A General Approach to Fully Linearize the Power Amplifiers in mMIMO with Less Complexity

The preprocessing using DPD has an inverse property to the nonlinear PA to mitigate the nonlinearties in the desired transmit signal [2]. From the state-of-the-art, mostly the linear parametric models have been used for the the DPD [3]. One of the methods to identify the DPD coefficients is least square (LS) due to its fast convergence [4, 5, 6]. But, despite mathematical simplicity, its computational complexity is high due to engagement of inverse operations of the matrices of large sizes that correspond to the estimation of large number of DPD coefficients. However, many works have proposed the algorithms to reduce the complexity for the identification of the DPD based on LS method [7, 8, 9, 10]. For example, the size of the matrix is reduced by normalization of the DPD basis functions (BFs) followed by their pruning [7]. Also, based on stationary random process, the time varying matrix associated with the DPD coefficients is replaced by a constant covariance matrix [8]. Further, in an iterative algorithm based on LS, the samples of the DPD coefficients (or the size of the matrix) can be reduced by considering the correlation in the observation errors between two iterations [9]. Besides, the matrix size can also be reduced using the eigenvalue decomposition and principal component analysis (PCA) that decreases the order of the memory polynomial model of the DPD [11, 10]. In eigenvalue decomposition, the number of DPD coefficients can be reduced by considering only the dominant eigenvectors. Whereas, in PCA, the reduction is achieved by converting the correlated BFs of the DPD into uncorrelated BFs.

Although, the above techniques help in reduction of the size of the matrices, but, for time varying and highly nonlinear PAs, still, the required number of DPD coefficients is large. Thus, it leads to an undesirable large matrix operations. Therefore, the recursive based algorithms like least mean square (LMS) [12], recursive least squares (RLS) [13, 14], and recursive prediction error method (RPEM) [15] are computationally more reliable at the cost of their slow convergence to the desired optimal value of the variables. Using LMS, the DPD adjusts its coefficients to minimize the mean square error (MSE) between the PA output and the desired signal. The coefficients are updated using stochastic gradient decent method that minimizes the instantaneous error in each iteration. However, LMS is quite unstable and it is very sensitive in the step size for the update [16]. In conventional LS estimation, a batch of input and output data samples of the PA are used to update the DPD coefficients. But, in RLS, using a set of equations, the LS estimation is represented recursively and the coefficients are updated accordingly for the obtained new data sample of the input and output. To discount the influence of older samples, it uses an exponential weighing known as forgetting factor. The chosen value of the forgetting factor gives a trade-off between the precision and convergence and its low value provides high fluctuation to noise. Therefore, the forgetting factor is improved further in RPEM by considering its variation with time [17]. In the existing works, mostly, these adaptive algorithms are applied to two types of DPD learning architectures: (i) direct learning architecture (DLA) [18, 19] and (ii) indirect learning architecture (ILA) [13, 14]. DLA has better performance in the presence of noise at the output the PA, but, ILA is more effective in the convergence rate [20]. Therefore, ILA is widely used for the identification of the DPD. Also, in our proposed work, we have considered RPEM algorithm in an ILA architecture for the DPD identification. Next, we describe the state-of-the-art for the linearization of the multi-antenna transmitters.

In the multi-antenna systems like MIMO or mMIMO, although the PAs in the transmitters are of same type, but, in practice, they have different nonlinearties due to their sensitivity to process, supply voltage, and temperature (PVT) [21, 22]. Therefore, a single DPD is not capable to linearize all the PAs [23] and ideally, each PA requires a separate DPD [24]. But, the ideal case provides undesirably high complexity in hardware implementation as well as in processing and even not feasible for a mMIMO transmitter where hundreds of PAs need to be linearized. Subsequently, instead of linerizing all PAs, a resultant single PA can be linearized whose output is the sum of the outputs of the PAs [25, 26]. However, it addresses the average nonlinerites of the PAs, thus, none of them is fully linearized. On the other hand, instead of sum, the beam-oriented (BO) output of the PAs in a given direction can be linearized using a single DPD [27, 28, 29]. As it addresses the nonlinearity of the BO output in the desired direction (main lobe), again, the PAs are not fully linearized. Thereby, they are not able to linearize the outputs in other directions that gives the nonlinear sidelobes and typically, their power level is only $10$ dB lesser than the linear main lobe [30]. This can be improved by frequently updating the DPD for different directions. Also, the number of DPD coefficients per update can be reduced using the pruning algorithms [31]. However, the frequent update is not reliable for online operations and it leads to high computational complexity. Moreover, at a time, the DPD is identified for a particular BO direction and the PAs are not fully linearized that still gives the comparable nonlinear sidelobes to the main lobe. If we assume the similar distribution of nonlinearites over the PAs, the side lobes can be reduced by optimally adjusting the amplitude of the phase shifters in the BO output [32]. However, in general, it cannot provide the full linearization of the PAs. The performance towards the full linearization can be improved by including extra tuning box to each PA. The tuning boxes compensate the nonlinear differences between the PAs such that the resultant nonlinearity is same for each PA, thereafter, using a single DPD, the PAs are fully linearized [33, 34, 35]. Nonetheless, for the compensation of the differences, each tuning box is modeled using a polynomial model which needs to be identified using a learning algorithm. Therefore, its complexity is approximately similar to incorporate separate DPD for each PA. Different from only DPD operations, the sidelobes in the BO output can be linearized more reliably using the two layer of operations: the DPD training followed by the post-weighting coefficients optimization that are multipled by the DPD output signal and distributed at the respective PAs’ input [36, 37]. In a simplified analysis, different post-weighting coefficients are assigned for each PA, but, in a branch of a PA, same post-weighting coefficient are multiplied to the BFs of the DPD. Thus, due to less degree of freedom per branch, it is less reliable in post-weighting linearization of the PAs [36]. Also, to distribute different signals to the branches of the PAs, separate RF-chain is needed for each PA that gives a high complexity in a mMIMO transmitter. Later, in our proposed work [37], we adopted an adaptive post-weighting architecture that increases the degree of freedom (DOF) per branch as well as reduces the number of RF-chains requirement. But, still, due to optimization of post-weighting coefficients for discrete range of directions, the PAs are not fully linearized.

As described earlier, the PAs in a multi-antenna transmitter can be fully linearized using identification of a separate DPD to each PA [24]. But, it leads to high complexity in the structure and in the computation to learn the coefficients. Also, for the distribution of the predistorted signals, it requires a separate RF-chain to each PA. Based on it, we propose a most general approach using a low-complexity DPD (LC-DPD) structure which approximates seperate DPD identification requirement as well as the reduction of the number of RF-chains as per the requirement in the mMIMO transmitters. The key contribution of this work is four-fold as follows. (i) First, we deduce the reduction in the number of coefficients for a given type of PAs in a subarray from the measurement data and the obtained numerical result of a system setting. Then, we propose a fully-featured DPD (FF-DPD) scheme to fully linearize the PAs in a subarray and describe its complexity in terms of number of multipliers, adders, and RF chains. (ii) Using the FF-DPD structure, we derive the less complex and non-trivial LC-DPD structure. The number of coefficients in it is reduced based on a geometric sequence and corresponding coefficients are represented in a block vector form. Based on the geometric sequence, we derive the expression of the number of multipliers, adders, and RF chains which are significantly reduced; thus, reduces the complexity. (iii) Next, for the training of the coefficients for the two schemes, we propose four algorithms based on indirect learning architecture based recursive prediction error method (ILA-RPEM): one for the FF-DPD scheme and three for the LC-DPD. Apart from the structural complexity of the FF-DPD, we also describe the computational complexity of its training. The performance of the three algorithms for the LC-DPD is determined based on the correlation of its common coefficients to the distinct coefficients in the structure. It is also shown that the complexities of the three algorithms are less than the algorithm for FF-DPD. Further, for various operations in the four algorithms, we define the four operators and describe their properties. (iv) Lastly, we obtain the numerical results in terms of power spectral density (PSD) and error vector magnitude (EVM) using the algorithms for the two schemes and obtain the various insights by comparing their performances.

As described earlier in this section, the predistortion signals for a given type of PAs can be obtained using the same set of BFs based on GMP (cf. Fig. 2). Thus, in FF-DPD, we consider the same set of BFs of order $P$ and memory length $M$ for all the PAs in the subarray. Therefore, the $l$th output $x_{l}$ from the FF-DPD can be obtained again using (1) after substituting $P_{l}=P$ and $M_{l}=M$. The total number of BFs in the set $\{s(n-m)|s(n-m)|^{p}\}$ for $p\in\{0,\cdots,P-1\}$ and $m\in\{0,\cdots,M-1\}$ is $PM$. However, as observed in Fig. 2, out of $PM$ BFs, some BFs have nonzero DPD coefficients for a given type of PAs. Therefore, in general, we represent $Q$ BFs as a vector $\bm{\Psi}=[\psi_{1},\cdots,\psi_{Q}]^{T}$ with their respective nonzero coefficient vector for the $l$th PA as $\bm{\Phi}_{l}=[\phi_{l,1},\cdots,\phi_{l,Q}]^{T}$, where $\psi_{i}$ is the $i$th BF³³3BF $\psi_{i}$ is a function of $s(n)$, is given by $\psi_{i}(s(n))$ for $i\in\{1,\cdots,Q\}$. and $\phi_{l,i}$ is its nonzero coefficient for $i\in\{1,\cdots,Q\}$. Using $\bm{\Psi}$ and $\bm{\Phi}_{l}$, $x_{l}$ in (1) can be expressed in matrix form as:

Using $x_{l}$, the output $y_{l}$ of the $l$th PA is obtained using the same process as in (2) where $x_{l}$ is multiplied by the phase shifter $w_{l}$ and inputted to the PA to get $y_{l}$, thus, output vector $\bm{Y}$ is obtained. Moreover, the coefficient vectors for the $S$ PAs in the subarray can be expressed in a block vector as: $\bm{\Phi}=[\bm{\Phi}_{1}^{T},\cdots,\bm{\Phi}_{S}^{T}]^{T}$. From Fig. 3(a), $S$ coefficients are multiplied by each BF. Thus, for $Q$ BFs, the FF-DPD structure has $QS$ coefficients. Besides, the number of multipliers, $N_{m}^{F}$ in the structure is same as the number of coefficients, i.e., $N_{m}^{F}=QS$. Further, using the structure of FF-DPD in Fig. 3(a), the number of adders, $N_{a}^{F}$ can be determined as follows⁴⁴4In this work, the numbers of adders for the different structures are determined for the assumption that an adder has two inputs and one output.. Each predistorted output ($x_{l}$; $l\in\{1,2,3,4\}$) is determined though the sum of the multiplications of the $Q$ coefficients by respective BFs. Therefore, in generation of each output, the number of adders is $(Q-1)$, i.e., one less than the number of coefficients. Thus, for $S$ outputs, the total number of adders, $N_{a}^{F}=(Q-1)S$. Moreover, the number of RF chains, $N_{RF}^{F}$ is same as the number of PAs, i.e., $N_{RF}^{F}=S$. For instance, Fig. 3(a) depicts the FF-DPD for $S=4$ and $Q=4$. It has $N_{m}^{F}=16$, $N_{a}^{F}=12$, and $N_{RF}^{F}=4$. If we compare Fig. 3(a) with Fig. 1, the ideal structure for the linearization of the subarray is same as the structure of FF-DPD except the same set of BFs has been used for all PAs in the later. Thus, the complexity of FF-DPD is still high in terms of multipliers, adders, and the RF chains. These complexities can be reduced using LC-DPD which is described next.

Using LC-DPD, we reduce the complexity to fully linearize the PAs as follows. As described earlier (cf. Fig. 2), coefficients of a dominant BF in the generation of the predistorted signals for different PAs has higher deviation in its value. Therefore, to reduce the number of coefficients in LC-DPD, the BFs in the vector $\bm{\Psi}=[\psi_{1},\cdots,\psi_{Q}]^{T}$ are arranged in decreasing order of their dominance. Then, more coefficients are multiplied by a higher dominant BFs than the lower dominant BFs to generate the predistorted signals. Therefore, different from FF-DPD, in LC-DPD, the number of coefficients are multiplied adaptively by the BFs according to their dominance. Therefore, we decrease the number of coefficients based on a geometric series as follows.

where $\nu\in\mathbb{Z}^{+}=\{0,1,\cdots\}$, $\sum_{i=1}^{g}n_{i}=Q$; $n_{i}\in\mathbb{P}=\{1,2,\cdots\}$ and the cases are: Case I: $Sr^{(\nu+q-1)}\geq 1$ and Case II: $\{Sr^{(\nu+g-2)}\geq 1\}\wedge\{Sr^{(\nu+g-1)}<1\}$. According to the sequence in (6), the $Q$ BFs are divided into $g$ groups, where each of the $n_{i}$ BFs in the $i$th group are multiplied by $Sr^{(\nu+i-1)}$ coefficients; thus, the total coefficients in the $i$th group is $n_{i}Sr^{(\nu+i-1)}$. Further, over the groups, the number multiplied coefficients per BF decreases as a geometric sequence with a common ratio $r$ $(<1)$. The sequence in Case II is same as in Case I except each BF in the last group $g$ is multiplied by one coefficient in Case II, because, $Sr^{(\nu+g-1)}<1$ and the number of coefficient per BF cannot be a fraction. Now, using this sequence, we define the coefficient vector $\bm{\overline{\Phi}}$ of the LC-DPD as below.

Example:  For LC-DPD in Fig. 3(b) where $S=4$ and $Q=4$, the BFs in the vector $\bm{\Psi}=[\underbrace{\psi_{1},\psi_{2}}_{n_{1}=2},\underbrace{\psi_{3},\psi_{4}}_{n_{2}=2}]^{T}$ are divided into $g=2$ groups and each group contains two BFs. For $\nu=1$ and $r=1/2$, the sequence of number of coefficients is: $\{n_{1}Sr^{\nu},n_{2}Sr^{\nu+1}\}=\{4,2\}$; thus, the total number of coefficients is $4+2=6$. Here, each of the BF in the first group is assigned by $Sr^{\nu}=2$ coefficients, while for the second group, it is $Sr^{\nu+1}=1$ as depicted in the figure. Further, for the LC-DPD coefficient vector $\bm{\overline{\Phi}}$, the Case I in (7a) is satisfied, thus, using (7), $\sigma_{1}=0$, and $\sigma_{2}=2$, the coefficient vector $\bm{\overline{\Phi}}=[\underbrace{\phi_{1,1},\phi_{3,1},\phi_{1,2},\phi_{3,2}}_{\text{elements of }\bm{\overline{\Phi}}_{1}},\underbrace{\phi_{1,3},\phi_{1,4}}_{\text{elements of }\bm{\overline{\Phi}}_{2}}]$. $\Box$

We study the processing behind the linearization of $l$th PA of the subarray using the ILA-RPEM in Fig. 4 as follows. Using the input $s(n)$, the predistorted output $x_{l}(n)$ followed by the PA output $y_{l}(n)$ are obtained using (3) and (2), respectively. To capture the inverse of the nonlinear behavior of the PA, the scaled output of the PA $y_{l}^{{}^{\prime}}(n)$ is inputted to the RPEM based learning block which generates the postdistorter signal $\tilde{x}_{l}(n)$ similar to (3) as:

where $\bm{\tilde{\Phi}}_{l}$ is the $l$th vector of the block vector $\bm{\tilde{\Phi}}$ and $\bm{\Psi}_{l}^{{}^{\prime}}$ is the vector of BFs defined using same polynomial terms as in $\bm{\Psi}$, only the difference is that instead of $s(n)$, $y_{l}^{{}^{\prime}}(n)$ is the input to the BFs in $\bm{\Psi}$. Thus, the $i$th element of the vector $\bm{\Psi}_{l}^{{}^{\prime}}$ is: $\psi_{l,i}^{{}^{\prime}}=\psi_{i}(y_{l}^{{}^{\prime}}(n))$. Now, the goal of the RPEM algorithm is to minimize the difference between the postdistorted signal $\tilde{x}_{l}$ in (10) and the predistorted signal $x_{l}$ in (3) iteratively by optimizing $\bm{\tilde{\Phi}}_{l}$. At the end of each iteration, $\bm{\tilde{\Phi}}_{l}$ is copied to $\bm{\Phi}_{l}$, i.e., $\bm{\Phi}_{l}=\bm{\tilde{\Phi}}_{l}$ and using it, the algorithm tries to capture the inverse of the nonlinear characteristics of the PA. Thus, using the estimated value $\bm{\widehat{\Phi}}_{l}$ at the end of the convergence, the FF-DPD generates the optimal predistored signal $\widehat{x}_{l}$ to linearize the $l$th PA.

Now, we describe the process to minimize the difference $e_{l}(n)=x_{l}(n)-\tilde{x}_{l}(n)$ in each iteration. The cost function $\mathcal{L}(\bm{\tilde{\Phi}}_{l})$ based on the average of the power content of $e_{l}(n)$ over the long horizon is defined as:

From [40], $\mathcal{L}(\bm{\tilde{\Phi}}_{l})$ in (11) can be minimized using the negative gradient of $e_{l}(n)$ in $\bm{\tilde{\Phi}}_{l}$. It can be obtained as:

Using the gradient in (12), the training block performs the below computations in (13) based on RPEM to get the trained coefficients $\bm{\tilde{\Phi}}_{l}^{(n)}$ for the $l$th PA in the $n$th iteration [40].

Here, in (13a), $e_{l}(n)$ is computed. Thereafter, the forgetting factor $\xi_{l}$ is determined recursively in (13b) using its value in the previous iteration and the rate of growth $\rho$. The initial value $\xi_{l}^{(0)}=\lambda_{0}$ and $\xi_{l}$ grows exponentially to $1$ with iterations. Using $\xi_{l}$, BF vector $\bm{\Psi}_{l}^{{}^{\prime}}$, and the covariance matrix $P_{l}$, the scalar $Z_{l}$ is computed in (13c). Initial value $P_{l}^{(0)}=\mu\bm{I}$, where $\bm{I}$ is the identity matrix and $\mu$ is a constant. Finally, matrix $P_{l}$ is obtained in (13d) followed by $\bm{\tilde{\Phi}}_{l}$ is determined in (13e) recursively at the end of the iteration. Moreover, RPEM is free from complex matrix inverse operations like in a LS estimation as $Z_{l}$ is scalar. Now, using the above study for the linearization of $l$th PA, we realize the full linearization of PAs of a subarray.

As described earlier, the structure of FF-DPD in Fig. 3(a) is similar to assigning individual DPD in Fig. 1 to each of the PAs. Its complexity is reduced only due to utilizing same set of $Q$ BFs for a given type of PAs. Therefore, using FF-DPD, the full linearization for $S$ PAs through ILA-RPEM is same as the parallel linearization of each of the $S$ PAs using the same process as for the $l$th PA (cf. previous paragraph). For the parallel operations, the different parameters are arranged in the matrix form as follows. We define, $\bm{E}=[e_{1},\cdots,e_{S}]^{T}$, $\bm{\xi}\triangleq\text{diag}(\xi_{1},\cdots,\xi_{S})$, $\bm{Z}\triangleq\text{diag}(Z_{1},\cdots,Z_{S})$, $\bm{\Upsilon}\triangleq\text{diag}(\bm{\Psi}_{1}^{{}^{\prime}},\cdots,\bm{\Psi}_{S}^{{}^{\prime}})$, $\bm{P}\triangleq\text{diag}(P_{1},\cdots,P_{S})$, and $\bm{\Xi}\triangleq\text{diag}(\xi_{1}\bm{I}_{Q},\cdots,\xi_{S}\bm{I}_{Q})$. Here, $\text{diag}(\cdot)$ is diagonal matrix constructed using the input scalar or matrix elements.

In Algorithm 1, the values are assigned to the independent parameters $\rho$, $\lambda_{0}$, and $\mu$. Then, the initial values of the covariance matrix $\bm{P}$ and the forgetting factor $\bm{\xi}$ are computed in Steps 3 and 4. Thereafter, the calculations from Step 9 to Step 15 are repeated until the coefficient vector $\bm{\tilde{\Phi}}$ converges. Lastly, we obtain the optimal coefficient vector $\bm{\widehat{\tilde{\Phi}}}$ in Step 18 which is copied to the FF-DPD, i.e., $\bm{\widehat{\Phi}}=\bm{\widehat{\tilde{\Phi}}}$.

To learn the LC-DPD coefficient vector $\bm{\overline{\Phi}}$, we exploit the learning process for FF-DPD. In this regard, vector $\bm{\overline{\Phi}}$ is converted into FF-DPD coefficient vector $\bm{\Phi}$ and vice versa. To realize the conversions mathematically, we define two operators $\bm{\mathfrak{M}}_{1}$ and $\bm{\mathfrak{M}}_{2}$.

Example:  From Fig. 3(b), the block vector $\bm{\overline{\Phi}}=[\bm{\overline{\Phi}}_{1}^{T},\bm{\overline{\Phi}}_{2}^{T}]^{T}$, where $\bm{\overline{\Phi}}_{1}=[\overline{\phi}_{1,1},\overline{\phi}_{3,1},\overline{\phi}_{1,2},\overline{\phi}_{3,2}]^{T}$, and $\bm{\overline{\Phi}}_{2}=[\overline{\phi}_{1,3},\overline{\phi}_{1,4}]^{T}$. Here, the parameters for LC-DPD are: $\nu=1$, $r=1/2$, $g=2$, $n_{1}=n_{2}=2$, $Q=4$, and $S=4$. To reshape $\bm{\overline{\Phi}}$ into the shape of $\bm{\Phi}=[\bm{\Phi}_{1}^{T},\bm{\Phi}_{2}^{T},\bm{\Phi}_{3}^{T},\bm{\Phi}_{4}^{T}]^{T}$ as for FF-DPD in Fig. 3(a), we use the operator $\mathfrak{M}_{1}$ as given below, where $\bm{\Phi}_{i}=[\phi_{i,1},\phi_{i,2},\phi_{i,3},\phi_{i,4}]^{T}$ for $i\in\{1,2,3,4\}$.

In (19), $\bm{\mathfrak{M}}_{1}$ satisfies the property (i) as each of its row vector contains one element as $1$ and remaining are $0$. Next, to verify the property (ii), for instance, the sum of the elements of the $5$th $(z=5)$ column vector is $4$ which entails that $5$th coefficient $\overline{\phi}_{1,3}$ of $\bm{\overline{\Phi}}$ repeats $4$ times in $\bm{\Phi}$. It can also be determined using the expression $r^{-(\nu+j-1)}$ where $j=2$ as the 5th $(z=5)$ column vector lies in the 2nd $(j=2)$ group and it gives $4$ after substitution of the value of the parameters. Besides, the block vector $\bm{\Phi}=[\bm{\Phi}_{1}^{T},\bm{\Phi}_{2}^{T},\bm{\Phi}_{3}^{T},\bm{\Phi}_{4}^{T}]^{T}$ is obtained using the computation in (14a), where $\bm{\Phi}_{1}=[\overline{\phi}_{1,1},\overline{\phi}_{1,2},\overline{\phi}_{1,3},\overline{\phi}_{1,4}]^{T}$, $\bm{\Phi}_{2}=[\overline{\phi}_{1,1},\overline{\phi}_{1,2},\overline{\phi}_{1,3},\overline{\phi}_{1,4}]^{T}$, $\bm{\Phi}_{3}=[\overline{\phi}_{3,1},\overline{\phi}_{3,2},\overline{\phi}_{1,3},\overline{\phi}_{1,4}]^{T}$, and $\bm{\Phi}_{4}=[\overline{\phi}_{3,1},\overline{\phi}_{3,2},\overline{\phi}_{1,3},\overline{\phi}_{1,4}]^{T}$. $\Box$

Example:  If we consider Fig. 3 for this example, from Fig. 3(a), the block vector $\bm{\Phi}$ for FF-DPD is expressed as $\bm{\Phi}=[\bm{\Phi}_{1}^{T},\bm{\Phi}_{2}^{T},\bm{\Phi}_{3}^{T},\bm{\Phi}_{4}^{T}]^{T}$, where $\bm{\Phi}_{j}=[\phi_{j,1},\phi_{j,2},\phi_{j,3},\phi_{j,4}]^{T}$ for $S=4$ and $Q=4$. To transform $\bm{\Phi}$ into the shape of $\bm{\overline{\Phi}}=[\bm{\overline{\Phi}}_{1}^{T},\bm{\overline{\Phi}}_{2}^{T}]^{T}$ as for the LC-DPD in Fig. 3(b), we use the operator $\mathfrak{M}_{2}$ as given below, where $\bm{\overline{\Phi}}_{1}=[\overline{\phi}_{1,1},\overline{\phi}_{3,1},\overline{\phi}_{1,2},\overline{\phi}_{3,2}]^{T}$, and $\bm{\overline{\Phi}}_{2}=[\overline{\phi}_{1,3},\overline{\phi}_{1,4}]^{T}$.