SBL Kalman Filter Based Hybrid Precoder/ Combiner for mmWave MIMO Systems: A Frequency Domain Approach

: The optimal design of hybrid precoder/ combiner for Millimetre Wave (mmWave) Multiple Input and Multiple Output Orthogonal Frequency Division Multiplexing (MIMO-OFDM ) system is developed presented. In the frequency domain approach, Sparse Bayesian Learning - Kalman Filter (SBL-KF) algorithm is used to design the optimal hybrid precoder/ combiner in mmWave MIMO OFDM systems. Sparse signal recovery problem for Single Measurement Vector (SMV) is discussed, that is close to the design of ideal digital baseband precoder by maximizing the mutual information from the hybrid precoder. SBL-KF scheme select the minimum number of active Radio Frequency (RF) chains based on the hyper parameter estimator. The minimum number of RF chain is approximate the ideal digital precoder / combiner design. Proposed SBL-KF scheme achieve low power consumption and enhanced spectral efficiency, when compared to the SBL, Orthogonal Matching Pursuit (OMP), Simultaneous OMP (SOMP) and Least Square schemes, which activate a fixed data streams and fixed number of RF chains.

of Arrivals (AoAs) and Angle of departures (AoDs) are considered as known, which is not viable in practical situations. The problem of optimal precoder is solved by using the greedy Simultaneous Orthogonal Matching Pursuit (SOMP). SOMP greedy framework leads to convergence errors because the performance of selection dictionary matrix and stopping criterion is more sensitive [10]. The sparse solution is obtained by selecting the global minima from the SBL cost function, which ensure the representation of sparse vector in the digital precoder [11]. Further, the Expectation-Maximization (EM) algorithm is suited for ideal hybrid precoder /combiner and it is convergence that the SBL technique to a fixed point log likelihood functions [12].
In the above works, mmWave MIMO architecture is considered as fixed. The number of RF chain and data streams are fixed and this can be adjusted depends on the Channel State Information (CSI). RF chains contain the ADC/DAC and power amplifier, which significantly increase the power consumption in mmWave MIMO hybrid system. The power budget is effectively reduced by activating the adaptive less RF Chains at the transceiver, but this is not satisfying the multiplexing gain of the system. SBL scheme is used in the hybrid precoder/combiner for frequency selective mmWave MIMO channel, which significantly improve the spectral and power efficiency of the mmWave MIMO systems [13][14][15].

Contributions
In the previous work, temporal correlation is not discussed which improve the system performance. Sparse channel estimation scheme is developed for time and wideband mmWave MIMO channel, which overcome the shortcoming of the existing literature [16][17][18][19][20]. SBL-KF based hybrid precoder / combiner design for mmWave MIMO channel is presented in the frequency domain. These exploit the time evolution and time varying spatial multipath component of the sparse mmWave MIMO channel vector. The proposed SBL-KF scheme selects the minimum number of active Radio Frequency (RF) chain based on the hyper parameter estimator. This strategy leads to improve significantly the higher spectral and power efficiency.
The organization of the paper is as follows: Section II, discuss about System Model for millimetre wave MIMO-OFDM system. In section III, design of hybrid precoder/ combiner using SBL-KF is proposed. In section IV, simulation results of the proposed SBL-KF scheme for hybrid precoder/combiner is discussed. Finally, conclusions and future work is presented in section V.

Notation:
Bold lower case a letters denotes the column vector, non-bold lower case a denotes the scalar values. Bold uppercase letters denotes the A matrices, A denotes the conjugate matrix of A ,

System Model For Millimeter Wave MIMO-OFDM System
mmWave MIMO-OFDM hybrid architecture with T N , R N and RF N denotes the number of antennas at the transmitter, number of antennas at the receiver and number of RF chains at the transceiver, respectively as shown in Fig.1 The channel model expressed in equation (2) is compactly written as The complex channel model for the k th subcarrier is expressed in the frequency domain and is given as

Design of Hybrid Precoder/ Combiner Using SBL-KF
The maximization of the mutual information gives the optimal precoders BB,k F and RF F . The optimal precoder is achieved using Gaussian signalling over the mmWave MIMO channel and the similar work discussed in [2], [5], as given below . These can be obtained from the singular value decomposition (SVD) of k H . From the transmit array vectors, select the columns of RF F RF precoder, which satisfy the constraint of unit magnitude elements and form a basis for the column space of the ideal digital MIMO precoder opt,k F . The maximization of mutual-information with respect to the precoder is approximated by the sparse matrix and it is formulated as below.
Let the quantized transmit array dictionary vector is defined as The problem of optimal transmit precoder design for mmWave MIMO system is solved by approximating the optimal ideal precoder opt,k F and is discussed below RAdenotes the optimal combiner and receive array response respectively.

Design of Precoder using SBL-KF
The SBL-KF technique starts by assigning the precoder matrix BB . The mean vector and covariance matrix is shown in equation (13) and equation (14)

Active RF chains at the precoder and combiner design
The ordered estimates of the hyperparameters are denoted as

Results
Simulation results of the proposed SBL-KF based hybrid precoder/ combiner scheme is discussed and compared with the existing schemes such as SBL, SOMP, GHP, OMP and LS for mmWave MIMO-OFDM systems. The antennas at the transmitter and at the receiver are assumed as T N 32  and R N 32  , respectively. The number of spatially active rays L is distributed uniformly and that is given as L ∼ U (Lmin, Lmax). The minimum value of active path Lmin = 6, maximum value of active path Lmax = 12 and average value of active path Lavg = 9 is considered for simulation. The total number of RF chains is considered as R=12 and the active RF chain is assumed as RF RF  The spectral efficiency performance for the proposed scheme is shown in Fig.2. The SBL-KF scheme based hybrid precoder/ combiner design achieve increased spectral efficiency when compared to the existing schemes. The proposed scheme result is close to the ideal optimal precoder and also gives the lower approximation error with feasible selection of data streams via active RF chains. NMSE versus SNR performance for proposed scheme are plotted in Fig. 3. The NMSE performance of the SBL-KF based hybrid precoder/ combiner design is close to the ideal digital precoder and it gives lower NMSE value compared to the other schemes. Number of paths vs NMSE for proposed and existing schemes is plotted in Fig.4. When the number of path is increases, the channel unknown parameters gets increased, thus the performance of NMSE increases for All schemes. NMSE vs SNR for various RF chain for proposed work is shown in Fig.5. Employing more number of RF chain at the transceiver gives the low estimation error in hybrid precoder/combiner systems. precoder/ combiner and compared with existing scheme is plotted in Fig.6. From the Fig. 6, the SBL-KF scheme achieves a lower approximation error when compared to the existing schemes based on hybrid precoder/ combiner design. From the value of angular grid Φ, the AoA/ AoDs are selected; with this the hybrid precoder design attains the approximation error close to zero. The proposed scheme result is close to the optimal digital precoder Fopt.

Conclusion
In the frequency domain approach, SBL-KF based hybrid precoder/ combiner design for mmWave MIMO systems is discussed. The proposed method achieves increased spectral efficiency and it is closer to the ideal digital MIMO precoder/ combiner method. The proposed method performs better compared to the LS, OMP, SOMP and SBL schemes. Active RF chain is selected by initiating the hyperparameter method by selecting the active RF chain, which reduce the required RF chain and leads to reduce the power consumption in mmWave MIMO systems.

Declarations
Funding The investigations presented in the manuscript are not funded by any external agencies.

Conficts of Interest
The author declare that there is no conflict of interest regarding the publication of this paper.

Availability of Data and Material Not applicable.
Authors' contributions SBL-KF based hybrid precoder / combiner design for mmWave MIMO channel is presented in the frequency domain.