Performance analysis of linear precoders with imperfect channel covariance information for multicell system

: Comparison and analysis of the regularised zero forcing precoder, rapid numerical algorithms-based precoder and the truncated polynomial expansion-based precoder are done for massive multiple-input multiple-output wireless system for multicell scenario. The analysis was done for the imperfect channel covariance information. The achievable signal-to- interference-and-noise ratio, spectral efficiency and energy efficiency were investigated. The simulated outcome of the rapid numerical algorithms, regularised zero forcing and truncated polynomial expansion precoders for multicell massive MIMO system was analysed. The rapid numerical algorithms-based precoder gave the best performance followed by the regularised zero forcing precoder, and the truncated polynomial expansion-based precoder had the lowest performance for the multicell massive MIMO system. The increase in spectral efficiency per cell can be attributed to the fact that the pre-log factor reduces with the increased number of pilots. Also, this leads to increased instantaneous signal-to-interference-and-noise ratio as the channel estimates become better with reduced pilot contamination. Again, for truncated polynomial expansion precoding there is a reduction in spectral efficiency because improvement in approximation quality do not overshadow the reduction in pre-log factor. The performance is evaluated for uncoordinated and coordinated massive MIMO.


Introduction
Multiple-input multiple-output (MIMO) procedures have increased impressive consideration in current wireless communication because it can altogether enhance the capacity and steadfast quality of wireless systems [1]. The embodiment of downlink multiuser MIMO is precoding, which implies that the antenna arrays are utilised to coordinate every information flag spatially towards its expected user terminal (UT). However, the precoding plan in multiuser MIMO needs extremely precise momentary channel state information (CSI) [2]which can be bulky to accomplish. In a multiuser MIMO, the channel fluctuates rapidly in the order of a ten of time and frequency. If coding is allowed athwart coherent intervals of many channels, it has been demonstrated that for system consisting of M antennas transmitting to K UTs, there is a linear increase in capacity in relation with the min (M, K) log 2 (1+SINR) [3,4]. This capacity scaling can be applied to massive MIMO systems with a base station (BS) having M transmitting antennas in communication with K UTs in the case of time-division duplexing (TDD).
The throughput can greatly be enhanced if BS communicating with the UT can do so within the same time-frequency resources. This can be achieved but at a price of enhanced inter-user interference, hence eroding the system performance. To mitigate this, dirty paper coding (DPC) is employed, where the transmitter jointly encodes the data symbols for all users [5,6]. The transmission scheme based on DPC comes with complexity that prohibits its practical realisation. This then necessitates the reliance on transmit-side preprocessing techniques called precoding schemes. It has been said that with massive MIMO, pilot contamination can be optimally mitigated with the simplest forms of precoding like eigen beamforming (BF) and matched filtering [7]. Later research have demonstrated that realisable BS antennas demand included linear precoding procedures, for example, MMSE [8]. We cannot overlook the complexity of computing the minimum mean square error (MMSE) precoding in massive MIMO network system, since MMSE relies on inverting of substantial matrices [9]. To circumvent this, a class of rapid numerical algorithms (RNA) and truncated polynomial expansion (TPE) precoders were discussed in [10] for the single cell multi-user case.
The multicell TPE-based precoder is discussed in [11] and compared to regularised zero forcing (RZF) precoding. The paper [12] develops a multicell MRC-BSC and ZF-BSC precoding to improve system downlink performance, and their influence on the received signal-to-interference-plus-noise ratio (SINR) of users in other cells.
This paper discusses multicell precoding using RNA-based, RZF and TPE-based precoders with and without BS cooperation. In our work, the proposed precoding schemes with and without BS cooperation are obtained by changing the matrix structure of single-cell RNA, RZF and TPE precoding, so they are both linear precoding and have low computational complexity. We consider the effects and analyse the received SINR of user in other cells. We use a similar analysis method as that employed in [12]. The SINR is used to calculate the capacity per cell for comparison purposes. We also look, at energy efficient based on these three precoding schemes for the case of uncoordinated and coordinated BSs and conclude by comparing the performance of these three precoders on overall. The channel model considered includes the imperfect covariance experienced by the BSs.
We begin by modelling the optimal multicell linear precoding and then tailor it to each of the multicell linear precoding schemes. The imperfect covariance channel is modelled before evaluating the SINR and data rate of imperfect channel covariance information is used to analyse the behaviour of the massive MIMO downlink network system over the modelled channels in multicell scenario.
Notation: lower-case and upper-case boldface letters denote vectors and matrices, respectively; (·) T , (·) H , (·) −1 and tr(·) denote the transpose, conjugate transpose, matrix inversion and trace, respectively; ℂ denotes the set of complex numbers, I N is the N × N identity matrix.
We assume that the TDD protocols are synchronised across cells to simultaneously transmit pilot signals and data to all cells. The received complex baseband signal Y j, k ∈ ℂ at the kth UT in the jth cell is where u ℓ ∈ ℂ M × 1 defines the transmit signal from the ℓth BS and h ℓ, j . k H ∈ ℂ M × 1 represents the channel vector from the ℓth BS to the mth UT in the jth cell, and n j, m ∼ CN 0, σ 2 accounts for the additive white Gaussian noise that has a variance σ 2 , at the receiver's input [11,[15][16][17][18].
The channel vectors are assumed to be Rayleigh fading and modelled as where Φ ℓ, j, m is the covariance matrix corresponding from the ℓth BS to the mth UT in the jth cell. The BSs use linear precoding with Gaussian codebooks in that the signal transmitted by the jth cell is represented by where F j = f j, 1 , …, f j, K ∈ ℂ M × K is the precoding matrix and x j = x j, 1 , …, x j, K ∈∼ CN 0, I K is the information vector with data symbols from all the UTs in the jth cell. The average power to be transmitted at BS j is constrained as Thus, the received signal vector at kth UT is expressed as

Model of imperfect covariance channel information
If we assume that the transmitter has imperfect information concerning the instantaneous channel realisation ĥ j, j, k of every UT. We can approximate the channel relating to kth UT in jth cell, by letting every BS to correlate the received signal with the pilot sequence of that user. This gives the processed received signal as where n j, m tr ∼ CN 0, I M and ρ tr > 0 gives the effective training SINR [8]. We use the MMSE estimate ĥ j, j, k of h j, j, k is With S j, k = (1/ ρ tr ) + ∑ ℓ = 1 L Φ ℓ, j, k −1 and Φ j, j, k is the channel covariance matrix of vector h j, j, k . The approximated channels from the jth BS to all UTs in jth cell are denoted Then we define R ℓ, j, k = Φ j, j, k S j, k Φ ℓ, j, k and we state here that ĥ j, j, k ∼ CN 0, R j, j, k is not dependent on the estimation error h j, j, k = ĥ j, j, k − h j, j, k as MMSE approximation is used.

RZF precoding
From the notation in [11], the RZF precoding matrix for the BS in the jth cell is expressed as where Z = ∑ k = 1 K Φ k − R k + (1/ ρ tr )I M and the scalar ϑ j > 0 is carefully chosen to satisfy the power constraint in the cell. We modify the coefficient of the precoding matrix by evoking norm minimisation scheme to improve the SINR performance of the transmission signal [22]. Then, the desired precoding matrix becomes

TPE precoding
Expanding on the idea of TPE, we presently give another class of low-complexity linear precoding plan for the multicell case. The Cayley-Hamilton hypothesis is used to specifically show that the inverse of a matrix A of dimension M can be composed as a weighted total of its first M powers [9,16]. If the jth BS utilises a truncation order J j , then the proposed TPE precoding matrix is expressed as where ω n, j , j = 0, …, J j − 1 are the J j scalar coefficients that are employed by the jth BS and the normalisation by K controls the energy of the precoding matrix. Then using the norm minimisation, the desired precoding matrix becomes For large-scale approximation of SINRs, it is shown that in the large-(M, K) regime, the SINR experienced by the mth UT served by the jth cell, can be estimated by a deterministic term, depending only on the channel statistics. Some extra notation are introduced to provide simpler representation of the SINR expression [16].
Then ω j = ω 0, j , …, ω J j − 1, j and let a j, m ∈ ℂ J j × 1 and Then the SINR associated with the mth UT in the jth cell is

RNA precoding
The ideal linear precoding is obscure in presence of imperfect CSI and involves wide-ranging optimisation procedures for the case of perfect CSI [23]. In this way, just heuristic precoding strategies are achievable in fading multi-cell networks. RNA is a state-of-the-art heuristic scheme with a simple closed-form precoding expression based on MMSE precoding [10]. It is stated in [10] that when the value of the power of the polynomial, J j , is low the performance of the TPE precoder is so poor, but as the value of J j increases, its bit-rates improve and nears that of RNA precoder. This comes at the cost of increase in the hardware needed for the implementation of the TPE precoding. RNA precoding will have relatively the same complexity as TPE precoding at the specified hardware implementation while giving a better performance. This means that the increase in hardware to improve the performance of TPE precoder will suffice the RNA precoder to reduce its complexity and come as close to the complexity of the TPE precoder but with a better performance in terms of throughput [9,10,24]. Using the notation of [18], the RNA precoding matrix used by the BS in the jth cell is where Z = ∑ k = 1 K Φ k − R k + (1/ ρ tr )I M and the scalar ϑ j > 0 is carefully chosen to satisfy the power constraint in the cell. Then using the norm minimisation, the desired precoding matrix becomes

Multi-cell linear precoding with coordination
In massive MIMO, the BSs are furnished with many antennas, this avails additional spatial dimensions which permit coordination of BF vectors athwart the BSs; this further improves the performance generally. This is possible due to the uplink-downlink duality in TDD massive MIMO systems. To enhance downlink attainable rate, the collaboration between BSs in various cells is evaluated, and it is expected that the jth BS knows the RNA, RZF or TPE precoding and channel approximation ĥ ℓ, j, k of every other single cell within the network. The new multi-cell linear precoding with BS cooperation can be expressed as follows [12]: These can be rewritten as in (5) to yield When compared with classical multicell precoding matrix in (5), the new multicell precoding matrix adds term h j, j, m H diag ĥ ℓ, j, k / diag ĥ j, j, m F ℓ (which embodies BS cooperation). Through this altering of the matrix structure of classical multicell RNA and TPE precoding, the inter-cell interference can be scaled down, and the downlink achievable sum rate can be enhanced. Even though the cooperation between BSs will scale up the complexity of the system, the data required can be effectively obtained by the BS switch which is associated with these BSs.
If we assume the channel hardening property, then (26) becomes (see (23)) . The second and the third terms on the RHS can be modified by use of the following identities: Using (25), the third term on the RHS of (23) can be written as Based on (26), we can rewrite (8) as (see (27)) , where the term and represents the BSs cooperation component.
Considering the mth UT in the jth cell for the RZF precoding, then SINR based on (12) and (27) will be expressed as where Considering the mth UT in the jth cell for the TPE precoding, then SINR based on (18) and (27) will be expressed as where Considering the mth UT in the jth cell for the RNA precoding, then SINR based on (27) will be expressed as where From both the precoding schemes, the BSs cooperation term works to reduce the inter-cell interference hence improving the SINR and consequently the sum rate.

Energy efficiency (EE)
The EE of the various precoding schemes is analysed with reference to a pragmatic consumption model of the circuit power. Then the tradeoff between the EE and the SINR is analysed. The EE is defined in [25,26] as EE = Throughput bits/s/cell Power consumption W/cell (31) and it is calculated in bits per joule. This can be viewed as a benefit-cost relationship, wherein the quality of service (throughput) is compared with the related costs (power consumption). Realistic evaluation of EE requires that power consumption (PC) be calculated based on effective transmit power (ETP) and the CP needed in powering the cellular network [26] PC = ETP + CP The CP consumption model for a general BS j in massive MIMO system can be expressed as [27][28][29][30][31][32] CP j = P FIX, j + P TC, j + P CE, j + P C/D, j + P BH, j + P SP, j where P FIX, j is a fixed power quantity to account for the power requirement of control signalling and load-independent power consumed by the backhaul infrastructure and the baseband processors and the power due to economic expenses. Also, P TC, j represents the power utilised in the transceiver chains, P CE, j accounts for power consumed in channel estimation, P C/D, j the power for channel encoding and decoding components, P BH, j accounts for power consumed in load-dependent backhaul signalling and P SP, j accounts for BS signal processing power.
The P FIX, j can be expressed as follows: where P FIX1, j is the power requirement of control signalling and load-independent power consumed by the backhaul infrastructure and the baseband processors and P FIX2, j is the power due to economic expenses [33] P FIX2, j = ∑ V = 1 V K r V U V, SE + C 0 + K c p bs (35) where V is the quantity of traffic classes, U V, SE is the actual chargeable information throughput corresponding to traffic class V, K r V is the income generated per unit data of traffic class V, p bs power expended by BSs during data transmission, K c is the cost of energy per joule and C 0 is the additional costs on top of the energy costs. The P TC, j for a cell j can be expressed as [28,29] P TC, j = M j P BS, j + P LO, j BS Circuit component while the P C/D, j for cell j is given as [30] P C/D, j = P COD + P DEC CT j The next component P BH, j is modelled in two parts the loadindependent and the load-dependent [31]. The earlier is taken care of under the P FIX, j while the second one is expressed as The channel estimation power P CE, j is then approximated according to the estimator employed where B is the bandwidth, L BS computational efficiency of BS, τ c is the coherence time and τ p is the length of the pilot sequence [34]. The power P SP, j used by BS j in receive combining and transmit precoding can be computed based on computational complexity of the schemes used. P SP, j can be decomposed as [34] P SP, j = P SP − R/T, j where P SP − R/T, j caters for overall power utilised in uplink (UL) reception and downlink (DL) transmission of information streams, is power needed to compute the combining vector and is power needed to compute the precoding vector at the jth BS. The P SP − R/T, j term is computed as while the P SP − C, j DL is evaluated as The P SP − C, j UL power consumption is dependent on the precoding scheme used.

Computation complexity of receive combining
It is assumed that the complex divisions and multiplications have the greatest impact on complexity neglecting the additions and subtractions [34]. The complexity is particularly impacted by the precoding matrix for the various precoding schemes. To evaluate the complexity, the lemmas in the Appendixare evoked.
The combining complexity of the RZF precoding can thus be written as [34] 3K From (14), the combining complexity of the TPE precoder can be evaluated based on the Appendix. Then the multiplication in (14), gives a complexity of 3K j 2 + K j M j /2 . But after the multiplication, there is the power n, in this case it was set to 2 because a J j of 3 was used. Then this yielded a complexity of K j 3 . The normalisation of the precoding vector needed in the decoding unit costs K j divisions for every BS. Thus, the total complexity was computed as follows: From (19), the combining complexity of the RNA precoder can be evaluated based on the Appendix. It was pointed out in [10] that the consideration of the initial three terms gives the quickest convergence of the iterative process for finding the inverse in RNA. The term Z j is presumed to be available for free at jth BS. Since RNA uses intercell channel estimates for ℓ ≠ j the complexity of computing these estimations is included. The multiplication complexity is expressed as Then to establish the complexity for the inversion, we follow the following procedure. From [10], the inverse for the first three terms (P = 3) in successive iterations can be written as And in general From R 1 the number of operations count to calculate it is From R 2 the number of operations count to calculate it is From R 3 the number of operations count to calculate it is Thus, to perform K iterations, the overall operations count for P = 3 can be expressed as Equation (52) gives us the inversion complexity of the RNA method. Then the multiplication after inversion gives M j τ p τ p − K j and the normalisation of the precoding vector needed in the decoding unit costs M j divisions for every BS. Thus, the complexity was computed as follows: Each iteration is assumed to correspond to the coherence time instance. The increase in number of iterations reduces the complexity of inversion in RNA precoding by a factor of M j 2 K − 1 .

Computation of receive combining power
From the computational complexity of receive combining, the receive combining power for various precoding schemes can be derived. For the RZF precoding, the receive combining power can be evaluated as Looking at the TPE precoding, the receive combining power can be expressed as For the case of RNA precoding, the receive combining power can be evaluated according to the following expression:

Energy efficiency and throughput
Based on the CP model developed, EE-CT analysis is performed to underline the importance of bandwidth in EE analysis. The EE in the jth cell is expressed as where CT j is the capacity for jth cell, ETP j is the ETP for jth cell and CP j is the CP for jth cell. Then the CT j is expressed as the The ETP j is then evaluated as follows in [34] ETP j = τ p where τ u is the uplink time and τ d is the downlink time, μ UE, jk is the power amplifier (PA) efficiency of the kth UT in the jth cell and μ BS, j is the PA at the jth BS.

Results and discussion
This section provides the comparison and analysis of RZF precoding, TPE precoding and the RNA-based precoding. First, we look at the comparison and analysis in performance between RZF precoding, TPE precoding and the RNA-based precoding in multicell massive MIMO with 20 cells. This comparison is carried out for varying from M = 16 to 160, with a step of 16 and K = 10 massive MIMO system. For the TPE precoder, the power of the polynomial, J j , is chosen to be 3 in this work. Fig. 1 compares the achievable SE per cell among RZF precoding, TPE precoding and the RNA-based precoding in multicell massive MIMO with a reuse factor of 1. Based on this figure, several observations can be made. The RNA-based precoding has the best achievable SE per cell, followed by the RZF precoding and then the TPE precoding. But it can also be noted that as the number of the BS antennas grow large, the TPE performance improves and matches that of the RZF precoding. The performance of TPE precoding is below the other two because it is just but an approximation of the RZF precoding. But as the number of the BS antennas grow, the approximation becomes better as it rides on the space-time diversity and the channel hardening. It is evident that the RNA-based precoder performs better than the TPE precoder and the RZF precoder under the same propagation condition.
From Figs. 2 and 3, we compare the RZF precoding, RNAbased precoding and the TPE precoding in the presence of different reuse factors. With a reuse factor of 2, the RNA-based precoding outperforms the RZF precoding and TPE precoding with the TPE precoding offering the lowest SE per cell. Furthermore, the RNAbased precoding and the RZF precoding exhibit an increase in SE per cell above that of a reuse factor of 1 but the TPE precoding exhibits a reduction in the SE per cell. This clearly shows that the TPE precoding can be applied for single cell case but its performance plummets in multicell massive MIMO system where practically reuse factors are introduced to optimally use the scarce radio resources. When the reuse factor is changed to 4, the increment in the SE per cell for both the RNA-based precoding and the RZF precoding increase, though that of the TPE precoder decreases further.
The increase in SE per cell can be attributed to the fact that the pre-log factor reduces with the increased number of pilots. Also, this leads to increased instantaneous SINR as the channel estimates become better with reduced pilot contamination. Again, for TPE precoding there is a reduction in SE because improvement in approximation quality do not overshadow the reduction in pre-log factor. This is a fact since the estimate is just evoked to enhance desired signal and not to mitigate interference. Thus, RNA based precoding has a good performance in presence of a high reuse factor as well as the RZF precoder and can render themselves easily for multicell massive MIMO systems. The performance of the precoding techniques for different reuse factors is tabulated in Table 1. Fig. 4 shows the impact of coordination in the multicell massive MIMO system. With the reuse factor of 4 selected, it can be observed that the coordination within multicell massive MIMO system improves the SE per cell for all the precoding schemes under consideration while holding the BS antennas and the UTs constant. This observation points to the fact that coordination in multicell massive MIMO further reduces the effect of intercell interference thereby improving the SE per cell in general. This enhancement in performance comes at the cost of increased complexity which simply points to increased hardware requirement. Fig. 5 shows the total CP against the number of BS antennas. The CP was computed as per the values in Tables 2 and 3. It can be observed that as the number of BS antennas increase, the total CP increases. Which is expected since the increase in BS antennas simply means more circuit to consume power. However, the RNA based precoding has a high CP consumption as compared to TPE precoding and RZF precoding which have equal CP.
Figs. 6 and 7 depict how the EE varies with the throughput. From the two figures, it can be observed that the EE increase with an increase in throughput up to a certain point then the EE starts decreasing with increasing throughput. The maximum EE achievable serves as an indicator on the optimal throughput that will deliver maximum EE. Consequently, this throughput corresponds to the number of BS antennas to achieve the desired EE. Thus, it can be stated that the use of all the BS antennas may not be optimal. This then brings in the idea of antenna selection method within the BS in massive MIMO system. This can even be     set to be dynamic to cater for the different fading condition for respective receive points (Table 3). Another observation is in terms of the reuse factor. The increase in reuse factor increases the maximum EE and the corresponding throughput. This is attributed to intercell interference reduction. The RNA precoder has the highest maximum EE followed by the RZF precoder. The TPE precoder has a poor EE and a very low throughput which reduces with the increase in the reuse factor. This limits its use as a precoding technique in massive MIMO.

Conclusion
The paper gives the performance analysis and comparison of the RNA-based precoder, RZF precoder and the TPE precoder for multicell downlink massive MIMO system. The performance of the three precoding schemes in terms of SE and the EE for imperfect CSI is studied. The SE and EE were derived theoretically for each of the precoding schemes under similar assumptions and for the wireless massive MIMO system.
From the simulation and the theoretical results, RNA-based precoding has higher SE and EE followed by the RZF precoding and the TPE precoder has the lowest. But when the BS antennas are increased, TPE precoder SE and EE nears that of the RZF precoder, when the reuse factor of 1 is used. With a higher reuse factor of 4, the SE and EE performance of TPE precoder is poor and cannot compare to the RNA based precoder and that of the RZF precoder. Thus, RZF precoder and RNA-based precoder have good performance at both lower and higher reuse factor as compared to TPE-precoder which only has improved performance at high number of BS antennas and a lower reuse factor of 1.
The CP consumed by the RNA based precoder is the highest but both RZF and TPE precoders have the same CP consumption. Though consuming the same CP as the RZF precoding, TPE precoder has poor SE and EE. This points out that TPE precoder is an inefficient technique to be realised in multicell massive MIMO systems. The RNA based precoder has high CP, but its performance is superior and can justify its implementation in multicell massive MIMO systems.