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IEICE TRANS. COMMUN., VOL.E88–B, NO.5 MAY 20051829

INVITED PAPER Special Section on Recent Progress in Antennas and Propagation Researches

MIMO Propagation Channel Modeling∗

Yoshio KARASAWA†a), Member

SUMMARY This paper provides an overview of research in channelmodeling for multiple-input multiple-output (MIMO) data transmission fo-cusing on a radio wave propagation. A MIMO channel is expressed asan equivalent circuit with a limited number of eigenpaths according to thesingular-value decomposition (SVD). Each eigenpath amplitude dependson the propagation structure not only of the path direction profiles for bothtransmission and reception points but also of intermediate regions. Inher-ent in adaptive control is the problem of instability as a hidden difficulty. Inthis paper these issues are addressed and research topics on MIMO from aradio wave propagation viewpoint are identified.key words: MIMO, singular-value decomposition, multipath fading, eigen-value distribution, spatial correlation, diversity, maximum ratio combining

1. Introduction

Research on information transmission using a multiple-input multiple-output (MIMO) configuration involving theuse of an array antenna for both transmission and recep-tion has gained momentum in recent years. This interest hasbeen spurred by the practical implementation of arrays ofterminals, as seen in wireless local area networks (LANs),providing the highly efficient transmission of information.

In terms of its scientific underpinnings, research onMIMO can be broadly divided into (i) research on arrayantennas and adaptive signal processing [1]–[8] for the im-plementation of antenna configurations and control meth-ods, (ii) research on information theory and coding schemes(space-time coding) [9]–[12] for the implementation of ef-ficient data transmission, and (iii) research on radio wavepropagation [13]–[18] for the modeling of MIMO channel.

From the viewpoint of radio wave propagation, MIMOis closely related to the space-domain signal processingtechniques, such as the space diversity and adaptive arraytechniques. Specifically, the arrival angle spreading of mul-tipath waves and the associated spatial correlation character-istics have a significant impact on the information transmis-sion capacity of MIMO. The modeling of a MIMO chan-nel, if attempted rigorously, requires sophisticated knowl-edge of numerical statistics [19]–[22]. Even with the use

Manuscript received December 10, 2004.Manuscript revised January 27, 2005.†The author is with the Department of Electronic Engineering,

The University of Electro-Communications, Chofu-shi, 182-8585Japan.

∗This article was originally published in the IEICE Trans-actions on Communications (Japanese Edition), vol.J86-B, no.9,pp.1706–1720, Sept. 2003.

a) E-mail: [email protected]: 10.1093/ietcom/e88–b.5.1829

of such tools, however, only a very limited number of mod-els, such as i.i.d. (independent identically distributed) chan-nels, have been solved mathematically. Against the back-ground of these research efforts, this paper addresses radiowave propagation as the key to the data transmission capac-ity of MIMO channels, and provides a general review of re-search on propagation channel modeling. After discussingthe basic principles necessary to understand the mechanismof MIMO data transmission, the latest research effort, suchas the modeling of channels that considers space correlationcharacteristics [23], a topic which we have pursued is pre-sented.

This paper is organized as follows: First, focusingon the angular spread of multipath waves, the radio wavepropagation environment for mobile communications is dis-cussed in Sect. 2. A propagation channel representation ofMIMO channels is then presented with respect to narrow-band and broadband signals in Sect. 3, and the critical role ofthe eigenvalue in the correlation matrices is discussed in thecontext of propagation channel representation. Based on theresults presented in Sect. 3, the statistical characteristics ofMIMO channels are clarified in Sect. 4 by addressing casesin which the correlation between branches can be ignoredand those in which it must be taken into consideration. Theproblem of the instability of weight control is also consid-ered in Sect. 5 as a unique feature of MIMO, and importantresearch topics for MIMO are finally identified from a radiowave propagation viewpoint in Sect. 6.

2. Radio Wave Propagation Environment for MobileCommunications

The radio wave propagation environment for mobile com-munications including portable terminals for use in wirelessLANs is a multipath environment characterized by reflec-tion, scattering, diffraction, and blocking. Focusing on theangular spread of the multipath environment, the propaga-tion environment is classified, as shown in Fig. 1, into twotypes: Model 1 is a propagation path model in which boththe transmitting and receiving stations exist in the same scat-tering area, and Model 2 is a propagation path model inwhich one station exists in a locally distributed scatteringarea and the other station is in a position with a clear viewof the scattering area. Representative examples of Model 1may be an indoor propagation environment (wireless LAN,etc.) and a propagation channel for mobile communicationswith low base-station-antenna-height systems. Typical cases

Copyright c© 2005 The Institute of Electronics, Information and Communication Engineers

1830IEICE TRANS. COMMUN., VOL.E88–B, NO.5 MAY 2005

Fig. 1 Classification of mobile propagation channels. (UT: user termi-nal, AP: access point, MT: mobile terminal, BS: base station)

of Model 2, on the other hand, may be propagation channelsinvolving mobile communications with high base-station-antenna-height systems.

Both access points (APs) and user terminals (UTs) inModel 1, and mobile stations (MSs) in Model 2, are suitablefor modeling if it is assumed that either the probability of thedirection of paths is uniformly distributed in the horizontalplane or the angular spread of the paths is sufficiently large.If the angles of arrival are uniformly distributed, the spatialcorrelation ρa can be expressed as

ρa(∆x)

(≡ 〈a

∗(x)a(x + ∆x)〉〈a∗(x)a(x)〉

)= J0(k∆x)

(for AP, UT, MS), (1)

where ∆x denotes the distance between antennas, a is a com-plex amplitude representing the channel response, J0 is azeroth-order Bessel function, k is the wavenumber of the ra-dio waves, and 〈〉 denotes the ensemble average. In Model2, under the assumption that the path-direction distributesnormally over a relatively narrow range (namely, the stan-dard deviation of angular spread, σθ, is much smaller thanunity) centered around the direction of the mobile terminal(angle from the frontal direction of the antenna: θ0), the spa-tial correlation can be expressed as [24]

ρa(∆x) = exp

jk∆x sin θ0 −k2∆x2 sin2 θ0σ

2θ

2

(for BS). (2)

Figure 2 shows the calculated values of the space corre-lation ρ(≡ |ρa|) represented by Eqs. (1) and (2) in Rayleighfading environment. Note that |ρa|2 with a different defi-nition is sometimes used in the literature (e.g., [24], [25])as a correlation coefficient. From the figure, it is evidentthat in an environment (AP, UT, or MS) in which the spa-tial correlation is expressed by Eq. (1), signal variations canbe treated as independent if the antennas are located at dis-tances greater than half the wavelength. At the base station

Fig. 2 Correlation characteristics in Rayleigh fading environment.

Fig. 3 Transfer function of multipath channel.

(BS) represented by Eq. (2), on the other hand, there existsa considerable spatial correlation depending on the angularspread of paths. For example, if σθ = 3◦, the correlation co-efficient will be approximately 0.8 for a distance of approxi-mately 2 wavelengths, and the correlation between antennasmay therefore be taken into account.

Figure 3 shows the frequency characteristics (transferfunction) of a multipath channel simulated assuming typicalmobile communications in an urban environment (Model 2),and illustrates the manner in which the frequency charac-teristics change as a function of spatial position [26]. Sev-eral dips (drops in transmission characteristics) are visible inthe 4 MHz band centered around 2 GHz. These dips, causedby the spread of delay in multipath waves, produce distor-tion of the transmission waveform with respect to the broad-band signals. As such characteristics change drastically withonly a slight change in position (of the order of 10 cm),the characteristics can fluctuate rapidly with time in high-speed moving bodies such as vehicles. The adaptive signalprocessing scheme for mobile communications can be re-garded as a technique that realizes reliable communicationsby overcoming the problem of irregularities in the frequency

KARASAWA: MIMO PROPAGATION CHANNEL MODELING1831

and time regions, as illustrated in Fig. 3.

3. Representation of MIMO Propagation Channels

For an intuitive understanding of spatial signal process-ing, we first consider signal transmission in free space us-ing a 4 × 4-element MIMO, (Fig. 4). If the distance be-tween antenna elements is sufficiently large and the dis-tance of transmission is relatively short (Fig. 4(a)), sev-eral methods of data transmission are possible. For ex-ample, successive signals s1s2s3s4s5s6s7s8 . . . can be di-vided among antennas 1-4 and transmitted in groups ofs1s5 . . . , s2s6 . . . , s3s7 . . . , and s4s8 . . . , respectively. Al-though the received signals are subject to some cross-interference, this problem can be solved through signal pro-cessing at the receiving site. Carrying this analysis a stepfurther, if the distance between arrays is made sufficientlylarge (Fig. 4(b)), information transmitted by dividing be-tween antenna elements cannot be separated spatially atthe receiving site, and the transmission mode depicted inFig. 4(a) no longer works. In this case, the only viable op-tion is to orientate each antenna beam with respect to eachopposite station. In terms of the propagation channel modelthat is discussed later, the former approach represents thesituation where approximately four eigenpaths of roughlythe same amplitude exist in the channel, whereas the lat-ter approach represents the situation where the paths havecollapsed into one. The former case permits various modi-fications of the data transmission scheme and is potentiallyuseful for high-speed data transmission. Since in multipathenvironments, a number of such eigenpaths exist even whenthe distance between arrays is large, MIMO data transmis-sion is characterized by the effective utilization of path mul-tiplicity.

Figure 5 shows the basic configuration of the datatransmission scheme in MIMO. To achieve data transmis-sion by exploiting environmental characteristics, which in-

Fig. 4 Data transmission scheme for MIMO channel in free space envi-ronment.

volves the use of multiple antennas, the transmitter performscoding in space and time regions, as shown in Fig. 4(a), andthe receiver performs the decoding. There are two categoriesfor data transmission whether the transmitting site has thechannel state information (CSI) or not. Case 1 in the fig-ure is that in which the transmitting site does not have CSI,while Case 2 is information is shared between the transmit-ter and receiver. The two cases involve different methods ofdata transmission. In the following section, the representa-tion of the propagation channel is discussed.

3.1 Propagation Channel Representation for Narrow-BandSignals

Consider a channel with M transmission antennas and N re-ception antennas. The channel response (CSI) between thearrays can be expressed by the following matrices:

A = [ a1 a2 . . . am . . . aM ] (3a)

am = [ a1m a2m . . . anm . . . aNm ]T , (3b)

where superscript T denotes transposition.The channel response matrix A can be expressed as fol-

lows in a singular-value decomposition (SVD) (e.g., [13]).

A = ErDEHt =

M0∑i=1

√λi er,ie

Ht,i, (4)

where

D ≡ diag[ √λ1

√λ2 . . .

√λM0

](5)

Et ≡ [et,1 et,2 . . . et,M0

](6)

Er ≡ [er,1 er,2 . . . er,M0

](7)

M0 ≡ min(M,N). (8)

Superscript H denotes a complex conjugate transposition,λi is the ith eigenvalue of the correlation matrix AAH (orAH A) for i = 1, 2, . . . ,M0 (in descending order), et,i is theeigenvector belonging to the eigenvalue λi of matrix AH A,and er,i is the eigenvector belonging to the eigenvalue λi ofmatrix AAH .

When decomposed in this manner, the propagation

Fig. 5 Data transmission scheme in MIMO configuration.


Fig. 6 Equivalent circuit of MIMO channel based on SVD.

Fig. 7 Parallel data transmission scheme through each eigenpath.

path for a MIMO channel can be represented as in Fig. 6,where (a) shows the equivalent circuit representing Eq. (3),and (b) is the equivalent circuit represented in SVD. It isclear that the MIMO channel comprises a maximum of M0

(the smaller of M and N) independent transmission paths.In this context, the term “maximum” refers to the maxi-mum possible number of actual paths. If there are multi-ple paths for which the eigenvalue is 0, the number of pathswill be reduced even further. Such virtual paths are referredto as eigenpaths. The amplitude gain for each unique pathis√λi, and it varies according to its eigenvalue. Thus, the

MIMO channel possesses the capacity to transmit M0 in-dependent streams of signals without cross-talk. Figure 7shows the specific configuration of an eigenmode transmis-sion scheme.

Next, consider transmission in which data transmit-ted through only the largest eigenpath is received withthe maximum signal-to-noise ratio (SNR) with transmis-

sion/reception weight control, that is, maximal ratio com-bining (MRC) transmission. Here, the transmission antennaweight vector is wt(≡ [wt,1, wt,2, . . . , wt,M]T ), the receiver an-tenna weight vector is wr(≡ [wr,1, wr,2, . . . , wr,N]T ), whichare normalized by wH

r wr = 1 and wHt wt = 1, the noise

component vector is n(t)(≡ [n1(t), n2(t), . . . , nN(t)]T ), andthe transmitting signal is s. The reception signal and its av-erage power are respectively given by

s(t) = wHr(Awt s(t) + n(t)

)(9)

〈|s(t)|2〉 = λ〈|s(t)|2〉 + 〈|n1(t)|2〉 (10)

λ = wHr Awtw

Ht AHwr = wH

t AHwrwHr Awt, (11)

where n1 is the noise component of antenna 1. Because theaverage power of noise for the various branches is equal, thequantity n1 is used as the representative component.

As λ in Eq. (11) is a gain factor of the SNR, setting wt

and wr such that λ becomes maximum will allow for trans-mission with the maximum SNR using the weight of the set(wt and wr). The set can be determined using Lagrange’smultiplier method, as

ϕ(wt,wr, λt, λr) = λ − λt(wHt wt − 1)

− λr(wHr wr − 1) (12)

∂ϕ

∂wr= 2Awtw

Ht AHwr − 2λrwr = 0 (13)

∂ϕ

∂wt= 2AHwrw

Hr Awt − 2λtwt = 0. (14)

The problem then reduces to the eigenvalue problemgiven by the following set of simultaneous equations:[

AwtwHt AH

]wr = λrwr (15)[

AHwrwHr A

]wt = λtwt. (16)

Equations (15) and (16) can be transformed into the form ofEq. (11) as follows.

wHr

[Awtw

Ht AH

]wr = wH

r λrwr = λr (17)

wHt

[AHwrw

Hr A

]wt = wH

t λtwt = λt. (18)

where both λr and λt are possible values of λ, as physicalquantities corresponding to the reception power or SNR.

The meaning of Eq. (15) can be interpreted as follows.Although λr is the eigenvalue of the matrix [Awtw

Ht AH],

as we are seeking the maximum value of λr, the maximumeigenvalue λr,max is the maximum value of the desired recep-tion power. The weight that produces the maximum value isthe eigenvector er,max of the maximum eigenvalue. How-ever, as this equation contains the quantity wt, it cannot besolved directly. Similarly, in Eq. (16), the maximum eigen-value λt,max is the maximum value of the desired receptionpower, and the weight that produces it is an eigenvectoret,max of the desired reception power.

Note that in Eq. (15), the matrix [AwtwHt AH] is com-

prised of the product of a vector Awt and its Hermitian trans-position. Therefore, its non-zero eigenvalues λr,max form a


matrix of rank 1. The eigenvector of that matrix will be avector that is obtained by normalizing the vector Awt. Sim-ilarly, in Eq. (16), in the matrix [AHwrw

Hr A], the nonzero

eigenvalues λt,max form a unique matrix of rank 1. There-fore, the eigenvector with eigenvalues λr,max and λt,max canbe represented by

er,max = wr =1√λr,max

Awt (19)

et,max = wt =1√λt,max

AHwr. (20)

From these two equations, we obtain

AH Awt =√λr,maxλt,max wt (21)

AAHwr =√λr,maxλt,max wr. (22)

This reduces the weight to be determined to the familiareigenvalue problem. Furthermore, the following equationcan be obtained from Eqs. (11), (17), and (18):

λr,max = λt,max ≡ λmax. (23)

Thus, the maximum value will be the maximum eigenvalueof the Hermitian matrix AH A (or AAH), and the weights forthe transmitter and receiver will be

wt = et,1, wr = er,1. (24)

This corresponds to an eigenmode transmission using thelargest eigenpath shown in Fig. 7, and the transmission inwhich all power is assigned solely to the eigenpath will real-ize MRC transmission. This transmission method, in whichthe transmission and reception processes are controlled op-timally with respect to the antenna pattern, is also referredto as eigen-beam-forming transmission.

The MRC transmission method represents transmis-sion via the primary path of channels. In contrast, multi-stream transmission [8] can be interpreted as transmissionusing the second or lower eigenvalue path, that is, minorpaths. Figure 8 shows a simple example of the latter ap-proach, where considerable benefits are gained through theuse of both the primary path and minor paths.

The figure illustrates the case where the primary andminor paths are of the same width, that is, where two eigen-paths with equal eigenvalues are available. Figure 8(a)shows the case where all power is assigned to one of thepaths and 4 bits per symbol are transmitted by 16-QAM.By contrast, Fig. 8(b) shows the case where information istransmitted via two paths by halving the power assigned toeach path by QPSK. There is a fourfold difference betweenQPSK and 16-QAM in terms of the carrier wave to noiseratio (CNR), at which the error rates are the same for thetwo paths. Thus, even when the transmission power is di-vided equally, the method described in Fig. 8(b) can achievea lower error rate. If the error rates are to be equalized, amodulation method (e.g., 8-QAM) that permits 3-bits-per-symbol transmission should be adequate if two paths are

Fig. 8 Is it possible to increase channel quality when utilizing all of theeigenpaths?

used. Thus, the figure illustrates that in transmission pathswith high SNR, the use of other eigenpaths (minor paths) ismore efficient than transmitting information solely over thelargest eigenpath (primary path).

However, what happens in cases with low SNR? Forexample, consider substituting QPSK, a 2-bit transmissionscheme, for the 16-QAM portion in Fig. 8(a). To achievedthe same result as in Fig. 8(b), BPSK is employed as themodulation method. However, in this case, given the re-lationship between the signal-to-noise ratio (SNR) and thebit error rate (BER), it is clear that the use of BPSK yieldslittle benefit. Thus, it is clear that the effect of MIMO issignificantly dependent not only on the number of antennasemployed, but also on the SNR of the propagation channel.

In cases where the transmitter does not possess infor-mation on the transmission path (Case 1 in Fig. 5), the keyto achieve adequate performance in the worst-case scenariowill be to use a transmission method that minimizes trans-mission deterioration. If the same signals (coherent sig-nals) are transmitted from the transmitter using a multiple-element antenna, null points occur in certain directions, asseen in an antenna pattern for an array antenna. The worstcase is the condition in which a path exists in such a null-point direction. Such a transmission method does not con-form to this paradigm.

For cases involving the use of a two-element antennaon the transmitter side, Alamouti devised a method inwhich, for information s1 and s2 to be transmitted, one an-tenna transmits s1 and −s∗2, and the other antenna transmitss2 and s∗1, and the receiver performs block decoding usingchannel state information (CSI) as prior knowledge. How-ever, due to the fact that the transmitter does not possesknowledge of the channel, the reception capacity suffers areduction in SNR equivalent to 3 dB, compared with theMRC used in Case 2 [9]. Transmission in which signalsare rearranged in both the space and time domains in thismanner is referred to as space-time block coding (STBC)transmission. Based on the transmission method proposed


by Alamouti, there has been active research interest in ex-panding the application of the method. Although furthertreatment of the STBC technique is beyond the scope ofthis paper, interested readers can obtain more informationin Refs. [9]–[12].

3.2 Wideband Signal

In the preceding section, an equivalent circuit representa-tion was given for narrow-band transmission paths in whichthe MIMO transmission path does not have frequency char-acteristics. In this section, the situation where frequency-selective characteristics within a band become irregular dueto the spread of delay in multipath waves, that is, a MIMOtransmission path representation with respect to frequency-selective fading, is considered.

Suppose that hnm denotes the impulse response of atransmission path (with, m, transmission antenna and, n, re-ception antenna, presented simply as h in the present dis-cussion) between arbitrary sets of antennas, and consider anapproximate representation of impulse response for the sit-uation in which the spreading of multipath delay is greaterthan the symbolic period Ts of the modulation signal. Thechannel impulse response is assumed to include the charac-teristics of a band-limited filter, and the receiver is assumedto perform several sampling operations per symbol. The de-scription of the model will therefore depend on system pa-rameters.

If the propagation paths represent a time-invariant sys-tem, impulse response h and impulse response h of a band-limited channel are respectively given as

h(τ) =∑

i

aiδ(τ − τi) (25)

h(τ) =∑

i

aig(τ − τi), (26)

where ai and τi denote the complex amplitude and theamount of delay of the elementary wave i, δ is the deltafunction, and g is the impulse response of the band-limitingfilter. The filter used for band-limiting is either a Nyquistor Gaussian filter. The bandwidth B of the filter is often de-signed such that BTs with respect to the symbol period Ts isapproximately 1.

The signal to be transmitted, assuming that it is an im-pulse that is generated in each symbolic period Ts, is givenby

s(t) =∞∑

k=−∞d(k)δ(τ − kTs), (27)

where d(k) is the time series of transmitted data, which istreated as an infinitely continuing series.

The reception waveform (exclusive of noise compo-nents) for the modulation signals subject to band restrictionsis given by

sR(t) = s(t) ⊗ h(t)

=

∞∑k=−∞

d(k)∑

i

aig(t − kTs − τi)

, (28)

where ⊗ denotes a convolution. If this signal is sampled attime t = kTs, the resulting output will be

sR(kTs) = · · · + d(k − 1)h(Ts) + d(k)h(0)

+ d(k + 1)h(−Ts) + · · · .(29)

In a system where sampling is performed with symbol pe-riod (Ts) timing, the impulse response of the transmissionpath is equivalent to the following equation with respect toimpulses at intervals of Ts:

he(τ) =∞∑

k′=−∞h(k′Ts)δ(τ − k′Ts). (30)

In one-sample-per-symbol operations, such as theequalization of multipath waveform distortion, the perfor-mance evaluation of adaptive arrays, or operation analysesbased on simulations, the number of multipath waves thatcan be incorporated is insufficient. To circumvent this prob-lem, two and four samples per symbol period are often em-ployed.

The equivalent impulse response for J samples persymbol can be expressed as

he(τ) =∞∑

k′=−∞

J∑j=1

h

{(k′ +

j − 1J

)Ts

}

· δ{τ −

(k′ +

j − 1J

)Ts

}.

(31)

Thus, in situations where the spread of the impulse responseof a propagation channel is greater than the symbol periodTs of the modulation signal, the impulse response can beconverted equivalently to an impulse response that is ren-dered discrete in Ts/J. In this case, the reception signal canbe determined as

sR

(Ts

J

)=

∞∑k=−∞

[d(k)h

{(−k +

J

)Ts

}], (32)

where is an integer that represents the temporal order ofsampling, and the range of k that provides a substantial con-tribution is the range equivalent to the spread of the delaycentered around /J.

In the case of a MIMO channel, the equivalent impulseresponse matrix He(τ) can be expressed by

He(τ) =∞∑

k′′=−∞

h11

(k′′J Ts

) . . .. . . hnm

(k′′J Ts

) . . .. . . hNM

(k′′J Ts

),

· δ

(τ − k′′Ts

J

), (33)


where hnm(τ) is a substitution of h in Eq. (26) for impulseresponse hnm of path nm(m = 1, 2, . . . ,M; n = 1, 2, . . . ,N).

In the equation, k′′ is an integer that represents discrete time,and the range in which k′′ makes a substantial contributionis equivalent to the spread of delay beginning in the vicinityof 0.

Another channel representation that can be employedwith respect to broadband signals is a frequency-domainrepresentation. In this case, it suffices to represent the chan-nel response characteristics of Eqs. (3)–(7) that representnarrow-band signals as a function of frequency. They canbe expressed as follows.

A( f ) =[anm( f )

](34)

anm( f ) =∫ ∞

−∞hnm(τ) exp(− j2π f τ)dτ

≈∫ ∞

−∞he,nm(τ) exp(− j2π f τ)dτ (35)

As a result, the correlation matrices, eigenvalues, andeigenvectors can all be represented as a function of fre-quency. The channel matrix A can then be processed bysingular-value decomposition as follows.

A( f ) = Er( f )D( f )EHt ( f )

=

M0∑i=1

√λi( f ) er,i( f )eH

t,i( f )(36)

Although the eigenvalues change continuously in theirrespective bands, in some cases, two eigenvalues, such asthe first and second eigenvalues, change in such a waythat they intersect. Even in that case, continuity can bemaintained within the band due to the fact that the max-imum eigenvalue λmax( f ) is always maximum. However,eigenvectors belonging to the maximum eigenvalue are ex-changed at the point where two eigenvalues become equaland intersect, thus resulting in a loss of continuity. Such asituation poses a risk of control instability, as will be dis-cussed in Sect. 5.

In a system such as that of orthogonal frequency-domain multiplexing (OFDM), where every signal in thetime series is expanded as a narrow-band signal in the fre-quency domain, the characteristic of each subchannel canbe defined as a carrier wave frequency corresponding to thesignal. Therefore, the expression in Eq. (36) can be directlyapplied to the analysis of the MIMO-OFDM informationtransmission method and the relevant transmission charac-teristics [6], [7].

To summarize the discussion above, Fig. 9 shows anexample of a broadband MIMO transmission path (a) interms of the time domain (b) based on Eq. (33) and the fre-quency domain (c) based on Eq. (34).

Fig. 9 Modeling of wideband MIMO propagation channel.

4. Statistical Representation of the MIMO Channel

4.1 Independent Identically Distributed Channels (Model1)

In mobile communications, the movement of terminalscauses multipath problems that manifest as temporal fluc-tuations. By contrast, in wireless LAN, where terminals arestationary desktops, the characteristics manifest as spatialchanges. In either case, it is critical to have a statistical un-derstanding of the multipath characteristics. In this section,the Model 1 environment depicted in Fig. 1 is considered.This case represents an environment in which there are alarge number of paths with sufficiently large angular spreadat both transmitting and receiving stations. In Fig. 2, if thedistance of separation between antennas is half the wave-length or greater, the correlation coefficient will be 0.5 orless, and thus the impact of correlation is negligible. Inthis environment, the amplitudes of all paths anm in Fig. 6(a)have the same Rayleigh distribution, and their fluctuationscan be treated as independent quantities. Such channelsare referred to as independent identically distributed (i.i.d.)channels.

The statistical characteristics of the i.i.d. channels, thatis, the probability distribution of eigenvalues of the correla-tion matrix, are examined as follows. In the multipath en-vironment, power fluctuations in the Rayleigh fading envi-ronment take the form of an exponential distribution. In the1-to-N maximum ratio combining diversity, the distributionbecomes a gamma distribution. MIMO represents a dimen-sional expansion of that distribution to M-to-N, for whichthe individual matrix elements and eigenvalues take the formof a Wishart distribution [19]–[22]. Figure 10 shows thecumulative distribution of eight eigenvalues of the corre-lation matrix AAH where M = N = 8. The distribution


Fig. 10 CDF of each eigenvalue for M = N = 8 in i.i.d. Rayleigh fadingchannel.

shown in the figure was obtained by computer simulations.The theory behind the determination of this distribution iswell established in the field of numerical statistics, and anequation for obtaining the cross-coupled probability distri-bution of all eigenvalues is available (not quoted here be-cause the equation is of little practical value for computationpurposes; the equation is indicated in Ref. [27] as quotedfrom Ref. [21]).

In the maximal ratio combining transmission (beam-forming transmission) of MIMO, the value of the largesteigenvalue, λmax, provides an index for the evaluation ofperformance. For the probability distribution of maximumeigenvalues, the following equations are obtained by inte-grating other eigenvalues from the cross-coupled probabilitydistribution of all eigenvalues as probability variables [22]:

f (λmax) =1P

P∑p=1

ϕp(λmax)2λQ−Pmax e−λmax (37a)

ϕk+1(λ) =

√k!

(k + Q − P)!LQ−P

k (λ) (37b)

(k = 0, 1, . . . , P − 1)

LQ−Pk (x) =

1k!

exxQ−P dk

dxk(e−xxQ−P+k) (37c)

P(≡ M0) ≡ min(M,N), Q ≡ max(M,N). (37d)

The actual computation of the above equations will notbe easy when the value of M, N, or Q is large. To circum-vent this problem, research on approximation techniquesis being pursued to determine a distribution by linking theM × N-configuration MIMO to a 1 × MN-configurationSIMO (single-input multiple-output) space diversity [27].With i.i.d. channels, determining the probability distributionof the largest eigenvalues of the M×N-configuration MIMOby computer simulation (for example, λ1 in Fig. 10), andcomparing the results with the SN ratio distribution (gammadistribution) for the maximum ratio combining for 1-to-MNspace diversity yields a high degree of agreement with re-

gard to distribution profiles [27]. This means that the prob-ability distribution of the first eigenvalues of the M × N-configuration MIMO can also be approximated by a gammadistribution. To complete this distribution as an algorithm,it is necessary to obtain an equation that gives the averageof the maximum eigenvalues. To this end, an approxima-tion formula has been proposed [27]. The following is analgorithm (gamma distribution, where f denotes a probabil-ity density function, and F denotes a cumulative distributionfunction) that gives the distribution of the maximum eigen-values obtained in this manner:

f (λmax) =1

(MN − 1)!λMN−1

max

ΓMN0

· exp

(−λmax

Γ0

)(38a)

F(λmax) = 1 − exp

(−λmax

Γ0

)·

MN∑j=1

(λmax/Γ0) j−1

( j − 1)!, (38b)

where Γ0 is the average of the maximum eigenvalues nor-malized by MN. It can effectively be approximated by thefollowing equation (for further details, see Ref. [27]):

Γ0 = 〈λmax〉/MN

≈( M + N

MN + 1

)2/3

for NM ≤ 250 (39a)

≈(√

M+√

N)2/MN

for NM > 250. (39b)

Comparing with values obtained by computer simulation,it has been found that the estimation using Eq. (38) is suf-ficiently accurate for practical purposes [27]. It should benoted that Eq. (39b) is derived as an asymptotic formula (oran equation that provides an upper bound on the averages)when both N and M are sufficiently large [13].

4.2 Distribution of Maximum Eigenvalues ConsideringSpatial Correlation

The Model 2 environment shown in Fig. 1 requires an anal-ysis that takes into account spatial correlation on the basestation side where the angular spread in the direction of thepath is relatively small. Figure 11 shows the average eigen-values, obtained by simulation, for the M = N-configurationMIMO, for which spatial correlation can be ignored (Model1: i.i.d. channel) and for the case where a correlation ex-ists only on one side (Model 2) [23]. An array of antennasplaced at equal intervals on a straight line is employed, andthe correlation between elements is set so as to yield the val-ues calculated according to Eq. (2) in all combinations. Inthe figure, the value ρ(≡ |ρa(d)|) of the correlation is definedas a correlation coefficient in terms of the nearest antennaelement distance d for the side that takes the correlation intoaccount. Figure 11(b) shows the case where this value is 0.9.The figure indicates that, in comparison to the case with nocorrelation, the largest eigenvalue is more dominant when


Fig. 11 Averaged value of each eigenvalue depending on spatial correla-tion.

there is correlation, and the second and subsequent eigen-values are relatively small. In the figure, the line connectedby ◦ represents the sum (= N2) of all eigenvalues.

Analyses of transmission characteristics taking correla-tion into account have been attempted [16], [17]. However,to date, no mathematical models have been established thatprovide a probability distribution of eigenvalues with a sim-ple form. For consistency with the approximation model de-scribed in Sect. 4.1, the diversity approximation model givenin reference [23] is considered hereafter.

Defining a new vector b as

b ≡[aT

1 aT2 . . . a

Tm . . . a

TM

]T(40)

to represent all paths in one vector using the column vectora in the channel response matrix A of Eq. (3), the correlationmatrix among paths is as follows:

Raa = 〈bbH〉 (41a)

≈

〈a1a

H1 〉 0 · · · 0

0 〈a2aH2 〉 · · · 0

......

. . ....

0 0 · · · 〈aMaHM〉

(model 2) (41b)

≈ INM (model 1). (41c)

where 〈〉 is the average over a sufficiently long time com-pared with the fading fluctuation cycles (or a sufficientlylong distance with respect to space changes when comparedwith the cycles of fluctuation), and INM denotes the unit ma-trix of NM × NM.

This matrix contains NM eigenvalues, which can besorted in descending order as [λ1 ∼ λNM]. In Model 1(Eq. (41c)), all eigenvalues are equal to 1. In Model 2(Eq. (41b)), if the MT side in Fig. 1(b) is assumed to haveM elements and the BS side has N elements, N sets of Mequal values are obtained.

In this case, the probability distribution of the maxi-mum eigenvalues λmax in AAH can be estimated by linkingthe distribution to Eq. (25) to give the probability distribu-tion of the SNR in a correlated 1-to-MN space diversity, asgiven by [23]

f (λmax) =1

MN∏j=1Γ j

MN∑j=1

exp(−λmax)/Γ j)MN∏k� jk=1

(1Γk− 1Γ j

) (42a)

F(λmax) = 1 −MN∑j=1

ΓMN−1j exp(−λmax/Γ j)

MN∏k� jk=1

(Γ j − Γk)(42b)

Γ j = (λ j + ε j)G(M,N; ρ)/MN (42c)

λ j � ε j, (42d)

where ε j denotes a very small quantity assigned to avoid thepossibility of Eqs. (42) becoming uncomputable if eigenval-ues are identical. Similarly, G(M,N; ρ) denotes the averageof the maximum eigenvalues λmax. In the case of Model 1,the following approximation formula exists for G(M,N; ρ)[23]:

G(M,N; ρ) = MN

{α +

( M + NMN + 1

)2/3

(1 − α)

}(43a)

α =

N∑

j=1λ j − N

MN − 1

3/√

M

. (43b)

Due to the fact that the above equations are based on


Eq. (39a) as the model, MN < 250 is the approximate crite-rion for applying the above equations.

This discussion has only considered estimation meth-ods based on the approximation of the largest eigenvaluesfor which a correlation exists. The establishment of a theo-retical or approximation model for the determination of theprobability distribution of all eigenvalues remains an impor-tant topic.

4.3 Channel Capacity

According to Shannon’s information theory, if a transmis-sion channel has an SNR of γ, the transmission channel ca-pacity C per second per Hz is given by

C = log2(1 + γ) [bits/s/Hz]. (44)

In MIMO, the following two cases, illustrated in Fig. 5,yield different channel capacities.Case 1: Only the receiver has information on a MIMO chan-nel response matrixCase 2: Both the transmitter and receiver share informationon the MIMO channel (based on prior measurements andother factors)

In Case 1, the transmission capacity is given by

C1 =

M0∑i=1

log2

(1 +λiγ0

M

), (45)

where γ0 denotes the SNR when all power from the trans-mitter is emitted from a single antenna and is received on asingle antenna after traversing paths with a path gain of 1.This means that the maximum capacity is achieved when thetransmitter does not know the conditions under which trans-mission is performed and the power is distributed uniformlyto all transmitting antennas (i.e., by avoiding a worst-casescenario where null points arise in a specific direction).

In Case 2, the transmission capacity is given by

C2 =

M0∑i=1

log2 (1 + λiγi) (46a)

M0∑i=1

γi = γ0. (46b)

In this case, a power proportional to γi is allocated to eacheigenpath, and the optimum allocation is based on the Wa-ter Filling (WF) theorem (for a specific allocation methodbased on the WF theorem, see Ref. [28]). The transmissioncapacity that is optimally allocated according to the WF the-orem is referred to as CWF .

In Case 2, the method in which all power is allocatedonly to the largest eigenpath (primary path) having the max-imum eigenvalue is an MRC method with controlled trans-mission/reception weights. In this case, the transmission ca-pacity is given by

CMRC = log2(1 + λ1γ0). (47)

The transmission path capacity of the MIMO channel inthe Rayleigh fading environment can be calculated as fol-lows, using the cross-coupled probability density functionof eigenvalues f (λ1, λ2, . . . , λM0 ):

〈C〉 =�

· · ·∫ ∞

0C(λ1, λ2, . . . , λM0 )

· f (λ1, λ2, . . . λM0 )dλ1 dλ2 . . . dλM0 .

(48)

However, including no-correlation cases, there is no sim-ple algorithm for calculating the cross-coupled probabilitydensity function f of eigenvalues. Therefore, the followingapproximation formula using the average of the eigenvaluesis often used:

〈C〉 = C{〈λ1〉, 〈λ2〉, . . . , 〈λM0〉

}. (49)

It has been determined, for MRC transmission calculatedsolely on maximum eigenvalues (i.e., calculated accordingto Eq. (47)), that the difference between the calculated val-ues based on the probability distribution of eigenvalues andthose calculated according to average values can be ignored[27]. Therefore, it is expected that Eq. (49) will also yield ahigh degree of accuracy.

Figure 12 shows the calculated averages of C1,CWF ,and CMRC for iid channels based on the algorithm(Eqs. (45)–(47)) for channel capacity. To permit a compar-ison of the three channel capacities on an equal basis, it isassumed that the total transmitting power is constant, andthat the channel bandwidth (or the bandwidth of transmit-ted signals) is fixed. In the figure, γ0 is used as a parame-ter, which is equivalent to the SNR. The figure shows threecases: γ0 = 0.1 (extremely poor SNR), γ0 = 1, and γ0 = 10

Fig. 12 Comparison of channel capacity for three transmission schemes.


Fig. 13 Effect of correlation on channel capacity CWF in higher SNRcase.

(high SNR). It can be seen from the figure that, (i) wherethe SNR of the transmission path is low, near-limit perfor-mance can be obtained for MRC by allocating all power tothe maximum eigenpath (the primary path), and (ii) wherethe SNR is high, a higher channel capacity can be obtainedby allocating some power to other eigenpaths (secondarypaths). It can also be appreciated that as the SNR increases,the difference between Case 1 (C1) and Case 2 (CWF) van-ishes. Figure 13 shows the channel capacity CWF includingcorrelation-rich cases, for γ0 = 10, which is a high SNRenvironment, by using the correlation coefficient as a pa-rameter. The figure indicates that when optimal transmis-sion is to be conducted, the impact of correlation results in adecline in channel capacity. The analysis of channel capac-ities indicates that it is important to understand the eigen-value distribution interms of a radio wave propagation char-acteristic. The same is true of the evaluation of bit errorrates (BERs). Therefore, MIMO propagation channel mod-eling, in essence, is the investigation of the characteristicsof eigenvalues of channel response matrices.

5. Impact of Radio Wave Propagation Environment onMIMO Transmission Characteristics

5.1 Case of Eigenpaths Reducing to One Path

Even in a multipath-rich environment where the area aroundthe transmitting station and the area around the receivingstation are surrounded by local scatterers, and where the an-gular spreading in the direction of the path is large, the sec-ond and subsequent eigenvalues reduce to zero, and onlythe first eigenvalue predominates in some cases as shown inFig. 14. This means that in the modeling of MIMO channels,in addition to both the propagation environment around thetransmitting and receiving stations, the intervening propa-

Fig. 14 Propagation environments consisting of only one eigenpath.

gation structure exerts a significant influence on the MIMOtransmission characteristics.

Figure 14(a) illustrates the case where the range of an-gles at which the receiving antenna has a clear view of thescattering area at the transmission point and the range ofangles at which the transmitting antenna has a clear viewof the scattering area at the reception point are both suf-ficiently smaller than the antenna beam width. Therefore,that in this environment, plane waves reach the destinationarea (the case in Fig. 4(b) also corresponds to this scenario).The same is true of the situation in Fig. 14(b) where sig-nals are relayed by means of a single antenna. This casehas previously been treated as a keyhole problem [15]. An-other example, as shown in Fig. 14(c), involves the arrivalof diffracted waves only from the edge of a single scatterer(building).

In all of these cases, the SVD representation of thechannel response matrix A takes the form of

A ≈ √λ1 er,1e

Ht,1, (50)

where the eigenvalues for the correlation matrix AAH (andAH A) collapse into one eigenvalue (i.e., the correlation ma-trix becomes a matrix of Rank 1). In this case, as may beclear from Fig. 14, because the multipath environment forthe area around the terminal is maintained, a space diversityeffect (improvement in combined gains and suppression offluctuations) can be expected. However, from a standpointof multistream transmission, the degree of freedom is lost.This situation can be understood as being a cascade connec-tion of a multiple-input single-output (MISO) channel to asingle-input multiple output (SIMO) channel.

From a high-efficiency information transmission stand-point, an optimal environment is one in which multiple in-dependent paths (eigenpaths), typically the indoor multipathenvironment, can exist. Therefore, wireless LAN representsan environment in which MIMO can operate to its full po-


tential.

5.2 Inherent Instability of MIMO Weight Control

In a situation where the propagation environment changesas a function of time, there is an inherent instability prob-lem in MIMO control when the weights of transmission andreception are controlled based on an adaptive algorithm, asillustrated in the following example.

Consider control involving MRC. Let λi(t)(i =

1, 2, . . . M0) be the eigenvalues of a correlation matrix, andassign numbers i to the eigenvalues that vary continuously(unlike the situation discussed in Sect. 2, the eigenvalues arenot in descending order). In this case, the MRC method is analgorithm with a weight equal to an eigenvector that alwaysproduces the maximum eigenvalue, that is,

λmax(t) = max{λ1(t), λ2(t), . . . , λM0 (t)

}(51)

Because the quantity λi changes continuously as a func-tion of time, λmax also changes continuously. However,the weight that implements the quantity loses continuityeach time the selection changes in Eq. (51). In addition,in situations where the transmitting and receiving stationsboth measure transmission characteristics (as in the TDDmethod, for example) and control weights are determinedindependently, as the difference between the maximumeigenvalue and the second eigenvalue becomes smaller, amismatch between transmitter and receiver weights is likelyto occer. Further more when the first and second eigenvaluesare equal by coincidence, the resulting eigenvector instabil-ity and transmission/reception mismatch increases the possi-bility of a significant decline in the reception signal strength[29]. Therefore, for practical control purposes, it is neces-sary to avoid this risk.

Although instability and discontinuity have been dis-cussed as problems of control in the time domain, similarproblems can occur between subchannels in the frequencydomain, such as the control of OFDM subchannels. Theissue of discontinuity as a result of null-forming by trans-mission/reception using an adaptive array is addressed inRef. [18]. This is a difficult problem that cannot be ignoredin systems in which adaptive control is performed by boththe transmitter and the eceiver.

6. Research Topics Relevant to MIMO Radio WavePropagation

Research in information theory, such as that on channelcapacity (Eqs. (45)–(47)) or the corresponding space-timecoding, will be substantially facilitated at the conceptuallevel if the propagation models can be organized as depictedin Fig. 6(b). Radio wave propagation, on the other hand, issubject to an extremely complex set of environmental fac-tors. Consequently, before a radio wave propagation modelinvolving MIMO channels can be established, many issuesmust be studied. Until now, knowledge of the radio wave

environment of the areas surrounding either the mobile sta-tion or base station has provided a sufficient foundation forthe development of a SIMO radio wave propagation mode.In MIMO, on the other hand, the propagation structure ofthe intervening paths also affects the distribution of corre-lation matrix eigenvalues, necessitating research by novelapproaches.

The research relevant to radio wave propagation thatremains to be undertaken includes the following topics.

(1) Analysis of how eigenvalues are distributed ina specific propagation environment (propagationmeasurement and propagation channel modeling).

(2) Analysis of how MIMO beam forming should beconfigured in concrete terms such as antenna con-figuration, and dual polarization, etc.

(3) Development of methods for antenna weight con-trol based on these topics (adaptive algorithm; inparticular, a robust control algorithm dealing withthe problem of discontinuity inherent in weightcontrol).

(4) Development of methods for a more efficient in-formation transmission scheme (space-time codedtransmission, power distribution by water-filling).

(5) Configuration and characteristic evaluation withapplication to wideband systems such as CDMAand OFDM.

(6) Analysis of how MIMO channel characteristics canbe determined in real time (measurement and esti-mation of channel characteristics).

In particular, for a propagation environment subject toextensive temporal fluctuations, solving problems (3) and(6) can be a challenging task, and many issues must be over-come. Furthermore, MIMO information transmission (5)over a wide-band channel, a topic that was limited to the rep-resentation of channels in this paper, involves an extremelycomplex treatment at both the mathematical and configura-tional levels, and requires study from the fundamentals.

MIMO is literally a multiple-input, multiple-outputtransmission method, and it is not necessary for array an-tennas to be employed. As stated in Item (2), any mode,for example, antenna/polarized wave/frequency slots, thatcomposes a diversity branch in the ordinary sense can beemployed. From a radio wave propagation standpoint,it may be interesting to incorporate orthogonal polarizedwaves into an M × N array to implement 2M × 2N space-polarization-combined MIMO transmission. To fully ex-ploit the capabilities of MIMO, it is necessary to keep trackof the behavior of the channel response matrix in real time(6). Thus, there remain many topics of research at the sys-tem level, such as whether the same frequency must be usedat the transmitting and receiving ends (i.e., TDD or FDD),and how to incorporate pilot signals for channel monitoring.

7. Conclusions

MIMO has the potential to surpass the capabilities of com-bined transmission and reception adaptive arrays. The out-


standing potential performance of MIMO can be summa-rized by the performance = “gains due to use of an arrayantenna for the transmitting station” × “gains due to the useof an array antenna for the receiving station” × “the numberof multiple streams.” MIMO can utilize not only the largestpath based on the largest eigenvalue in the equivalent cir-cuit, but also secondary paths based on the second and lowereigenvalues, although the latter effect depends on the com-plex interplay of both the propagation environment includ-ing Tx and Rx site environments and intervening propaga-tion structure and the number of antennas deployed. In thispaper, typical examples operating in a Rayleigh fading en-vironment were presented. The question as to whether thebenefits offered by MIMO (Fig. 12) are significant or rela-tively trivial must be judged, taking the difficulty of imple-mentation into account.

Starting from fundamental examples such as the nar-row band, i.i.d. Rayleigh fading, time invariance, and theabsence of interference waves, we have touched upon topicsat the forefront of research, such as wideband representa-tions and interpath correlations. Research efforts based onradio wave propagation must therefore begin from the topicscovered in this paper, and a number of issues await explo-ration. It is hoped that this paper will provide some valuablesuggestions for future research and development efforts inMIMO technology.

Finally, the author thanks Dr. T. Taniguchi of the Uni-versity of Electro-Communications and Dr. Y. Zhang of Vil-lanova University (USA) for valuable discussion. Gratitudeis also extended to Dr. K. Sakaguchi of the Tokyo Insti-tute of Technology for providing valuable information onthe Wishert distribution. The assistance of Mr. M. Tsurutaand Miss J. Fu, graduate students of the Karasawa Labora-tories of the University of Electro-Communications, in thepreparation of some of the drawings is also appreciated.

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Yoshio Karasawa received B.E. degreefrom Yamanashi University in 1973 and M.S.and Dr. Eng. Degrees from Kyoto University in1977 and 1992, respectively. He joined KDDR&D Labs. in 1977. From July 1993 to July1997, he was a Department Head of ATR Op-tical and Radio Communications Res. Labs.and ATR Adaptive Communications Res. Labs.,both in Kyoto. Currently, he is a professor of theUniversity of Electro-Communications, Tokyo.Since 1977, he has been engaged in studies on

wave propagation and radio communication antennas, particularly on theo-retical analysis and measurements for wave-propagation phenomena, suchas multipath fading in mobile radio systems, tropospheric and ionosphericscintillation, and rain attenuation. His recent interests are in frontier regionsbridging “wave propagation” and “digital transmission characteristics” inwideband mobile radio systems such as MIMO. Dr. Karasawa received theYoung Engineers Award from the IECE of Japan in 1983 and the Merito-rious Award on Radio from the Association of Radio Industries and Busi-nesses (ARIB, Japan) in 1998. He is a member of the IEEE, SICE andURSI.

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