Severely Fading MIMO Channels: Models and Mutual Information

Seung Ho Choi and Peter Smith Ben Allen and Wasim Q. Malik Mansoor Shafi Department of Electrical and Department of Engineering Science Telecom New Zealand Ltd Computer Engineering University of Oxford, Oxford, UK PO Box 293, Wellington, University of Canterbury Email: [email protected], New Zealand Christchurch, New Zealand [email protected] Email: mansoor.shafi@telecom.co.nz Email: [email protected], [email protected]

Abstract— In most wireless communications research, the spheric radio links [4]. Despite the range of applications where channel models considered experience less severe fading than the severe fading may be encountered, relatively little attention classic Rayleigh fading case. In this work, however, we investigate has been given to such channels, with the exception of the MIMO channels where the fading is more severe. In these envi- ronments, we show that the coefficient of variation of the channel Nakagami model. For severely fading MIMO channels, there amplitudes is a good predictor of the link mutual information, is even less information available in literature, partly due for a variety of models. We propose a novel channel model for to the difficulty of performing statistical analysis of random severely fading channels based on the complex multivariate t Nakagami matrices. distribution. For this model, we are able to compute exact results In this paper, we investigate MIMO channels where the for the ergodic mutual information and approximations to the outage probabilities for the mutual information. Applications fading is more severe than Rayleigh fading. We propose of this work include wireless sensors, RF tagging, land-mobile, a novel channel model for this scenario which is simple, indoor-mobile, ground-penetrating radar, and ionospheric radio flexible, lends to closed form analysis, and is a direct extension links. Finally, we point out that the methodology can also be of the Rayleigh model to the severe fading region. Here, extended to evaluate the mutual information of a cellular MIMO fading severity is defined in terms of the probability of deep link and the performance of various MIMO receivers in a cellular scenario. In these cellular applications, the channel itself is not fading events. It does not concern the level crossing rate, the severely fading but the multivariate t distribution can be applied average fade duration, or Doppler effects, and the propagation to model the effects of inter-cell interference. channel is assumed to be narrowband. We show that the coefficient of variation of the channel amplitudes yields a I. INTRODUCTION good prediction of the mutual information. We also propose Typical wireless communications channels experience con- using the complex multivariate t distribution to model the siderable multipath propagation which causes signal fading at channel amplitudes for a variety of channels. We compute the the receiver. Several amplitude probability distributions have associated ergodic mutual information and approximate outage been used in the literature to describe random fading. These probabilities. Finally, we show how the methodology is easily include the Rician distribution for a range of line-of-sight extended to evaluate the mutual information of cellular MIMO and non-line-of-sight channels, the Nakagami distribution for and the performance of various MIMO receivers. For these wideband channels, and the for propa- scenarios, the multivariate t distribution is used to model the gation in non-homogeneous media. The inter-cell interference. is commonly used to model the fading in strongly dispersive linear channels comprising homogeneous media, such as some II. A COMPARISON OF SEVERE FADING CHANNEL urban wireless channels. Based on the Jakes model, this MODELS description assumes a large number of overlapping taps with A. System Model uniformly distributed phases and angles-of-arrival. Results on Consider a single user (nT , nR) MIMO system with nT the mutual information of Rayleigh-faded MIMO channels are transmit antennas and nR receive antennas. The system equa- widely available in the literature [1]. tion is given by [5], [6] It is well known that the Nakagami-m distribution can model the amplitude for cases when the fading is more severe than r = Hs + n (1) Rayleigh. These scenarios arise in a variety of applications. For example, measurements reported in [2] were considerably where r is the nR 1 received signal vector, s is the complex × more severe than Rayleigh for an RF tagging application. nT 1 transmitted signal vector, n is an nR 1 additive white × × Other areas where severe fading models are useful include complex Gaussian noise vector with unit modulus , wireless sensors [3], land-mobile, indoor-mobile and iono- and H is an nR nT complex channel matrix. In this section × we consider the channel coefficients, hrs = (H)rs, to be of an extra parameter. The 2 parameters are necessary since one 2 the form, hrs = Ars exp(jθrs), where θrs are independent is fixed by the normalization E(A ) = 1 and the remaining and identically distributed (iid) uniform phase variables over parameter is varied to give a range of CV values. This allows [0, 2π] and the Ars are iid amplitudes drawn from a range us to investigate the effect of CV on ergodic MI for a fixed of distribution types. As is standard practice, eg. [5], [6], we type of normalized amplitude distribution. For example, if A is normalize the amplitude distributions so that the mean power uniform over [a, b], for 0 < a < b, then 0 < a < 1 is required 2 is unity, E(Ars) = 1. This allows a fair comparison across to satisfy the normalization condition, which can be written distribution types. Classical models for the amplitude prob- as E(A2) = (a2 + b2 + ab)/3 = 1. Solving this constraint ability distribution include Rayleigh, Ricean and Nakagami. for b gives, b = a/2 + √12 3a2/2. Now, we compute the − − We also consider a range of alternative distributions including mean amplitude as, E(A) = (a+b)/2 = a/4+ √12 3a2/4. − lognormal, gamma, beta, uniform, truncated-t and truncated- As a varies from 0 to 1, the mean varies from its minimum Gaussian. Both the truncated distributions are truncated at zero value (E(A) = √3/2) to the maximum value (E(A) = 1). to give positive amplitudes. The distributions are chosen to Inserting these values in (2) gives a possible range for the CV give a wide variation in the type of amplitude variable and to values in the interval [0, 1/√3]. Hence, for uniform amplitudes cover severe fading scenarios. Note that the selected models we are only able to consider the variation in ergodic MI over include short tailed (Gaussian), long tailed (lognormal), finite 0 < CV < 1/√3. Several of the other distributions also have support (uniform) and infinite support (Nakagami) distribu- restrictions on the range of CV values that can be achieved. tions. The Weibull distribution is another distribution that is A thorough description of all these distributions can be found worth consideration in a more comprehensive study, since it in [8]. also covers the severe fading region. Hence, there are considerable differences between the am- plitude distributions considered in this work, covering a large variety of channel propagation conditions. D. Results

B. Coefficient of Variation and Mutual Information For a range of amplitude distributions and a wide range The fading severity of a channel model can be measured by of CV values the ergodic MI was simulated from (3) using the coefficient of variation (CV), which is defined by [7] 100,000 replicates. Results for a (4,2) MIMO system with a SNR of 18 dB and 0 dB are shown in Figs. 1 and V ar(A) 1 2, respectively. Note that the legends in these figures refer CV = = 1 2 (2) to ‘t’ and ‘Gaussian’, whereas truncated versions of these E(A) sE(A) − p distributions were actually used. Figure 1 shows that at high for the case of amplitudes, A, normalized to give E(A2) = 1. SNR the CV is an excellent predictor of the ergodic MI which Another measure of fading is the amount of fading (AF), which is not greatly affected by the precise amplitude distribution. is defined as AF = CV 2 = V ar(A)/[E(A)]2. Clearly, CV Figure 2 shows that this conclusion begins to break down at and AF are closely related, and we will use the CV for the low SNR where the ergodic MI is more sensitive to the type analysis in the rest of this paper. of distribution. Four of the distributions considered can only In this section, we study the ergodic mutual information give CV values less than 1. Hence the MI curves for these of the system in (1) as a function of the CV. The mutual distributions are overlaid by the others and are difficult to see. information (MI) is defined by [5], [6] As a reference point, the CV for a Rayleigh channel is 0.526. Clearly, for small CV values, all the amplitude distributions I ρ HH† give very similar results. The top curve, which diverges from = log det nR + b/s/Hz (3) I 2 n   T  the others, corresponds to the . This achieves where ρ is the average SNR per receiver branch, and † denotes the higher CV values by becoming bimodal. Hence the am- the complex conjugate transpose. plitude distribution is very different to physical reality and Note that the MI in (3) is often described as the capacity for this case is an extreme example. In Fig. 2, where the curves the scenario when the transmitter has no channel information. diverge, the short-tailed distributions (Nakagami and gamma) We investigate the sensitivity of the ergodic MI to the type of are similar and the long-tailed distributions (lognormal and amplitude distribution and whether the CV is a good measure truncated-t) are similar. Hence, at high SNR all the curves are of the effect of the fading severity on the ergodic MI. similar, whereas at low SNR, distributions which are similar in nature show similar patterns. C. Amplitude Distributions These results motivate the development of a simple channel The amplitude distributions considered here are all 2- model which can cover the full range of severely fading parameter distributions. Distributions such as the uniform channels, CV > CVRayleigh, and act as an approximation to (U[a, b]) and Beta (Beta[p, q]) have 2 parameters in their other severely fading channels with the same CV. In the next standard form. Others, such as the t, have only 1 parameter section we propose the complex multivariate t model and show in their standard form but can be scaled by a constant to give that it has the desired properties. 2.5 13 r=1.1 CMT 12 zero CV 2 Rayleigh log−normal 11 gamma beta uniform 1.5 10 t r=1.3 Ergodic MI (bits/s/Hz) Nakagami Gaussian 9 1 r=3 0 0.5 1 1.5 2 Coefficient of Variation (CV)

Fig. 1. Ergodic MI vs CV for a range of amplitude distributions in a (4,2) Probability Density Function 0.5 MIMO system with SNR=18dB

0 0 0.5 1 1.5 2 2.5 3 3.5 3 x

2.5 Fig. 3. A comparison of the amplitude distributions for the CMT model zero CV (r=1.1, 1.3 and 3) with the Rayleigh case. log−normal gamma 2 beta uniform t gamma function. Note that the division of the iid matrix H Ergodic MI (bits/s/Hz) Nakagami 1.5 Gaussian by a single, common random variable results in dependent 0 0.5 1 1.5 2 entries for HCMT. However, the independence of the H matrix Coefficient of Variation (CV) preserves the uncorrelated nature of the entries. With this H Fig. 2. Ergodic MI vs CV for a range of amplitude distributions in a (4,2) definition, the elements of CMT are uncorrelated, identically MIMO system with SNR=0dB distributed, complex t variables with uniform phase over [0, 2π]. When r , the model collapses to the baseline → ∞ iid complex Gaussian matrix and the elements of HCMT have III. THE COMPLEX MULTIVARIATE T CHANNEL MODEL amplitudes with CV equal to CVRayleigh. For finite r, the A. Introduction amplitudes are longer tailed than the Rayleigh case and give severe fading scenarios with a CV easily derived as Motivated by the desire to study severe fading in MIMO channels and the results in section III, we seek a model for the elements of the channel matrix which is simple, Γ(r)2 analytically tractable and covers the full range of cases in CVCMT = 1 (5) sΓ(r 1/2)2Γ(3/2)2(r 1) − the region CV > CVRayleigh. Such a model is the complex − − multivariate t (CMT) distribution [9], [10]. The CMT is one From (5) it is straightforward to show that CV of the family of elliptical multivariate distributions [9] and CMT → CV as r and CV as r 1. The hence a wide body of knowedge is available. Also, the CV Rayleigh → ∞ CMT → ∞ → of the amplitudes resulting from the use of the CMT satisfies probability density functions for the amplitudes of the entries H the requirement that CVCMT > CVRayleigh. Furthermore, the of the CMT matrix are shown in Fig. 3. Three densities are marginal distributions for any channel matrix element are shown, corresponding to r = 1.1, r = 1.3 and r = 3. These complex t distributed and this includes the baseline case of are compared to the Rayleigh density and it can be seen that complex Gaussian entries as a special case. Hence the CMT the severe fading cases put more of the probability close to model generalizes the classical complex Gaussian matrix to zero, but balance this by a longer tail to keep the average the severe fading scenario. power at unity. Note that the Nakagami distribution achieves The simplest description of the CMT channel matrix, HCMT, larger CV values by the rather non-physical mechanism of is to express it as the ratio of an iid complex Gaussian matrix, placing a peak at the origin. Hence, the Nakagami distribution H and the square root of an independent scaled χ2 variable gives severe fading channels where amplitudes close to zero [9], [10]. This formulation is: have the largest probability. This is in contrast to the CMT channel where the basic shape of the density is similar to the H Rayleigh but pushed towards the origin. HCMT = (4) S2 Representations for HCMT, including the probability density r−1 function, can be obtained from the viewpoint of multivariate q where S2 has a complex χ2 distribution with r degrees statistical theory [9], [10] but the representation in (4) is of freedom and probability density function fS2 (x) = sufficient for our purposes and leads to a simple analytical xr−1 exp( x)/Γ(r), where r > 1, x > 0 and Γ( ) is the method for computing both MI and system performance. − · B. Mutual Information 18 Consider the link model (1) with the CMT channel matrix r=40 H 16 CMT. We use the representation for the ergodic MI, E( ), r=3 I r=2 defined by [6] 14 ρ 12 E( ) = E m log2 1 + λCMT (6) I nT    10 where λCMT is an arbitrary, non-zero eigenvalue of H H† CMT CMT and m = min(nR, nT ). Using (4), we can write 8 (6) as 6 ρ (r 1)λ E( ) = E m log 1 + − (7) Mean Mutual Information (bits/s/Hz) I 2 n S2 4   T  2 where λ is an arbitrary, non-zero eigenvalue of the usual 2 4 6 8 10 12 14 16 18 20 Wishart matrix, HH†. The expectation in (7) is over both SNR (dB) λ and S2. Note that the effect of the CMT model is to leave the form of solution for an iid Rayleigh fading MIMO channel Fig. 4. Ergodic Mutual Information vs SNR for a (3,3) MIMO system with = 2 3 40 unchanged, but to replace the fixed SNR, ρ by a variable SNR, the CMT channel and r , , . Lines represent analytical results and circles denote simulations. ρ(r 1)/S2. This formulation allows us to derive the ergodic − MI exactly, as shown below. where cj is defined as [11] λ E( ) = E m log2 1 + δ 2 m−1 m−1 j+m−n 2k−2l 2n−2m+2l I S 1 ( 1) (2l)! k−l 2l−j−m+n  ∞   c = − m j m 22k−j−m+nl!(j + m n)!(n m + l)! = log(1 + δ x) f (x) dx (8) i k=l   log 2 X l=Xd 2 e X − − Z0 (14) 2 where δ is defined as ρ(r 1)/nT and X = λ/S has density and i = j + m n with representing the ceiling function. − − d e fX (x). Note that log, with no subscript, refers to the natural After evaluating the integral in (13), a closed-form expres- logarithm. We derive the density of X below. sion for the MI of the CMT channel is derived as Consider the pair of transformations X = λ/S2, Y = S2. n+m−2 Standard transformation theory for random variables gives: m E( ) = cj Γ(r + j + 1) I Γ(r) log 2 × f (x, y) = f 2 (xy, y) J j=n−m X,Y λ,S (9) X i=0 y x j 1 where J = det . Hence ( 1)j−i 0 1 i − (r + j i)2 × j   X   − 1 fX,Y (x, y) = fλ(xy) fS2 (y) y (10) F 1, r + j i, r + j i + 1, (15) 2 1 − − − θ n+m−2   cj j r+j −(1+x)y = x y e dy (11) where 2F1 is a Gauss hypergeometric function and θ is given Γ(r) j=n−m by θ = δ/(1 δ). Note that when r is an integer, (15) can be X − where we have used the pdf for λ given in [11] and n = further simplified since 2F1 can be written in closed form. In Fig. 4, the ergodic MI is computed via (15) for a (3, 3) MIMO max(nR, nT ). Hence, fX (x) is evaluated as system in a CMT channel with r = 2, 3, 40 and a range of SNR values. The analytical results are verified by simulation. n+m−2 ∞ cj j r+j −(1+x)y The ergodic MI is shown to increase with r, i.e., as the CMT fX (x) = x y e dy Γ(r) channel approaches a Rayleigh channel, which matches the j=n−m   Z0 X results in Figs. 1 and 2. n+m−2 j cj x Γ(r + j + 1) = (12) Γ(r) (1 + x)r+j+1 C. Mutual Information Outage j=n−m X For a fixed value of S2, the MI is known to be well- Substituting (12) in (8), E( ) is written as approximated by a Gaussian random variable [12], [13]. I Hence, we can approximate the MI outage of the CMT model n+m−2 m c Γ(r + j + 1) ∞ log(1 + δ x) xj by E( ) = j dx I log 2 Γ(r) (1 + x)r+j+1 ∞ j=n−m 0 x µI(y) Z P ( < x) = Φ − f 2 (y)dy (16) X (13) I σ (y) S Z0  I  IV. APPLICATIONS TO CELLULAR MIMO 1

0.9 A. Mutual Information

0.8 0 dB 10 dB 20 dB The approach taken in Section III is to average the well known results for an iid Rayleigh fading channel over a 0.7 variable SNR. This is exactly the scenario considered in [15] 0.6 where a cellular MIMO system is investigated and the system

0.5 equation is written as

0.4 r = √ΓHs + n. (19) CDF (Prob(MI

7 S12F3 ACKNOWLEDGMENT 6 The authors would like to thank Howard Huang for his

5 assistance in reproducing the cellular simulations in [15].

4 REFERENCES Mean Mutual Information (bits/s/Hz) 3 S3F1 [1] A. J. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless S12F1 Communications. Cambridge, UK: Cambridge University Press, 2003 & S6F1 2 [2] D. Kim and M.A. Ingram, “Measurements of small-scale fading and path 2 4 6 8 10 12 14 16 loss for long range RF tags,” IEEE Trans. Antennas Propagat., vol. 51, Average SINR (dB) pp. 1740-1749, Aug. 2003. [3] J. Galbreath and J. Frolik, “Channel allocation strategies for wireless Fig. 6. Ergodic MI vs average SNR for a (2,2) MIMO system and 6 cellular sensors statically deployed in multipath environments,” Proc. IPSN, pp. scenarios. Stars represent the cellular simulations and circles denote the CMT 334-341, 2006. approximation. [4] H. Bai, H. Aerospace and M. Atiquizzaman, “Error modeling scheme for fading channels in wireless communications: A survey,” IEEE Commun. Surveys, vol. 5, no.2, Fourth quarter, 2003. [5] G.J. Foschini and M.J. Gans, “On limits of wireless communication in a fading environment when using multiple antennas,” Wireless Pers. P (SNR) = E (P (SNR)) (22) Commun., vol.6, no.3, pp. 311-335, Mar. 1998. e,CMT e [6] I.E. Telatar, “Capacity of multi-antenna Gaussian channels,” European = ρ(r 1)/S2 Trans. Telecommun., vol.10, no.6, pp. 586-595, Nov./Dec. 1999. where SNR , and the expectation is over the [7] R. U. Nabar, Signaling for General MIMO Channels, PhD Thesis, 2 − variable S . For reasons of space, results are not presented Stanford University, Feb. 2003. here, but a preliminary analysis shows that several systems can [8] N. L. Johnson, S. Kotz and N. Balakrishnan Continuous Univariate Distributions, Volume 1/2, 2nd ed. New York, USA: John Wiley & be handled in this way, e.g., linear MIMO receivers using zero- Sons, 1994/1995. forcing or minimum-mean-squared-error (MMSE) combiners. [9] T. W. Anderson, An Introduction to Multivariate Statistical Analysis, In section IV, we have shown that the CMT model can be 2nd ed. New York, USA: John Wiley & Sons, 1984. [10] A. Dogandzic, A. Nehorai, and J. Wang, “Maximum likelihood estima- applied to cellular MIMO systems and allows a closed form tion of compound-Gaussian clutter and target parameters,” Proc. Annual evaluation of both MI and performance. Workshop on Adaptive Sensor Array Processing, Lincoln Laboratory, Lexington, MA, Mar. 2004. V. CONCLUSION [11] H. Shin and J. H. Lee, “On the capacity of MIMO wireless channels,” IEICE Trans. Commun., vol. E87-B, no. 3, pp. 671-677, Mar. 2004. Fading severity is often measured by the CV of the ampli- [12] P. J. Smith and M. Shafi, “An approximate capacity distribution for tudes and we have shown that for a fixed CV, the ergodic MI MIMO systems,” IEEE Trans. Commun., vol. 52, no. 6, pp. 887-890, of a MIMO channel is relatively insensitive to the exact type 2004. [13] Z. Wang, G. B. Giannakis, “Outage mutual information of space-time of amplitude distribution. This conclusion is especially valid MIMO channels,” IEEE Trans. Inform. Theory, vol. 50, no. 4, pp. 657- when the fading is not too severe or at high SNR. In the severe 662, 2004. [14] A. M. Tulino, S. Verdu, “Asymptotic Outage Capacity of Multiantenna fading region, i.e., CV > CVRayleigh, we have proposed a novel Channels,” IEEE Int. Conf. Acoustics, Speech Sig. Proc., Philadelphia, model which is simple and analytically tractable, based on Mar. 2005. the complex multivariate t distribution. For this model, exact [15] H. Huang and R. A. Valenzuela, “Fundamental Simulated Performance ergodic MI results are derived as well as approximations to of Downlink Fixed Wireless Cellular Networks with Multiple Antennas,” IEEE PIMRC, vol. 1, pp. 161- 165, Sept. 2005. the MI outage. The model has applications to wireless sensors, RF tagging, land-mobile, indoor-mobile and ionospheric radio