only once, and can be kept constant for all channel realizations.
The detector can be implemented digitally as a table look-up procedure using channel knowledge and quantizer output.On the other hand, this feature is lost in joint phase and amplitude modulation schemes such as QAM.The quantizer needs to be dynamically updated for each channel realization in low-resolution ADC based QAM systems.This is because the fading channel amplitude may vary over a wide range, but the phase always varies over [−π, π) .
However, phase modulation is historically known to be optimum only up to modulation order 16 under peak power limitations [43].Hence, it is a notable future research direction to extend the results of this paper to higher order phase and amplitude modulations by taking practical design considerations into account.
A major result of this paper is the discovery of a ternary SEP behaviour, indicating the where to place the diversity combiner (before or after quantizer or detector) and its type is needed when multiple diversity branches are available for data reception.

APPENDIX A PROOF OF THEOREM 1
Let us consider a class of hypothetical genie-aided detectors g : that has the knowledge of channel noise W ∈ C, fading coefficient H ∈ C and quantizer output for particular realizations of H = h and W = w.We first observe that since n < log 2 M, there exists at least one quantization region R k (depending on w and h) such that S w,h, k contains at least M 2 n signal points.We note that M 2 n is always an integer greater than 2 since M is assumed to be an integer power of 2.Then, the conditional SEP of any detector g given W = w and H = h, which we will denote by p g (SNR, h, w), can be lower-bounded as By averaging with respect to w and h, we have p g (SNR) ≥ M −2 n 2 n p min , where p g (SNR) is the average SEP corresponding to detector g.This concludes the proof since the obtained lower bound does not depend on the choice of modulation scheme, quantizer structure and detector rule, and hence holds for detectors not utilizing the knowledge of W for any choice of modulation scheme and quantizer structure.

APPENDIX B PROOF OF THEOREM 2
To prove Theorem 2, we will first obtain the following result.First, we will consider the case in which both µ 1 and µ 2 lie outside R • , where R • is the set of interior points of R.This is the case shown in Fig. 9.To start with, we will assume 0 ≤ Arg (µ 1 ) ≤ Arg (µ 2 ) < π.Then, for any y ∈ R, the angle between the line segments L Oy and L Oµ 1 is smaller than the one between the line segments L Oy and L Oµ 2 . 5ence, applying the cosine rule for the triangle formed by O, y and µ 1 , and for the triangle formed by O, y and µ 2 , it can be seen that |y − µ 1 | ≤ |y − µ 2 | for all y ∈ R. 6 Therefore, ). Next, we assume Arg (µ 2 ) ∈ [−π, 0) and 0 ≤ Arg (µ 1 ) ≤ |Arg (µ 2 )| ≤ π.Let W be the auxiliary random variable distributed according to CN (μ, 1) with μ = re |Arg(µ 2 )| , i.e., μ is the reflection of µ 2 around the real line.Symmetry around the real line implies that g (µ 2 ) is equal to g (μ) = Pr W ∈ R , which is less than g (µ 1 ) due to our arguments above.For Arg (µ 1 ) ∈ [−π, 0) , the same analysis still holds after reflecting µ 1 around the real line, leading to g (µ 1 ) ≥ g (µ 2 ) for all Second, we consider the case where µ 1 ∈ R • but µ 2 / ∈ R • .This is the case shown in Fig. 10.It is enough to establish the desired result only for 0 ≤ Arg (µ 1 ) ≤ Arg (µ 2 ) < π.
When µ 1 or µ 2 has a negative phase angle, the same analysis below still holds after reflecting the mean with negative phase around the real line.Let W be the auxiliary random variable distributed according to CN (μ, 1) with μ = re α , i.e., μ is located at the upper boundary of R. Our analysis in the first case shows that g (μ) = Pr W ∈ R ≥ g (µ 2 ) since both μ and ).This establishes the desired results for all Finally, we will consider the third case where both µ 1 and µ 2 lie inside R • .This is the case shown in Fig. 11.Similar to the first two cases, it is enough to focus only on 0 ≤ Arg (µ 1 ) ≤ Arg (µ 2 ) ≤ α.We divide R into four disjoint regions given by Using the symmetry in the problem, we have ) , where r = |h|.Further, x (k, h) is unique with probability one due to the continuity assumption of the fading distribution.Consider now the semi-circle We will show that all the terms in ( 37) are equal to each other.Next, we define The same idea extends to any D k , and we define We will use the sets defined in (38) to show that all the terms in (37) are equal.
To complete the proof, we let p since multiplication with e θ ′ i rotates the ith signal point to xM 2 and multiplication with e θ ′′ k removes the effect of partition selection for h.Secondly, we observe that when h ∈ D k , the all bisectors to which √ SNRhx i is closest for this range of h values.Hence, the following chain of equalities hold: where (a) follows from the independence of W , H and X, and (b) follows from above observations and the circular symmetry property of W .Let us now define z = e θ ′′ k h in (39).Since multiplication with a unit magnitude complex number is a unitary transformation (i.e., rotation) over the complex plane, we have where (a), (b) and (c) follow from the circular symmetry of H [30], [31] and the corresponding [45].Switching to polar coordinates, and using the identities rf H (r cos λ, r sin λ) = By using the change of variables θ = π M + λ, we conclude the proof.

APPENDIX D PROOF OF LEMMA 2
The proof of part (i) follows immediately from Definition 1: For the proof of part (ii), given any ǫ > 0, let c > 0 be such that for all SNR ≥ c and i ∈ We start with the case M = 4 and n = 2, and obtain upper and lower bounds on p 1 (SNR) that will lead to the same exponential equality.For the upper bound, we write p 1 (SNR) as as SNR → ∞.Using the fact that the line 1 − 2 π θ is a lower bound for cos θ for θ ∈ 0, π 2 , we have for large values of SNR.Equation (42) shows that p 1 (SNR) e ≤ SNR − 1 2 for all m ≥ 1 2 when M = 4 and n = 2.For the other direction, we obtain a lower bound on p 1 (SNR) as below.Γ(m+ 1 2 ) Γ(m+1) .We first consider the case m = 1 2 .Then, as SNR → ∞, we have For m > 1 2 , we define θ * (SNR) as above and lower bound p 1 (SNR) for large values of SNR as Using ( 43) and ( 44), we conclude that p 1 (SNR) We observe that p 4 (SNR) ≤ p 5 (SNR) since the integrands are always positive and the integral with respect to w is over the whole real line for p 5 (SNR, h).Thus, it will be enough to show

Figure 1 :
Figure 1: The receiver architecture with low-resolution quantization.The signal detector observes only the n-bit quantized versions of Y to estimate the transmitted signal.

Figure 2 :
Figure 2: An illustration of average SEP calculations.If the noise does not drag the original M -PSK constellation point rotated by the channel h beyond the region E (shaded area), there will not be any errors in decoding.

Figure 3 :
Figure 3: An illustration of the centering property for QPSK modulation with 2-bit and 3-bit quantization.Original signal points are indicated by '⋄', whereas the rotated ones after multiplication with √ SNR and h are indicated by '•'.Quantization region boundaries and the corresponding bisectors are indicated in solid black lines and green dash lines, respectively.The shaded area represents the region of attraction of the transmitted symbol x 3 .

Definition 1 :Lemma 2 :
We say a function f is exponentially equal to SNR d if lim SNR→∞ log f (SNR) log SNR = d for some d ∈ R. We write f (SNR) e = SNR d to indicate exponential equality whenever this limit exists.Similarly, we also write f (SNR) e ≤ SNR d and f (SNR) e ≥ SNR d if lim SNR→∞ log f (SNR) log SNR ≤ d and lim SNR→∞ log f (SNR) log SNR ≥ d, respectively.The following lemma establishes two important properties for exponential equality.Let f (SNR) e = SNR d and f i (SNR) e = SNR d i for i ∈ [1 : N].Then, (i) For any α > 0, αf (SNR) e = SNR d (i.e., invariance with scaling property).(ii) N i=1 f i (SNR) e = SNR dmax , where d max = max i∈[1:N ] d i (i.e., summation property).Proof: See Appendix D.The next two lemmas establish the decay rates for p 1 (SNR) and p 2 (SNR) in Theorem 3 in terms of exponential equalities.
Since it is positive and increases to the limiting function g ∞ (θ, β) = sin 2 θ sin 2 β −m

Fig. 4 Figure 5 :
Fig. 4 plots the average SEP as a function of SNR for QPSK modulation with n = 2, 3, 4bit quantization under Nakagami-m fading with shape parameter m = 1 and 2. The simulated

Figure 7 :
figure also confirms that the decay exponents of both L (SNR) and U (SNR) are the same as that of the p (SNR).Next, in Fig.7, we plot the simulated average SEP curves as a function of SNR for 8-PSK modulation with 2-bit quantization and 16-PSK modulation with 2 and 3-bit quantization.We consider equi-probable transmitted symbols and Nakagami-m fading channel model with m = 0.5, 1 and 2. The simulated results are again generated by using Monte Carlo simulations.We can clearly observe an error floor for high SNR values when n < log 2 M, as established by Theorem 1.In particular, the average SEP for 8-PSK has a lower bound of 0.5 with 2-

sufficiency of log 2 M
+ 1 bits for achieving asymptotically optimum M-ary communication reliability.Hence, without modifying the conventional RF chain, we can use one extra bit and still achieve the asymptotically optimum communication performance.Another important future research direction is to compare and contrast the backward-compatible receiver design approach of using one extra bit proposed in this paper with other approaches that can potentially modify the conventional RF chain and manipulate the received signals in the waveform domain by introducing extra analog components.This study needs to be done in detail by considering accuracy and agility of analog domain operations, energy consumption of analog and digital circuit components, different modulation schemes and the average SEP performance curves resulting from different low-resolution ADC based receiver architectures.Similarly, utilizing the results of this paper, a further detailed study on the receiver architecture design to determine
Note that E i contains all H k 's (i.e., bisectors of quantization regions) to which x i is the closest signal point since x i 's are uniformly spaced on the unit circle in C. Furthermore, this statement continues to be true for √ SNRhx i as long as Arg (h) ∈ − π 2 n , π