Network densification by increasing the densities of base stations (BSs)/access points (APs) deployed, and millimeter-wave (mmWave) communication by exploiting large unused bandwidth in the higher frequencies of the radio spectrum, are two key strategies for achieving the orders of magnitude data rate improvement required for future wireless communication networks . On one hand, cloud radio access network (CRAN), in which distributed low-complexity remote radio heads (RRHs) are deployed close to the users, and coordinated by a central unit (CU) for joint processing, provides a cost-effective way of achieving network densification. Thanks to the centralized resource allocation and joint signal processing for the RRHs at the CU, CRAN achieves significant improvements in both the spectral efficiency and energy efficiency compared to the conventional cellular network [2, 3, 4, 5, 6, 7, 8, 9], while RRHs can be simple relay nodes that transmit and receive quantized/compressed baseband signals to/from the CU over their fronthaul links. On the other hand, with advances in radio frequency (RF) circuits, wireless communication over mmWave bands has emerged as a promising technology to achieve high-rate communications, due to the large bandwidth available and the beamforming gain brought about by the possibility of deploying a large number of antennas at the transceivers thanks to the small wavelengths [10, 11]. Thus, CRAN when integrated with mmWave communication, achieves the double goals of network densification and ample bandwidth at the same time for future wireless networks. However, in such a dense CRAN operating over a large mmWave bandwidth, the digital fronthaul links would be easily saturated by the large volume of sampled and quantized/compressed baseband signals that need to be transmitted between the CU and RRHs; thus, it is crucial to find cost-effective solutions to reduce the transmission rate required for each fronthaul link.
In CRANs operating over the conventional cellular frequency bands, a considerable body of prior work (e.g. [2, 4, 3, 5]) has investigated various techniques for data compression at the RRHs to achieve fronthaul rate reduction. Further, channel estimation in networks with coordinated multi-antenna BSs has also been considered in prior work [12, 13]. In 
, random matrix theory was used to derive an approximate lower bound on the uplink ergodic achievable rate in a system with multiple single-antenna users and multi-antenna BSs, while considered an estimate-compress-forward approach for the uplink of a CRAN and proposed various schemes to optimize the ergodic achievable sum rate subject to backhaul constraints. Most of the above techniques for compression and/or channel estimation are applicable for relatively small bandwidth compared to that in mmWave, and typically involve complex signal processing and cooperative signal compression across the RRHs, which are difficult to implement for mmWave systems due to practical cost and complexity considerations. A low-complexity training sequence design for CRAN was considered in , where the problem of minimizing the training length, while maintaining local orthogonality among the training sequences of the users, was considered; however, the fronthaul constraints at the RRHs were ignored. For frequency-selective mmWave channels, channel estimation for hybrid precoding was considered in [15, 16]. The approach is to represent the channel taps in the angular domain corresponding to a set of quantized angles, and then use sparse signal processing techniques to estimate the relevant parameters.
In this paper, we propose a new architecture for mmWave CRAN by leveraging the use of advanced lens antenna arrays [17, 18, 19, 20, 21, 22] at the RRHs. A full-dimensional lens antenna array  consists of an electromagnetic (EM) lens with energy focusing capability integrated with an antenna array whose elements are located on the focal surface of the lens (see Fig. 1). The amplitude response of a lens array can be expressed as a “sinc”-type function in terms of the angles of arrival/departure of plane waves incident on/transmitted from it, and the locations of the antenna elements [18, 19]. Hence, by appropriately designing the locations of the antenna elements on the focal surface, the lens array is capable of focusing most of the energy from a uniform plane wave arriving in a particular direction onto a specific antenna element or subset of elements. Moreover, due to the multi-path sparsity of mmWave channels , a lens array can be used to achieve the capacity of a point-to-point mmWave channel with multiple antennas via a technique called path division multiplexing , which uses simple single-carrier modulation even for transmission over wideband frequency-selective channels, and has low signal processing complexity.
With lens antenna arrays, the angular domain111Also known as beamspace channel . sparsity of mmWave channels is transformed to the spatial domain, which then enables lower complexity signal processing for channel estimation and data transmission. Beamspace channel estimation based on compressed sensing techniques was considered in  for a mmWave system with a multi-antenna BS and multiple users. However, the channel was assumed to be frequency non-selective, unlike the general freqeuncy-selective channel in this paper. Moreover, in a CRAN, since the angles of arrival of the signals from different users are typically independent, and thus different at each RRH, the user signals are effectively separated over a small number of focusing antennas that can be selected for signal sampling and quantization. Thus, the use of lens antenna arrays can potentially achieve significant reduction in both the fronthaul rate requirement for transmitting the quantized signals to the CU and the interference among the users for the joint decoding at the CU in the uplink transmission. However, due to the finite fronthaul rate constraint at each RRH, it is crucial to design antenna selection and signal quantization schemes to maximize the achievable user rates at the CU. Since the RRHs in a CRAN are typically simple relay nodes, in this paper, we consider that simple uniform scalar quantization (SQ) of the sampled baseband signals is performed independently over the antennas selected at each RRH. The major contributions of this paper are summarized as follows.
We propose a simple energy-detection based antenna selection at each RRH and a low-complexity quantization bit allocation algorithm over the selected antennas to minimize the total quantization noise power based only on the estimates of the received signal power at different antennas, subject to the fronthaul rate constraint.
When the CU has perfect channel state information (CSI), we show that the proposed system with lens antenna arrays can achieve better sum-rate performance compared to a conventional CRAN using uniform planar arrays (UPAs) at the RRHs and orthogonal frequency division multiplexing (OFDM) transmission by the users, when the same quantization algorithm is employed in both cases.
Under imperfect CSI, we propose a reduced-size, approximate MMSE beamforming, by exploiting the energy-focusing property of lens antenna arrays. In the proposed scheme, for each user, only the data streams where the estimated channel gains are larger than a certain threshold are chosen for beamforming, while the interference on these streams is also approximated by thresholding the channel gain estimates.
With the proposed bit allocation at the RRHs, channel estimation at the CU, and data transmission with single-carrier modulation, we compare the proposed system with the conventional CRAN with UPAs via simulations, and show that when the fronthaul is constrained, the proposed system can achieve significant sum-throughput gains at much lower signal processing complexity and training overhead compared to the benchmark.
The rest of this paper is organized as follows. In Section II we present the system model for the proposed mmWave CRAN with lens arrays, and carry out an analysis of the achievable rates under perfect and imperfect CSI, with our proposed bit allocation and channel estimation schemes. Section III gives a brief description of the benchmark CRAN system with UPAs and OFDM transmission. In Section IV, we compare our proposed system with the benchmark system via simulations. Finally, Section V concludes the paper.
Notation: In this paper, denotes equality by definition, and means “distributed as”. The cardinality of a finite set is denoted by , while denotes the elements in that are not in . Sets of real, complex, and integer matrices are denoted by , and respectively, while , , and denote the set of non-negative real numbers, non-negative integers, and positive integers, respectively. denotes the Kronecker delta function. The normalized sinc function is defined as for and if . The imaginary unit is denoted by with . Scalars are denoted by lower-case letters, e.g.
, while vectors and matrices are denoted by bold-face lower-case and upper-case letters, e.g.and , respectively. For , denotes the smallest integer greater than or equal to , and denotes the largest integer less than or equal to . For , denotes its magnitude and denotes its phase in radian. For a vector , denotes its Euclidean norm. A vector with all elements equal to is denoted by , where the dimension is implied from the context. For vectors and matrices, denotes transpose, and denotes conjugate transpose (Hermitian). For matrix , denotes the sum of its diagonal elements (trace). For with linearly independent columns, denotes the Moore-Penrose pseudo-inverse. denotes the Kronecker product, while
denotes the identity matrix of dimension. A diagonal matrix with elements on the main diagonal is denoted by , and a block diagonal matrix by . denotes a circularly symmetric complex Gaussian (CSCG) distribution centered at with covariance , and
denotes a uniform distribution over the interval.
Ii System Model
Before introducing the system model, we first summarize all the notations used in this paper in Table I for ease of reference.
|Set of RRHs|
|Set of sectors at each RRH|
|; respectively (resp.),||Set of antennas at sector of each RRH with lens antenna arrays (resp. UPAs)|
|(resp. )||Set of all antennas at each RRH with lens array (resp. UPA)|
|with (resp. with||Set of antennas selected at RRH with lens antenna array (resp. UPA)|
|Total number of selected antennas|
|Set of all streams (selected antennas)|
|Set of users|
|Set of paths from user to RRH|
|or||Rectangular dimensions of EM lens or UPA in - plane (normalized by wavelength)|
|Maximum gain path from user for stream|
|(resp. )||Tap delay (resp. estimate) of maximum gain path from user for stream|
|(resp. )||Time-domain channel coefficient (resp. estimate) for stream from user at time|
|(resp. )||Time-domain data (resp. pilot) symbol transmitted by user at time|
|(resp. )||Time-domain quantized data (resp. pilot) signal for stream and time|
|(resp. )||Time-domain delay compensated quantized data signal according to delay (resp. estimated delay ) of maximum gain path from user|
|(resp. )||Sum of channel coefficients corresponding to paths from user for stream , which have a delay difference (resp. ) with (resp. )|
|Thresholded estimates of|
|(resp. )||Frequency-domain channel coefficient (resp. estimate) for stream from user on SC|
|Frequency-domain data symbol transmitted by user on SC|
|(resp. )||Frequency-domain quantized data (resp. pilot) signal for stream , and SC|
We study the uplink transmission in a mmWave-based dense CRAN cluster (see Fig. 2) with multi-antenna RRHs, denoted by . The cluster is sectorized, where each RRH has sectors covering each, and each sector is served by a full-dimensional lens antenna array with antenna elements denoted by the set , where denotes the set of sectors. Further, we use without the superscript to denote the set of all antenna elements in all sectors of an RRH, i.e., , and . Each RRH is connected to the CU via an individual fronthaul link of finite capacity in bits per second (bps). There are single-antenna users in the CRAN cluster, denoted by . All the users and RRHs share the same bandwidth for communication. Depending on a user’s location, its signals are incident on at most one sector of an RRH, and we treat any inter-sector/inter-cluster leakage interference as additive Gaussian noise over the bandwidth of interest. We further assume that the total number of sectors in the CRAN is greater than or equal to the number of users, i.e., .
The users transmit over a mmWave, frequency-selective, block-fading channel of bandwidth Hz. The channel between user and RRH 222Since each user’s signals are received by only one sector of every RRH, we refer to the channel between the user and the RRH, instead of the particular sector of the RRH, for convenience. has paths denoted by . Then the discrete-time channel coefficient vector at time index between user and RRH can be expressed using a geometric channel model as
where and denote, respectively, the complex gain and delay in symbol periods corresponding to path , and is the array response for the elevation and azimuth angles of arrival of path denoted by and , respectively.
We consider that for each sector, each RRH is equipped with a rectangular EM lens in the plane, with dimensions normalized by the wavelength333The dimensions are assumed to be same at all RRHs for convenience. along the - and -axes, respectively. The EM lens is followed by a full-dimensional antenna array with elements placed on the focal surface of the lens, which is a hemisphere around the lens’ center (taken to be the origin in Fig. 1) with radius equal to the focal length of the lens (see Fig. 1). Let each antenna element be indexed by a pair of indexes , where denotes the index in the elevation direction along the focal surface of the lens and denotes the index in the azimuth direction along the focal surface. Now, for a ray drawn from the center of the lens to an antenna element , let denote the azimuth angle made by the ray, where are the maximum azimuth angles in the negative and positive -directions, respectively. Similarly, let denote the elevation angle made by the ray, where are the maximum elevation angles covered by the antenna array in the negative and positive -directions, respectively. Then, the antennas are placed such that the indexes run from the integers to . Thus, the elevation angles of the antenna elements are related to the indexes by . Next, for each index , the index runs from the integers to , so that the azimuth angles of the antenna elements are related to the indexes by . Then the amplitude response in (1) of the lens array element , to a uniform plane wave incident at elevation and azimuth angles , can be expressed as 
Since the RRHs have fronthaul links of finite capacities, they must quantize the received signals before forwarding to the CU. Moreover, the RRHs are typically low-cost nodes with limited processing capability. Thus, we consider that they perform uniform SQ independently on each antenna, with the SQ bit allocation adapted in each channel coherence interval. As shown in Fig. 3, the RRHs perform bit allocation based on the estimated received power levels at each antenna element, which can be obtained before converting the signals to the baseband, either using feedback from the automatic gain control (AGC) circuitry, or by means of analog power estimators , which can be implemented using band-pass filters and envelope detectors.
We consider a frame-based transmission by the users, with frame duration in symbol periods, where denotes the minimum coherence time among all the user-RRH channels. Each frame is further divided into the following three stages as shown in Fig. 4:
A power probing stage of duration , where the users transmit constant amplitude signals in order to enable the RRHs to perform bit allocation and antenna selection;
A channel training stage of duration , where the users transmit pilot symbols, and the CU performs channel estimation for the selected antennas using the quantized signals forwarded by the RRHs; and
A data transmission stage of duration , where the users transmit their data, which is quantized and forwarded by the RRHs for decoding at the CU.
The above stages are separated by guard intervals of symbols (see Fig. 4), where denotes the maximum delay spread of all the user-RRH channels. Note that . In the following, we describe each stage in detail.
Ii-a Uniform Scalar Quantization (SQ) and Bit Allocation at RRHs
During the power-probing stage, the users transmit constant amplitude signals for a duration , and each RRH obtains an estimate of the average received power on each antenna either from the AGC circuitry, or using analog power estimators. We consider that each RRH performs uniform SQ independently on the real and imaginary components of the complex baseband samples received at an antenna , using bits.444This can be performed using low-cost, low-resolution analog-to-digital converters (ADCs). If , the symbols on antenna are not forwarded to the CU by RRH (i.e., this antenna is not selected for subsequent channel training and data transmission). Following the design in , the resulting quantized samples can be expressed as
The quantization error is assumed to be uncorrelated with , and as the SQ is performed independently at each antenna and for each sample, we have for any or , . With Nyquist rate sampling, the transmission rate required to forward the quantized signals over all the antennas is bps, which must not exceed the fronthaul capacity .
Each RRH uses the estimate of the average received power computed from the received signals in the power-probing stage to perform the SQ in the subsequent channel training and data transmission stages. We consider the design of quantization bit allocation to minimize the total SQ noise power over all the antennas subject to the fronthaul capacity constraint at the RRH, as captured by the following optimization problem555We assume that the objective function is defined for as well.
Since the ’s are integers, the above problem is non-convex. However, if the variables are relaxed so that , we have the following relaxed problem
which is convex since the objective function is convex, while the constraint (5a) is linear. Thus, we have the following proposition.
The optimal solution to problem (6) is given by
where is such that .
Please refer to the appendix. ∎
Let denote the set of antennas with non-zero allocation according to (7). With the optimal solution to problem (6), we proceed to construct a feasible integer solution for the original problem (5) by rounding as follows666We construct a feasible integer solution such that if for some in (7), then as well.
where is an appropriate threshold. Notice that as is decreased, most of the ’s would be rounded above, making it more difficult to satisfy the constraint in (5) and vice versa. Hence, a suitable can be found by bisection on the interval , each time evaluating the constraint in (5) with in (8) and updating accordingly.
From (7), we observe that the optimal solution to problem (6) allocates more bits to the antenna with higher estimated power , which is desirable, since the antennas that receive stronger signals from the users are more likely to have useful information to be decoded. The algorithm is summarized in Table II, and involves two instances of bisection search. The number of iterations in the first instance to find is , and the number of iterations in the second instance to find a suitable is . The total number of iterations for the algorithm is thus . If all the ’s are normalized such that , and , the total number of iterations is , which is negligible for typical values of , like . Thus, based on the received power estimates in the power probing stage, each RRH selects the set of antennas denoted by with and forwards the received symbols on these antennas to the CU after SQ in the subsequent channel training and data transmission stages. We assume that the fronthaul capacity at each RRH is such that , so that the number of selected antennas is at least , i.e. . Then, , and since , it is always feasible to recover all users’ signals at the CU via linear processing with independent channels. Let denote the total number of selected antennas in the network. For the ease of exposition, we refer to the selected antennas as “streams” and use the index to denote each stream so that the stream corresponds to the antenna in sector of RRH .
Ii-B Path Delay Compensation and Achievable Sum-Rate with Perfect CSI
In this subsection, we describe the operations at the CU, assuming perfect CSI. If the RRHs are equipped with lens antenna arrays, we propose that the users transmit simultaneously with single-carrier modulation, so that the data symbols transmitted by user are given by , where is the transmit power and is the complex data symbol of user . Using creftypeplural 3 and 1, the quantized symbols received at the CU corresponding to stream can be expressed as
We assume that the CU decodes the users’ data symbols via linear processing, after path delay compensation  on the received signals for each stream as described below. Let denote the strongest path among all those arriving on stream (selected antenna) from user . Due to the response of the lens array, the delayed versions of a particular user’s signals with different angles of arrival are focused on different antenna elements at each RRH in general. Thus, to ensure that the symbols of each user that have undergone the strongest path gain on each stream are combined at the CU in a synchronized manner, each stream of quantized symbols , corresponding to antenna of RRH , is advanced by the delay corresponding to the path , to obtain the delay compensated signal for each user , as given by
In order to write the summation over the paths in (10) in terms of the delay differences with the delay of the maximum gain path , we define for each antenna and user pair , the new channel coefficient
which is equivalent to the channel coefficient (or sum of coefficients) corresponding to the path(s) from user to antenna , which has (have) a delay difference of with the maximum gain path of user to the same antenna. Then, (10) can be expressed as
where and , with and defined in (4) denote the AWGN and quantization noise samples shifted by symbol periods. Note that in (12) depends on user whose maximum gain path is used as reference. The second and third terms in (12) represent user ’s own delayed symbols, and the interfering symbols from other users, respectively. Collecting the signals from all the streams, (12) can be written in vector form as
where all the vectors are of dimension . We consider that the CU performs linear receive beamforming on with the beamforming vector to construct the estimate of user ’s symbol. Treating the inter-symbol and inter-user interference in (12) as Gaussian noise, the signal to interference-plus-noise ratio (SINR) for decoding is given by creftype 14 on the top of the next page.
denotes the covariance of the noise and interference terms in (13). In this case, the SINR in (14) becomes , and by assuming the worst-case CSCG distribution for the quantization noise, a lower bound on the achievable sum rate over all users is thus given by
in bps/Hz. In the next subsection, we consider channel estimation at the CU, and extend the achievable rate analysis to the case with imperfect CSI.
Ii-C Channel Estimation and Achievable Sum-Rate with Imperfect CSI
Since the CU does not know the CSI a priori, it needs to estimate the CSI using pilot signals sent by the users and then quantized and forwarded by the RRHs in the channel estimation stage (see Fig. 4). The users transmit known pilot symbols given by . Let denote the vector of time-domain channel taps from user to antenna . Then, the vector of quantized symbols received at the CU during the channel estimation stage can be expressed as
is a Toeplitz matrix constructed from consecutive shifts of the pilot symbols of the users, while with denotes the AWGN, and with denotes the quantization noise. Also, , and in (17). It is sufficient to estimate in (18
), to find both the channel coefficients and their corresponding tap delays. We do not assume any prior knowledge of the probability distribution function (pdf) of the elements ofat the CU, i.e. they are treated as unknown constants. Then, the least-squares (LS) estimate of is given by
where we define . Since each component of the vector is a linear combination of the independent zero-mean random variables in , the second term in , , can be modeled as a CSCG random vector, with mean and covariance
, due to the central limit theorem. Thus,in (19). Note that must be satisfied for the solution in (19) to exist. The MSE of the estimate is given by
which is minimized if , where is a constant [27, Example 4.3]. From the construction of , this translates to the condition , which is satisfied if each user’s training sequence is orthogonal to that of every other user’s training sequence, and has the “ideal” auto-correlation property . This can be ensured by using e.g., unit-amplitude Zadoff-Chu sequences [28, 29] for , with equal transmit power . In this case, so that