Gaussian Multiple Access Channels with One-Bit Quantizer at the Receiver †,‡

The capacity region of a two-transmitter Gaussian multiple access channel (MAC) under average input power constraints is studied, when the receiver employs a zero-threshold one-bit analogue-to-digital converter (ADC). It is proven that the input distributions of the two transmitters that achieve the boundary points of the capacity region are discrete. Based on the position of a boundary point, upper bounds on the number of the mass points of the corresponding distributions are derived. Furthermore, a lower bound on the sum capacity is proposed that can be achieved by time division with power control. Finally, inspired by the numerical results, the proposed lower bound is conjectured to be tight.


I. INTRODUCTION
The energy consumption of an analog-to-digital converter (ADC) (measured in Joules/sample) grows exponentially with its resolution (in bits/sample) [1], [2].When the available power is limited, for example, for mobile devices with limited battery capacity, or for wireless receivers that operate on limited energy harvested from ambient sources [3], the receiver circuitry may be constrained to operate with low resolution ADCs.The presence of a low-resolution ADC, in particular a one-bit ADC at the receiver, alters the channel characteristics significantly.Such a constraint not only limits the fundamental bounds on the achievable rate, but it also changes the nature of the communication and modulation schemes approaching these bounds.For example, in a real additive white Gaussian noise (AWGN) channel under an average power constraint on the input, if the receiver is euuipped with a K-bit ADC front end, the capacity-achieving input distribution is discrete with at most K + 1 mass points [4].This is in contrast with the optimality of the Gaussian input distribution when the receiver has infinite resolution.
Especially with the adoption of massive multiple-input multiple-output (MIMO) receivers and the millimeter wave (mmWave) technology enabling communication over large bandwidths, communication systems with limitedresolution receiver front ends are becoming of practical importance.Accordingly, there have been a growing research interest in understanding both the fundamental information theoretic limits and the design of practical communication protocols for systems with finite-resolution ADC front ends.In [5], the authors show that for a Rayleigh fading channel with a one-bit ADC and perfect channel state information at the receiver (CSIR), quadrature phase shift Borzoo Rassouli, Morteza Varasteh and Deniz Gündüz are with the Intelligent Systems and Networks group of Department of Electrical and Electronics, Imperial College London, United Kingdom.emails: {b.rassouli12, m.varasteh12, d.gunduz}@imperial.ac.uk.keying (QPSK) modulation is capacity-achieving.In case of no CSIR, [6] shows that (QPSK) modulation is optimal when the signal-to-noise (SNR) ratio is above a certain threshold, which depends on the coherence time of the channel, while for SNRs below this threshold, on-off QPSK achieves the capacity.For the point-to-point multipleinput multiple-output (MIMO) channel with a one-bit ADC front end at each receive antenna and perfect CSIR, [7] shows that QPSK is optimal at very low SNRs, while with perfect channel state information at the transmitter (CSIT), upper and lower bounds on the capacity are provided in [8].
To the best of our knowledge, the existing literature on communications with low-resolution ADCs focus exclusively on point-to-point systems.Our goal in this paper is to understand the impact of low-resolution ADCs on the capacity region of a multiple access channel (MAC).In particular, we consider a two-transmitter Gaussian MAC with a one-bit quantizer at the receiver.The inputs to the channel are subject to average power constraints.
We show that any point on the boundary of the capacity region is achieved by discrete input distributions.Based on the slope of the tangent line to the capacity region at a boundary point, upper bounds on the cardinality of the support of these distributions are proposed.
The paper is organized as follows.Section II introduces the system model.In section III, the capacity region of a general two-transmitter memoryless MAC under input average power constraints is investigated.Through an example, it is shown that when there is input average power constraint, it is necessary to consider the capacity region with the auxiliary random variable U in general.The main result of the paper is given in section ?? followed by its proof in section IV.Finally, section V concludes the paper.
Notations.Random variables are denoted by capital letters, while their realizations with lower case letters.

II. SYSTEM MODEL AND PRELIMINARIES
We consider a two-transmitter memoryless Gaussian MAC (as shown in Figure 1) with a one-bit quantizer Γ at the receiver front end.Transmitter j = 1, 2 encodes its message W j into a codeword X n j and transmits it over the shared channel.The signal received by the decoder is given by where {Z i } n i=1 is an independent and identically distributed (i.i.d.) Gaussian noise process, also independent of the channel inputs X n 1 and X n 2 with Z i ∼ N (0, 1), i ∈ [1 : n].Γ represents the one-bit ADC operation given by This channel can be modelled by the triplet (X 1 × X 2 , p(y|x 1 , x 2 ), Y), where X 1 , X 2 (= R) and Y (= {0, 1}), Fig. 1: A two-transmitter Gaussian MAC with a one-bit quantizer at the receiver.
respectively, are the alphabets of the inputs and the output.The conditional pmf of the channel output Y conditioned on the channel inputs X 1 and X 2 (i.e.p(y|x 1 , x 2 )) is characterized by where Upon receiving the seuuence Y n , the decoder finds the estimates ( Ŵ1 , Ŵ2 ) of the messages.
We assume that the message pair The average probability of error is defined as Average power constraints are imposed on the channel inputs as where x j,i (w j ) denotes the i th element of the codeword x n j (w j ).A rate pair (R 1 , R 2 ) is said to be achievable for this channel if there exists a seuuence of (2 nR1 , 2 nR2 , n) codes (satisfying the average power constraints) such that lim n→∞ P (n) e = 0.The capacity region C (P 1 , P 2 ) of this channel is the closure of the set of achievable rate pairs (R 1 , R 2 ).

III. MAIN RESULTS
Proposition 1.The capacity region C (P 1 , P 2 ) of a two-transmitter memoryless MAC with average power constraints P 1 and P 2 is the set of non-negative rate pairs (R 1 , R 2 ) that satisfy for some Proof.The capacity region of the discrete memoryless (DM) MAC with input cost constraints has been addressed in Exercise 4.8 of [9].If the input alphabets are not discrete, the capacity region is still the same because: 1) the converse remains the same if the inputs are from a continuous alphabet; 2) the region is achievable by coded time sharing and the discretization procedure (see Remark 3.8 in [9]).Therefore, it is sufficient to show the cardinality bound |U | ≤ 5.
Let P be the set of all product distributions on R 2 .Let g : P → R 5 be a vector-valued mapping defined element-wise as Let G ⊂ R 5 be the image of P under the mapping g (i.e., G = g(P)).Note that although g is a continuous mapping, G is not necessarily compact1 (since g 4 , g 5 cannot be bounded).Given an arbitrary (U, X 1 , X 2 ) ∼ F U F X1|U F X2|U , we obtain the vector r as Therefore, r is in the convex hull of G ⊂ R 5 .By Carathéodory's theorem [10], r is a convex combination of 6 (= 5 + 1) or fewer points in G , which states that it is sufficient to consider |U| ≤ 6.Since P is a connected set and the mapping g is continuous, G is a connected subset of R 5 .Therefore, connectedness of G refines the cardinality of U to |U| ≤ 5.
Lemma 1.For the boundary points of C (P 1 , P 2 ) that are not sum-rate optimal, it is sufficient to have |U| ≤ 4.
Proof.Any point on the boundary of the capacity region that does not maximize R 1 + R 2 , is either of the form In other words, it is one of the corner points of the corresponding pentagon in (3).As in the proof of Proposition 1, define the mapping g : P → R 4 , where g 1 and g 2 are the coordinates of this boundary point conditioned on U = u, and g 3 , g 4 are the same as g 4 and g 5 in (4), respectively.The sufficiency of |U| ≤ 4 in this case follows similarly to the proof of Proposition 1.
When there is no input cost constraint, the capacity region of the MAC can be characterized either through the convex hull operation as in [9, Theorem 4.2], or with the introduction of an auxiliary random variable as in [9,Theorem 4.3].The following remark states that when there is an input cost constraint, the capacity region has only the computable characterization with the auxiliary random variable.
, where the union is over all product distributions that satisfy the average power constraints.
Let R 2 (P 1 , P 2 ) be the set of non-negative rate pairs we can conclude that R 2 (P 1 , P 2 ) ⊆ C (P 1 , P 2 ).This follows from the fact that in the region R 2 (P 1 , P 2 ), the average power constraint E[X 2 j |u] ≤ P j holds for every realization of the auxiliary random variable U , which is a stronger condition than E[X 2 j ] ≤ P j used in the capacity region.The following example shows that R 1 (P 1 , P 2 ) and R 2 (P 1 , P 2 ) can be strictly smaller than C (P 1 , P 2 ).
Consider the same Gaussian MAC with one-bit quantizer at the receiver (as depicted in Figure 1) with the following changes: i) we also impose per codeword average cost constraints as The capacity region of this channel is the set of non-negative rate pairs (R 1 , R 2 ) such that (3) holds for some Also, let R 1 be the rate region in Remark 2 with the additional constraints E[X j ] = 0, j = 1, 2.
In order to show that R 1 can be strictly smaller than the capacity region, we show that there exists a point in the capacity region which is not in R 1 .We have, max ≤ max ≤ max ≤ max = max where ( 5) is due to the fact that X 1 + X 2 is a function of the pair (X 1 , X 2 ), and the following Markov chain holds: ≤ 2, since X 1 and X 2 are independent and zero mean.Also, the channel from X to Y is characterized by the conditional distribution p Y |X (y|x) ∼ Bern(Q(x)).( 7) is due to [4], where the maximum is shown to be achieved by the CDF For this joint distribution on )), which results in (8).
In what follows, it is proved that the ineuuality in ( 6) is strict.In other words, the sum rate of cannot be obtained by any rate pair in R 1 , while it is in the capacity region.Let X = X 1 + X 2 , where X 1 and X 2 are two zero-mean independent random variables on X 1 (= X 2 ) satisfying the average power constraint We show that the minimum Lévy distance 2 between F * X (x) and all the distributions F X (x) 2 The Lévy distance between two distributions F, G : R → [0, 1] is defined as The distributions of X * and X.
The same applies to F X2 with parameter p (≤ 1 4 ).The distribution of X induced by F X1 F X2 is given by where p * p p(1 − p ) + p (1 − p) denotes the binary convolution operation.Let F be the set of all distributions on X obtained in this way.It can be easily verified that (see Figure 2) for any given p, p ≤ 1  4 , the Lévy distance between F X and F * X is Subseuuently, This shows that there is a neighborhood of F * X whose intersection with F is empty.Note that any neighborhood with radius less than 1  16 has this property.Combined with the facts that the mutual information is continuous and F * X is the uniuue solution, it proves that the ineuuality in ( 6) is strict.Therefore, R 1 (= R 2 ) is smaller than the capacity region in general.
The main result of this paper is provided in the following theorem.It bounds the cardinality of the support set of the capacity achieving distributions.
Theorem 1.Let P be an arbitrary point on the boundary of the capacity region C (P 1 , P 2 ) of the memoryless MAC with a one-bit ADC front end (as shown in Figure 1) achieved by F P U (u)F P X1|U (x 1 |u)F P X2|U (x 2 |u).Let l P be the slope of the line tangent to the capacity region at this point.For any u ∈ U, the conditional input distributions F P X1|U (x 1 |u) and F P X1|U (x 1 |u) have at most n 1 and n 2 points of increase 3 , respectively, where Proof.The proof is provided in Section IV.
Proposition 1, Lemma 1 and Theorem 1 establish upper bounds on the number of mass points of the distributions that achieve a boundary point.The significance of this result is that once it is known that the optimal inputs are discrete with at most certain number of mass points, the capacity region along with the optimal distributions can be obtained via computer programs.

IV. PROOF OF THEOREM 1
Since the capacity region is a convex space, it can be characterized by its supporting hyperplanes.In other words, any point on the boundary of the capacity region, denoted by (R b 1 , R b 2 ), can be written as for some λ > 0.
Any rate pair (R 1 , R 2 ) ∈ C (P 1 , P 2 ) is within the pentagon defined by (3) for some F U F X1|U F X2|U that satisfies the power constraints.Therefore, due to the structure of the pentagon, the problem of finding the boundary points is euuivalent to the following maximization problem.
where on the right hand side (RHS) of ( 11), the maximizations are over all F U F X1|U F X2|U that satisfy the power constraints.It is obvious that when λ = 1, the two lines in (11) are the same, which results in the sum capacity.
For any product of distributions F X1 F X2 and the channel in (1), let I λ be defined as With this definition, (11) can be written as where the maximization is over product distributions of the form p U (u) Proposition 2. For a given F X1 and any λ > 0, I λ (F X1 F X2 ) is a concave, continuous and weakly differentiable function of F X2 .In the statement of this Proposition, F X1 and F X2 could be interchanged.
Proof.The proof is provided in Appendix A.
Proposition 3. Let P 1 , P 2 be two arbitrary non-negative finite real numbers.For the following problem max the optimal inputs F * X1 and F * X2 , which are not uniuue in general, have the following properties, (i) The support sets of F * X1 and F * X2 are bounded subsets of R. (ii) F * X1 and F * X2 are discrete distributions that have at most n 1 and n 2 points of increase, respectively, where Proof.We start with the proof of the first claim.Assume that 0 < λ ≤ 1, and F X2 is given.Consider the following optimization problem: < +∞, since for any λ > 0, from (12), From Proposition 2, I λ is a continuous, concave function of F X1 .Also, the set of all CDFs with bounded second moment (here, P 1 ) is convex and compact 4 .Therefore, the supremum in ( 14) is achieved by a uniuue distribution.
Since for any F X1 (x) = s(x − x 0 ) with |x 0 | 2 < P 1 , we have E[X 2 1 ] < P 1 , the Lagrangian theorem and the Karush-Kuhn-Tucker conditions state that F * X1 achieves the maximum if and only if there exists a θ 1 ≥ 0 such that and Lemma 1.The Lagrangian multiplier θ 1 is nonzero.
Proof.Having a zero Lagrangian multiplier means the power constraint is inactive.In other words, if θ 1 = 0, (14) and (15) imply that sup We prove that (18) does not hold by showing that its left hand side (LHS) is strictly lower than 1, while its RHS euuals 1.The details are provided in Appendix B. 4 The compactness follows from [11, Appendix I].
I λ (F X1 F X2 ) (0 < λ ≤ 1) can be written as where we have defined and p(y; F X1 F X2 ) is nothing but the pmf of Y with the emphasis that it has been induced by F X1 and F X2 .Likewise, p(y; F X1 |x 2 ) is the conditional pmf p(y|x 2 ) when X 1 is drawn according to F X1 .From (19), ĩλ (x 1 ; F X1 |F X2 ) can be considered as the density of I λ over F X1 when F X2 is given.i λ (x 2 ; F X2 |F X1 ) can be interpreted in a similar way.
(15) is an unconstrained optimization problem over the set of all CDFs.Note that since x 2 dF X1 (x) is linear and weakly differentiable in F X1 , so is the objective function in (15).Therefore, a necessary condition for optimality of where ĩλ (x 1 ; F X1 |F X2 ) is the density of I λ over F X1 when F X2 is given (see ( 21)).( 23) can be verified to be euuivalent to with euuality if and only if x 1 is a point of increase of F * X1 .In what follows, we prove that in order to satisfy (24), F * X1 must have a bounded support by showing that the LHS of (24) goes to −∞ with x 1 .The following lemma is useful in the seuuel for taking the limit processes inside the integrals.Lemma 2. Let X 1 and X 2 be two independent random variables satisfying E[X 2  1 ] ≤ P 1 and E[X 2 2 ] ≤ P 2 , respectively (P 1 , P 2 ∈ [0, +∞)).Considering the conditional pmf in (1), the following ineuualities hold.
Proof.The proof is provided in Appendix C.
Since any point of increase of F * X1 must satisfy (24) with euuality, and Similarly, for a given F X1 , the optimization problem boils down to the following necessary condition for the optimality of F * X2 , which holds with euuality if and only if x 2 is a point of increase of F * X2 .However, there are two main differences between (34) and (24).First is the difference between i λ and ĩλ .Second is the fact that we do not claim θ 2 to be nonzero, since the approach used in Lemma 1 cannot be readily applied to θ 2 .
Nonetheless, the boundedness of the support of F * X2 can be proved by inspecting the behaviour of the LHS of (34) when |x 2 | → +∞.

It can be easily verified that
If θ 2 > 0, the LHS of (34) goes to −∞ with |x 2 | which proves that X * 2 is bounded.For the case of θ 2 = 0, we rely on the boundedness of X 1 , and use the fact that i λ approaches its limit in (35) from below.In other words, there is a real number K such that i λ (x 2 ; This establishes the boundedness of X * 2 .By rewriting i λ , we have Note that without loss of generality, we have assumed where A 1 , A 2 are some non-negative real numbers.Also, we restrict our attention to x 2 → +∞, since the case x 2 → −∞ follows similarly.It is obvious that the first term in the right hand side of (36) approaches −λ log p Y (1; It is also obvious that the remaining terms go to zero when x 2 → +∞.Hence, it is sufficient to show that the second line of (36) approaches zero from below, which is shown in Appendix E. This proves that X * 2 has a bounded support.Remark 3. We remark here that the order of showing the boundedness of the supports is important.First, for a given F X2 (not necessarily bounded), it is proved that F * X1 is bounded.Then, for a given bounded F X1 , it is shown that F * X2 is also bounded.The order is reversed when λ > 1, and it follows the same steps as in the case of λ ≤ 1.Therefore, it is omitted.
We next prove the second claim in Proposition 3. We assume that 0 < λ < 1, and a bounded F X1 is given.We already know that for a given bounded F X1 , F * X2 has a bounded support denoted by [A 1 , A 2 ].Let P * 2 denote the average power of X * 2 .Also, let S 2 denote the set of all probability distributions on the Borel sets of [A 1 , A 2 ] and p * 0 = p Y (0; F X1 F * X2 ) be the probability of the event Y = 0, induced by F * X2 and the given F X1 .The set is the intersection of S 2 with two hyperplanes 6 .We can write points (three mass points) when λ < 1 (when λ = 1).When λ > 1, the order is reversed and it follows the same steps as in the case of λ < 1.Therefore, it is omitted.

V. CONCLUSION
In this paper, the capacity region of a two-transmitter Gaussian MAC under average input power constraints and one-bit ADC front end at the receiver is considered.It is shown that an auxiliary random variable is necessary for characterizing the capacity region.An upper bound for the cardinality of this auxiliary variable is derived, and it is proved that the distributions that achieve the boundary points of the capacity region, are finite and discrete.
APPENDIX A PROOF OF PROPOSITION 2

A. Concavity
When 0 < λ ≤ 1, we have For a given Therefore, I λ is a concave function of F X2 .For a given The same reasoning applies to the case λ > 1.

B. Continuity
When λ ≤ 1, the continuity of the three terms on the right hand side of (42) is investigated.Let {F X2,n } be a seuuence of distributions which is weakly convergent 7 to F X2 .For a given F X1 , we have = p(y; F X1 |x 0 2 ), (45) 7 The weak convergence of {Fn} to F (also shown as Fn(x) for all continuous and bounded functions ψ( which proves the continuity of H(Y |X 2 ) in F X2 .In a similar way, it can be verified that H(Y x 2 )dF X1 (x 1 ) is a bounded and continuous function of x 2 which guarantees the continuity of Therefore, for a given F X1 , I λ is a continuous function of F X2 .Exchanging the roles of F X1 and F X2 and also the case λ > 1 can be addressed similarly, and are omitted for the sake of brevity.

C. Weak Differentiability
For a given F X1 , the weak derivative of I λ at F 0 X2 is given by if the limit exists.It can be verified that In a similar way, for a given F X2 , the weak derivative of The case λ > 1 can be addressed similarly.

APPENDIX B PROOF OF LEMMA 1
We have sup where (47) is from the assumption that 0 < λ ≤ 1; (48) is justified by the fact that since the U function is monotonically decreasing and the sign of the inputs does not affect the average power constraints, X 1 and X 2 can be assumed non-negative (or alternatively non-positive) without loss of optimality; in (49), we use the fact that , and for t ∈ [0, 1  2 ], H b (t) ≥ t; (50) is based on the convexity and monotonicity of the function Q , which is shown in Appendix D. Therefore, the left hand side of (18) is strictly lower than 1.
Since X 2 has a finite second moment (E[X 2 2 ] ≤ P 2 ), from Chebyshev ineuuality, we have . By this choice of F X1 , we get where ( 53) is due to the fact that ) is maximized at x 2 = 0. (53) shows that I λ (≤ 1) can become arbitrarily close to 1 given that M is large enough.Hence, its supremum over all distributions F X1 is 1.This means that (18) cannot hold, and θ 1 = 0. min p Y (0; F X1 F X2 ), p Y (1; where ( 57) is based on the convexity and monotonicity of the function where (58) is due to convexity of Q(α + √ x) in x for α ≥ 0.
(27) is obtained as follows.After some manipulation, (73) becomes equivalent to where H b (t) = log( 1−t t ) is the derivative of the binary entropy function; (75) is due to the fact that H b (t) is a decreasing function.

∞ 1 y=0 1 +Fig. 3 :
Fig. 3: The figure used in the evaluation of (66) •) on R. Note that Fn(x) (x) if and only if d L (Fn, F ) → 0.where (44) is due to the fact that the U function can be dominated by 1, which is an absolutely integrable function over F X1 .Therefore, p(y; F X1 |x 2 ) is continuous in x 2 , and combined with the weak convergence of {F X2,n }, we X1 F X2 ) log p(y; F X1 F X2 ),which proves the continuity of H(Y ) in F X2 .H(Y |X 2 = x 2 ) is a bounded (∈ [0, 1]) continuous function of x 2 ,since it is a continuous function of p(y; F X1 |x 2 ), and the latter is continuous in x 2 (see (45)).Therefore, w → F X1 |x 2 ) ≥ min p(0; F X1 |x 2 ), p(1; F X1 |x 2 )