Optimal Bit Allocation-Based Hybrid Precoder-Combiner Design Techniques for mmWave MIMO-OFDM Systems

This work conceives techniques for the design of hybrid precoders/combiners for optimal bit allocation in frequency selective millimeter wave (mmWave) multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems, toward transmission rate maximization. Initially, the optimal fully digital ideal precoder/ combiner design is derived together with a closed-form expression for the optimal bit allocation in the above system. This is followed by the development of a framework for optimal transceiver design and bit allocation in a practical mmWave MIMO-OFDM implementation with a hybrid architecture. It is demonstrated that the pertinent problem can be formulated as a multiple measurement vector (MMV)-based sparse signal recovery problem for joint design of the RF and baseband components across all the subcarriers, and an explicit algorithm is derived to solve this using the simultaneous orthogonal matching pursuit (SOMP). To overcome the shortcomings of the SOMP-based greedy approach, an MMV sparse Bayesian learning (MSBL)-based state-of-the-art algorithm is subsequently developed, which is seen to lead to improved performance due to the superior sparse recovery properties of the Bayesian learning framework. Simulation results verify the efficacy of the proposed designs and also demonstrate that the performance of the hybrid transceiver is close to that of its fully-digital counterpart.


I. INTRODUCTION
Millimeter-wave (mmWave) wireless communication has attracted significant attention due to the availability of vast spectral bands that can in turn enable ultra-high data rates in 5G networks [1]- [3]. This has heralded a new epoch in the development of wireless communication technology, and cellular systems. However, mmWave bands suffer from much higher propagation losses [4]- [6] when compared to conventional sub-6 GHz systems in the lower frequency bands. Recent research demonstrates that large antenna arrays The associate editor coordinating the review of this manuscript and approving it for publication was Irfan Ahmed . at both the transmitter and receiver ends i.e., multipleinput-multiple-output (MIMO) techniques, rendered possible due to the very small wavelengths, can provide significant beamforming gain [7]- [9] to successfully overcome the high propagation losses. Also, since mmWave systems, most likely, operate on wideband channels with a large bandwidth, orthogonal frequency division multiplexing (OFDM) coupled with MIMO has been acknowledged as a promising technology to overcome the inter-symbol interference (ISI) arising due to the frequency selective nature of the channel [10]- [12]. Furthermore, in the mmWave MIMO regime, the traditional transceiver design, based on a fully-digital implementation of the precoder and combiner, is highly VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ infeasible as it necessitates a dedicated radio frequency (RF) chain, DAC/ ADC for each antenna element [13], [14]. This arises due to the increased cost, area, and complexity of RF components operating in the mmWave regime, coupled with the high power consumption of the ADCs that are required to operate at a very high sampling rate. This has led to massive research in alternative signal processing paradigms for efficient implementation of mmWave MIMO communication. Recently, hybrid architectures that spread the signal processing operations over the analog and digital domains, have gained much popularity as they warrant a substantially lower number of RF chains. Such a hybrid beamformer can achieve a very high spectral efficiency, while simultaneously limiting the power consumption in comparison to the traditional MIMO architecture [15]- [19]. The RF precoder and combiner in such a system are implemented via constant magnitude analog phase shifters to compensate for the large scale path loss and shadowing effects at mmWave bands, while the digital precoder and combiner can provide the necessary flexibility to perform spatial multiplexing using only a very small number of RF chains. A brief review of the various works in the existing literature on the design of hybrid signal processing techniques for mmWave MIMO systems is presented next.

A. RELATED WORKS
A major fraction of the prior works have been devoted to investigate hybrid precoding and combing algorithms in narrowband mmWave channels [14], [20], [21]. Extensions to wideband hybrid precoder and combiner designs have been explored in [5], [12], [22]. Authors in [5] study the feasibility of attaining gigabit-per-second data rates for distances up to 1 km using millimeter-wave mobile broadband (MMB) links in an urban environment. In [22], Kim et al. present and thoroughly study a multi-beam diversity scheme for single-stream transmission in MIMO-OFDM systems. In [23], the authors proposed an iterative hybrid beamformer algorithm to maximize the average spectral efficiency in mmWave MIMO-OFDM systems with the aid of classical block coordination descent technique. The authors in [24] explored the hybrid beamforming optimization problem for mmWave MIMO-OFDM systems considering beam squint effect. In a radically different approach [14], [21], [25], [26] exploit the sparse scattering characteristic of mmWave MIMO channels for hybrid transceiver design, and the associated analog beamformers are chosen from predetermined dictionaries, such as the array response vectors over an angular grid or the well-known discrete Fourier transform (DFT) matrix. An innovative and practically appealing scheme for hybrid beamforming with single-stream transmission in MIMO-OFDM systems was discussed in [22], which employs an exhaustive search over the RF and baseband codebooks. However, no specific criterion was suggested for the design of the pertinent codebooks. As an improvement over the earlier approach, the authors in [12] derived a scheme for the optimal design of the baseband precoders by employing a predefined RF codebook. Coming next to the signal processing strategies that can be employed in such systems, recent research has also shown that sparse signal recovery algorithms are eminently suited for mmWave MIMO systems due to the sparse characteristic of multi-path signal transmission. In order to leverage this important characteristic feature of such systems, reference [14] conceives an orthogonal matching pursuit (OMP)based approach, which exhibits an excellent improvement in performance over traditional approaches. Authors in [27] present a novel iterative technique based on the OMP algorithm for designing the hybrid precoder. For ease of practical implementation, [28] proffers a transceiver design that has a much lower complexity. The authors in reference [29], [30] propose a codebook-based joint hybrid transceiver formulation considering the transmission of multiple streams in mmWave MIMO systems. The work in [14], develops an equivalent multiple measurement vector (MMV) problem for sparse signal reconstruction for optimal transmit precoder and receive combiner design, which was subsequently solved employing the greedy simultaneous orthogonal matching pursuit (SOMP) technique. Further, the SOMP technique is a greedy algorithm and its performance is sensitive to the selection of the dictionary matrix and stopping condition. These aspects often lead to convergence errors and poor performance of the precoder combiner thus determined. The sparse Bayesian learning (SBL) technique, described in the seminal work in [31], offers an excellent alternative to avoid this obstacle. The SBL algorithm determines the sparsest representation of the digital precoding/combining transceiver, characterized by the array response vectors, owing to the fact that the global minimum of the SBL cost function is attained at the sparsest solution. Moreover, the SBL algorithm converges to a fixed point of the log-likelihood cost function due to the advantage offered by the EM algorithm, thus ensuring robust performance with limited complexity, independent of the choice of initialization. Its performance guarantees coupled with ease of implementation make it perfectly applicable for the design of the precoder and combiner in mmWave MIMO-OFDM systems.
Various optimization criteria have been successfully explored for optimal transceiver design in conventional MIMO systems. In [32], the authors developed an optimal transceiver design for MIMO single carrier and multi carrier block transmission systems that minimizes the weighted combination of symbol estimation errors under the transmit power constraint. The optimal transceiver for OFDM systems described in [33] has been formulated to minimize the uncoded bit error rate (BER) with a unitary constraint imposed upon the precoder. The work in [34] develops an optimal transceiver filterbank structure for redundant block transmission system for two different formulations, viz., maximizing the output signal to noise ratio (SNR) under the zero-forcing (ZF) constraint and minimizing the mean squared error (MSE) subject to a total transmit power constraint. The authors in [35] developed an optimal MIMO transceiver that minimizes the uncoded BER subject to both individual and total power constraints using principles of convex optimization and majorization theory. The work detailed in reference [36] designs the MIMO transceiver by minimizing the MSE and BER under the total as well as peak transmit power constraints. A minimum BER design with a channel independent transmitter is derived in [37]. A ZF solution that minimizes the total transmit power for a given BER is developed in [38]. In [39], [40], Palomar et al. designed the optimal MIMO transceiver under an exhaustive set of quality of service (QoS) constraints such as MSE, signal to interference noise ratio (SINR), BER, further explored the problem of symbol constellation optimization. Vaidyanathan et al. described a novel approach in [41] for joint transceiver optimization and bit allocation toward transmission rate maximization. To the best of our knowledge, as can also be seen from the table above, none of the existing works have considered the problem of hybrid precoder/ combiner design with optimal bit allocation for mmWave MIMO-OFDM systems toward bit-rate maximization, which therefore forms the focus of this work. For ease of reading, a quick comparative visual summary of the various contributions of the salient works reviewed above is given in Table 1. The contributions of this work are listed below.

B. CONTRIBUTIONS OF THIS WORK
A brief itemized description of the novel contributions of this work follows.
• This work considers hybrid precoder and combiner design for a single user wideband frequency selective spatially sparse mmWave MIMO-OFDM system. We begin with developing a procedure for joint design of the digital precoder and combiner by considering the zero forcing (ZF) design principle, which significantly lowers the signal processing complexity at the receiver. In addition, the proposed design maximizes the transmission rate by attaining the optimal bit allocation. An important aspect of the proposed design is that the optimal fully digital transceiver and the bit allocation are obtained in a closed form.
• This framework is subsequently extended to mmWave MIMO-OFDM implementation with a hybrid precoding/ combining transceiver, in which the analog precoder and combiner are constrained to have constant magnitude elements. Owing to the sparse multi-path scattering in a typical mmWave MIMO system, the resulting channel is spatially sparse. This useful property is exploited to develop a simplistic procedure for hybrid transceiver design by employing the multicarrier Simultaneous Orthogonal Matching Pursuit (SOMP) technique. The beam squint effect has also been considered in the SER performance of the proposed MSBL scheme to demonstrate its performance in a practical wideband channel.
• Furthermore, the SOMP algorithm, although efficient and yields acceptable performance, is often plagued by issues related to convergence due to its sensitivity to the choice of the dictionary matrix. To overcome this, a multicarrier transceiver design is presented for mmWave MIMO-OFDM systems, based on the stateof-the-art Multiple Sparse Bayesian Learning (M-SBL) algorithm for simultaneous sparse signal recovery. This is seen to achieve a better performance in comparison to the SOMP scheme in terms of improved symbol error rate (SER) and reduced number of RF chains.
• Subsequently, an integer bit allocation algorithm that assigns bits in an optimal fashion across input symbol streams of each subcarrier is developed to maximize the bit rate as well as to enhance the power efficacy.
• Exhaustive simulation results demonstrate the improved transmission rate performance of the proposed designs and also that the performance of the proposed hybrid precoder combiner is close to that of its ideal fully digital counterpart. This is significant since it evidences the fact that one can achieve a performance close to the ideal system with a only fraction of the number of RF chains, thus significantly lowering the power consumption and cost of hardware implementation.
The organization of the paper and notation are described next.

C. ORGANIZATION
The rest of the paper is organized as follows: Section II introduces the mmWave MIMO-OFDM system and channel model followed by formulation of the optimization problem for bit-rate maximization. Section III determines the ideal VOLUME 9, 2021 fully-digital precoder/combiner designs with the QoS constraint pertaining to the symbol error rate (SER) and total transmit power. Section IV develops the design of a practical hybrid transceiver for a mmWave MIMO-OFDM system followed by SOMP-based greedy approach. Section V then describes an equivalent MSBL algorithm for improved hybrid precoder/combiner design in the above system followed by a discussion on optimal bit allocation. The performance of the proposed scheme is validated via the simulation results presented in section VI followed by our overall assessment and concluding remarks in section VII. For reading convenience, the proofs of some of the propositions have been moved to the appendices at the end of the paper. Notation: The following notation is employed across this paper. The operators (·) T , (·) * and (·) H , denote the transpose, conjugate and Hermitian of a matrix, respectively. C M ×N is the set of M ×N matrices comprising of complex entries. E {·} represents the statistical expectation operator. I N denotes an N ×N identity matrix. |·|, · , · F and · 0 denote the scalar magnitude, vector norm, matrix Frobenius norm and l 0 -norm respectively. M(i, :) and M(:, j) denote the ith row and jth column of a matrix M respectively. The standard Gaussian Q function, defined as Q(x) = 1 √ 2π ∞ x e −y 2 /2 dy is denoted by Q(x).

II. mmWave MIMO-OFDM SYSTEM MODEL AND PROBLEM FORMULATION
Consider an OFDM based mmWave hybrid MIMO system, as described in [13], [14], [45], with N T transmit antennas, N R receive antennas and N RF RF chains, where N RF ≤ min(N R , N T ) at the transmitter as well as at the receiver. Consider N s ≤ N RF parallel input symbol streams for K subcarriers in the mmWave MIMO-OFDM system. A block diagram of the mmWave hybrid MIMO-OFDM systems is shown in Fig. 1. Here, F BB [k] ∈ C N RF ×N s represents the frequency-selective baseband precoder for the kth subcarrier and F RF ∈ C N T ×N RF denotes the frequency-flat analog domain RF precoder. Thus, the frequency-selective hybrid precoder F[k] ∈ C N T ×N s corresponding to the kth subcarrier is given as F[k] = F RF F BB [k]. Similarly, at the receiver, W BB [k] ∈ C N RF ×N s represents the frequency-selective baseband combiner corresponding to the kth subcarrier, whereas W RF ∈ C N R ×N RF denotes the analog domain RF combiner, which is frequency-flat. Hence, the hybrid combiner W[k] ∈ C N R ×N s at the receiver is given as Employing the hybrid precoder and combiner, as described above, the received signal y[k] ∈ C N s ×1 at the output of the baseband combiner is expressed as where s[k] ∈ C N s ×1 denotes the baseband quadrature amplitude modulated (QAM) symbol vector and n[k] ∼ CN 0, σ 2 n I N R represents additive white Gaussian noise (AWGN). Furthermore, the symbols of the vector s[k] are assumed to be uncorrelated with zero mean, and thus, the covariance matrix . . , N s − 1 on its principal diagonal. The quantity H[k] ∈ C N R ×N T represents the mmWave MIMO-OFDM channel matrix corresponding to the kth subcarrier. It is important to note that the RF precoder F RF and combiner W RF are implemented using analog phase-shifters, whose phase elements are digitally controlled, while amplitudes are constant. Thus, their elements are restricted as [ , ∀ i, j, without loss of generality. The next subsection describes the wideband fequency-selective mmWave MIMO-OFDM channel model.

A. mmWave MIMO-OFDM CHANNEL MODEL
The dth delay tap of the wideband mmWave MIMO channel, denoted as H d ∈ C N R ×N T , d = 0, 1, . . . , N c − 1, can be expressed as [42], [45] where α l , θ R l and θ T l represent the complex path gain, angle of arrival (AoA) and angle of departure (AoD), respectively, associated with the lth spatial multipath component of the mmWave MIMO channel, and L denotes the number of multipath components. The quantity, p rc (·) represents the combination of pulse-shaping and other low pass filters, T s represents the sampling period and τ l denotes the delay corresponding to the lth multipath component. The vectors a R (θ R l ) ∈ C N R ×1 and a T (θ T l ) ∈ C N T ×1 denote the receive and transmit uniform linear array (ULA) response vectors respectively [9], [13], corresponding to the lth multipath, and are expressed as where λ represents the wavelength of the transmitted signal, d R and d T stand for the inter-antenna spacings at the receiver and at the transmitter ends, respectively. ULAs have been chosen for the proposed wideband mmWave MIMO system owing to their higher RF processing speed, lower complexity feeding circuits and substantially lower inter-element coupling losses [46], [47]. The mmWave MIMO channel tap H d described in (2) can be succinctly expressed as where the matrices A R (θ R ) ∈ C N R ×L and A T (θ T ) ∈ C N T ×L denote the transmit and receive array response matrices described as The quantity D d denotes a diagonal matrix of size L × L with the diagonal elements being the complex path gains Finally, the frequency-domain representation of the mmWave MIMO-OFDM channel matrix H[k] corresponding to kth subcarrier is expressed as where The next section describes the design of the optimal fully-digital precoder and combiner for mmWave hybrid MIMO-OFDM systems.

B. PROBLEM FORMULATION
The key aim of this subsection is to jointly develop a design of the hybrid ZF transceiver and optimally allocate input bits to maximize the number of bits transmitted per block with a given symbol error rate (SER) constraint for the mmWave MIMO-OFDM system. The transmitted signal x[k] ∈ C N T ×1 at the kth subcarrier is given as , whose transmit power is expressed as The combined noise vector n[k] ∈ C N s ×1 at the kth subcarrier obeys Thus, the power of each element of the combined noise n[k], denoted by σ 2 n i,k , is given as Let SER denote the target QoS and b i,k denote the number of bits carried by the ith symbol stream at the kth subcarrier. For the QAM modulated symbol vector s[k], one can express b i,k as where the quantity obeys [48]. Furthermore, for sufficiently high bit rate, i.e, 2 b i,k 1, the expression for b i,k can be approximated as Using the above result, the objective function for bit-rate maximization in mmWave MIMO-OFDM transceiver design can be formulated as However, one can readily observe that this cost function is non-convex in nature, which renders it intractable. Therefore, this work follows a two-step procedure based on the standard integer relaxation approach [49]- [52] to simplify the design. The first step solves the optimization problem without considering the floor operation, followed by assigning the integer bits appropriately to the input symbol streams loaded over the sub-carriers, as described later in Algorithm 3. Therefore, the bit-rate maximization problem for mmWave MIMO-OFDM transceiver design can be formulated as where P 0 denotes the total transmit power. The first constraint limits the total transmit power, the second one represents the ZF condition for transceiver design, whereas the last constraint represents the constant magnitude restriction on the elements of the RF precoder and combiner. Furthermore, the ZF constraint simplifies the signal processing complexity significantly at the receiver, since the received signal The optimal fully-digital precoder F[k] and combiner W[k] designs are described in the next section, followed by the hybrid precoder and combiner designs in the subsequent section.

III. OPTIMAL FULLY-DIGITAL PRECODER/WWWWW/COMBINER DESIGN WITH A QUALITY OF SERIVCE (QoS) CONSTRAINT
The optimization problem above is solved in two steps. The first step determines the optimal power allocation, i.e., σ 2 s i,k for each input symbol stream i and subcarrier k that maximizes the bit rate for a given precoder F[k] and combiner W[k]. This is subsequently employed in the next step for the design of the optimal fully-digital precoder and combiner.
In order to derive the optimal power allocation, the Lagrangian for the optimization problem in Eq. (14) can be formulated as where α and β i,k are the Lagrange multipliers. Upon employing the Karush-Kuhn-Tucker (KKT) conditions, as described in Appendix-A, the optimal σ 2 s i,k is expressed as Now, substituting Eq. (73) in the optimization objective, one can reformulate the bit rate expression as The design of optimal fully-digital ZF transceiver that maximizes the bit rate above is described next. Let the singular value decomposition (SVD) of the mmWave MIMO-OFDM channel matrix H[k] be given as where where Proof: Proof is given in Appendix-C. Note that the hybrid precoder design requires channel state information at the transmitter (CSIT) in order to derive the optimal fully-digital precoder F [k] [1], [14], [53]. However, in practical scenarios, the CSI is estimated at the receiver and fed back to the transmitter using a limited feedback link. Designing the hybrid precoder using limited/ imperfect CSIT is a challenging problem, which may be considered as a possible future work.
where A[k] is given in Eq. (19).
Proof: Proof is given in Appendix-D Employing Lemma-2, the noise variance σ 2 n i,k is given as Next, substituting the expression of F[k] from Lemma-1 and the quantity σ 2 n i,k from the above result, the bit rate b in Eq. (17) can be rewritten as From the above, one can readily observe that maximizing the bit rate b with respect to A[k] is equivalent to minimizing 54114 VOLUME 9, 2021 Employing the Hadamard inequality [54], one can now lower bound the quantity in (23) as .
Since, the above result is independent of the diagonal matrix D[k], one can choose D[k] = I N s , without loss of generality. Therefore, the bit rate in Eq. (22) can be modified as Hence, in order to design the optimal fully-digital precoder and combiner, one only needs to focus on finding the matrix where λ i (Y) represents the ith smallest eigenvalue of the matrix Y.
Proof: Proof is given in [54]. Since the diagonal elements of the matrix [k] are arranged in decreasing order, employing the above theorem, one can write Therefore,  (23), and in turn, maximizes (17), as Employing the above key result, the optimal fully-digital precoder F[k] and combiner W[k] for the kth subcarrier can be derived using Lemma-1 and 2 as follows: The optimization problem in Eq. (14) can also be solved without taking the approximation in Eq. (12) to derive the bit allocation given by b new i,k , as shown in Appendix-B, and also reproduced below: However, the expression for b new i,k renders the optimization problem for hybrid transceiver design intractable. Thus, the high bit-rate approximation of Eq. (12) is essentially employed to obtain closed form expressions of the fully-digital precoder and combiner, whereas the optimal integer bit allocation procedure developed in Algorithm-3 employs the exact bit-rate expression, thus emphasizing the practical suitability of the design procedure. Moreover, it can be readily observed that at high SNR, b new i,k −→ b i,k , which is identical to the solution obtained with the high-bit rate assumption.

IV. HYBRID PRECODER/WWWWW/COMBINER DESIGN FOR mmWave MIMO OFDM SYSTEMS
This section develops hybrid precoder and combiner designs for mmWave MIMO-OFDM systems. We begin by describing the design of the hybrid precoder. The problem for design of the baseband precoder F BB [k] and the RF precoder F RF can be formulated as Concatenating the optimal digital precoders F[k] obtained from (32) and the baseband precoders F BB [k] across all the K subcarriers as and noting the fact that the RF precoder F RF is frequency-flat, one can recast Eq. (32) as One can now employ the following key observations to simplify the hybrid precoder design procedure. As derived in (8) (38) where R(·) and C(·) represent the row and column spaces, respectively, of a matrix. Furthermore, from Eq. (32), it follows that Finally, employing the SVD of the mmWave MIMO-OFDM channel matrix H[k] described in (18), together with (35), (38) and (39), one can conclude that Therefore, in order to simplify the hybrid precoder design, the columns of the RF precoder F RF can be suitably selected from the columns of the transmit array response matrix A T (θ T ). Interestingly, this choice also satisfies the non-convex constraint in (37). Thus, the mmWave hybrid precoder design problem can be reformulated as follows. Let F BB ∈ C L×KN s denote an intermediate baseband precoding matrix, whose N RF non-zero rows constitute the desired baseband precoder F BB . Then, we can recast (37) as arg min where the constraint (41) arises since only N RF rows are allowed to be non-zero in the matrix F BB . From the solution of the above optimization problem, which will be described in the next subsection and in Section-V, one can obtain the RF precoder F RF by extracting the columns of A T (θ T ), whose indices correspond to those of the non-zero rows of F BB .
In practical scenarios, where the true transmit array response matrix A T (θ T ) is unknown in (41), one can employ a transmit array response dictionary matrix constructed from a G Tquantized angular-grid as described next. Let the set T = θ 1,T , θ 2,T , . . . , θ G T ,T denote the quantized angular-grid spanning the AoD space. The transmit array response dictionary matrix A T ,D ( T ) ∈ C N T ×G T is constructed as The equivalent hybrid precoder design problem for the mmWave MIMO-OFDM systems is given as arg min where F BB,D ∈ C G T ×KN s denotes the intermediate baseband precoder matrix corresponding to the transmit dictionary A T ,D (θ T ). It is important to note that the matrix F BB,D , owing to the constraint in (43), can only have N RF non-zero rows, which implies that it is block-sparse in nature. Also, the resulting optimization problem is popularly known as an MMV-based sparse signal recovery problem, and the techniques to solve this are described in the later part of this section.
Along similar lines, one can formulate the hybrid combiner design problem for a mmWave MIMO-OFDM system as arg min where W BB,D ∈ C N RF ×KN s denotes the intermediate baseband combiner, whereas the matrix W ∈ C N R ×KN s is the concatenated optimal digital combiner defined as The quantity A R,D ( R ) ∈ C N R ×G R represents the receive array response dictionary matrix constructed as where the set R = θ 1,R , θ 2,R , . . . , θ G R ,R denotes the quantized angular-grid spanning the AoA space. Note that the design problem in (44)

Algorithm 1 Simultaneous Orthogonal Matching Pursuit (SOMP) Algorithm to Design Hybrid Precoder in mmWave MIMO-OFDM Systems
Required: Concatenated optimal fully-Digital precoder F, transmit array response dictionary matrix A T ,D ( T ), number of RF chains N RF .

V. MSBL FOR HYBRID PRECODER/WWWWW/ COMBINER DESIGN IN mmWave MIMO-OFDM SYSTEMS
The proposed MSBL technique is a Bayesian method of sparse signal recovery, which assigns the parameterized Gaussian prior below to the ith row of the intermediate baseband precoder F BB,D [31] p F BB,D (i, : From the prior assignment as seen above, it can be noted that the hyperparameter ζ i , 1 ≤ i ≤ G T , parameterizes the covariance matrix of the multivariate prior associated with the ith row of the matrix F BB,D . Let Z = diag ζ 1 , ζ 2 , . . . , ζ G T ∈ R G T ×G T represent the hyperparameter matrix. The prior assignment for the intermediate baseband precoder F BB,D is given as The MMSE estimate F BB,D ∈ C G T ×KN s of the intermediate baseband precoder F BB,D can be determined as [55] where the associated error covariance matrix ∈ C G T ×G T is obtained as The quantity σ 2 e in the above expressions denotes the variance of each element of the approximation error F − A T ,D ( T ) F BB,D . It can be readily observed that the MMSE estimate F BB,D in (49) depends on the hyperparameter matrix Z, through the associated error covariance matrix . In addition, it can be observed from (47) that as the hyperparameter ζ i → 0, the associated ith row of the intermediate baseband precoder F BB,D (i, :) → 0 KN s ×1 [31]. Therefore, estimating the block-sparse matrix F BB,D translates to estimating the corresponding hyperparameter matrix Z. For superior MMV-based sparse signal recovery, the proposed MSBL approach obtains the estimate Z, which maximizes the log-Bayesian evidence log [p(F; Z)], evaluated as where the quantity F = σ 2 e I N T + A T ,D ( T )ZA H T ,D ( T ) ∈ C N T ×N T and the constant κ 1 = −KN s log(π). As described in [31], the log-Bayesian evidence maximization above with respect to the hyperparameter matrix Z has several maximas, which results in a non-concave problem. Thus, direct maximization of the log-Bayesian evidence for the estimation of Z becomes infeasible. The well-known expectationmaximization (EM) [55], in such a scenario, is ideally suited for maximizing the cost function in each iteration, which guarantees convergence to a local optima. Therefore, the proposed MSBL-based framework employs the EM technique for estimation of the hyperparameters, and in turn, the blocksparse intermediate baseband precoder F BB,D . Furthermore, it has a geometric convergence rate as discussed in [56]. The key steps of the EM procedure are described next.
Let the complete information set be constructed as F, F BB,D . Let Z (m−1) denote the estimate of the hyperparameter matrix Z in the (m−1)th iteration. The EM procedure to update the estimate Z (m) in the mth iteration is described below. It involves two steps, the expectation (E-step) and the maximization (M-step). In the E-step, the log-likelihood function L Z| Z (m−1) of the complete information set is evaluated as Subsequently, the maximization-step (M-step) maximizes L Z| Z (m−1) with respect to the hyperparameter matrix Z. It can be readily observed that the first term E log p F| F BB,D in (52) can be simplified as which is independent of the hyperparameter matrix Z. Ignoring this term, the hyperparameter estimation in the M-step is equivalently expressed as Substituting log p F BB,D ; Z from (48), the equivalent optimization problem for estimation of Z is given as It can be readily observed from (55) that the estimation of the hyperparameter matrix Z is decoupled with respect to each hyperparameter ζ i . Differentiating the objective function with respect to ζ i and setting the resulting expression equal to zero yields the update ζ where the a posteriori mean F (m) BB,D ∈ C G T ×KN s and the associated covariance matrix (m) ∈ C G T ×G T are obtained by setting Z = Z (m−1) in (49) and (50) Substituting (58) into (56), the hyperparameter update is given as The EM procedure described above are repeated for a max- ≤ , whichever is achieved earlier, where the stopping parameters and M max are suitably chosen. As demonstrated in [31], the global minima of the MSBL-cost function guarantees the sparsest representation of F using the columns of the transmit array response dictionary matrix A T ,D ( T ).
Similarly, the RF precoder F MSBL RF can be extracted from A T ,D ( T ) by columns indexed by the set K as A step-by-step procedure describing the MSBL-based hybrid precoder design is presented in Algorithm-2. A similar approach can also be leveraged to design the baseband and RF combiner W BB [k], 0 ≤ k ≤ K − 1, and W RF , respectively, formulated in (44) for mmWave MIMO-OFDM systems. The hybrid precoder obtained from the MSBL-based design can now be employed for optimal bit allocation, as described in the next subsection.

Algorithm 2 MSBL Algorithm to Design Hybrid Precoder in mmWave MIMO-OFDM Systems
Input: Concatenated optimal fully-Digital precoder F, transmit array response dictionary matrix A T ,D ( T ), number of RF chains N RF , variance of the approximation error σ 2 e , stopping parameters and M max Initialization: ζ bits allocated per block b, as described in (14), is maximized. To this end, employing the MSBL-based hybrid precoders F MSBL RF and F MSBL BB [k], together with (11) and (73), the number of bits b i,k can be obtained as . It is noteworthy that the quantity b i,k might not always turn out to be an integer. A simple, yet inefficient, solution to deal with this problem is to use integer number of bits given by b i,k , where · denotes the floor value. However, when the quantity b i,k = 0. In such scenarios, the power σ 2 s i,k assigned to the ith input stream of the kth subcarrier is left unused. In order to tackle this power wastage during the bit allocation, one can implement the recursive bit allocation procedure described in Algorithm-3, which assigns the power only to those streams that correspond to the non-zero values of b i,k .

B. COMPLEXITY ANALYSIS
This section presents a brief analysis of the computational complexities for the Algorithms 1, 2 and 3. As described in Table 1, the computational complexity of the proposed SOMP algorithm turns out to be O (KG T N T N s ), which arises from the computation of the matrix at each iteration. In Table 2, the computational cost of the proposed MSBL scheme is derived. It can be seen that the matrix inversion involved in calculating the matrix (m) in the E-step results in the complexity of order O G 3 T . Finally, the complexity of the integer bit allocation in Algorithm 3 turns out to be O N 2 s N T .

VI. SIMULATION RESULTS
This section presents the simulation results to demonstrate the performance of the proposed multicarrier optimal bit allocation scheme for frequency selective mmWave MIMO-OFDM systems. The symbol error rate (SER) performance of the proposed multicarrier ZF transceiver is also illustrated for the same system. The transmission rate of the proposed multicarrier optimal bit allocation algorithm based on the fully-digital optimal precoder/ combiner is compared with that of the hybrid precoder/ combiner designs obtained using the multicarrier SOMP and MSBL approaches outlined in sections Algorithm 1 and V. The SER performance is also presented for the fully-digital, MSBL and SOMP based precoders/ combiners. Finally, the variation in the performance of multicarrier MSBL and SOMP algorithms for the hybrid precoder/ combiner design with the number of RF chains N RF has also been demonstrated through simulation results. The following configurations are considered to comprehensively illustrate the performance for various systems. In System-I,   is assumed to be zero mean circularly symmetric complex Gaussian with covariance matrix E{nn H } = I N R , where σ 2 n = 1. In the multicarrier MSBL based precoder and combiner design approach, the hyperparameter vector is initialized as with the maximum number of EM iterations set to M max = 150 and the stopping parameter as = 10 −9 . Fig. 2(a) shows the total number of bits allocated per block or per OFDM symbol for both System-I and System-II in the SNR range 0 − 10 dB, averaged over 10 4 random realizations of the frequency selective mmWave MIMO channel. The SER for each input stream over each subcarrier is fixed as = 10 −5 . The simulation results show that the hybrid precoder and combiner designed using the MSBL approach is closer to that of the optimal fully-digital precoder and combiner compared to the SOMP design for both the systems. Here it is important to realize that although the SOMP leads to a higher bit allocation than the ideal transceiver, this leads to the poor SER as seen next. The SER performance corresponding to the two systems is depicted in Fig. 2(b). The SNR for evaluating the bit allocation in both the systems is set as 12dB. It can be readily observed from the results that the SER performance of the MSBL-based hybrid transceiver design is significantly lower than that of its SOMP counterpart and closer to that of the ideal fully digital design. Therefore, the simulation results in Fig. 2 comprehensively demonstrate that the MSBL approach for hybrid precoder and combiner design better approximates the optimal fully-digital precoder and combiner in each system in comparison to the SOMP algorithm. This arises due to the fact that the MSBL has improved sparse signal recovery properties in comparison to the SOMP. Furthermore, the performance of the latter scheme is highly sensitive to the choice of the dictionary matrix and stopping criterion.
Results are now described to characterize the performance of the hybrid precoder and combiner design schemes with varying number of RF chains N RF . Fig. 3(a) shows the number of bits per block for System-I, for various values of the number of RF chains N RF . The SNR is once again set as 12dB with the SER = 10 −5 and the results are averaged over 10 4 random instances of the frequency selective mmWave MIMO channel. The results of a similar study are shown in Fig. 3(b) for System-II. The results demonstrate that the bits per block for the precoders and combiners designed using the MSBL approach is closest to that of the ideal design when compared to the SOMP algorithm for various values of the number of RF chains. Furthermore, the performance of the SOMP algorithm approaches the optimal number of bits allocated only as the number of RF chains N RF increases. This can be attributed to the increased number of array steering vectors chosen from the corresponding transmit and receive array response dictionary matrices, which in turn leads to a better approximation of the ideal transceiver. On similar lines, Fig. 4(a) and Fig. 4(b) compare the resulting SER performance of hybrid precoders and combiners designed employing the MSBL and SOMP approaches for various values of the number of RF chains N RF . The results once again demonstrate the improved SER performance of the former design, thus reinforcing the trend seen in previous figures. Fig. 5(a) plots the SER performance of the proposed algorithms upon incorporating the beam squint effect, as described in [23], [24], [57], in the wideband MIMO-OFDM channel. The beam squint effect is referred to the frequency-selectivity of the array response vectors, which results in the diffusion of AoAs/ AoDs broadening the beamwidth of the desired signal in the spatial domain. To incorporate this effect, the array response vectors of Eq. (3) and Eq. (4) for the frequency f are replaced by following expressions [57] a R (θ R l , f ) where f c is the carrier frequency. Employing this, the mmWave MIMO-OFDM CFR of the kth subcarrier can be expressed as where and f k f c + k − K +1 2 B K , ∀k. It can be readily observed that the beam squint effect translates into sharing non-identical supports of the beamspace channel across the subcarriers. Fig. 5(a) plots the SER of the proposed MSBL based hybrid precoder/combiner design for two different bandwidths and number of RF chains. It can be observed that increasing the bandwidth leads to slightly poor SER owing to the dominance of the beam squint effect. However, this performance loss can be compensated via increasing the number of RF chains N RF , as can be verified from the figure. Furthermore, we have also plotted the oracle LS scheme as a performance benchmark, which assumes the support of the wideband beamspace channel of the mmWave MIMO-OFDM systems to be perfectly known across all the subcarriers. The SER performance of the proposed MSBL technique with N RF = 8 is seen to closely approach that of the Oracle LS scheme without considering the knowledge of the support of the beam space channel. This demonstrates the efficacy of the proposed MSBL scheme. Fig. 5   error is seen to reduce upon increasing the number of RF chains N RF .

VII. CONCLUSION
The work derived a framework for design of the optimal ideal fully-digital and hybrid precoder/combiner together with optimal bit allocation in a frequency selective mmWave MIMO-OFDM system toward transmission rate maximization with a bit error rate-based QoS constraint. This was followed by the development of more practical transceiver designs for a hybrid RF-baseband architecture that also exploits the spatially sparse scattering characteristics of the wideband mmWave MIMO channel. Toward this end, an equivalent MMV-based simultaneous sparse recovery problem was formulated to compute the hybrid precoder/combiner. Two explicit procedures were subsequently described for transceiver design, based on the simplistic SOMP approach as well as the more sophisticated MSBL technique for multiple sparse signal estimation. The proposed MSBL algorithm for the hybrid transceiver design is seen to outperform the conventional SOMP technique both in terms of SER and the required number of RF chains. The integer bit allocation algorithm optimally allocates bits across different i/p symbol streams and subcarriers to maximize the total number of bits allocated per block. Simulation results comprehensively demonstrated the performance of the proposd schemes for a mmWave MIMO-OFDM system and also negligible performance loss of the hybrid precoder combiner architecture. The latter observation can have significant implications for practical implementation of mmWave MIMO-OFDM systems due to the lower cost and complexity associated with hybrid transceivers. Future works may extend this framework for hybrid precoder design in a wideband mmWave MIMO-OFDM system by considering the beam-squint effect as well as imperfect CSIT in both single-user and multi-user scenarios.

APPENDIX A SOLUTION TO THE RELAXED OPTIMIZATION PROBLEM IN (14)
The KKT Conditions for the optimization problem in (15)  In the above, α * , β * s i,k , σ 2 * s i,k denote the optimal values of the Lagrangian parameters and input symbol power. From iv, we obtain Using iii it follows that Upon formulating the Lagrangian for the optimization problem above, followed by the KKT framework, the optimal σ 2,new s i,k can be derived as Substituting the above value of σ 2new into the exact bit-rate expression of Eq. (11)  Hence, the noise variance for the ith input stream is given as