On the MIMO Channel Capacity Predicted by Kronecker and Müller Models

Wireless Pers Commun (2008) 47:91–100DOI 10.1007/s11277-007-9393-x

On the MIMO Channel Capacity Predictedby Kronecker and Müller Models

Müge Karaman Çolakoglu · Mehmet Safak

Published online: 5 October 2007© Springer Science+Business Media, LLC. 2007

Abstract This paper presents a comparison between the outage capacity of MIMO chan-nels predicted by the Kronecker and Müller models as a function of the number of scatterers,transmit and receive antennas. The Müller model is based on the singly-scattered rays betweenarrays of transmit- and receive antennas, while the Kronecker model considers only doublescattering. The channel capacity predictions by the Müller model were observed to be higherthan those by the Kronecker model. Moreover, Müller model is simpler since it is character-ized by fewer parameters, and accounts for frequency selective fading whilst the Kroneckermodel is valid only for frequency flat fading.

Keywords Antenna arrays · Communication system performance · Diversity methods ·Eigenvalues and eigenfunctions · Fading channels · Information rates · MIMO systems

1 Introduction

MIMO systems provide significant improvement in the capacity of wireless communicationsystems. However, an accurate model of the MIMO channel is needed for a realistic assess-ment of the capacity. The MIMO channel capacity may be described in terms of the numberand the spacing of transmit and receive antennas and the scattering richness (which is relatedto the number and geographical distribution of the scatterers). Since rays undergo significantamounts of attenuation each time they are scattered, the channel capacity is also influencedby the number of times the transmitted rays are scattered in the propagation process.

M. K. ÇolakogluMST Division, ASELSAN Inc., P.O. 101, Yenimahalle, Ankara 06172, Turkeye-mail: [email protected]

M. Safak (B)Department of Electrical and Electronics Engineering, Hacettepe University, Beytepe,Ankara 06800, Turkeye-mail: [email protected]

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92 M. K. Çolakoglu, M. Safak

This paper presents a comparative study of two models for the MIMO channels. In theKronecker model, which applies for a flat-fading MIMO channel, the rays are assumed to bescattered twice before arriving at the receiver; thus direct and the single-scattered rays areignored [1,2]. In this model, fading correlations are separated at transmit and receive antennaarrays. The second model, which will be referred to as the Müller model, is an asymptoticrandom channel model for a frequency-selective fading MIMO channel, where only sin-gly scattered rays are considered [3]. The Müller model is characterised by the number ofscatterers, and the transmit- and receive antennas; these parameters uniquely determine thechannel capacity. It is simpler and emphasizes the importance of scattering richness in MIMOchannels.

2 MIMO Channel Models

Consider a flat-fading MIMO system with a transmit array of NT antennas and a receivearray of NR antennas. The NR-dimensional received signal vector y may be written as

y = Hx + n, (1)

where x denotes the NT -dimensional transmitted complex signal vector, and n is an NR-dimensional noise vector with zero-mean independent and identically distributed (i.i.d.) com-plex Gaussian entries, with real and imaginary parts having equal variances. H denotes theNR × NT channel matrix with complex elements {hi, j }, describing the channel gain betweenthe j th transmitting antenna and the i th receiving antenna.

The capacity of a MIMO channel when the transmitter has no channel state information(CSI) is given by

C = log2 det

(I + γ

NTHHH

)(bits/s/Hz) (2)

where γ is the average signal-to-noise ratio (SNR) per receive antenna, found by equallydividing the total transmit power E[xH x] into NT transmit antennas, and I is an NR × NR

identity matrix. The superscript .H and E[.] denote, respectively, conjugate transpose, andthe mean value. Note that the capacity C in (2) is a random variable since the channel matrixH has random entries.

There are various methods for finding the average capacity. The simplest approach con-sists of replacing the mean value of HHH in (2) by a deterministic correlation matrix and,consequently to find the resulting eigenvalues and the average capacity [4,5]. Note that theentries of the channel matrix may be correlated with each other depending on the antennaspacing, angular spread of the transmitted and the received signals, and the scattering rich-ness. Since it is impossible to reduce fading correlations simultaneously between all arrayelements, the validity of the model chosen for the correlation matrix is critical in determiningthe channel capacity.

The second approach is based on the observation that, when the elements of H are zero-mean complex Gaussian random variables, HHH is a central Wishart matrix. The probabilitydensity function (pdf) of the ordered eigenvalues of a complex Wishart matrix is well-known[6]. Ergodic and outage capacities can then be determined by using the joint pdf of the orderednon-zero eigenvalues of HHH [6,7]. In this model, it may not be easy to determine the mar-ginal pdf’s of the eigenvalues and the capacity analytically, especially for large numbers oftransmit- and receive antennas. Here, we will focus our attention on the Kronecker and theMüller models.

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On the MIMO Channel Capacity Predictedby Kronecker and Müller Models 93

2.1 Kronecker Model

This model applies to a flat-fading MIMO channel, where the transmitter has no CSI, whilstthe receiver has perfect CSI. Only the scatterers located in the vicinity of transmit and receiveantenna arrays are assumed to contribute to the propagation mechanism. These scatterers areassumed to be in the far-fields of the corresponding arrays. The remote scatterers are ignored,since path losses will limit their contributions. The reader is referred to [1,2] for a detaileddescription of this model.

As shown in Fig. 1, dt and dr denote the element spacing of a uniform linear array ofNT antennas at the transmitter and of NR antennas at the receiver, respectively. The antennaelements are assumed to be isotropic. The S scatterers close to the transmit array (transmitscatterers) are assumed to be at least at a distance of Rt0 from the transmit array and to beconfined to a scattering radius of Dt from the line-of-sight (LOS). The exact locations of thescatterers are not needed in this model. Corresponding parameters for the receiving side areRr0 and Dr , respectively. The number of isotropic scatterers S on both sides is assumed tobe sufficiently large (typically S >10) so that random fading conditions apply. The distancebetween transmit and receive scatterers is denoted by R where R >> Rr0 and Rt0.

Rays transmitted by the transmit array are assumed to be scattered firstly from a trans-mit scatterer, with an angular spread of θt = 2 tan−1 (Dt/Rt0), and then from a receivescatterer, with an angular spread of θr = 2 tan−1 (Dr/Rr0), before arriving at the receivearray. Receive scatterers may be considered to consist of S virtual antennas with an averagespacing of 2Dr/S, and having an angular spread of θS = 2 tan−1 (Dt/R). Note that directand singly-scattered rays are ignored and the scattering is assumed to take place only in thevicinity of the transmitter and the receiver.

The NR × NT MIMO channel matrix is given by [1]

H = 1√S

R1/2θr ,dr

Gr R1/2θs ,2Dr /S Gt R1/2

θt ,dt(3)

NR × NR matrix Rθr ,dr denotes the correlation matrix governing the fading correlationsbetween the receive antennas, while NT × NT matrix Rθt ,dt accounts for the fading cor-relations between the transmit antennas. The S×S matrix Rθs ,2Dr /S determines the fadingcorrelations between the signals received, with an angular spread of θs , by the receive scat-terers, separated from each other with an average spacing of 2Dr/S. The S × NT matrixGt has i.i.d. complex Gaussian entries and accounts for the propagation between the NT

transmit antennas and the S receive scatterers. Similarly, the NR × S matrix Gr describes the

Fig. 1 Kronecker model for a fading MIMO channel

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propagation between S receive scatterers and NR receive antennas. A factor 1/√

S is usedto normalize the channel energy, i.e., to make the channel energy independent of the numberof scatterers.

Factorization of the channel matrix as shown in (3) allows the separation of the fadingcorrelations at each stage of the propagation process. This may help the design and opti-mization of MIMO systems. This model is reported to fail in predicting the MIMO systemperformance under certain circumstances and to be more accurate in channels with low fadingcorrelations [8,9].

2.2 Müller Model

This is an analytical asymptotic model for a frequency selective fading channel. In this model,the singular values of a random channel matrix H shows fewer random fluctuations and even-tually become deterministic as its size goes to infinity. The singular value distributions ofasymptotically large matrices can be calculated analytically and only the surviving physicalparameters show significant influence on the singular value distributions. Though the asymp-totic distribution of singular values is only an approximation to the distribution of finite-sizematrices, the asymptotic singular value distribution thus found can help identifying the dom-inant physical parameters which characteize a MIMO channel [3,7,10]. The random matrixtheory is widely used in the analysis of MIMO systems. The reader is referred to [3,11] fordetails.

Consider arrays of NT transmit and NR receive antennas in a channel with a total numberof S scatterers. As shown in Fig. 2, transmit and receive antenna arrays are located at the fociof concentric ellipsoidal (equi-delay) surfaces on which scatterers are located. The scattererslocated on the surfaces of outer ellipsoids correspond to longer delays for the received rays.The channel richness is defined as the number of scatterers that can be distinguished fromeach other in delay and space coordinates. Each ray is assumed to be scattered only oncebefore arriving at the receive antenna array, i.e., direct (LOS) ray and multiple scattering

Fig. 2 Delay and space coordinates for the Müller model

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is ignored. Note that multiple scattering implies weaker received rays compared to singlescattering.

Consider the scattering from the surface of the lth ellipsoid. The propagation coefficientbetween µth transmit antenna and νth receive antenna is given by

hv,µ,l =Sl∑

κ=1

Aκ,l ej (ϑκ,µ,l + ϕκ,v,l ) (4)

Here, the summation with respect to κ accounts for the contributions from all the Sl scattererslocated on the surface of the lth ellipsoid. ϕκ,υ,l , ϑκ,µ,l , Aκ,l and Sl show, respectively, therelative carrier phase at νth receive antenna due to κth scatterer, the relative carrier phase atκth scatterer due to µth transmit antenna, the attenuation of the κth path, and the number ofscatterers at delay l.

The multipath intensity profile of this frequency-selective MIMO channel is defined by theaverage received power from rays with Rayleigh distributed amplitudes, Aκ,l , correspondingto the lth delay:

Pl�= 1

Sl

Sl∑κ=1

E[∣∣Aκ,l

∣∣2], l = 1, 2, . . . L (5)

NR-dimensional received signal vector y at time k may be written as [3]

y [k] =L∑

l=1

Hlx [k − l − 1], (6)

where x denotes the NT -dimensional transmitted signal vector and the NR × NT channelmatrix Hl corresponds to lth delay. The eigenvalue distribution of HlHH

l , 1 < l < L , and of

H , where �= ∑L

l=1 Hl is the space-time channel matrix, are identical and converge toidentical asymptotic limits as NT , NR go to infinity whilst NT /NR remains fixed [3]. Thisimplies that a channel with a multipath intensity profile (frequency-selective fading channel),defined by (5), may be considered as a channel with a single delay (flat-fading channel) ifthe richness per delay, Sl/NR , is replaced by the total (channel) richness, S/NR :

ρ�=

L∑l=1

Sl

NR= S

NR(7)

Consequently, the asymptotic eigenvalue distribution of the space-time channel matrix doesnot change if delay times of particular paths vary. Hence, there is no need to distinguishbetween the distributions of path attenuations conditioned on different delays. This impliesthat the multipath intensity profile, hence the frequency-selectivity of the MIMO channeldoes not alter the asymptotic eigenvalue distribution. To summarize, the channel is suffi-ciently characterized by the channel richness, ρ = S/NR , the system load, β = NT /NR , andthe distribution of attenuations Aκ , 1 < κ < S as S goes to infinity. Here, E

[|Aκ |2] = 1 isassumed for all κ values; this implies a uniform multipath intensity profile.

Note that the Kronecker model explicitly accounts for the fading correlations in the prop-agation process. However, the fading correlations in the channel are implicitly accountedfor by the Müller model through the diversity provided by the asymptotically large channelrichness and the numbers of transmit and receive antennas.

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3 Results

In this section, we compare the outage capacity performance of the Kronecker and the Müllermodels. The parameters used to characterize the Kronecker model are listed in Table 1 [1].The frequency of operation was chosen to be 2 GHz (λ=0.15 m).

Noting that dt = dr = λ and angular spreads for arrival and departure, θr = θt = 900,are sufficiently large, fading correlations between antennas in transmit and receive arrays arenot very strong. The parameter values in Table 1 also suggest that the operation is in the farfields of both scatterers and antennas.

For a fixed value of β = NT /NR = 2 and S = 21, the outage capacity by the Kroneckermodel (KM) shows a steady improvement with increasing values of NT and NR (see Fig. 3).However, outage capacity predictions by the Müller model (MM) were observed to be rela-tively insensitive to the values of NT and NR as long as their ratio is fixed. This confirms theasymptotic character of the Müller model.

Figure 4 shows the effects of increase in the number of scatterers, transmit- and receiveantennas on the outage capacity for β = NT /NR = 1/3 and ρ = S/NR = 3. Outage capac-ity predictions by both models increase monotonously with increasing values of NT , NR

and S. The predictions by the Müller model was observed to be higher than those by theKronecker model. For higher values of the channel richness, ρ = S/NR , the capacity wasobserved to improve slightly.

Irrespective of the value of β = NT /NR , an increase in NT leads to an improvement inthe Kronecker model (see Fig. 5). As for the Müller model, the outage capacity is highestfor NT = 2 but it gets lower with increasing values of NT . It was observed that, below athreshold value of approximately 8.1 bits/s/Hz, the outage capacity was observed to degradewith increasing values of NT until β = 1 and it begins to improve again for β > 1. Above the

Table 1 Parameter values usedfor the Kronecker model

dt (m) dr (m) Rt0 (m) Rr0 (m) Dt (m) Dr (m) R (m)

0.15 0.15 50 50 50 50 50,000

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity in bits/sec/Hz

Pro

b. c

apac

ity<

absc

issa

S=21, SNR=10dB

NT=2,N

R=1,KM

NT=2,N

R=1,MM

NT=4,N

R=2,KM

NT=4,N

R=2,MM

NT=6,N

R=3,KM

NT=6,N

R=3,MM

Fig. 3 The effect of the number of transmit and receive antennas on the outage capacity, predicted byKronecker (KM) and Müller models (MM), for β = NT /NR = 2, S = 21 and SNR = 10 dB

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0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

b. c

apac

ity<

absc

issa

SNR=10dB

NT=1,N

R=3,S=9,KM

NT=1,N

R=3,S=9,MM

NT=3,N

R=9,S=27,KM

NT=3,N

R=9,S=27,MM

NT=5,N

R=15,S=45,KM

NT=5,N

R=15,S=45,MM

Fig. 4 The effects of NT , NR , and S on the outage capacity for NT /NR = 1/3, S/NR = 3, and SNR=10 dB

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

b. c

apac

ity <

abs

ciss

a

NR

=4, S=21, SNR=10dB

NT=2, KM

NT=2, MM

NT=4, KM

NT=4, MM

NT=8, KM

NT=8, MM

Fig. 5 The effect of NT , the number of transmit antennas, on the outage capacity predictions by Kroneckerand Müller models for NR = 4, S = 21, and SNR=10 dB

8.1 bits/s/Hz threshold, the outage capacity degrades monotonically with increasing values ofNT , though with increasing slopes. This may be explained by the fact that, when the channelrichness and the receiver diversity are sufficiently high, transmit diversity does not provideany additional diversity advantage [12] and it may even cause a degradation in the outagecapacity, since the transmit power is divided by the increasing number of transmit antennas.In the range of capacity values below threshold, Müller model may underestimate the outagecapacity compared to the Kronecker model.

Figure 6 shows the effects of the number of receive antennas on the outage capacity forβ = NT /NR < 1. In this case as well, capacity improves with NR in both methods, thoughthe improvement becomes less significant for higher values of NR . Predictions by the Müllermethod are more optimistic compared to those of the Kronecker model.

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3 4 5 6 7 8 9 10 11 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

b. c

apac

ity<

absc

issa

NT =2,S=21, SNR=10dB

N

R=4,KM

NR

=4,MM

NR

=6,KM

NR

=6,MM

NR

=8,KM

NR

=8,MM

Fig. 6 The effect of NR , the number of receive antennas, on the outage capacity predictions by Kroneckerand Müller models for β < 1, NT = 2, S = 21, and SNR=10 dB

2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

b. c

apac

ity<

absc

issa

NT=2,N

R=4,SNR=10dB

S=11,KMS=11,MMS=21,KMS=21,MMS=31,KMS=31,MMS=41,KMS=41,MM

Fig. 7 The effect of the channel richness (number of scatterers) on the outage capacity predicted by Kroneckerand Müller models for NT = 2, NR = 4, and SNR=10 dB

Figure 7 shows the effect of the channel richness on the outage capacity. The channelrichness has a significant impact on the capacity predicted by both methods. Irrespective ofthe number of scatterers, the capacity predictions by the Kronecker model are lower thanthose by the Müller method.

To have an idea about the relative significance of the three major parameters, NT , NR

and S, in Müller’s model, let us first consider the case for NT = 4, NR = 2, S = 21 andSNR=10 dB in Fig. 2, where the 50% outage capacity is 7.3 bps/Hz. However, for NT = 2and NR = 4 (the total number of antennas remains the same), Fig. 6 depicts 8.3 bps/Hz asthe 50% outage capacity. This confirms that NR is more effective than NT in improvingthe outage capacity. When the number of receive antennas is doubled, i.e., NR = 8, Fig. 6shows that the 50% outage capacity increases from 8.3 bps/Hz to 9.5 bps/Hz. On the other

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hand, again for NT = 2 and NR = 4, the 50% outage capacity is 7.4 bps/Hz for S =11,8.3 bps/Hz for S =21 and 9.3 bps/Hz for S =41 (see Fig. 7). This shows that doubling thescattering richness, S, and the number of receive antennas, NR , has similar effects on theoutage capacity, at least for the values of the considered parameters.

4 Conclusion

MIMO systems are used to improve the performance of wireless telecommunication sys-tems. However, one needs an accurate model of the MIMO channel for this purpose. Thispaper presents a comparative study of the outage capacity predictions by the Müller and theKronecker models.

The two models are then compared as a function of the number of scatterers, transmit- andreceive antennas and the SNR per receive antenna. Additional parameters needed for char-acterizing the Kronecker model are assumed fixed for comparison purposes. The capacitypredictions by both methods were observed to improve with increasing values of the numberof scatterers and the SNR ratio.

The Kronecker model applies to a flat fading channel, and separates fading correlationsat transmit and receive antenna arrays and in the channel. Since this model considers onlythe rays scattered from both transmit and receive scatterers, it may be more appropriate forradio propagation in urban areas. However, it may lead to pessimistic predictions in subur-ban channels, where direct- and/or singly-scattered rays may also reach the receiver. Somemeasurement results are reported to show that this model fails under certain circumstances.

The Müller model is simpler, valid for a frequency-selective fading MIMO channel andcharacterized by the scattering richness and the numbers of transmit and receive antennas.Since it considers only the singly-scattered rays, this model may describe a suburban chan-nel more accurately. Noting that the singly-scattered rays undergo less attenuation comparedwith multiple-scattering, the capacity predictions by the Müller model may be higher com-pared with the Kronecker model. It may even overestimate the capacity in urban channelswhere rays undergo mostly multiple-scattering. In this model, the scattering richness playsan important role, which is usually ignored in other models.

An increase in the number of receive antennas leads to an improvement in the outagecapacity predicted by both methods but the rate of improvement tapered with increasingvalues of the number of receive antennas.

The Kronecker model shows a steady improvement in the outage capacity as the transmitarray size increases, while the Müller model behaves differently for different values of theoutage capacity. When the transmit array size is higher than the receive array size, the chan-nel capacity, predicted by the Müller model, was observed to increase with the increasingnumber of transmit antennas for low capacity values, but to decrease for higher capacityvalues. When the transmit array size is less than the receive array size, the outage capacitywas observed to decrease with increasing transmit array size.

These results need to be supported by measurements in realistic scenarios.

Acknowledgements This work has been partially funded by the Scientific and Technological ResearchCouncil of Turkey (TUBITAK) under Grant 102E039 (COST 289) and was carried out in the framework ofEuropean COST Action 289 Spectrum and Power Efficient Broadband Communications.

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Authors Biographies

Müge Karaman Çolakoglu received her B.Sc. and M.Sc. degreesin electrical and electronics engineering from Hacettepe University,Ankara, Turkey, in 1999 and 2006, respectively. She is currently workingin ASELSAN Inc., Ankara, Turkey.

Mehmet Safak is with the Hacettepe University, Ankara, Turkey. Heis the head of the Telecommunications Group in Hacettepe Universityand the chairman of the Management Committee of COST Action 289“Spectrum and Power Efficient Broadband Communications”. His cur-rent research interests include the areas of wireless and mobile commu-nications, channel modeling, OFDM and MIMO techniques.

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On the MIMO Channel Capacity Predicted by Kronecker and Müller Models

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