On the Required Number of Antennas in a PointtoPoint LargebutFinite MIMO System: OutageLimited Scenario
Abstract
This paper investigates the performance of the pointtopoint multipleinputmultipleoutput (MIMO) systems in the presence of a large but finite numbers of antennas at the transmitters and/or receivers. Considering the cases with and without hybrid automatic repeat request (HARQ) feedback, we determine the minimum numbers of the transmit/receive antennas which are required to satisfy different outage probability constraints. Our results are obtained for different fading conditions and the effect of the power amplifiers efficiency on the performance of the MIMOHARQ systems is analyzed. Moreover, we derive closedform expressions for the asymptotic performance of the MIMOHARQ systems when the number of antennas increases. Our analytical and numerical results show that different outage requirements can be satisfied with relatively few transmit/receive antennas.
I Introduction
The next generation of wireless networks must provide data streams for everyone everywhere at any time. Particularly, the data rates should be orders of magnitude higher than those in the current systems; a demand that creates serious power concerns because the data rate scales with power monotonically. The problem becomes even more important when we remember that currently the wireless network contributes of global emissions and its energy consumption is expected to increase every year [1].
To address the demands, the main strategy persuaded in the last few years is the network densification [2]. One of the promising techniques to densify the network is to use many antennas at the transmit and/or receive terminals. This approach is referred to as massive or large multipleinputmultipleoutput (MIMO) in the literature.
In general, the more antennas the transmitter and/or the receiver are equipped with, the better the data rate/link reliability. Particularly, the capacity increases and the required uplink/downlink transmit power decreases with the number of antennas. Thus, the trend is towards asymptotically high number of antennas. This is specially because millimeter wave communication [3, 4], which are indeed expected to be implemented in the next generation of wireless networks, makes it possible to assemble many antennas at the transmit/receive terminals. However, large MIMO implies challenges such as hardware impairments and signal processing complexity which may limit the number of antennas in practice. Also, one of the main bottlenecks of large MIMO is the channel state information (CSI) acquisition, specifically at the transmitter. Therefore, it is interesting to use efficient feedback schemes such as hybrid automatic repeat request (HARQ) whose feedback overhead does not scale with the number of antennas.
The performance of HARQ protocols in singleinputsingleoutput (SISO) and MIMO systems is studied in, e.g., [5, 6, 7, 8] and [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], respectively. MIMO transmission with many antennas is advocated in [21, 22] where the timedivision duplex (TDD)based training is utilized for CSI feedback^{1}^{1}1The results of [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38] are mostly on multiuser MIMO networks, as opposed to our work on pointtopoint systems. However, because many of the analytical results in [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38] are applicable in pointtopoint systems as well, these works are cited.. Also,[23, 24, 25, 26, 27, 28, 29] introduce TDDbased schemes for large systems. In the meantime, frequency division duplex (FDD)based massive MIMO has recently attracted attentions and lowoverhead CSI acquisition methods were proposed [30, 31, 32]. Considering imperfect CSI, [33] derives lower bounds for the uplink achievable rate of the MIMO setups with large but finite number of antennas. Finally, [34] (resp. [35]) studies the zeroforcing based TDD (resp. TDD/FDD) systems under the assumption that the number of transmit antennas and the singleantenna users are asymptotically large while their ratio remains bounded (For detailed review of the literature on massive MIMO, see [36, 37, 38]).
To summarize, a large part of the literature on the pointtopoint and multiuser large MIMO is based on the assumption of asymptotically many antennas. Then, a natural question is how many transmit/receive antennas do we require in practice to satisfy different qualityofservice requirements. The interesting answer this paper establishes is relatively few, for a large range of outage probabilities.
Here, we study the outagelimited performance of pointtopoint MIMO systems in the cases with large but finite number of antennas. We derive closedform expressions for the required number of transmit and/or receive antennas satisfying various outage probability requirements (Theorem 1). The results are obtained for different fading conditions and in the cases with or without HARQ. Furthermore, we analyze the effect of the power amplifiers (PAs) efficiency and the antennas spatial correlation on the system performance (Sections IV.A and V.B, respectively) and study the outage probability in the cases with adaptive power allocation between the HARQ retransmissions (Section IV.B). Finally, we study the asymptotic performance of MIMO systems. Particularly, denoting the outage probability and the number of transmit and receive antennas by and respectively, we derive closedform expressions for the normalized outage factor which is defined as when the number of transmit and/or receive antennas increases (Theorem 2).
As opposed to [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], we consider large MIMO setups and determine the required number of antennas in outagelimited conditions. Also, the paper is different from [21, 22, 23, 24, 25, 26, 27, 29, 28, 30, 31, 32, 33, 35, 34, 36, 37, 38] because we study the outagelimited scenarios in pointtopoint systems, implement HARQ and the number of antennas is considered to be finite. The differences in the problem formulation and the channel model makes the problem solved in this paper completely different from the ones in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 28, 30, 31, 32, 33, 35, 34, 36, 37, 38], leading to different analytical/numerical results, as well as to different conclusions. Finally, our discussions on the asymptotic outage performance of the MIMO setups and the effect of PAs on the performance of MIMOHARQ schemes have not been presented before.
Our analytical and numerical results indicate that:

Different qualityofservice requirements can be satisfied with relatively few transmit/receive antennas. For instance, consider a SIMO (S: single) setup without HARQ and transmission signaltonoise ratio (SNR) dB. Then, with the data rate of 3 natsperchannel use (npcu), the outage probabilities , and are guaranteed with and receive antennas, respectively (Fig. 5a). Also, the implementation of HARQ reduces the required number of antennas significantly (Fig. 3).

Considering the moderate/high SNRs, the required number of transmit (resp. receive) antennas scales with and linearly, if the number of receive (resp. transmit) antennas is fixed. Here, is the maximum number of HARQ retransmission rounds, is the number of channel realizations experienced in each round, denotes the SNR, is the outage probability constraint and represents the inverse Gaussian function. These scaling laws are changed drastically, if the numbers of the transmit and receive antennas are adapted simultaneously (see Theorem 1 and its following discussions for details).

For different fading conditions, the normalized outage factor converges to constant values, unless the number of receive antennas grows large while the number of transmit antennas is fixed (see Theorem 2). Also, for every given number of transmit/receive antennas, the normalized outage factor increases with linearly.

There are mappings between the performance of MIMOHARQ systems in quasistatic, slow and fastfading conditions, in the sense that with proper scaling of the channel parameters the same outage probability is achieved in these conditions. This point provides an appropriate connection between the papers considering one of these fading models.

Adaptive power allocation between the HARQ retransmissions leads to marginal antenna requirement reduction, while the system performance is remarkably affected by the inefficiency of the PAs. Finally, the spatial correlation between the antennas increases the required number of antennas while, for a large range of correlation conditions, the same scaling rules hold for the uncorrelated and correlated fading scenarios.
Ii System Model
Consider a pointtopoint MIMO setup with transmit antennas and receive antennas. We study the blockfading conditions where the channel coefficients remain constant during the channel coherence time and then change to other values based on their probability density function (PDF). In this way, the received signal is given by
(1) 
where is the fading matrix, is the transmitted signal and denotes the independent and identically distributed (IID) complex Gaussian noise matrix. The results are mainly given for IID Rayleighfading channels where each element of the channel matrix H follows a complex Gaussian distribution (To analyze the effect of the antennas spatial correlation, see Fig. 7 and Section V.B). The channel coefficients are assumed to be known at the receiver which is an acceptable assumption in blockfading channels [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. On the other hand, there is no CSI available at the transmitter except the HARQ feedback bits. The feedback channel is supposed to be delay and errorfree.
As the most promising HARQ approach leading to highest throughput/lowest outage probability [5, 6, 14, 9], we consider the incremental redundancy (INR) HARQ with a maximum of retransmissions, i.e., the message is retransmitted a maximum of times. Note that setting represents the cases without HARQ, i.e., openloop communication. Also, a packet is defined as the transmission of a codeword along with all its possible retransmissions. We investigate the system performance for three different fading conditions:

Fastfading. Here, it is assumed that a finite number of channel realizations are experienced within each HARQ retransmission round.

Slowfading. In this model, the channel is supposed to change between two successive retransmission rounds, while it is fixed for the duration of each retransmission.

Quasistatic. The channel is assumed to remain fixed within a packet period.
Fastfading is an appropriate model for fastmoving users or users with long codewords compared to the channel coherence time [39, 14]. On the other hand, slowfading can properly model the cases with users of moderate speeds or frequencyhopping schemes [10, 11, 12, 15, 16, 17]. Finally, the quasistatic represents the scenarios with slowmoving or stationary users, e.g., [5, 18, 19, 20, 9, 17].
Iii Problem Formulation
Considering the INR HARQ with a maximum of retransmissions, information nats are encoded into a parent codeword of length channel uses and the codeword is divided into subcodewords of length . In each retransmission round, the transmitter sends a new subcodeword and the receiver combines all signals received up to the end of that round. Thus, the equivalent rate at the end of round is npcu where denotes the initial transmission rate. The retransmissions continue until the message is correctly decoded by the receiver or the maximum permitted retransmission round is reached.
Let us denote the determinant and the Hermitian of the matrix X by and , respectively. Assuming fastfading conditions with independent fading realizations in the th round and an isotropic Gaussian input distribution over all transmit antennas, the results of, e.g., [40, Chapter 15], [41, Chapter 7], can be used to find the outage probability of the INRbased MIMOHARQ scheme as
(2) 
Here, is the total transmission power and is the transmission power per transmit antenna (in dB, we have which, because the noise variance is set to 1, represents the SNR as well). Also, represents the identity matrix.
Considering in (2), the outage probability is rephrased as
(3) 
in a slowfading channel. Also, setting the outage probability in a quasistatic fading channel is given by
(4) 
Using (2)(4) for given initial transmission rate and SNR, the problem formulation of the paper can be expressed as
(5) 
Here, denotes an outage probability constraint and are the minimum numbers of transmit/receive antennas that are required to satisfy the outage probability constraint. In the following, we study (5) in four distinct cases:

Case 1: is large but is given.

Case 2: is given but is large.

Case 3: Both and are large and the transmission SNR is low.

Case 4: Both and are large and the transmission SNR is high.
It is worth noting that the three first cases are commonly of interest in large MIMO systems. However, for the completeness of the discussions, we consider Case 4 as well. Moreover, in harmony with the literature [34, 35]^{2}^{2}2In [34, 35], which study multiuser MIMO setups, and are supposed to follow (6) while, as opposed to our work, they are considered to be asymptotically large., we analyze Cases 34 under the assumption
(6) 
with being a constant. However, as seen in the following, it is straightforward to extend the results of the paper to the cases with other relations between the numbers of antennas.
Iv Performance Analysis
To solve (5), let us first introduce Lemma 1. The lemma is of interest because it represents the outage probability as a function of the number of antennas, and simplifies the performance analysis remarkably.
Lemma 1: Considering Cases 14, the outage probability of the INRbased MIMOHARQ system is given by
(7) 
where is the Gaussian function and for different cases and are given in (8).
Proof.
The proof is based on (2)(4) and [42, Theorems 13], where considering Cases 14 the random variable converges in distribution to a Gaussian random variable which, depending on the numbers of antennas, has the following characteristics
(8) 
In this way, from (2) and for different cases, the outage probability in fastfading condition is given by
(9) 
where, because is the average of independent Gaussian random variables , we have . Consequently, using the cumulative distribution function (CDF) of the Gaussian random variables, the outage probability of the fastfading condition is given by (7.i). The same arguments can be applied to derive (7.iiiii) in the slowfading and quasistatic conditions. ∎
Lemma 1 leads to the following corollaries:

For Cases 14, using INR MIMOHARQ in the quasistatic, slow and fastfading channels leads to scaling the variance of the equivalent random variable by , and , respectively. That is, using HARQ, there exists mappings between the quasistatic, the slow and the fastfading conditions in the sense that with proper scaling of in (7) they lead to the same outage probability.

With asymptotically large numbers of transmit and/or receive antennas, the optimal data rate which leads to zero outage probability and maximum throughput is given by with derived in (8); Interestingly, the result is independent of the fading condition. Also, with asymptotically high number of antennas and no HARQ is needed because the message is decoded in the first round (with probability 1).
Using Lemma 1, the minimum numbers of antennas satisfying different outage probability constraints are determined as stated in Theorem 1.
Theorem 1: The minimum numbers of the transmit and/or receive antennas in an INRbased MIMOHARQ system that satisfy the outage probability constraint are given by
(10) 
if the channel is fastfading. Here, and denote the inverse function and the Lambert W function, respectively. For the slowfading and quasistatic conditions, the minimum numbers of the antennas are obtained by (10) where the term is replaced by and , respectively.
Proof.
Considering Lemma 1 and a fastfading condition, (5) is rephrased as
(11) 
which for different cases leads to
Case 1:
(12) 
Case 2:
(13) 
Case 3: where
(14) 
In (IV), is obtained by using the approximation for large ’s and variable transform Also, denotes the Lambert W function defined as [43]. Note that the Lambert W function has an efficient implementation in MATLAB and MATHEMATICA.
For Case 4, we consider two scenarios and use the following approximations.
Case 4 with : Then, and
(15) 
Here, is obtained by implementing the Riemann integral approximation in the first three summation terms. Then, follows from some manipulations, the fact that is assumed large, and for large ’s.
Case 4 with : Then, and
(16) 
where and are obtained with the same procedure as in (IV)^{3}^{3}3We can follow the same procedure as in (IV)(IV) to write (17) in the cases with which can be solved numerically via, e.g., “’fsolve” function of MATLAB or by different approximation schemes. However, for simplicity and because it is a special condition, we do not consider as a separate case.. Finally, note that with slowfading and quasistatic fading channel (11) is rephrased as
(18) 
and
(19) 
respectively. Therefore, as stated in the theorem, with slowfading and quasistatic channel the required numbers of the transmit and/or receive antennas are determined by (10) while the term is replaced by and , respectively. ∎
According to Theorem 1, the following conclusions can be drawn:

Using the tight approximation in (10), the required number of receive antennas in Case 1 is rephrased as
(20) where the last approximation holds for moderate/high SNRs. Thus, at moderate/high SNR regimes, the required number of receive antennas increases with linearly. On the other hand, the required number of receive antennas is inversely proportional to the number of experienced fading realizations the number of transmit antennas , and . Interestingly, we can use (10.Case 2) to show that at high SNRs the same scaling laws hold for Cases 1 and 2. That is, in Case 2, the required number of transmit antennas decreases (resp. increases) with , and (resp. ) linearly.

The same scaling laws are valid in Cases 3 and 4, i.e., when the numbers of transmit and receive antennas increase simultaneously. For instance, the required number of antennas increases with and the code rate semilinearly^{4}^{4}4The variable is semilinear with if for given constants and (see (10.Cases 34)). At hard outage probability constraints, i.e., small values of , the required number of antennas decreases with the number of retransmissions according to On the other hand, the number of antennas decreases with linearly when the outage constraint is relaxed, i.e., increases. The only difference between Cases 3 and 4 is that in Case 3 (resp. Case 4) the number of antennas decreases with (resp. ) linearly.

It has been previously proved that at low SNRs the same performance is achieved by the MIMO systems using INR and repetition time diversity (RTD) HARQ [14, Section V.B]. Thus, although the paper concentrates on the INR HARQ, the same number of antennas are required in the MIMORTD setups, as long as the SNR is low.
As the number of antennas increases, the CDF of the accumulated mutual information, e.g., in fastfading conditions, tends towards the step function. Therefore, depending on the SNR and the initial rate, the outage probability rapidly converges to either zero or one as the number of transmit and/or receiver antennas increases. To further elaborate on this point and investigate the effect of the number of antennas, we define the normalized outage factor as
(21) 
Intuitively, (21) gives the negative of the slope of the outage probability curve plotted versus the product of the numbers of transmit/receive antennas. Also, (21) follows the same concept as in the diversity gain [9, Eq. 14] which is an efficient metric for the asymptotic analysis of the MIMO setups. Theorem 2 studies the normalized outage factor in more details.
Theorem 2: For Cases 14, different fading conditions and appropriate initial rates/SNR, the normalized outage factor is approximated by (24).
Proof.
With given initial transmission rate and SNR , we use the approximation
(22) 
for large ’s and (7) to rewrite the normalized outage factor (21) as
(23) 
Then, from (8), the normalized outage factor in different cases is found as
(24) 
where is found by following the same approach as in (IV)(IV). Finally, note that to use (22) the initial rate and the SNR should be such that for the considered number of antennas. Otherwise, the outage probability converges to 1 and ∎
Interestingly, the theorem indicates that:

The normalized outage factor becomes constant in all cases, except Case 1 with a given (resp. large) number of transmit (resp. receive) antennas. Intuitively, this is because in all cases (except Case 1) the power per transmit antenna decreases by increasing the number of transmit antennas. Therefore, there is a tradeoff between increasing the diversity and reducing the power per antenna and, as a result, the normalized outage factor converges to the values given in (24). In Case 1, however, the message decoding probability is always increased by increasing the number of receive antennas and, as seen in Theorem 2, the normalized outage factor increases with monotonically, as long as .

In Case 3, the normalized outage factor becomes independent of the transmission SNR as long as . In Cases 1, 2 and 4, on the other hand, the normalized outage factor scales with the SNR according to if the SNR is high.

In cases 34, the normalized outage factor does not depend on the initial transmission rate. Moreover, in Case 3 the normalized outage factor is independent of the ratio between the number of transmit and receive antennas.
Iva On the Effect of Power Amplifiers
As the number of the transmit antennas increases, it is important to take the efficiency of radiofrequency PAs into account [36, 37, 38]. For this reason, we use Lemma 1 to investigate the system performance in the cases with nonideal PAs as follows.
It has been previously shown that the PA efficiency can be written as [44, 45], [46, Eq. (3)] and [47, Eq. (3)]
(25) 
Here, and are the output, the maximum output, and the consumed power of the PA, respectively, denotes the maximum power efficiency achieved at , and is a parameter that, depending on the PA classes, varies between . In this way, and because the INRbased MIMOHARQ setup can be mapped into an equivalent SISOHARQ system (see Lemma 1 and its following discussions), the equivalent mean and variances (8) are rephrased as
(26) 
in the cases with nonideal PAs. This is the only modification required for the nonideal PA scenario and the rest of the analysis remains the same as before.
IvB On the Effect of Power Allocation
Throughout the paper, we studied the system performance assuming a peak power constraint at the transmitter. However, the system performance is improved if the transmission powers are updated in the HARQ retransmission rounds.
Let the transmission power in the th round be . Then, the outage probability in the fastfading condition^{5}^{5}5For simplicity, the results of this part are given mainly for the fastfading condition. It is straightforward to extend the results to the cases with other fading models., i.e., (2), is rephrased as
(27) 
where and are obtained by replacing into (8). Here, is obtained by . Also, is based on the fact that the sum of independent Gaussian random variables is a Gaussian random variable with the mean and variance equal to the sum of the variables means and variances, respectively.
If the message is correctly decoded in the th round, the total transmission energy and the total number of channel uses are and , respectively. Also, the total transmission energy and the number of channel uses are and if an outage occurs, where all possible retransmission rounds are used. Thus, we can follow the same procedure as in [5, 14, 6] to find the average power, defined as the expected transmission energy over the expected number of channel uses, as