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Finite-SNR outage analysis for MIMO channels with imperfect channel state information
Accepted Manuscript Finite-SNR outage analysis for MIMO channels with imperfect channel state information Nandita Lavanis, Devendra Jalihal, Arun Pachai Kannu, Srikrishna Bhashyam PII: DOI: Reference:
S1874-4907(17)30010-1 http://dx.doi.org/10.1016/j.phycom.2016.12.005 PHYCOM 354
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Physical Communication
Received date: 17 March 2016 Revised date: 9 November 2016 Accepted date: 28 December 2016 Please cite this article as: N. Lavanis, D. Jalihal, A.P. Kannu, S. Bhashyam, Finite-SNR outage analysis for MIMO channels with imperfect channel state information, Physical Communication (2017), http://dx.doi.org/10.1016/j.phycom.2016.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Finite-SNR Outage Analysis for MIMO Channels with Imperfect Channel State Information Nandita Lavanisa,1,∗, Devendra Jalihala , Arun Pachai Kannua , Srikrishna Bhashyama a
Dept. of Electrical Engineering, IIT Madras, Chennai, India
Abstract In this paper, a point-to-point multiple-input multiple-output (MIMO) channel with imperfect channel state information (CSI) at the receiver and no CSI at the transmitter is considered. Using Monte Carlo simulations, we compute the optimum number of active antennas required at the transmitter (topt ) to minimize the outage probability. We show that, apart from the number of transmit antennas, topt depends on the signal to noise ratio (SNR), multiplexing gain, coherence time, and the number of receive antennas. Our results give insights on the behavior of topt with respect to these parameters. Specifically, we show that as the multiplexing gain increases, the value of topt increases from one, and as the multiplexing gain reaches its maxima, the value of topt equals the minimum of the number of transmit and receive antennas. The intermediate behavior of topt with respect to multiplexing gain depends on the MIMO channel configuration. topt for the MIMO channel with perfect CSI at the receiver follows a similar pattern as that with imperfect CSIR. For a multiple-input single-output (MISO) channel with imperfect CSIR, we obtain a tight upper bound on the outage probability. Using this analytical upper bound, we can calculate topt for any fixed channel configuration. For a MISO channel with imperfect CSIR and fixed SNR, topt reduces as multiplexing gain increases; however, for fixed multiplexing gain and fixed SNR, topt monotonically increases with increase in coherence time of the channel. Corresponding author Email addresses: [email protected] (Nandita Lavanis), [email protected] (Devendra Jalihal), [email protected] (Arun Pachai Kannu), [email protected] (Srikrishna Bhashyam) 1 The author is currently a faculty at SSN College of Engineering, Chennai, India ∗
Preprint submitted to Physical Communication
November 9, 2016
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Keywords: Channel estimation, Non-coherent communication, Optimum number of antennas, Outage probability, Training signals 1. Introduction Multiple antennas are quite common in modern wireless systems due to their advantages of higher spectral efficiency and higher reliability compared to single antenna systems. Ergodic capacity is a metric of relevance in fast fading scenarios where the coherence time of the channel is small compared to the time duration over which communication happens. Ergodic capacity of multiple-input multiple-output (MIMO) channels is analyzed in [1] and [2] assuming perfect channel state information (CSI) at the receiver and no CSI at the transmitter for i.i.d. Rayleigh channel model. It is assumed in [1] that the transmitter knows the distribution of channel statistics. In [1], a formula for MIMO capacity is derived with perfect CSI and i.i.d Rayleigh fading, and the concept of outage is introduced. For slow fading channels, the outage probability and diversity-multiplexing-tradeoff (DMT) are relevant metrics. When the data rate of communication exceeds channel capacity for a given realization of the fading channel, the probability of detection error becomes high and the outage event occurs. In realistic communication environments, it is difficult to obtain perfect CSI. This leads to the case in which both the transmitter as well as the receiver do not have CSI, but the receiver is allowed to estimate CSI. This is also termed as the non-coherent communication scenario [3]. In the literature, a block-fading assumption is used. Quasi-static or block-fading channel implies that the channel fading is constant for a block and independently changes in the next block. Since capacity achieving distributions are not i.i.d., three kinds of scenarios are considered in the literature [4] for noncoherent analysis. In the first scenario, unitary space-time modulation is considered [5], and a lower bound on non-coherent capacity is obtained. In the second scenario, either the high-SNR or low-SNR limit is considered and the non-coherent capacity is obtained. High-SNR approximations are provided in [3] and [6]. In [7], the outage probability analysis is performed using a high-SNR assumption. In [8], number of transmit antennas required to maximize the diversity multiplexing tradeoff is analyzed. In the low SNR regime, [9] computes the capacity and analyzes its dependence on various 2
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parameters such as the coherence time, number of antennas etc. Similarly, in [10], capacity and optimal input are provided in the low SNR case. In the last scenario, channel estimation is performed and achievable rate and analytical bounds are computed for the non-coherent channel using the specific estimation scheme. The non-coherent capacity analysis is performed in [3] and [11] to provide channel estimates under a block-fading assumption by assuming a similar training-based channel estimation scheme. Though no closed-form expression of non-coherent capacity is available in literature, [3] provides a high-SNR expression of the same, while [11] computes a lower bound using the particular channel estimation scheme and optimize it. While [11] uses one-shot channel estimates, the channel estimation and hence the lower bound on the capacity is improved by [4] in a large MIMO setting by using successive decoding strategy for performing channel estimation. A lower bound on the spectral efficiency of a non-coherent MIMO channel is evaluated in a large-system limit in [12] using a randomly-biased QPSK signalling. In this paper, we consider non-coherent MIMO channels belonging to the third scenario where the training-based scheme from [11] is used to perform channel estimation. In [11], Hassibi and Hochwald obtained a lower bound on the capacity of non-coherent channels by bounding the achievable rate of a training-based scheme. The training sequences, training power, and training duration were optimized to maximize a capacity lower bound. It was observed in [11] and [3] that at high SNR it is optimal to use min(MT , MR , ⌊ T2 ⌋) antennas to maximize the capacity lower bound and diversity gain, respectively. In [4, 12], a training-based scheme that is different from the scheme in [11] is studied and the capacity is again studied in the large MIMO setting. Although we use the same training-based scheme as in [11], our work on the optimal number of antennas in non-coherent MIMO systems differs from prior work in the following ways: (1) We consider minimizing outage probability instead of maximizing capacity. (2) We consider the finite moderate SNR regime instead of the high or low SNR regimes studied in prior work. (3) We use the finite-SNR multiplexing gain as defined in [13]. Our results show that at finite SNR, the optimal number of antennas can be different from the high SNR result of min(MT , MR , ⌊ T2 ⌋) antennas. The dependence of this optimal number of antennas on multiplexing gain, SNR, and coherence time is studied here. Our contributions in this paper are as follows: 1) We show that in the finite-SNR regime, the optimal number of antennas can be different from 3
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M ∗ = min(MT , MR , ⌊ T2 ⌋), with varying SNR, as against [8] in which it is shown that these remain fixed at M ∗ in the asymptotically high-SNR regime. 2) Using Monte Carlo simulations, we obtain the optimal number of active transmit antennas (topt ) that minimizes the outage probability and we analyze the same for multiple-input single-output (MISO) and MIMO channels with/without perfect CSIR. 3) For non-coherent MIMO channels, our results show the following: (a) For MT > MR , as the multiplexing gain increases, the value of topt initially increases from unity to maximum and then decreases before settling to tMIMO = MR when the multiplexing gain tends to its maxopt imum value. (b) For MT ≤ MR and with increasing multiplexing gain, the value of topt increases and settles to tMIMO = MT when the multiplexing gain opt tends to its maximum value. 4) For non-coherent MISO channels, we provide an analytical expression for an upper bound on the outage probability of a communication scheme employing training-based channel estimation. The analytical bound is shown to be tight for moderate to high SNR values. The paper is organized as follows: In Section 2, we provide the details of the training-based communication scheme. In Section 3, we derive the analytical expression for an upper bound on the outage probability. In Section 4, we provide numerical results characterizing the outage probability, its analytical upper bound and the behavior of optimal number of active transmit antennas. In Section 5, we present the conclusions. 2. System model and related definitions A quasi-static, frequency-flat, MIMO channel with MT transmit and MR receive antennas with Rayleigh fading is described by √ ρ Y= HX + W. (1) MT In (1), the channel coefficients in the matrix H ∈ C MR ×MT are zero-mean, i.i.d., circularly symmetric complex Gaussian with unit variance. X ∈ C MT ×T denotes the transmitted signal, Y ∈ C MR ×T denotes the received signal and W ∈ C MR ×T denotes the noise. The transmitted signal X is normalized to have average transmit power at each antenna to be unity. T denotes the coherence interval over which the channel coefficients are assumed to remain unchanged, i.e., the block fading model is used. The entries in the noise matrix W are zero-mean, i.i.d., circularly symmetric complex Gaussian with 4
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unit variance. For this model, it follows that ρ is the average SNR at each receive antenna. It should be noted that (1) serves as a model for MIMO channels with/without perfect CSIR. In case of imperfect CSIR, a training scheme is considered as described in the next subsection. 2.1. Details of the training scheme The channel estimation scheme from [11] designed for the MIMO channel in (1) is used. The assumptions at the transmitter are as follows. Since CSIT is not available, the transmitter does not have knowledge of the instantaneous SNR or the channel gains, whereas it has knowledge of the average value of received SNR and the probability density function (pdf) of the channel coefficients. Equal power allocation is assumed across all the transmit antennas. The block of T symbols is divided such that a deterministic training signal Xτ ∈ C MT ×Tτ is sent in the initial Tτ duration and the subsequent Td = (T −Tτ )+ duration is used for sending data Xd ∈ C MT ×Td . Using the observations corresponding to the training duration Yτ ∈ C MT ×Tτ , the receiver determines the minimum mean squared error (MMSE) estimate of the chanˆ = E{H|Yτ , Xτ }. Since H and W are Gaussian and independent, nel as H ˆ is the MMSE estimator is identical to the linear MMSE estimator. Since H ˆ ∗ H)] ˆ ˆ E[tr(H 2 ¯ zero mean, the variance of each of its entries is σHˆ = MT MR . Let H = σHˆ . H The training signal matrix Xτ is assumed to be a multiple of a matrix with orthonormal columns. In this case, it has been shown in [11] that the MSE is ¯ are i.i.d. complex Gaussian with unit variance. minimal and the entries in H ¯H ¯ ∗ is the This result will be used in further analysis. Furthermore, W = H ∗ ¯ H ¯ is the Wishart matrix for Wishart matrix for MR < MT and W = H MT < MR . 2.2. Outage probability of the training Scheme Let the power allocated in the training phase and the data phase be ρτ and ρd , respectively. The fraction of energy allocated to the data signal is denoted by α, hence ρd =
αρT , Td
ρT = ρτ Tτ + ρd Td .
(2)
ˆ we perform the coherent minimum distance Using the channel estimate H, decoding, ignoring the channel estimation error. The data rate that can be
5
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achieved using this strategy is specified by the following lower bound on the mutual information [14, 11] ˆ = ILMIMO (X; Y|H)
(T − Tτ ) ( ρ˜ ¯ ∗ ¯ ) log det(IMT + H H) , T MT
where ρ˜ is the effective SNR defined as ρd ρτ . ρ˜ = 1 + ρd + ρτ
(3)
(4)
Let R be the data rate of communication. Since apriori channel state information (CSI) is not available, it is possible that R is greater than the mutual information bound in (3). We define the outage probability as Pout,MIMO = P(ILMIMO < R).
(5)
The multiplexing gain r at finite-SNR [13] with data rate R, SNR ρ, array gain g is R r= . (6) log2 (1 + gρ) For the MIMO system, g is chosen equal to MR [13]. The multiplexing gain r gives an indication about the sensitivity of the rate adaptation policy w.r.t SNR. Note that outage probability in (5) depends on ρ, r, MT , MR and T . For brevity in notations, we do not show this dependency explicitly. The diversity gain d(r, ρ) at a multiplexing gain r and an SNR ρ, as defined in [13], is ∂ d(r, ρ) = −ρ ln Pout,MIMO (r, ρ), (7) ∂ρ where we have explicitly denoted the dependence of outage probability on r and ρ. 2.3. MIMO channel with perfect CSIR In case of perfect CSIR, assuming an uncorrelated input on all the antennas, the mutual information [1] is ( ρ ∗ ) H H) . (8) IMIMO,per (X; Y|H) = log det(IMT + MT
The corresponding outage probability from (5) and (6) is
Pper out,MIMO = P(IMIMO,per < r log2 (1 + gρ)). 6
(9)
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3. Outage Probability Analysis Previous studies of non-coherent MIMO channels in the high SNR regime have shown that the optimal number of active transmit antennas which maximizes the ergodic channel capacity [11] or the diversity-multiplexing tradeoff [7] depends on various parameters such as coherence interval and number of receive antennas. In this paper, we are interested in minimizing the outage probability Pout,MIMO in the finite SNR regime. In this regard, let t ∈ {1, · · · , MT } denote the number of active antennas used by our training scheme. In order to estimate the channel coefficients from t transmit antennas properly (i.e., guarantee zero estimation error in the absence of noise), we need that the training duration Tτ ≥ t. However, increasing the training duration beyond t decreases the data duration. Hence, we choose Tτ = t. In [11], it has been shown that the choice of Tτ = t maximizes the ergodic achievable rates E(ILMIMO ). We also use the power allocation fraction α given below, which maximizes L E(IMIMO ) [11], √ ν − ν(ν − 1), for Td > t; 1 , for Td = t; (10) αo = 2 √ ν + ν(ν − 1), for Td < t, where ν =
(t+ρT ) . ρT (1− Tt )
Subsequently, we analyze the behavior of Pout,MIMO
d
with respect to various parameters such as r, ρ, t, MR and T . For the coherent MT × MR MIMO with perfect CSI, the pre-log factor of the ergodic channel capacity is min(MT , MR ). For the non-coherent case, we spend only T − Tτ duration for the data. With t active transmit antennas, Tτ = t and hence the multiplexing gain in our non-coherent MIMO model is bounded as (T − t) 0 ≤ r ≤ min(t, MR ) ≡ rmax . (11) T We define, q ≡ min{t, MR }, n ≡ max{t, MR }. Substituting αo (optimal fraction of power allocation) from (10) in (2), the effective SNR in (4) is written as ζ1 ρ2 , ζ1 = αo (1 − αo )T 2 , ζ2 = T 2 − T t − αo T 2 + 2tαo T. (12) tTd + ζ2 ρ ) ( rT We define threshold SNR as ρth ≡ (1 + gρd ) (q(T −t)) − 1 , using ρd from (2) ρ˜(ρ, T, t) =
7
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the threshold SNR is ) ( gT αo ρ (q(TrT−t)) ) −1 . ρth (ρ, r, T, t, MR ) = (1 + T −t
(13)
Using (12) and (13), define ϕ as the ratio of threshold SNR to effective SNR as ϕMIMO (ρ, r, t, T, MR ) ≡
ρth . ρ˜
(14)
We utilize the incomplete gamma function γ(a, x) [15] in the following theorem, which is written as a−1 j ∑ ( x ) γ(a, x) = Γ(a) 1 − e−x . j! j=0
(15)
Theorem 1. With t active antennas and MR = 1, the outage probability defined in (5) is upper bounded as Pout,MISO ≤ PUout,MISO where U Pout,MISO =
γ(t, tϕMISO ) . Γ(t)
(16)
Proof: For t transmit antennas, the lower bound on the MIMO mutual information in (3) is T − t( ρ˜ ¯ ∗ ¯ ) T − t ( ρ˜ ¯ ¯ ∗ ) log det(It + H H) = log det(IMR + H H) . T t T t (17) MR ×t ¯ ∗ t×MR ¯ In (17), H ∈ C , H ∈C . Define { ¯H ¯ ∗ , if MR < t then W ∈ C MR ×MR ; H (18) W= ¯ ∗ H, ¯ if MR ≥ t then W ∈ C t×t . H
L ˆ = IMIMO (X; Y|H)
W has a complex Wishart distribution [1], [16]. From (18), W ∈ C q×q . From (17) and (18), the lower bound on the MIMO mutual information is rewritten as, ( ) L ˆ = (T − t) log det(Iq + ρ˜W) (X; Y|H) IMIMO T t 8
(19)
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q
(T − t) ∑ ( ρ˜ ) = log 1 + λi , (20) T t i=1 ( ) ρe = log det(It + h∗ h) , for MR = 1. (21) t In (20), λ′i s denote the ordered eigenvalues of the complex Wishart matrix W( [16] with λq being the smallest. From (5) )and (20), we have Pout,MIMO = ( ) ∑ oT ) . The minimum eigenvalue of P (TT−t) qi=1 log 1 + ρt˜ λi < r log(1 + gρα T −t the complex Wishart matrix W denoted by λq is used to compute an upper bound of the outage probability. ϕMIMO defined in (14) is used. ( (T − t) ( ρ˜ ) gραo T ) Pout,MIMO ≤ Pr q log 1 + λq < r log(1 + ) T t T −t (( ) ) (q(TrT−t)) ( ) oT t 1 + gρα − 1 ( ) T −t ≤ Pr λq < ≤ Pr λq < tϕMIMO . ρ˜ (22) For the MISO channel, from (21), we have (( ) ( ρ˜ rT αo ρT )) U Pout,MISO = Pr log det(It + h∗ h) < log 1 + g , t T −t T −t
(23)
where ρd is defined in (2). In (23), h∗ h has a χ2 -distribution with 2t degrees of freedom. Following [15], t−1
U Pout,MISO (r, ρ, t, T, α) =
∑ (tΦM ISO )k γ(t, tΦM ISO ) = 1 − e−tΦM ISO , (24) Γ(t) k! k=0
where ΦM ISO = ΦM IM O |MR =1 . In (24), γ(a, x) is an incomplete gamma function defined in (15).
Remark 1. For the non-coherent single-input multiple-output (SIMO) channel with MT = 1, the outage probability is derived in a similar way, and is given by PUout,SIMO (ρ, r, T, MR ) =
γ(MR , ϕSIMO ) , Γ(MR )
where ϕSIMO is obtained by substituting MT = 1 in (14). 9
(25)
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Remark 2. For the non-coherent MIMO channel, we define the optimum number of transmit antennas as the active number of antennas which minimizes the outage probability as tMIMO (ρ, r, T, MR , MT ) = arg min Pout,MIMO , opt 1≤t≤MT
(26)
where Pout,MIMO is defined in (5). Similarly, using (9) for the MIMO channel with perfect CSIR, we define the optimum number of transmit antennas as the active number of antennas as per tMIMO opt,per (ρ, r, MR , MT ) = arg min Pout,MIMO , 1≤t≤MT
(27)
where Pper out,MIMO is defined in (9). Remark 3. For the MIMO channel, from (20) we compute two upper bounds on the outage probability. They are based on the following bounds based on the minimum and maximum eigenvalue. The bound based on the minimum eigenvalue (λq ) is computed from (20) using the following inequality q
( (T − t) ∑ ( ρ˜ ) q(T − t) ρ˜ ) log 1 + λi ≥ log 1 + λq . T t T t i=1
(28)
Further, from (22) using the pdf of the minimum eigenvalue of a Wishart matrix, an upper bound on the outage probability is computed as ( ) γ n − q + 1, tq2 ϕMIMO U1 Pout,MIMO (r, ρ, t, T, MR ) = . (29) Γ(n − q + 1) 1 PUout,MIMO is not tight due to bounding of the eigenvalues by q times the minimum eigenvalue. Similarly, the bound based on the maximum eigenvalue (λ1 ) is computed from (20) using the following inequality q
( ρ˜ ) (T − t) ∑ ( ρ˜ ) (T − t) log 1 + λ1 . log 1 + λi ≥ T t T t i=1
(30)
Using the pdf the maximum eigenvalue of a Wishart matrix given by [17], the upper bound on the outage probability is computed as 2 PUout,MIMO (r, ρ, t, T, MR ) = cn,q det G(ϕMIMO ).
10
(31)
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In (31), G(ϕMIMO ) is q × q Hankel(∏ matrix given∏by G(ϕMIMO ) ) = [γ(n − q + q n q MIMO i + j − 1, ϕ )]i,j=1 and cn,q = i=1 (q − i)! i=1 (n − i)! . It is observed that the upper bounds in (29), (31) plotted in Fig. 5 are not tight for t > 1. This is because the lower bounds in (28) and (30) are not tight. U1 2 We can find a bound PUout,MIMO = min{Pout,MIMO , PUout,MIMO }. PUout,MIMO is also plotted in Fig. 5 along with (29), (31) and it is observed that this minimum outage bound significantly improves for lower multiplexing gains. However, PUout,MIMO is still not tight and hence is not used to compute topt in the MIMO case. Instead Monte Carlo simulation of (5) is used to compute topt . 4. Numerical Results and Discussion We numerically study the behavior of outage probability bound PUout,MISO and PUout,MIMO for various values of multiplexing gain r, SNR (ρ), and coherence interval T . We also characterize the behavior of the optimal number of active antennas tMIMO that minimizes the outage probability with/without opt perfect CSIR. Monte Carlo simulations are used to compute the outage probability in (5), the minimization of which leads to tMIMO . For performing opt simulations, either a 2 × 4 or a 4 × 2 MIMO system operating on an SNR range of 5 to 30 dB is considered. The coherence interval is assumed to be in the range 0 < T ≤ 200 and the constraint of T > 2MT is used as given by Hassibi in [11]. 4.1. MISO channel U For the non-coherent 4 × 1 MISO channel, Pout,MISO in (23) is numerically computed and plotted for various multiplexing gains in Fig. 1 with the number of active antennas varied from 1 to 4. For comparison, we also plot the Monte-Carlo simulation of outage probability Pout in (5). We see that our outage probability upper bound is tight even for moderate values of SNR (20 dB). From this plot, we also note that the optimal number of active transmit antennas changes with the value of the multiplexing gain. We explicitly plot versus multiplexing gain in Fig. 2. As the r increases, the value of tMISO opt U the number of transmit antennas which minimizes Pout,MISO monotonically decreases. This can be interpreted in the following way. For MISO channels, the pre-log factor of the ergodic channel capacity (or the multiplexing gain) does not increase with the number of transmit antennas. On the other hand, the training overhead increases with the number of transmit antennas. Hence, 11
U
Outage probability Pout,MISO(r, ρ, T, α, t)
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−1
10
t=1 t=2 t=3 t=4 Monte Carlo simulation t=1 Monte Carlo simulation t=2 Monte Carlo simulation t=3 Monte Carlo simulation t=4
−2
10
0.1
0.2
0.3
0.4 0.5 0.6 Multiplexing gain (r)
0.7
0.8
0.9
U Figure 1: Pout,MISO and Pout,MISO versus r for ρ = 20 dB, T = 10, MT = 4, MR = 1 and for various t.
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as the multiplexing gain approaches rmax = T T−1 , the optimal value tMISO opt approach unity. U The behavior of Pout,MISO versus r in Fig. 1 and Fig. 2 is very similar to the behavior of MISO channels shown by Telatar in [1]. Telatar explains that for a MISO channel with perfect CSIR the number of antennas which minimizes the probability of outage is large for smaller data rate, whereas the number is small for larger data rate. From (6), we know that for a fixed SNR the multiplexing gain r is directly proportional to the data rate. The same behavior is observed for the MISO channel with imperfect CSIR as seen in Fig. 2, where the optimum number of antennas which minimize the outage probability reduces as r increases. It is maximum for small r and becomes minimum for large r. Hence, the observations in Fig. 2 agree with the Teletar’s conjecture [1], proved in [18], that the number of active transmit antennas decreases as the rate increases. In Fig. 3, we plot tMISO versus the channel coherence time interval T for opt fixed multiplexing gain r = 0.88 and various SNR values from moderate to high. We note that for fixed r and ρ, tMISO increases monotonically with T . opt In the Fig. 3, Tij denotes the value of T at which the topt jumps from i to j (where j is equal to i + 1). 4.2. MIMO channels 4.2.1. MT ≤ MR A comparison of optimal number of active antennas with and without perfect CSIR is plotted in Fig. 4. tMIMO obtained from the Monte Carlo opt simulations of (5) using (3) and (6) is plotted as a function of r in Fig. 4 for a 2 × 4 MIMO channel. tMIMO opt,per given by (27) is also plotted and compared. From Fig. 4, we note that tMIMO increases with increase in r in steps of one, opt and reaches the maximum value of MT . This trend is exhibited for a range of SNR values. Note that this trend is exactly the opposite of that seen in the MISO case (see Fig. 2) since tMISO reduces with increase in r. For the opt MIMO MIMO channel with perfect CSIR, topt,per , defined in (27), follows a similar pattern as that with imperfect CSIR. For the MIMO channel with imperfect CSIR, the observations can be interpreted as follows. In the 2 × 4 MIMO case, increasing the number of active antennas increases the ergodic capacity. Recall that the maximum multiplexing gain with t active transmit antennas is equal to min(t, MR ) (TT−t) which is equal to t(TT−t) when MT < MR . For large values of T , TT−t ≈ 1 and hence, to support higher values of r, higher number of active antennas are required. 13
4.5
ρ = 5 dB ρ = 10 dB ρ = 30 dB
4 opt
Optimum number of transmit antennas (tMISO)
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3.5 3 2.5 2 1.5 1 0.5 0
0.1
0.2
0.3
0.4 0.5 0.6 Multiplexing gain (r)
0.7
0.8
Figure 2: tMISO versus r for ρ = 10, 30 dB, T = 10, MT = 4, MR = 1. opt
14
0.9
3
2.5 Optimum number of antennas (topt)
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ρ = 10 dB ρ = 20 dB ρ = 30 dB
2 T23 1.5
1 T12 T 12 0.5 0
10
20
T12
30 40 50 Channel block length (T)
60
70
80
Figure 3: tMISO versus T for MT = 4, MR = 1, r = 0.88 and various SNR. opt
15
3
Optimum number of antennas (topt)
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2.5
2
1.5
1 SNR = 10 dB (perfect CSIR) SNR = 25 dB (perfect CSIR) SNR = 10 dB (imperfect CSIR) SNR = 25 dB (imperfect CSIR)
0.5
0 0
0.5
1 Multiplexing gain (r)
1.5
2
Figure 4: tMIMO versus r for T = 20, MT = 2, MR = 4 and various SNR compared with opt tMIMO obtained for perfect CSIR. opt,per
16
Fig. 5 plots the outage probability for a 2 × 4 MIMO system obtained from Monte-Carlo simulations and compares it with the two upper bounds on the outage probability obtained in Remark 3 in (29), (31). These upper bounds are obtained using the pdf of minimum and maximum eigenvalue of a Wishart matrix. Apart from these bounds, the minima of the upper bounds, PUout,MIMO , is also plotted. Fig. 5 shows that the upper bound for t = 1, i.e. the MISO case, is tight. However, it is not tight for t = 2. Hence, PUout,MIMO is not used to compute topt in the MIMO case. Instead, Monte Carlo simulation of (5) is used to compute topt . 0
10 Outage probability (PU ,P ) out,MIMO out,MIMO
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−2
10
t=1 Monte Carlo t=2 Monte Carlo t=1, UB using λ
−4
10
min
t=2, UB using λ
min
−6
10
t=1, UB using λ
max
t=2 , UB using λmax t=2, min PU out
−8
10
0
0.2
0.4
0.6 0.8 1 1.2 Multiplexing gain (r)
1.4
1.6
U1 U2 Figure 5: Pout,MIMO , Pout,MIMO and Pout,MIMO versus r for ρ = 10 dB, T = 20, MT = 2, MR = 4 and for various t.
4.2.2. MT > MR A 4×2 MIMO channel is considered for the plots described below. In Fig. as a function of r for T = 200 and with perfect as well as 6, we plot tMIMO opt imperfect CSIR at various SNR. With imperfect CSIR, it is observed that at increases as r increases, reaches maximum, reduces and then T = 200, tMIMO opt 17
5 4.5 Optimum number of antennas (topt)
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4 3.5 3 2.5 2 1.5 SNR = 10 dB (perfect CSIR) SNR = 30 dB (perfect CSIR) SNR = 10 dB SNR = 30 dB
1 0.5 0 0
0.5
1 Multiplexing gain (r)
1.5
2
Figure 6: tMIMO versus r for T = 200, MT = 4, MR = 2 and various SNR compared with opt tMIMO with perfect CSIR. opt,per
18
5 4.5 Optimum number of antennas (tMIMO ) opt
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4 3.5 3 2.5 2 1.5 1
r=0.71 r=1.01 r=1.51
0.5 0 5
10
15
20
25
30
SNR (ρ) Figure 7: tMIMO versus SN R for T = 20, MT = 4, MR = 2 and various r. opt
19
stabilizes at MR as r → rmax . It is observed that irrespective of the SNR, as r → rmax , tMIMO = MR . The behavior of tMIMO as a function of r for MT > opt opt MR MIMO systems is similar to that of the MISO systems, especially after r exceeds a particular value, wherein it can be observed that the optimum number of active antennas that minimizes the outage probability reduces as multiplexing gain increases. 0
out,MIMO
)
10
Outage probability (P
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−2
10
−4
10
t=1, MC t=2, MC t=3, MC t=4, MC
−6
10
−8
10
0
0.5
1 1.5 Multiplexing gain (r)
2
Figure 8: Pout,MIMO versus r for ρ = 20 dB, T = 20, MT = 4, MR = 2 and for various t.
Fig. 7 plots tMIMO versus ρ for various and fixed r. These observations are opt same as that in Fig. 6. The behavior of tMIMO versus SNR is clearly observed opt in this figure since ρ is varied in the range of 5 to 30 dB. Specifically, for a remains constant as SNR is increased only for 4 × 2 MIMO system, tMIMO opt = MR . certain range of r which is around 1.5 < r < 2. For 1.5 < r < 2, tMIMO opt MIMO Fig. 6 shows variation of topt with SNR for some values of r. Fig. 8 plots the outage probability for a 4 × 2 MIMO system obtained from Monte-Carlo simulations of Pout defined in (5).
20
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5. Conclusion In this paper, we performed outage analysis of a training-based communication scheme for non-coherent MIMO channels. We studied the behavior of optimal number of active transmit antennas (topt ) that minimizes the outage probability using Monte Carlo simulations with respect to various parameters such as MT , MR , ρ, multiplexing gain and coherence interval. We derived an analytical expression to upper bound the outage probability for the MISO channels and it was observed to be tight for moderate to high SNR values. 6. References [1] E. Telatar, Capacity of multi-antenna Gaussian channels,, European transactions on telecommunications 10, (6) (1999) 585–595. [2] G. J. Foschini, M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless personal communications 6 (3) (1998) 311–335. [3] L. Zheng, D. N. Tse, Communication on the grassmann manifold: A geometric approach to the noncoherent multiple-antenna channel, Information Theory, IEEE Transactions on 48 (2) (2002) 359–383. [4] K. Takeuchi, R. R. Muller, M. Vehkapera, T. Tanaka, On an achievable rate of large rayleigh block-fading MIMO channels with no csi, Information Theory, IEEE Transactions on 59 (10) (2013) 6517–6541. [5] B. M. Hochwald, T. L. Marzetta, Unitary space-time modulation for multiple-antenna communications in rayleigh flat fading, IEEE transactions on Information Theory 46 (2) (2000) 543–564. [6] W. Yang, G. Durisi, E. Riegler, On the capacity of large-MIMO blockfading channels, Selected Areas in Communications, IEEE Journal on 31 (2) (2013) 117–132. [7] L. Zheng, D. N. Tse, Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels, Information Theory, IEEE Transactions on 49 (5) (2003) 1073–1096.
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[8] L. Zheng, Diversity-multiplexing tradeoff: A comprehensive view of multiple antenna systems, Ph.D. thesis, University of California at Berkeley (2002). URL http://web.mit.edu/lizhong/www/papers.html [9] S. Ray, M. M´edard, L. Zheng, On noncoherent MIMO channels in the wideband regime: Capacity and reliability, Information Theory, IEEE Transactions on 53 (6) (2007) 1983–2009. [10] L. Zheng, D. N. Tse, M. M´edard, Channel coherence in the low-SNR regime, Information Theory, IEEE Transactions on 53 (3) (2007) 976– 997. [11] B. Hassibi, B. M. Hochwald, How much training is needed in multipleantenna wireless links?, Information Theory, IEEE Transactions on 49 (4) (2003) 951–963. [12] K. Takeuchi, R. R. Muller, M. Vehkapera, T. Tanaka, Practical signaling with vanishing pilot-energy for large noncoherent block-fading mimo channels, in: 2009 IEEE International Symposium on Information Theory, IEEE, 2009, pp. 759–763. [13] R. Narasimhan, Finite-SNR diversity multiplexing tradeoff for correlated Rayleigh and Rician MIMO channels, Information Theory, IEEE Transactions on 52 (9) (2006) 3965–3979. [14] H. Weingarten, Y. Steinberg, S. Shamai, Gaussian codes and weighted nearest neighbor decoding in fading multiple-antenna channels, IEEE Trans. on Information Theory 50 (8) (2004) 1665–1686. [15] M. Abramowitz, I. Stegen, Handbook of Mathematical Functions, ninth Edition, Dover Publication, New York, USA, 1970, pp. 260–262. [16] A. Edelman, Eigenvalues and condition numbers of random matrices, Ph.D. thesis, Massachusetts Institute of Technology (1989). URL http://www-math.mit.edu/~ edelman/Edelman/publications.htm [17] C. Khatri, Distribution of the largest or the smallest characteristic root under null hypothesis concerning complex multivariate normal populations, The Annals of Mathematical Statistics 35 (4) (1964) 1807–1810. 22
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[18] E. Abbe, S.-L. Huang, E. Telatar, Proof of the outage probability conjecture for MISO channels, Information Theory, IEEE Transactions on 59 (5) (2013) 2596–2602.
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*Brief Biography Click here to download Brief Biography: NL_bio.pdf
Nandita Lavanis received her B.E. in Electronics Engineering from Visvesvaraya National Institute of Technology, Nagpur, India and M.E. in Electrical Communication Engineering from Indian Institute of Science (IISc), Bangalore, India. She was employed with Tata Elxsi India Ltd. and subsequently with Philips Semiconductors, part of Philips Innovation Campus, Bangalore where she worked on ADSL modem and image compression algorithms. She earned her Ph. D from the Department of Electrical Engineering at Indian Institute of Technology (IIT), Madras. She is faculty in SSN College of Engineering, Chennai. Her research interests are MIMO systems and signal processing applied to wireless communication.
*Author Photo Click here to download high resolution image
S1874-4907(17)30010-1 http://dx.doi.org/10.1016/j.phycom.2016.12.005 PHYCOM 354
To appear in:
Physical Communication
Received date: 17 March 2016 Revised date: 9 November 2016 Accepted date: 28 December 2016 Please cite this article as: N. Lavanis, D. Jalihal, A.P. Kannu, S. Bhashyam, Finite-SNR outage analysis for MIMO channels with imperfect channel state information, Physical Communication (2017), http://dx.doi.org/10.1016/j.phycom.2016.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
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Finite-SNR Outage Analysis for MIMO Channels with Imperfect Channel State Information Nandita Lavanisa,1,∗, Devendra Jalihala , Arun Pachai Kannua , Srikrishna Bhashyama a
Dept. of Electrical Engineering, IIT Madras, Chennai, India
Abstract In this paper, a point-to-point multiple-input multiple-output (MIMO) channel with imperfect channel state information (CSI) at the receiver and no CSI at the transmitter is considered. Using Monte Carlo simulations, we compute the optimum number of active antennas required at the transmitter (topt ) to minimize the outage probability. We show that, apart from the number of transmit antennas, topt depends on the signal to noise ratio (SNR), multiplexing gain, coherence time, and the number of receive antennas. Our results give insights on the behavior of topt with respect to these parameters. Specifically, we show that as the multiplexing gain increases, the value of topt increases from one, and as the multiplexing gain reaches its maxima, the value of topt equals the minimum of the number of transmit and receive antennas. The intermediate behavior of topt with respect to multiplexing gain depends on the MIMO channel configuration. topt for the MIMO channel with perfect CSI at the receiver follows a similar pattern as that with imperfect CSIR. For a multiple-input single-output (MISO) channel with imperfect CSIR, we obtain a tight upper bound on the outage probability. Using this analytical upper bound, we can calculate topt for any fixed channel configuration. For a MISO channel with imperfect CSIR and fixed SNR, topt reduces as multiplexing gain increases; however, for fixed multiplexing gain and fixed SNR, topt monotonically increases with increase in coherence time of the channel. Corresponding author Email addresses: [email protected] (Nandita Lavanis), [email protected] (Devendra Jalihal), [email protected] (Arun Pachai Kannu), [email protected] (Srikrishna Bhashyam) 1 The author is currently a faculty at SSN College of Engineering, Chennai, India ∗
Preprint submitted to Physical Communication
November 9, 2016
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Keywords: Channel estimation, Non-coherent communication, Optimum number of antennas, Outage probability, Training signals 1. Introduction Multiple antennas are quite common in modern wireless systems due to their advantages of higher spectral efficiency and higher reliability compared to single antenna systems. Ergodic capacity is a metric of relevance in fast fading scenarios where the coherence time of the channel is small compared to the time duration over which communication happens. Ergodic capacity of multiple-input multiple-output (MIMO) channels is analyzed in [1] and [2] assuming perfect channel state information (CSI) at the receiver and no CSI at the transmitter for i.i.d. Rayleigh channel model. It is assumed in [1] that the transmitter knows the distribution of channel statistics. In [1], a formula for MIMO capacity is derived with perfect CSI and i.i.d Rayleigh fading, and the concept of outage is introduced. For slow fading channels, the outage probability and diversity-multiplexing-tradeoff (DMT) are relevant metrics. When the data rate of communication exceeds channel capacity for a given realization of the fading channel, the probability of detection error becomes high and the outage event occurs. In realistic communication environments, it is difficult to obtain perfect CSI. This leads to the case in which both the transmitter as well as the receiver do not have CSI, but the receiver is allowed to estimate CSI. This is also termed as the non-coherent communication scenario [3]. In the literature, a block-fading assumption is used. Quasi-static or block-fading channel implies that the channel fading is constant for a block and independently changes in the next block. Since capacity achieving distributions are not i.i.d., three kinds of scenarios are considered in the literature [4] for noncoherent analysis. In the first scenario, unitary space-time modulation is considered [5], and a lower bound on non-coherent capacity is obtained. In the second scenario, either the high-SNR or low-SNR limit is considered and the non-coherent capacity is obtained. High-SNR approximations are provided in [3] and [6]. In [7], the outage probability analysis is performed using a high-SNR assumption. In [8], number of transmit antennas required to maximize the diversity multiplexing tradeoff is analyzed. In the low SNR regime, [9] computes the capacity and analyzes its dependence on various 2
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parameters such as the coherence time, number of antennas etc. Similarly, in [10], capacity and optimal input are provided in the low SNR case. In the last scenario, channel estimation is performed and achievable rate and analytical bounds are computed for the non-coherent channel using the specific estimation scheme. The non-coherent capacity analysis is performed in [3] and [11] to provide channel estimates under a block-fading assumption by assuming a similar training-based channel estimation scheme. Though no closed-form expression of non-coherent capacity is available in literature, [3] provides a high-SNR expression of the same, while [11] computes a lower bound using the particular channel estimation scheme and optimize it. While [11] uses one-shot channel estimates, the channel estimation and hence the lower bound on the capacity is improved by [4] in a large MIMO setting by using successive decoding strategy for performing channel estimation. A lower bound on the spectral efficiency of a non-coherent MIMO channel is evaluated in a large-system limit in [12] using a randomly-biased QPSK signalling. In this paper, we consider non-coherent MIMO channels belonging to the third scenario where the training-based scheme from [11] is used to perform channel estimation. In [11], Hassibi and Hochwald obtained a lower bound on the capacity of non-coherent channels by bounding the achievable rate of a training-based scheme. The training sequences, training power, and training duration were optimized to maximize a capacity lower bound. It was observed in [11] and [3] that at high SNR it is optimal to use min(MT , MR , ⌊ T2 ⌋) antennas to maximize the capacity lower bound and diversity gain, respectively. In [4, 12], a training-based scheme that is different from the scheme in [11] is studied and the capacity is again studied in the large MIMO setting. Although we use the same training-based scheme as in [11], our work on the optimal number of antennas in non-coherent MIMO systems differs from prior work in the following ways: (1) We consider minimizing outage probability instead of maximizing capacity. (2) We consider the finite moderate SNR regime instead of the high or low SNR regimes studied in prior work. (3) We use the finite-SNR multiplexing gain as defined in [13]. Our results show that at finite SNR, the optimal number of antennas can be different from the high SNR result of min(MT , MR , ⌊ T2 ⌋) antennas. The dependence of this optimal number of antennas on multiplexing gain, SNR, and coherence time is studied here. Our contributions in this paper are as follows: 1) We show that in the finite-SNR regime, the optimal number of antennas can be different from 3
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M ∗ = min(MT , MR , ⌊ T2 ⌋), with varying SNR, as against [8] in which it is shown that these remain fixed at M ∗ in the asymptotically high-SNR regime. 2) Using Monte Carlo simulations, we obtain the optimal number of active transmit antennas (topt ) that minimizes the outage probability and we analyze the same for multiple-input single-output (MISO) and MIMO channels with/without perfect CSIR. 3) For non-coherent MIMO channels, our results show the following: (a) For MT > MR , as the multiplexing gain increases, the value of topt initially increases from unity to maximum and then decreases before settling to tMIMO = MR when the multiplexing gain tends to its maxopt imum value. (b) For MT ≤ MR and with increasing multiplexing gain, the value of topt increases and settles to tMIMO = MT when the multiplexing gain opt tends to its maximum value. 4) For non-coherent MISO channels, we provide an analytical expression for an upper bound on the outage probability of a communication scheme employing training-based channel estimation. The analytical bound is shown to be tight for moderate to high SNR values. The paper is organized as follows: In Section 2, we provide the details of the training-based communication scheme. In Section 3, we derive the analytical expression for an upper bound on the outage probability. In Section 4, we provide numerical results characterizing the outage probability, its analytical upper bound and the behavior of optimal number of active transmit antennas. In Section 5, we present the conclusions. 2. System model and related definitions A quasi-static, frequency-flat, MIMO channel with MT transmit and MR receive antennas with Rayleigh fading is described by √ ρ Y= HX + W. (1) MT In (1), the channel coefficients in the matrix H ∈ C MR ×MT are zero-mean, i.i.d., circularly symmetric complex Gaussian with unit variance. X ∈ C MT ×T denotes the transmitted signal, Y ∈ C MR ×T denotes the received signal and W ∈ C MR ×T denotes the noise. The transmitted signal X is normalized to have average transmit power at each antenna to be unity. T denotes the coherence interval over which the channel coefficients are assumed to remain unchanged, i.e., the block fading model is used. The entries in the noise matrix W are zero-mean, i.i.d., circularly symmetric complex Gaussian with 4
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unit variance. For this model, it follows that ρ is the average SNR at each receive antenna. It should be noted that (1) serves as a model for MIMO channels with/without perfect CSIR. In case of imperfect CSIR, a training scheme is considered as described in the next subsection. 2.1. Details of the training scheme The channel estimation scheme from [11] designed for the MIMO channel in (1) is used. The assumptions at the transmitter are as follows. Since CSIT is not available, the transmitter does not have knowledge of the instantaneous SNR or the channel gains, whereas it has knowledge of the average value of received SNR and the probability density function (pdf) of the channel coefficients. Equal power allocation is assumed across all the transmit antennas. The block of T symbols is divided such that a deterministic training signal Xτ ∈ C MT ×Tτ is sent in the initial Tτ duration and the subsequent Td = (T −Tτ )+ duration is used for sending data Xd ∈ C MT ×Td . Using the observations corresponding to the training duration Yτ ∈ C MT ×Tτ , the receiver determines the minimum mean squared error (MMSE) estimate of the chanˆ = E{H|Yτ , Xτ }. Since H and W are Gaussian and independent, nel as H ˆ is the MMSE estimator is identical to the linear MMSE estimator. Since H ˆ ∗ H)] ˆ ˆ E[tr(H 2 ¯ zero mean, the variance of each of its entries is σHˆ = MT MR . Let H = σHˆ . H The training signal matrix Xτ is assumed to be a multiple of a matrix with orthonormal columns. In this case, it has been shown in [11] that the MSE is ¯ are i.i.d. complex Gaussian with unit variance. minimal and the entries in H ¯H ¯ ∗ is the This result will be used in further analysis. Furthermore, W = H ∗ ¯ H ¯ is the Wishart matrix for Wishart matrix for MR < MT and W = H MT < MR . 2.2. Outage probability of the training Scheme Let the power allocated in the training phase and the data phase be ρτ and ρd , respectively. The fraction of energy allocated to the data signal is denoted by α, hence ρd =
αρT , Td
ρT = ρτ Tτ + ρd Td .
(2)
ˆ we perform the coherent minimum distance Using the channel estimate H, decoding, ignoring the channel estimation error. The data rate that can be
5
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achieved using this strategy is specified by the following lower bound on the mutual information [14, 11] ˆ = ILMIMO (X; Y|H)
(T − Tτ ) ( ρ˜ ¯ ∗ ¯ ) log det(IMT + H H) , T MT
where ρ˜ is the effective SNR defined as ρd ρτ . ρ˜ = 1 + ρd + ρτ
(3)
(4)
Let R be the data rate of communication. Since apriori channel state information (CSI) is not available, it is possible that R is greater than the mutual information bound in (3). We define the outage probability as Pout,MIMO = P(ILMIMO < R).
(5)
The multiplexing gain r at finite-SNR [13] with data rate R, SNR ρ, array gain g is R r= . (6) log2 (1 + gρ) For the MIMO system, g is chosen equal to MR [13]. The multiplexing gain r gives an indication about the sensitivity of the rate adaptation policy w.r.t SNR. Note that outage probability in (5) depends on ρ, r, MT , MR and T . For brevity in notations, we do not show this dependency explicitly. The diversity gain d(r, ρ) at a multiplexing gain r and an SNR ρ, as defined in [13], is ∂ d(r, ρ) = −ρ ln Pout,MIMO (r, ρ), (7) ∂ρ where we have explicitly denoted the dependence of outage probability on r and ρ. 2.3. MIMO channel with perfect CSIR In case of perfect CSIR, assuming an uncorrelated input on all the antennas, the mutual information [1] is ( ρ ∗ ) H H) . (8) IMIMO,per (X; Y|H) = log det(IMT + MT
The corresponding outage probability from (5) and (6) is
Pper out,MIMO = P(IMIMO,per < r log2 (1 + gρ)). 6
(9)
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3. Outage Probability Analysis Previous studies of non-coherent MIMO channels in the high SNR regime have shown that the optimal number of active transmit antennas which maximizes the ergodic channel capacity [11] or the diversity-multiplexing tradeoff [7] depends on various parameters such as coherence interval and number of receive antennas. In this paper, we are interested in minimizing the outage probability Pout,MIMO in the finite SNR regime. In this regard, let t ∈ {1, · · · , MT } denote the number of active antennas used by our training scheme. In order to estimate the channel coefficients from t transmit antennas properly (i.e., guarantee zero estimation error in the absence of noise), we need that the training duration Tτ ≥ t. However, increasing the training duration beyond t decreases the data duration. Hence, we choose Tτ = t. In [11], it has been shown that the choice of Tτ = t maximizes the ergodic achievable rates E(ILMIMO ). We also use the power allocation fraction α given below, which maximizes L E(IMIMO ) [11], √ ν − ν(ν − 1), for Td > t; 1 , for Td = t; (10) αo = 2 √ ν + ν(ν − 1), for Td < t, where ν =
(t+ρT ) . ρT (1− Tt )
Subsequently, we analyze the behavior of Pout,MIMO
d
with respect to various parameters such as r, ρ, t, MR and T . For the coherent MT × MR MIMO with perfect CSI, the pre-log factor of the ergodic channel capacity is min(MT , MR ). For the non-coherent case, we spend only T − Tτ duration for the data. With t active transmit antennas, Tτ = t and hence the multiplexing gain in our non-coherent MIMO model is bounded as (T − t) 0 ≤ r ≤ min(t, MR ) ≡ rmax . (11) T We define, q ≡ min{t, MR }, n ≡ max{t, MR }. Substituting αo (optimal fraction of power allocation) from (10) in (2), the effective SNR in (4) is written as ζ1 ρ2 , ζ1 = αo (1 − αo )T 2 , ζ2 = T 2 − T t − αo T 2 + 2tαo T. (12) tTd + ζ2 ρ ) ( rT We define threshold SNR as ρth ≡ (1 + gρd ) (q(T −t)) − 1 , using ρd from (2) ρ˜(ρ, T, t) =
7
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the threshold SNR is ) ( gT αo ρ (q(TrT−t)) ) −1 . ρth (ρ, r, T, t, MR ) = (1 + T −t
(13)
Using (12) and (13), define ϕ as the ratio of threshold SNR to effective SNR as ϕMIMO (ρ, r, t, T, MR ) ≡
ρth . ρ˜
(14)
We utilize the incomplete gamma function γ(a, x) [15] in the following theorem, which is written as a−1 j ∑ ( x ) γ(a, x) = Γ(a) 1 − e−x . j! j=0
(15)
Theorem 1. With t active antennas and MR = 1, the outage probability defined in (5) is upper bounded as Pout,MISO ≤ PUout,MISO where U Pout,MISO =
γ(t, tϕMISO ) . Γ(t)
(16)
Proof: For t transmit antennas, the lower bound on the MIMO mutual information in (3) is T − t( ρ˜ ¯ ∗ ¯ ) T − t ( ρ˜ ¯ ¯ ∗ ) log det(It + H H) = log det(IMR + H H) . T t T t (17) MR ×t ¯ ∗ t×MR ¯ In (17), H ∈ C , H ∈C . Define { ¯H ¯ ∗ , if MR < t then W ∈ C MR ×MR ; H (18) W= ¯ ∗ H, ¯ if MR ≥ t then W ∈ C t×t . H
L ˆ = IMIMO (X; Y|H)
W has a complex Wishart distribution [1], [16]. From (18), W ∈ C q×q . From (17) and (18), the lower bound on the MIMO mutual information is rewritten as, ( ) L ˆ = (T − t) log det(Iq + ρ˜W) (X; Y|H) IMIMO T t 8
(19)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
q
(T − t) ∑ ( ρ˜ ) = log 1 + λi , (20) T t i=1 ( ) ρe = log det(It + h∗ h) , for MR = 1. (21) t In (20), λ′i s denote the ordered eigenvalues of the complex Wishart matrix W( [16] with λq being the smallest. From (5) )and (20), we have Pout,MIMO = ( ) ∑ oT ) . The minimum eigenvalue of P (TT−t) qi=1 log 1 + ρt˜ λi < r log(1 + gρα T −t the complex Wishart matrix W denoted by λq is used to compute an upper bound of the outage probability. ϕMIMO defined in (14) is used. ( (T − t) ( ρ˜ ) gραo T ) Pout,MIMO ≤ Pr q log 1 + λq < r log(1 + ) T t T −t (( ) ) (q(TrT−t)) ( ) oT t 1 + gρα − 1 ( ) T −t ≤ Pr λq < ≤ Pr λq < tϕMIMO . ρ˜ (22) For the MISO channel, from (21), we have (( ) ( ρ˜ rT αo ρT )) U Pout,MISO = Pr log det(It + h∗ h) < log 1 + g , t T −t T −t
(23)
where ρd is defined in (2). In (23), h∗ h has a χ2 -distribution with 2t degrees of freedom. Following [15], t−1
U Pout,MISO (r, ρ, t, T, α) =
∑ (tΦM ISO )k γ(t, tΦM ISO ) = 1 − e−tΦM ISO , (24) Γ(t) k! k=0
where ΦM ISO = ΦM IM O |MR =1 . In (24), γ(a, x) is an incomplete gamma function defined in (15).
Remark 1. For the non-coherent single-input multiple-output (SIMO) channel with MT = 1, the outage probability is derived in a similar way, and is given by PUout,SIMO (ρ, r, T, MR ) =
γ(MR , ϕSIMO ) , Γ(MR )
where ϕSIMO is obtained by substituting MT = 1 in (14). 9
(25)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Remark 2. For the non-coherent MIMO channel, we define the optimum number of transmit antennas as the active number of antennas which minimizes the outage probability as tMIMO (ρ, r, T, MR , MT ) = arg min Pout,MIMO , opt 1≤t≤MT
(26)
where Pout,MIMO is defined in (5). Similarly, using (9) for the MIMO channel with perfect CSIR, we define the optimum number of transmit antennas as the active number of antennas as per tMIMO opt,per (ρ, r, MR , MT ) = arg min Pout,MIMO , 1≤t≤MT
(27)
where Pper out,MIMO is defined in (9). Remark 3. For the MIMO channel, from (20) we compute two upper bounds on the outage probability. They are based on the following bounds based on the minimum and maximum eigenvalue. The bound based on the minimum eigenvalue (λq ) is computed from (20) using the following inequality q
( (T − t) ∑ ( ρ˜ ) q(T − t) ρ˜ ) log 1 + λi ≥ log 1 + λq . T t T t i=1
(28)
Further, from (22) using the pdf of the minimum eigenvalue of a Wishart matrix, an upper bound on the outage probability is computed as ( ) γ n − q + 1, tq2 ϕMIMO U1 Pout,MIMO (r, ρ, t, T, MR ) = . (29) Γ(n − q + 1) 1 PUout,MIMO is not tight due to bounding of the eigenvalues by q times the minimum eigenvalue. Similarly, the bound based on the maximum eigenvalue (λ1 ) is computed from (20) using the following inequality q
( ρ˜ ) (T − t) ∑ ( ρ˜ ) (T − t) log 1 + λ1 . log 1 + λi ≥ T t T t i=1
(30)
Using the pdf the maximum eigenvalue of a Wishart matrix given by [17], the upper bound on the outage probability is computed as 2 PUout,MIMO (r, ρ, t, T, MR ) = cn,q det G(ϕMIMO ).
10
(31)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
In (31), G(ϕMIMO ) is q × q Hankel(∏ matrix given∏by G(ϕMIMO ) ) = [γ(n − q + q n q MIMO i + j − 1, ϕ )]i,j=1 and cn,q = i=1 (q − i)! i=1 (n − i)! . It is observed that the upper bounds in (29), (31) plotted in Fig. 5 are not tight for t > 1. This is because the lower bounds in (28) and (30) are not tight. U1 2 We can find a bound PUout,MIMO = min{Pout,MIMO , PUout,MIMO }. PUout,MIMO is also plotted in Fig. 5 along with (29), (31) and it is observed that this minimum outage bound significantly improves for lower multiplexing gains. However, PUout,MIMO is still not tight and hence is not used to compute topt in the MIMO case. Instead Monte Carlo simulation of (5) is used to compute topt . 4. Numerical Results and Discussion We numerically study the behavior of outage probability bound PUout,MISO and PUout,MIMO for various values of multiplexing gain r, SNR (ρ), and coherence interval T . We also characterize the behavior of the optimal number of active antennas tMIMO that minimizes the outage probability with/without opt perfect CSIR. Monte Carlo simulations are used to compute the outage probability in (5), the minimization of which leads to tMIMO . For performing opt simulations, either a 2 × 4 or a 4 × 2 MIMO system operating on an SNR range of 5 to 30 dB is considered. The coherence interval is assumed to be in the range 0 < T ≤ 200 and the constraint of T > 2MT is used as given by Hassibi in [11]. 4.1. MISO channel U For the non-coherent 4 × 1 MISO channel, Pout,MISO in (23) is numerically computed and plotted for various multiplexing gains in Fig. 1 with the number of active antennas varied from 1 to 4. For comparison, we also plot the Monte-Carlo simulation of outage probability Pout in (5). We see that our outage probability upper bound is tight even for moderate values of SNR (20 dB). From this plot, we also note that the optimal number of active transmit antennas changes with the value of the multiplexing gain. We explicitly plot versus multiplexing gain in Fig. 2. As the r increases, the value of tMISO opt U the number of transmit antennas which minimizes Pout,MISO monotonically decreases. This can be interpreted in the following way. For MISO channels, the pre-log factor of the ergodic channel capacity (or the multiplexing gain) does not increase with the number of transmit antennas. On the other hand, the training overhead increases with the number of transmit antennas. Hence, 11
U
Outage probability Pout,MISO(r, ρ, T, α, t)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
−1
10
t=1 t=2 t=3 t=4 Monte Carlo simulation t=1 Monte Carlo simulation t=2 Monte Carlo simulation t=3 Monte Carlo simulation t=4
−2
10
0.1
0.2
0.3
0.4 0.5 0.6 Multiplexing gain (r)
0.7
0.8
0.9
U Figure 1: Pout,MISO and Pout,MISO versus r for ρ = 20 dB, T = 10, MT = 4, MR = 1 and for various t.
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
as the multiplexing gain approaches rmax = T T−1 , the optimal value tMISO opt approach unity. U The behavior of Pout,MISO versus r in Fig. 1 and Fig. 2 is very similar to the behavior of MISO channels shown by Telatar in [1]. Telatar explains that for a MISO channel with perfect CSIR the number of antennas which minimizes the probability of outage is large for smaller data rate, whereas the number is small for larger data rate. From (6), we know that for a fixed SNR the multiplexing gain r is directly proportional to the data rate. The same behavior is observed for the MISO channel with imperfect CSIR as seen in Fig. 2, where the optimum number of antennas which minimize the outage probability reduces as r increases. It is maximum for small r and becomes minimum for large r. Hence, the observations in Fig. 2 agree with the Teletar’s conjecture [1], proved in [18], that the number of active transmit antennas decreases as the rate increases. In Fig. 3, we plot tMISO versus the channel coherence time interval T for opt fixed multiplexing gain r = 0.88 and various SNR values from moderate to high. We note that for fixed r and ρ, tMISO increases monotonically with T . opt In the Fig. 3, Tij denotes the value of T at which the topt jumps from i to j (where j is equal to i + 1). 4.2. MIMO channels 4.2.1. MT ≤ MR A comparison of optimal number of active antennas with and without perfect CSIR is plotted in Fig. 4. tMIMO obtained from the Monte Carlo opt simulations of (5) using (3) and (6) is plotted as a function of r in Fig. 4 for a 2 × 4 MIMO channel. tMIMO opt,per given by (27) is also plotted and compared. From Fig. 4, we note that tMIMO increases with increase in r in steps of one, opt and reaches the maximum value of MT . This trend is exhibited for a range of SNR values. Note that this trend is exactly the opposite of that seen in the MISO case (see Fig. 2) since tMISO reduces with increase in r. For the opt MIMO MIMO channel with perfect CSIR, topt,per , defined in (27), follows a similar pattern as that with imperfect CSIR. For the MIMO channel with imperfect CSIR, the observations can be interpreted as follows. In the 2 × 4 MIMO case, increasing the number of active antennas increases the ergodic capacity. Recall that the maximum multiplexing gain with t active transmit antennas is equal to min(t, MR ) (TT−t) which is equal to t(TT−t) when MT < MR . For large values of T , TT−t ≈ 1 and hence, to support higher values of r, higher number of active antennas are required. 13
4.5
ρ = 5 dB ρ = 10 dB ρ = 30 dB
4 opt
Optimum number of transmit antennas (tMISO)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
3.5 3 2.5 2 1.5 1 0.5 0
0.1
0.2
0.3
0.4 0.5 0.6 Multiplexing gain (r)
0.7
0.8
Figure 2: tMISO versus r for ρ = 10, 30 dB, T = 10, MT = 4, MR = 1. opt
14
0.9
3
2.5 Optimum number of antennas (topt)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
ρ = 10 dB ρ = 20 dB ρ = 30 dB
2 T23 1.5
1 T12 T 12 0.5 0
10
20
T12
30 40 50 Channel block length (T)
60
70
80
Figure 3: tMISO versus T for MT = 4, MR = 1, r = 0.88 and various SNR. opt
15
3
Optimum number of antennas (topt)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
2.5
2
1.5
1 SNR = 10 dB (perfect CSIR) SNR = 25 dB (perfect CSIR) SNR = 10 dB (imperfect CSIR) SNR = 25 dB (imperfect CSIR)
0.5
0 0
0.5
1 Multiplexing gain (r)
1.5
2
Figure 4: tMIMO versus r for T = 20, MT = 2, MR = 4 and various SNR compared with opt tMIMO obtained for perfect CSIR. opt,per
16
Fig. 5 plots the outage probability for a 2 × 4 MIMO system obtained from Monte-Carlo simulations and compares it with the two upper bounds on the outage probability obtained in Remark 3 in (29), (31). These upper bounds are obtained using the pdf of minimum and maximum eigenvalue of a Wishart matrix. Apart from these bounds, the minima of the upper bounds, PUout,MIMO , is also plotted. Fig. 5 shows that the upper bound for t = 1, i.e. the MISO case, is tight. However, it is not tight for t = 2. Hence, PUout,MIMO is not used to compute topt in the MIMO case. Instead, Monte Carlo simulation of (5) is used to compute topt . 0
10 Outage probability (PU ,P ) out,MIMO out,MIMO
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
−2
10
t=1 Monte Carlo t=2 Monte Carlo t=1, UB using λ
−4
10
min
t=2, UB using λ
min
−6
10
t=1, UB using λ
max
t=2 , UB using λmax t=2, min PU out
−8
10
0
0.2
0.4
0.6 0.8 1 1.2 Multiplexing gain (r)
1.4
1.6
U1 U2 Figure 5: Pout,MIMO , Pout,MIMO and Pout,MIMO versus r for ρ = 10 dB, T = 20, MT = 2, MR = 4 and for various t.
4.2.2. MT > MR A 4×2 MIMO channel is considered for the plots described below. In Fig. as a function of r for T = 200 and with perfect as well as 6, we plot tMIMO opt imperfect CSIR at various SNR. With imperfect CSIR, it is observed that at increases as r increases, reaches maximum, reduces and then T = 200, tMIMO opt 17
5 4.5 Optimum number of antennas (topt)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
4 3.5 3 2.5 2 1.5 SNR = 10 dB (perfect CSIR) SNR = 30 dB (perfect CSIR) SNR = 10 dB SNR = 30 dB
1 0.5 0 0
0.5
1 Multiplexing gain (r)
1.5
2
Figure 6: tMIMO versus r for T = 200, MT = 4, MR = 2 and various SNR compared with opt tMIMO with perfect CSIR. opt,per
18
5 4.5 Optimum number of antennas (tMIMO ) opt
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
4 3.5 3 2.5 2 1.5 1
r=0.71 r=1.01 r=1.51
0.5 0 5
10
15
20
25
30
SNR (ρ) Figure 7: tMIMO versus SN R for T = 20, MT = 4, MR = 2 and various r. opt
19
stabilizes at MR as r → rmax . It is observed that irrespective of the SNR, as r → rmax , tMIMO = MR . The behavior of tMIMO as a function of r for MT > opt opt MR MIMO systems is similar to that of the MISO systems, especially after r exceeds a particular value, wherein it can be observed that the optimum number of active antennas that minimizes the outage probability reduces as multiplexing gain increases. 0
out,MIMO
)
10
Outage probability (P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
−2
10
−4
10
t=1, MC t=2, MC t=3, MC t=4, MC
−6
10
−8
10
0
0.5
1 1.5 Multiplexing gain (r)
2
Figure 8: Pout,MIMO versus r for ρ = 20 dB, T = 20, MT = 4, MR = 2 and for various t.
Fig. 7 plots tMIMO versus ρ for various and fixed r. These observations are opt same as that in Fig. 6. The behavior of tMIMO versus SNR is clearly observed opt in this figure since ρ is varied in the range of 5 to 30 dB. Specifically, for a remains constant as SNR is increased only for 4 × 2 MIMO system, tMIMO opt = MR . certain range of r which is around 1.5 < r < 2. For 1.5 < r < 2, tMIMO opt MIMO Fig. 6 shows variation of topt with SNR for some values of r. Fig. 8 plots the outage probability for a 4 × 2 MIMO system obtained from Monte-Carlo simulations of Pout defined in (5).
20
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5. Conclusion In this paper, we performed outage analysis of a training-based communication scheme for non-coherent MIMO channels. We studied the behavior of optimal number of active transmit antennas (topt ) that minimizes the outage probability using Monte Carlo simulations with respect to various parameters such as MT , MR , ρ, multiplexing gain and coherence interval. We derived an analytical expression to upper bound the outage probability for the MISO channels and it was observed to be tight for moderate to high SNR values. 6. References [1] E. Telatar, Capacity of multi-antenna Gaussian channels,, European transactions on telecommunications 10, (6) (1999) 585–595. [2] G. J. Foschini, M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless personal communications 6 (3) (1998) 311–335. [3] L. Zheng, D. N. Tse, Communication on the grassmann manifold: A geometric approach to the noncoherent multiple-antenna channel, Information Theory, IEEE Transactions on 48 (2) (2002) 359–383. [4] K. Takeuchi, R. R. Muller, M. Vehkapera, T. Tanaka, On an achievable rate of large rayleigh block-fading MIMO channels with no csi, Information Theory, IEEE Transactions on 59 (10) (2013) 6517–6541. [5] B. M. Hochwald, T. L. Marzetta, Unitary space-time modulation for multiple-antenna communications in rayleigh flat fading, IEEE transactions on Information Theory 46 (2) (2000) 543–564. [6] W. Yang, G. Durisi, E. Riegler, On the capacity of large-MIMO blockfading channels, Selected Areas in Communications, IEEE Journal on 31 (2) (2013) 117–132. [7] L. Zheng, D. N. Tse, Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels, Information Theory, IEEE Transactions on 49 (5) (2003) 1073–1096.
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[8] L. Zheng, Diversity-multiplexing tradeoff: A comprehensive view of multiple antenna systems, Ph.D. thesis, University of California at Berkeley (2002). URL http://web.mit.edu/lizhong/www/papers.html [9] S. Ray, M. M´edard, L. Zheng, On noncoherent MIMO channels in the wideband regime: Capacity and reliability, Information Theory, IEEE Transactions on 53 (6) (2007) 1983–2009. [10] L. Zheng, D. N. Tse, M. M´edard, Channel coherence in the low-SNR regime, Information Theory, IEEE Transactions on 53 (3) (2007) 976– 997. [11] B. Hassibi, B. M. Hochwald, How much training is needed in multipleantenna wireless links?, Information Theory, IEEE Transactions on 49 (4) (2003) 951–963. [12] K. Takeuchi, R. R. Muller, M. Vehkapera, T. Tanaka, Practical signaling with vanishing pilot-energy for large noncoherent block-fading mimo channels, in: 2009 IEEE International Symposium on Information Theory, IEEE, 2009, pp. 759–763. [13] R. Narasimhan, Finite-SNR diversity multiplexing tradeoff for correlated Rayleigh and Rician MIMO channels, Information Theory, IEEE Transactions on 52 (9) (2006) 3965–3979. [14] H. Weingarten, Y. Steinberg, S. Shamai, Gaussian codes and weighted nearest neighbor decoding in fading multiple-antenna channels, IEEE Trans. on Information Theory 50 (8) (2004) 1665–1686. [15] M. Abramowitz, I. Stegen, Handbook of Mathematical Functions, ninth Edition, Dover Publication, New York, USA, 1970, pp. 260–262. [16] A. Edelman, Eigenvalues and condition numbers of random matrices, Ph.D. thesis, Massachusetts Institute of Technology (1989). URL http://www-math.mit.edu/~ edelman/Edelman/publications.htm [17] C. Khatri, Distribution of the largest or the smallest characteristic root under null hypothesis concerning complex multivariate normal populations, The Annals of Mathematical Statistics 35 (4) (1964) 1807–1810. 22
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*Brief Biography Click here to download Brief Biography: NL_bio.pdf
Nandita Lavanis received her B.E. in Electronics Engineering from Visvesvaraya National Institute of Technology, Nagpur, India and M.E. in Electrical Communication Engineering from Indian Institute of Science (IISc), Bangalore, India. She was employed with Tata Elxsi India Ltd. and subsequently with Philips Semiconductors, part of Philips Innovation Campus, Bangalore where she worked on ADSL modem and image compression algorithms. She earned her Ph. D from the Department of Electrical Engineering at Indian Institute of Technology (IIT), Madras. She is faculty in SSN College of Engineering, Chennai. Her research interests are MIMO systems and signal processing applied to wireless communication.
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