Compute-and-Forward Strategies for Cooperative Distributed Antenna Systems

# Compute-and-Forward Strategies for Cooperative Distributed Antenna Systems

\authorblockNSong-Nam Hong,  and Giuseppe Caire,  \authorblockADepartment of Electrical Engineering, University of Southern California, Los Angeles, CA, USA \authorblockA(e-mail: {songnamh, caire}usc.edu) This research was supported in part by the KCC (Korea Communications Commission), Korea, under the R&D program supervised by the KCA (Korea Communications Agency) (KCA-2011-11921-04001).
###### Abstract

We study a distributed antenna system where antenna terminals (ATs) are connected to a Central Processor (CP) via digital error-free links of finite capacity , and serve user terminals (UTs). This model has been widely investigated both for the uplink (UTs to CP) and for the downlink (CP to UTs), which are instances of the general multiple-access relay and broadcast relay networks. We contribute to the subject in the following ways: 1) for the uplink, we apply the “Compute and Forward” (CoF) approach and examine the corresponding system optimization at finite SNR; 2) For the downlink, we propose a novel precoding scheme nicknamed “Reverse Compute and Forward” (RCoF); 3) In both cases, we present low-complexity versions of CoF and RCoF based on standard scalar quantization at the receivers, that lead to discrete-input discrete-output symmetric memoryless channel models for which near-optimal performance can be achieved by standard single-user linear coding; 4) For the case of large , we propose a novel “Integer Forcing Beamforming” (IFB) scheme that generalizes the popular zero-forcing beamforming and achieves sum rate performance close to the optimal Gaussian Dirty-Paper Coding.

The proposed uplink and downlink system optimization focuses specifically on the ATs and UTs selection problem. In both cases, for a given set of transmitters, the goal consists of selecting a subset of the receivers such that the corresponding system matrix has full rank and the sum rate is maximized. We present low-complexity ATs and UTs selection schemes and demonstrate, through Monte Carlo simulation in a realistic environment with fading and shadowing, that the proposed schemes essentially eliminate the problem of rank deficiency of the system matrix and greatly mitigate the non-integer penalty affecting CoF/RCoF at high SNR. Comparison with other state-of-the art information theoretic schemes, such as “Quantize reMap and Forward” for the uplink and “Compressed Dirty Paper Coding” for the downlink, show competitive performance of the proposed approaches with significantly lower complexity.

Compute and Forward, Reverse Compute and Forward, Lattice Codes, Distributed Antenna Systems, Multicell Cooperation.

## I Introduction

A cloud base station is a Distributed Antenna System (DAS) formed by a number of simple antenna terminals (ATs) [1], spatially distributed over a certain area, and connected to a central processor (CP) via wired backhaul [2, 3, 4]. Cloud base station architectures differ by the type of processing made at the ATs and at the CP, and by the type of wired backhaul. At one extreme of this range of possibilities, the ATs perform just analog filtering and (possibly) frequency conversion, the wired link are analog (e.g., radio over fiber [5]), and the CP performs demodulation to baseband, A/D and D/A conversion, joint decoding (uplink) and joint pre-coding (downlink). At the other extreme we have “small cell” architectures where the ATs perform encoding/decoding, the wired links send data packets, and the CP performs high-level functions, such as scheduling, link-layer error control, and macro-diversity packet selection.

In this paper we focus on an intermediate DAS architecture where the ATs perform partial decoding (uplink) or precoding (downlink) and the backhaul is formed by digital links of fixed rate . In this case, the DAS uplink is an instance of a multi-source single destination layered relay network where the first layer is formed by the user terminals (UTs), the second layer is formed by the ATs and the third layer contains just the CP (see Fig. 1). The corresponding DAS downlink is an instance of a broadcast layered relay network with independent messages.

In our model, analog forwarding from ATs to CP (uplink) or from CP to ATs (downlink) is not possible. Hence, some form of quantization and forwarding is needed. A general approach to the uplink is based on the Quantize reMap and Forward (QMF) paradigm of [6] (extended in [7] where it is referred to as Noisy Network Coding). In this case, the ATs perform vector quantization of their received signal at some rate . They map the blocks of quantization bits into binary words of length by using some randomized hashing function (notice that this corresponds to binning if ), and let the CP perform joint decoding of all UTs’ messages based on the observation of all the (hashed) quantization bits.111The information-theoretic vector quantization of [6], [7] can be replaced by scalar quantization with a fixed-gap performance degradation [8]. It is known [6] that QMF achieves a rate region within a bounded gap from the cut-set outer bound [9], where the bound depends only on the network size and on , but it is independent of the channel coefficients and of the operating SNR. For the broadcast-relay downlink, a general coding strategy has been proposed in [10] based on a combination of Marton coding for the general broadcast channel [11] and a coding scheme for deterministic linear relay networks, “lifted” to the Gaussian case. Specializing the above general coding schemes to the the DAS considered here, for the uplink we obtain the scheme based on quantization, binning and joint decoding of [12], and for the downlink we obtain the Compressed Dirty-Paper Coding (CDPC) scheme of [13]. From an implementation viewpoint, both QMF and CDPC are not practical, the former requiring vector quantization at the ATs and joint decoding of all UT messages based on the hashed quantization bits at the CP, and the latter requiring Dirty-Paper Coding (notoriously difficult to implement in practice) and vector quantization at the CP.

A lower complexity alternative strategy for general relay networks was proposed in [14] and goes under the name of Compute and Forward (CoF). CoF makes use of lattice codes, such that each relay can reliably decode a linear combination with integer coefficient of the interfering codewords. Thank to the fact that lattices are modules over the ring of integers, this linear combination translates directly into a linear combination of the information messages defined over a suitable finite field. CoF can be immediately used for the DAS uplink. The performance of CoF was examined in [15] for the DAS uplink in the case of the overly simplistic Wyner model [16]. It was shown that CoF yields competitive performance with respect to QMF for practically realistic values of SNR.

This paper contributes to the subject in the following ways: 1) for the DAS uplink , we consider the CoF approach and examine the corresponding system optimization at finite SNR for a general channel model including fading and shadowing (i.e., beyond the nice and regular structure of the Wyner model); 2) For the downlink, we propose a novel precoding scheme nicknamed Reverse Compute and Forward (RCoF); 3) For both uplink and downlink, we present low-complexity versions of CoF and RCoF based on standard scalar quantization at the receivers. These schemes are motivated by the observation that the main bottleneck of a digital receiver is the Analog to Digital Conversion (ADC), which is costly, power-hungry, and does not scale with Moore’s law. Rather the number of bit per second produced by an ADC is roughly a constant that depends on the power consumption [17, 18]. Therefore, it makes sense to consider the ADC as part of the channel. The proposed schemes, nicknamed Quantized CoF (QCoF) and Quantized RCoF (RQCoF), lead to discrete-input discrete-output symmetric memoryless channel models naturally matched to standard single-user linear coding. In fact, QCoF and RQCoF can be easily implemented using -ary Low-Density Parity-Check (LDPC) codes [19, 20, 21] with and prime, yielding essentially linear complexity in the code block length and polynomial complexity in the system size (minimum between number of ATs and UTs).

The two major impairments that deteriorate the performance of DAS with CoF/RCoF are the non-integer penalty (i.e., the residual self-interference due to the fact that the channel coefficients take on non-integer values in practice) and the rank-deficiency of the resulting system matrix over the -ary finite field. In fact, the wireless channel is characterized by fading and shadowing. Hence, the channel matrix from ATs to UTs does not have any particularly nice structure, in contrast to the Wyner model case, where the channel matrix is tri-diagonal [15]. Thus, in a realistic setting, the system matrix resulting from CoF/RCoF may be rank deficient. This is especially relevant when the size of the finite field is small (e.g., it is constrained by the resolution of the A/D and D/A conversion). The proposed system optimization counters the above two problems by considering power allocation, network decomposition and antenna selection at the receivers (ATs selection in the uplink and UTs selection in the downlink). We show that in most practical cases the AT and UT selection problems can be optimally solved by a simple greedy algorithm. Numerical results show that, in realistic networks with fading and shadowing, the proposed optimization algorithms are very effective and essentially eliminate the problem of system matrix rank deficiency, even for small field size .

A final novel contribution of this paper consists of the Integer-Forcing Beamforming (IFB) downlink scheme, targeted to the case where is large, and therefore the DAS downlink reduces to the well-known vector Gaussian broadcast channel. In this case, a common and well-known low-complexity alternative to the capacity-achieving Gaussian DPC scheme consists of Zero-Forcing Beamforming (ZFB), which achieves the same optimal multiplexing gain, at the cost of some performance loss at finite SNR. IFB can be regarded both as a generalization of ZFB and as the dual of Integer-Forcing Receiver (IFR), proposed in [22] for the uplink multiuser MIMO case. We demonstrate that IFB can achieve rates close to the information-theoretic optimal Gaussian DPC, and can significantly outperform conventional ZFB. This gain can be explained by the fact that IFB is able to reduce the power penalty of ZFB, due to non-unitary beamforming.

The paper is organized as follows. In Section II we define the uplink and downlink DAS system model, summarize some definitions on lattices and lattice coding, and review CoF. In Section III we consider the application of CoF to the DAS uplink and introduce the (novel) concept of network decomposition to improve the CoF sum rate. Section IV considers the DAS downlink and presents the RCoF scheme. In Section V we introduce the low-complexity “quantized” versions of CoF and RCoF. Section VI focuses on the symmetric Wyner model and presents a simple power allocation strategy to alleviate the impact of non-integer penalty. In the case of a realistic DAS channel model including fading, shadowing and pathloss, a low-complexity greedy algorithm for ATs selection (uplink) and UTs selection (downlink) is presented in Section VII. Finally, Section VIII considers the case of large backhaul rate and presents the IFB scheme. Some concluding remarks are provided in Section IX.

## Ii Preliminaries

In this section we provide some basic definitions and results that will be extensively used in the sequel.

### Ii-a Distributed Antenna Systems: Channel Model

We consider a DAS with ATs and UTs, each of which is equipped with a single antenna. The ATs are connected to the CP via digital backhaul links of rate (see Fig. 1). A block of channel uses of the discrete-time complex baseband uplink channel is described by

 Y––=HX––+Z––, (1)

where we use “underline” to denote matrices whose horizontal dimension (column index) denotes “time” and vertical dimension (row index) runs across the antennas (UTs or ATs), the matrices

 X––=⎡⎢ ⎢⎣x––1⋮x––K⎤⎥ ⎥⎦andY––=⎡⎢ ⎢ ⎢⎣y––1⋮y––L⎤⎥ ⎥ ⎥⎦

contain, arranged by rows, the UT codewords and the AT channel output vectors , for , and , respectively. The matrix contains i.i.d. Gaussian noise samples , and the matrix contains the channel coefficients, assumed to be constant over the whole block of length and known to all nodes.

Similarly, a block of channel uses of the discrete-time complex baseband downlink channel is described by , where we use “tilde” to denote downlink variables, contains the AT codewords, contain the channel output and Gaussian noise at the UT receivers, and is the downlink channel matrix.

Since ATs and UTs are separated in space and powered independently, we assume a symmetric per-antenna power constraint for both the uplink and the downlink, given by for all and by for all , respectively.

### Ii-B Nested Lattice Codes

Let be the ring of Gaussian integers and be a Gaussian prime. 222A Gaussian integer is called a Gaussian prime if it is a prime in . A Gaussian prime satisfies exactly one of the following conditions [Stillwell]: 1) ; 2) one of is zero and the other is a prime number in of the form (with a nonnegative integer); 3) both of are nonzero and is a prime number in of the form . In this paper, is assumed to be a prime number congruent to modulo , which is an integer Gaussian prime according to condition 2).. Let denote the addition over , and let be the natural mapping of onto . We recall the nested lattice code construction given in [14]. Let be a lattice in , with full-rank generator matrix . Let denote a linear code over with block length and dimension , with generator matrix . The lattice is defined through “construction A” (see [23] and references therein) as

 Λ1=p−1g(C)T+Λ, (2)

where is the image of under the mapping (applied component-wise). It follows that is a chain of nested lattices, such that and .

For a lattice and , we define the lattice quantizer , the Voronoi region and . For and given above, we define the lattice code with rate . Construction A provides a natural labeling of the codewords of by the information messages . Notice that the set is a system of coset representatives of the cosets of in . Hence, the natural labeling function is defined by .

### Ii-C Compute and Forward

We recall here the CoF scheme of [14]. Consider the -user Gaussian multiple access channel (G-MAC) defined by

 y––=K∑k=1hkx––k+z––, (3)

where , and the elements of are i.i.d. . All users make use of the same nested lattice codebook , where has second moment . Each user encodes its information message into the corresponding codeword and produces its channel input according to

 x––k=[t–k+d––k]modΛ, (4)

where the dithering sequences ’s are mutually independent across the users, uniformly distributed over , and known to the receiver. Notice that, as in many other applications of nested lattice coding and lattice decoding (e.g.,[Zamir, 23, 27]), random dithering is instrumental for the information theoretic proofs, but a deterministic dithering sequence that scrambles the input and makes it zero-mean and uniform over the shaping region can be effectively used in practice, without need of common randomness. The decoder’s goal is to recover a linear combination with integer coefficient vector . Since is a -module (closed under linear combinations with Gaussian integer coefficients), then . Letting be decoded codeword (for some decoding function which in general depends on and ), we say that a computation rate is achievable for this setting if there exists sequences of lattice codes of rate and increasing block length , such that the decoding error probability satisfies .

In the scheme of [14], the receiver computes

 ^y–– = [αy––−K∑k=1akd––k]modΛ (5) = [v––+z––\tiny{eff}(h,a,α)]modΛ,

where

 z––\tiny{eff}(h,a,α)=K∑k=1(αhk−ak)x––k+αz–– (6)

denotes the effective noise, including the non-integer self-interference (due to the fact that in general) and the additive Gaussian noise term. The scaling, dither removal and modulo- operation in (5) is referred to as the CoF receiver mapping in the following. By minimizing the variance of with respect to , we obtain

 σ2(h,a) = minασ2z\tiny{eff}(h,a,α) (7) = SNR(∥a∥2−SNR|hHa|21+SNR∥h∥2) \lx@stackrel(a)= aH(SNR−1I+hhH)−1a

where follows from the matrix inversion lemma [24]. Since is uniquely determined by and , it will be omitted in the following, for the sake of notation simplicity. From [14], we know that by applying lattice decoding to given in (5) the following computation rate is achievable:

 R(h,a,SNR)=log+(SNRaH(SNR−1I+hhH)−1a), (8)

where .

The computation rate can be maximized by minimizing with respect to . The quadratic form (7) is positive definite for any , since the matrix has eigenvalues

 λi={SNR/(1+∥h∥2SNR)i=1SNRi>1 (9)

By Cholesky decomposition, there exists a lower triangular matrix such that . It follows that the problem of minimizing over is equivalent to finding the ”shortest lattice point” of the -dimensional lattice generated by . This can be efficiently obtained using the complex LLL algorithm [25, 26]  possibly followed by Phost or Schnorr-Euchner enumeration (see [27]) of the non-zero lattice points in a sphere centered at the origin, with radius equal to the shortest vector found by complex LLL. Algorithm summarizes the procedures used in this paper to find the optimal integer vector .

## Iii Compute and Forward for the DAS Uplink

In this section we apply CoF to the DAS uplink and further improve its sum rate by introducing the idea of network decomposition. The scheme is illustrated in Fig. 2, where CoF is used at each AT receiver. For simplicity of exposition, we restrict to consider the same number of UTs and ATs. The notation, however, applies also to the case of addressed in Section VII, when considering AT selection. The UTs make use of the same lattice code of rate , and produce their channel input , , according to (4). Each AT decodes the codeword linear combination , for a target integer vector determined according to Algorithm 1, independently of the other ATs. If , where the latter denotes the computation rate of the G-MAC formed by the UTs and the -th AT, taking on the form given in (8), the decoding error probability at AT can be made as small as desired. Letting denote the information message corresponding to the target decoded codeword , the code linearity over and the -module structure of yield

 u––ℓ=K⨁k=1qℓ,kw––k, (10)

where . After decoding, each AT forwards the corresponding information message to the CP via wired links of fixed . This can be done if . The CP collects all the messages for and forms the system of linear equations over

 ⎡⎢ ⎢⎣^u––1⋮^u––L⎤⎥ ⎥⎦=Q⎡⎢ ⎢⎣^w––1⋮^w––K⎤⎥ ⎥⎦, (11)

where we define and the system matrix . Provided that has rank over , the CP obtains the decoded messages by Gaussian elimination. Assuming this full-rank condition and for all , the error probability can be made arbitrarily small for sufficiently large . The resulting achievable rate per user is given by [15]:

 R=min{R0,minℓ{R(hℓ,aℓ,SNR)}}. (12)
###### Remark 1

Since each AT determines its coefficients vector in a decentralized way, by applying Algorithm 1 independently of the other ATs’ channel coefficients, the resulting system matrix may be rank-deficient. If , requiring that all ATs can decode reliably is unnecessarily restrictive: it is sufficient to select a subset of ATs which can decode reliably and whose coefficients form a full-rank system matrix. This selection problem will be addressed in Section VII.

The sum rate of CoF-based DAS can be improved by network decomposition with respect to the system matrix . Although the elements of are non-zero, the corresponding may include zeros, since some elements of the vectors may be zero modulo . Because of the presence of zero elements, the system matrix may be put in block diagonal form by column and row permutations. If the permuted system matrix has diagonal blocks, the corresponding network graph decomposes into independent subnetworks and CoF can be applied separately to each subnetwork such that taking the minimum of the computation rates over the subnetworks is not needed. Hence, the sum rate is given by the sum (over the subnetworks) of the sum rates of each network component. In turns, the common UT rate of each indecomposable subnetwork takes on the form (12). For given , the disjoint subnetwork components can be found efficiently using depth-first or breadth-first search [28]. This also essentially reduces the computation complexity of Gaussian elimination, which is performed independently for each subnetwork. We assume that, up to a suitable permutation of rows and columns, can be put in block diagonal form with diagonal blocks for , where we use the following notation: for a matrix with rows index set and column index set , denotes the submatrix obtained by selecting the rows in and the columns in . The following results are immediate:

###### Lemma 1

If is a full-rank matrix, the diagonal blocks are full-rank square matrices for every . ∎

###### Theorem 1

CoF with network decomposition, applied to a DAS uplink with channel matrix , achieves the sum rate

 R\tiny{CoF}(H,A) = S∑s=1|As|min{R0,min{R(hk,ak,SNR):k∈As}}, (13)

where is the matrix of CoF integer coefficients, and where the system matrix has full rank over and can be put in block diagonal form by rows and columns permutations, with diagonal blocks for . ∎

## Iv Reverse Compute and Forward for the DAS Downlink

In this section we propose a novel downlink precoding scheme nicknamed “Reverse” CoF (RCoF). Again, we restrict to the case although the notation applies to the case of , treated in Section VII. In a DAS downlink, the role of the ATs and UTs can be reversed with respect to the uplink. Each UT can reliably decode an integer linear combination of the lattice codewords sent by the ATs. However, the UTs cannot share the decoded codewords as in the uplink, since they have no backhaul links. Instead, the “interference” in the finite-field domain can be totally eliminated by zero-forcing precoding (over the finite field) at the CP. RCoF has a distinctive advantage with respect to its CoF counterpart viewed before: since each UT sees only its own lattice codeword plus the effective noise, each message is rate-constrained by the computation rate of its own intended receiver, and not by the minimum of all computation rates across all receivers, as in the uplink case. In order to achieve different coding rates while preserving the lattice -module structure, we use a family of nested lattices , obtained by a nested construction A as described in [14, Sect. IV.A]. In particular, we let with and with denoting the linear code over generated by the first rows of a common generator matrix , with . The corresponding nested lattice codes are given by , and have rate . We let , where denotes the integer coefficients vector used at UT for the modulo- receiver mapping (see (5)), and we let denote the downlink system matrix, assumed to have rank . Then, RCoF scheme proceeds as follows (see Fig. 3):

• The CP sends independent messages to UTs (if , then a subset of UTs is selected, as explained in Section VII). We let denote the UT destination of the -th message, encoded by at rate .

• The CP forms the messages by appending zeros to each -th information message of symbols, so that all messages have the same length .

• The CP produces the precoded messages

 ⎡⎢ ⎢ ⎢ ⎢⎣~\boldmathμ–––––––––––––1⋮~\boldmathμ–––––––––––––L⎤⎥ ⎥ ⎥ ⎥⎦=~Q−1⎡⎢ ⎢⎣~w––1⋮~w––L⎤⎥ ⎥⎦. (14)

(notice: if then is replaced by the submatrix ).

• The CP forwards the precoded message to AT for all , via the digital backhaul link.

• AT locally produces the lattice codeword (the densest lattice code) and transmits the corresponding channel input according to (4). Because of linearity, the precoding and the encoding over the finite field commute. Therefore, we can write , where and .

• Each UT applied the CoF receiver mapping as in (5), with integer coefficients vector and scaling factor , yielding

 ^~y––kℓ = ⎡⎢ ⎢ ⎢ ⎢⎣~aTkℓ⎡⎢ ⎢ ⎢ ⎢⎣~\boldmathν–––––––––––––1⋮~\boldmathν–––––––––––––L⎤⎥ ⎥ ⎥ ⎥⎦+~z––\tiny{eff}(~hkℓ,~akℓ,αkℓ)⎤⎥ ⎥ ⎥ ⎥⎦modΛ (15) = \lx@stackrel(a)= ⎡⎢ ⎢⎣([~aTkℓB]modp\bb Z[j])⎡⎢ ⎢⎣~t–1⋮~t–L⎤⎥ ⎥⎦+~z––\tiny{eff}(~hkℓ,~akℓ,αkℓ)⎤⎥ ⎥⎦modΛ \lx@stackrel(b)= [~t–ℓ+~z––\tiny{eff}(~hkℓ,~akℓ,αkℓ)]modΛ,

where (a) is due to the fact that for any codeword , and (b) follows from the following result:

###### Lemma 2

Let . Assuming invertible over , if , then:

 [~AB]modp\bb Z[j]=I. (16)
###### Proof 1

Using , we have:

 [~AB]modp\bb Z[j] = [([~A]modp\bb Z[j])B]modp% \bb Z[j] (17) = [g(~Q)g(~Q−1)]modp\bb Z[j] (18) = [g(~Q~Q−1)]modp\bb Z[j] (19) = I. (20)

From (15) we have that RCoF induces a point-to-point channel at each desired UT , where the the integer-valued interference is eliminated by precoding, and the remaining effective noise is due to the non-integer residual interference and to the channel Gaussian noise. The scaling coefficient and the integer vector are optimized independently by each UT using (7) and Algorithm 1. It follows that the desired message can be recovered with arbitrarily small probability of error if , where the latter takes on the form given in (8). Including the fact that the precoded messages can be sent from the CP to the ATs if , we arrive at:

###### Theorem 2

RCoF applied to a DAS downlink with channel matrix achieves the sum rate

 R\tiny{RCoF}(~H,~A) = L∑ℓ=1min{R0,R(~hℓ,~aℓ,SNR)}.

###### Remark 2

When the channel matrix has the property that each row is a permutation of the first row (e.g., in the case is circulant, as in the Wyner model [16]), each UT has the same computation rate and hence a single lattice code is sufficient.

## V Low-Complexity Schemes

This section considers low-complexity versions of the schemes of Sections III and IV, using one-dimensional lattices and scalar quantization. Our approach is suited to the practically relevant case where the receivers are equipped with ADCs of fixed finite resolution, such that scalar quantization is included as an unavoidable part of the channel model. In this case, CoF and RCoF, as well as QMF and CDPC, are not possible since lattice quantization requires to have access to the unquantized (soft) signal samples.

The quantized versions of CoF and RCoF follow as a special cases, by choosing the generator matrix of the shaping lattice to be , with in order to satisfy the per-antenna power constraint with equality. The resulting lattice code is with and , for a linear code over of rate . Furthermore, we introduce a scalar quantization stage as part of each receiver. This is defined by the function , applied component-wise. Since is the -dimensional complex cubic lattice, also the modulo- operations in CoF/RCoF are performed component-wise. Hence, we can restrict to a symbol-by-symbol channel model instead of considering -vectors as before.

Consider the same G-MAC setting of Section II-C. Given the information message , encoder produces the codeword and the corresponding lattice codeword . The -th component of its channel input is given by

 xk,i=[tk,i+dk,i]modτ\bb Z[j], (21)

where the dithering samples are i.i.d. across users and time dimensions, and uniformly distributed over the square region . The received signal is given by (3). The receiver selects the integer coefficients vector and produces the sequence with components

 ui = g−1(pτ([Q(τ/p)\bb Z[j](αyi−K∑k=1akdk,i)]modτ\bb Z% [j])) (22) = g−1([Q\bb Z[j](pτ(K∑ℓ=1aktk,i+ξi(h,a,α)))]modp\bb Z[j]), (23)

for , where

 ξi(h,a,α)=K∑k=1(αhk−ak)xk,i+αzi. (24)

Since by construction, and using the obvious identity with and , we arrive at

 (25)

where and where the components of the discrete additive noise are given by . This shows that the concatenation of the lattice encoders, the G-MAC and the receiver mapping (22) reduces to an equivalent discrete linear additive-noise finite-field MAC (FF-MAC) given by (25).

###### Remark 3

Notice that is obtained from the channel output by component-wise analog operations (scaling by and translation by ), scalar quantization and modulo reduction. In fact, the scalar quantization and the modulo lattice operations commute, i.e., the modulo operation can be performed directly on the analog signals by wrapping the complex plane into the Voronoi region of , and then the scalar quantizer can be applied to the wrapped samples. This corresponds to the analog sawtooth transformation, followed by scalar quantization, applied to the real and imaginary parts of the complex baseband signal, as shown in Fig. 4.

The marginal pmf of can be calculated numerically, and it is well approximated by assuming . In Appendix A, we obtain an accurate and easy way to calculate the pmf of the effective noise component based on such Gaussian approximation. The optimal choice of and for the discrete channel (25) consists of minimizing the entropy of the discrete additive noise . However, this does not lead to a tractable numerical method. Instead, we resort to the minimization of the unquantized effective noise variance , which leads to the same expression (7) and integer search of Algorithm 1. We assume that and are determined in this way, independently, by each receiver, and omit from the notation.

In the following, we will present coding schemes for the induced FF-MAC in (25) and for the corresponding Finite-Field Broadcast Channel (FF-BC) resulting from the downlink, by exchanging the roles of ATs and UTs. We follow the notation used in Sections III and IV and let and denote the system matrix for the uplink and for the downlink, respectively.

### V-a QCoF and LQF for the DAS Uplink

In this section we present two schemes referred to as Quantized CoF (QCoF) and Lattice Quantize and Forward (LQF), which differ by the processing at the ATs. QCoF is a low-complexity quantized version of CoF. The quantized channel output at AT is given by

 u––ℓ=v––ℓ⊕% \boldmathζ–––––––––––––––(hℓ,aℓ), (26)

where, by linearity, is a codeword of . This is a point-to-point channel with discrete additive noise over . AT can successfully decode if . This is an immediate consequence of the well-known fact that linear codes achieve the capacity of symmetric discrete memoryless channels[29]. If , each AT can forward the decoded message linear finite-field combination to the CP, so that the original UT messages can be obtained by Gaussian elimination (see Section III). With the same notation of Theorem 1, including network decomposition which applies verbatim here, we have:

###### Theorem 3

QCoF with network decomposition, applied to a DAS uplink with channel matrix , achieves the sum rate

 (27)

Next, we consider the LQF scheme, which may provide an attractive alternative in the case , i.e., when is large and a small value of is imposed by the ADC complexity and/or power consumption constraints. In LQF, the UTs encode their information messages by using independently generated, not nested, random linear codes over , in order to allow for different coding rates . In this case, the fine lattice for UT is and the symbol by symbol quantization maps the channel into an additive MAC channel over , with discrete additive noise. Hence, independently generated random linear codes are optimal for this channel (this is easily seen form the fact that the channel is additive over the finite field). In LQF, the ATs forwards its quantized channel observations directly to the CP without local decoding. Hence, LQF can be seen as a special case of QMF without binning. From (26), the CP sees a FF-MAC with -dimensional output:

 ⎡⎢ ⎢⎣u––1⋮u––L⎤⎥ ⎥⎦=Q⎡⎢ ⎢⎣c––1⋮c––K⎤⎥ ⎥⎦⊕⎡⎢ ⎢ ⎢⎣\boldmathζ–––––––––––––(h1,a1)⋮\boldmathζ–––––––––––––(hL,aL)⎤⎥ ⎥ ⎥⎦. (28)

The following result provides an achievable sum rate of LQF subject to the constraint .

###### Theorem 4

Consider the FF-MAC, defined by as in (28). If has rank , the following sum rate is achievable by linear coding

 R\tiny{FF-MAC}=2Klogp−K∑k=1H(ζ(hk,ak)). (29)
###### Proof:

See Appendix B. \qed

The relative merit of QCoF and LQF depends on , , and on the actual realization of the channel matrix . In symmetric channel cases (i.e., Wyner model [16]), where the AT have the same computation rate, QCoF beats LQF by making sufficiently large. On the other hand, if the modulation order is predetermined as in a conventional wireless communication system, and this is relatively small with respect to , LQF outperforms QCoF by breaking the limitation of the minimum computation rate over the ATs.

### V-B RQCoF for the DAS Downlink

Exchanging the roles of AT s and UTs and using (25), the DAS downlink with quantization at the receivers is turned into the FF-BC

 ⎡⎢ ⎢⎣~u––1⋮~u––K⎤⎥ ⎥⎦=~Q⎡⎢ ⎢⎣~c––1⋮~c––L⎤⎥ ⎥⎦⊕⎡⎢ ⎢ ⎢⎣\boldmathζ–––––––––––––(~h1,~a1)⋮\boldmathζ–––––––––––––(~hK,~aK)⎤⎥ ⎥ ⎥⎦. (30)

The following result yields that simple matrix inversion over can achieve the capacity of this FF-BC. Intuitively, this is because there is no additional power cost with Zero-Forcing Beamforming (ZFB) in the finite-field domain (unlike ZFB in the complex domain).

###### Theorem 5

Consider the FF-BC in (30) for . If has rank , the sum capacity is

 C\tiny{FF-BC}=2Llogp−L∑ℓ=1H(ζ(~hℓ,~aℓ)). (31)

and it can be achieved by linear coding.

###### Proof:

See Appendix C. \qed

Motivated by Theorem 5, we present the RQCoF scheme using finite-field matrix inversion precoding at the CP. As for RCoF, we use nested linear codes where has rate and let denote the UT destination of the -th message, encoded by . The CP precodes the zero-padded information messages as in (14) and sends the precoded message to AT for all , via the digital backhaul link. AT generates the codeword , and the corresponding transmitted signal according to (21), with Each UT produces its quantized output according to the scalar mapping (22) and obtains:

 ~u––kℓ = ⎛⎜ ⎜⎝~qTkℓ⎡⎢ ⎢⎣~c––1⋮~c––L⎤⎥ ⎥⎦⎞⎟ ⎟⎠⊕\boldmathζ–––––––––––––(~hkℓ,~akℓ) (32) = = ~v––ℓ⊕% \boldmathζ–––––––––––––––(~hkℓ,~akℓ)

where is a codeword of . Thus, UT can recover its desired message if