IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003
`
`281
`
`From Theory to Practice: An Overview of MIMO
`Space–Time Coded Wireless Systems
`
`David Gesbert, Member, IEEE, Mansoor Shafi, Fellow, IEEE, Da-shan Shiu, Member, IEEE,
`Peter J. Smith, Member, IEEE, and Ayman Naguib, Senior Member, IEEE
`
`Tutorial Paper
`
`Abstract—This paper presents an overview of recent progress
`in the area of multiple-input–multiple-output (MIMO) space–time
`coded wireless systems. After some background on the research
`leading to the discovery of the enormous potential of MIMO
`wireless links, we highlight the different classes of techniques
`and algorithms proposed which attempt to realize the various
`benefits of MIMO including spatial multiplexing and space–time
`coding schemes. These algorithms are often derived and analyzed
`under ideal independent fading conditions. We present the state
`of the art in channel modeling and measurements, leading to a
`better understanding of actual MIMO gains. Finally, the paper
`addresses current questions regarding the integration of MIMO
`links in practical wireless systems and standards.
`Index Terms—Beamforming, channel models, diversity, mul-
`tiple-input–multiple-output (MIMO), Shannon capacity, smart
`antennas, space–time coding, spatial multiplexing, spectrum
`efficiency, third-generation (3G), wireless systems.
`
`area networks (WLAN), third-generation (3G)1 networks and
`beyond.
`MIMO systems can be defined simply. Given an arbitrary
`wireless communication system, we consider a link for which
`thetransmittingendas wellasthereceivingendisequippedwith
`multiple antenna elements. Such a setup is illustrated in Fig. 1.
`The idea behind MIMO is that the signals on the transmit (TX)
`antennas at one end and the receive (RX) antennas at the other
`end are “combined” in such a way that the quality (bit-error rate
`or BER) or the data rate (bits/sec) of the communication for each
`MIMO user will be improved. Such an advantage can be used to
`increase both the network’s quality of service and the operator’s
`revenues significantly.
`A core idea in MIMO systems is space–time signal
`processing in which time (the natural dimension of digital com-
`munication data) is complemented with the spatial dimension
`inherent in the use of multiple spatially distributed antennas.
`As such MIMO systems can be viewed as an extension of the
`so-called smart antennas, a popular technology using antenna
`arrays for improving wireless transmission dating back several
`decades.
`A key feature of MIMO systems is the ability to turn multi-
`path propagation, traditionally a pitfall of wireless transmission,
`into a benefit for the user. MIMO effectively takes advantage
`of random fading [1]–[3] and when available, multipath delay
`spread [4], [5], for multiplying transfer rates. The prospect of
`many orders of magnitude improvement in wireless communi-
`cation performance at no cost of extra spectrum (only hardware
`and complexity are added) is largely responsible for the suc-
`cess of MIMO as a topic for new research. This has prompted
`progress in areas as diverse as channel modeling, information
`theory and coding, signal processing, antenna design and mul-
`tiantenna-aware cellular design, fixed or mobile.
`This paper discusses the recent advances, adopting succes-
`sively several complementing views from theory to real-world
`network applications. Because of the rapidly intensifying
`efforts in MIMO research at the time of writing, as exemplified
`by the numerous papers submitted to this special issue of
`JSAC, a complete and accurate survey is not possible. Instead
`this paper forms a synthesis of the more fundamental ideas
`presented over the last few years in this area, although some
`very recent progress is also mentioned.
`1Third-generation wireless UMTS-WCDMA.
`0733-8716/03$17.00 © 2003 IEEE
`
`I. INTRODUCTION
`
`DIGITAL communication using multiple-input–multiple-
`
`output (MIMO), sometimes called a “volume-to-volume”
`wireless link, has recently emerged as one of the most sig-
`nificant technical breakthroughs in modern communications.
`The technology figures prominently on the list of recent
`technical advances with a chance of resolving the bottleneck of
`traffic capacity in future Internet-intensive wireless networks.
`Perhaps even more surprising is that just a few years after its
`invention the technology seems poised to penetrate large-scale
`standards-driven commercial wireless products and networks
`such as broadband wireless access systems, wireless local
`
`Manuscript received June 1, 2002; revised December 5, 2002. The work of
`D. Gesbert was supported in part by Telenor AS, Norway.
`D. Gesbert is with the Department of Informatics, University of Oslo, Blin-
`dern, 0316 Oslo, Norway (e-mail: gesbert@ifi.uio.no).
`M. Shafi is with Telecom New Zealand, Wellington, New Zealand (e-mail:
`Mansoor.Shafi@telecom.co.nz).
`D. Shiu is with Qualcomm, Inc., Campbell, CA 95008 USA (e-mail:
`dashiu@qualcomm.com).
`P. J. Smith is with the Department of Electrical and Computer Engi-
`neering, University of Canterbury, Christchurch, New Zealand (e-mail:
`p.smith@elec.canterbury.ac.nz).
`A. Naguib was with Morphics Technology, Inc., Campbell, CA 95008 USA.
`He is now with Qualcomm, Inc., Campbell, CA 95008 USA.
`Digital Object Identifier 10.1109/JSAC.2003.809458
`
`Authorized licensed use limited to: University of Canterbury. Downloaded on November 18, 2008 at 20:46 from IEEE Xplore. Restrictions apply.
`
`IPR2018-01476
`Apple Inc. EX1016 Page 1
`
`

`

`282
`
`IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003
`
`Fig. 1. Diagram of a MIMO wireless transmission system. The transmitter and receiver are equipped with multiple antenna elements. Coding, modulation, and
`mapping of the signals onto the antennas may be realized jointly or separately.
`
`The article is organized as follows. In Section II, we attempt
`to develop some intuition in this domain of wireless research,
`we highlight the common points and key differences between
`MIMO and traditional smart antenna systems, assuming the
`reader is somewhat familiar with the latter. We comment on a
`simple example MIMO transmission technique revealing the
`unique nature of MIMO benefits. Next, we take an information
`theoretical stand point in Section III to justify the gains and
`explore fundamental limits of transmission with MIMO links in
`various scenarios. Practical design of MIMO-enabled systems
`involves the development of finite-complexity transmission/re-
`ception signal processing algorithms such as space–time
`coding and spatial multiplexing schemes. Furthermore, channel
`modeling is particularly critical
`in the case of MIMO to
`properly assess algorithm performance because of sensitivity
`with respect to correlation and rank properties. Algorithms
`and channel modeling are addressed in Sections IV and V,
`respectively. Standardization issues and radio network level
`considerations which affect the overall benefits of MIMO im-
`plementations are finally discussed in Section VI. Section VII
`concludes this paper.
`
`II. PRINCIPLES OF SPACE-TIME (MIMO) SYSTEMS
`Consider the multiantenna system diagram in Fig. 1. A com-
`pressed digital source in the form of a binary data stream is fed
`to a simplified transmitting block encompassing the functions
`of error control coding and (possibly joined with) mapping to
`complex modulation symbols (quaternary phase-shift keying
`(QPSK), M-QAM, etc.). The latter produces several separate
`symbol streams which range from independent to partially
`redundant to fully redundant. Each is then mapped onto one
`of the multiple TX antennas. Mapping may include linear
`spatial weighting of the antenna elements or linear antenna
`space–time precoding. After upward frequency conversion,
`filtering and amplification, the signals are launched into the
`wireless channel. At the receiver, the signals are captured by
`possibly multiple antennas and demodulation and demapping
`operations are performed to recover the message. The level of
`intelligence, complexity, and a priori channel knowledge used
`in selecting the coding and antenna mapping algorithms can
`vary a great deal depending on the application. This determines
`the class and performance of the multiantenna solution that is
`implemented.
`In theconventionalsmartantenna terminology,onlythetrans-
`mitter or the receiver is actually equipped with more than one
`element, being typically the base station (BTS), where the extra
`
`cost and space have so far been perceived as more easily af-
`fordable than on a small phone handset. Traditionally, the in-
`telligence of the multiantenna system is located in the weight
`selection algorithm rather than in the coding side although the
`development of space–time codes (STCs) is transforming this
`view.
`Simple linear antenna array combining can offer a more re-
`liable communications link in the presence of adverse propa-
`gation conditions such as multipath fading and interference. A
`key concept in smart antennas is that of beamforming by which
`one increases the average signal-to-noise ratio (SNR) through
`focusing energy into desired directions, in either transmit or re-
`ceiver. Indeed, if one estimates the response of each antenna
`element to a given desired signal, and possibly to interference
`signal(s), one can optimally combine the elements with weights
`selected as a function of each element response. One can then
`maximize the average desired signal level or minimize the level
`of other components whether noise or co-channel interference.
`Another powerful effect of smart antennas lies in the concept
`of spatial diversity. In the presence of random fading caused
`by multipath propagation, the probability of losing the signal
`vanishes exponentially with the number of decorrelated antenna
`elements being used. A key concept here is that of diversity
`order which is defined by the number of decorrelated spatial
`branches available at the transmitter or receiver. When com-
`bined together, leverages of smart antennas are shown to im-
`prove the coverage range versus quality tradeoff offered to the
`wireless user [6].
`As subscriber units (SU) are gradually evolving to become
`sophisticated wireless Internet access devices rather than just
`pocket telephones, the stringent size and complexity constraints
`are becoming somewhat more relaxed. This makes multiple an-
`tenna elements transceivers a possibility at both sides of the link,
`even though pushing much of the processing and cost to the net-
`work’s side (i.e., BTS) still makes engineering sense. Clearly,
`in a MIMO link, the benefits of conventional smart antennas are
`retained since the optimization of the multiantenna signals is
`carried out in a larger space, thus providing additional degrees
`of freedom. In particular, MIMO systems can provide a joint
`transmit-receive diversity gain, as well as an array gain upon
`coherent combining of the antenna elements (assuming prior
`channel estimation).
`In fact, the advantages of MIMO are far more fundamental.
`The underlying mathematical nature of MIMO, where data is
`transmitted over a matrix rather than a vector channel, creates
`new and enormous opportunities beyond just the added diver-
`sity or array gain benefits. This was shown in [2], where the
`
`Authorized licensed use limited to: University of Canterbury. Downloaded on November 18, 2008 at 20:46 from IEEE Xplore. Restrictions apply.
`
`IPR2018-01476
`Apple Inc. EX1016 Page 2
`
`

`

`GESBERT et al.: FROM THEORY TO PRACTICE: AN OVERVIEW OF MIMO SPACE–TIME CODED WIRELESS SYSTEMS
`
`283
`
`Fig. 2. Basic spatial multiplexing (SM) scheme with three TX and three RX antennas yielding three-fold improvement in spectral efficiency. Ai, Bi, and Ci
`represent symbol constellations for the three inputs at the various stages of transmission and reception.
`
`author shows how one may under certain conditions transmit
`independent data streams simultaneously over the
`eigenmodes of a matrix channel created by
`TX and
`RX an-
`tennas. A little known yet earlier version of this ground breaking
`result was also released in [7] for application to broadcast dig-
`ital TV. However, to our knowledge, the first results hinting at
`the capacity gains of MIMO were published by Winters in [8].
`Information theory can be used to demonstrate these gains
`rigorously (see Section III). However, intuition is perhaps best
`givenby a simpleexample of such a transmission algorithm over
`MIMO often referred to in the literature as V-BLAST2 [9], [10]
`or more generically called here spatial multiplexing.
`In Fig. 2, a high-rate bit stream (left) is decomposed into
`three independent
`-rate bit sequences which are then trans-
`mitted simultaneously using multiple antennas, thus consuming
`one third of the nominal spectrum. The signals are launched
`and naturally mix together in the wireless channel as they use
`the same frequency spectrum. At the receiver, after having
`identified the mixing channel matrix through training symbols,
`the individual bit streams are separated and estimated. This
`occurs in the same way as three unknowns are resolved from a
`linear system of three equations. This assumes that each pair
`of transmit receive antennas yields a single scalar channel
`coefficient, hence flat fading conditions. However, extensions
`to frequency selective cases are indeed possible using either a
`straightforward multiple-carrier approach (e.g., in orthogonal
`frequency division multiplexing (OFDM),
`the detection is
`performed over each flat subcarrier) or in the time domain by
`combining the MIMO space–time detector with an equalizer
`
`2Vertical-Bell Labs Layered Space–Time Architecture.
`
`(see for instance [11]–[13] among others). The separation is
`possible only if the equations are independent which can be
`interpreted by each antenna “seeing” a sufficiently different
`channel in which case the bit streams can be detected and
`merged together to yield the original high rate signal. Iterative
`versions of this detection algorithm can be used to enhance
`performance, as was proposed in [9] (see later in this paper for
`more details or in [14] of this special issue for a comprehensive
`study).
`A strong analogy can be made with code-division
`multiple-access (CDMA)
`transmission in which multiple
`users sharing the same time/frequency channel are mixed upon
`transmission and recovered through their unique codes. Here,
`however, the advantage of MIMO is that the unique signatures
`of input streams (“virtual users”) are provided by nature in a
`close-to-orthogonal manner (depending however on the fading
`correlation) without frequency spreading, hence at no cost of
`spectrum efficiency. Another advantage of MIMO is the ability
`to jointly code and decode the multiple streams since those are
`intended to the same user. However, the isomorphism between
`MIMO and CDMA can extend quite far into the domain of
`receiver algorithm design (see Section IV).
`Note that, unlike in CDMA where user’s signatures are
`quasi-orthogonal by design, the separability of the MIMO
`channel relies on the presence of rich multipath which is
`needed to make the channel spatially selective. Therefore,
`MIMO can be said to effectively exploit multipath. In contrast,
`some smart antenna systems (beamforming, interference rejec-
`tion-based) will perform better in line-of-sight (LOS) or close
`to LOS conditions. This is especially true when the optimiza-
`tion criterion depends explicitly on angle of arrival/departure
`
`Authorized licensed use limited to: University of Canterbury. Downloaded on November 18, 2008 at 20:46 from IEEE Xplore. Restrictions apply.
`
`IPR2018-01476
`Apple Inc. EX1016 Page 3
`
`

`

`284
`
`IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003
`
`parameters. Alternatively, diversity-oriented smart antenna
`techniques perform well in nonline-of-sight (NLOS), but they
`really try to mitigate multipath rather than exploiting it.
`In general, one will define the rank of the MIMO channel
`as the number of independent equations offered by the above
`mentioned linear system. It is also equal to the algebraic rank of
`the
`channel matrix. Clearly, the rank is always both less
`than the number of TX antennas and less than the number of
`RX antennas. In turn, following the linear algebra analogy, one
`expects that the number of independent signals that one may
`safely transmit through the MIMO system is at most equal to
`the rank. In the example above, the rank is assumed full (equal
`to three) and the system shows a nominal spectrum efficiency
`gain of three, with no coding. In an engineering sense, however,
`both the number of transmitted streams and the level of BER on
`each stream determine the link’s efficiency (goodput3 per TX
`antenna times number of antennas) rather than just the number
`of independent input streams. Since the use of coding on the
`multiantenna signals (a.k.a. space–time coding) has a critical
`effect on the BER behavior, it becomes an important component
`of MIMO design. How coding and multiplexing can be traded
`off for each other is a key issue and is discussed in more detail
`in Section IV.
`
`III. MIMO INFORMATION THEORY
`In Sections I and II, we stated that MIMO systems can
`offer substantial improvements over conventional smart an-
`tenna systems in either quality-of-service (QoS) or transfer
`rate in particular through the principles of spatial multiplexing
`and diversity. In this section, we explore the absolute gains
`offered by MIMO in terms of capacity bounds. We summarize
`these results in selected key system scenarios. We begin with
`fundamental results which compare single-input–single-output
`(SISO), single-input–multiple-output (SIMO), and MIMO ca-
`pacities, then we move on to more general cases that take
`possible a priori channel knowledge into account. Finally, we
`investigate useful limiting results in terms of the number of
`antennas or SNR. We bring the reader’s attention on the fact
`that we focus here on single user forms of capacity. A more
`general multiuser case is considered in [15]. Cellular MIMO
`capacity performance has been looked at elsewhere, taking into
`account the effects of interference from either an information
`theory point of view [16], [17] or a signal processing and
`system efficiency point of view [18], [19] to cite just a few
`example of contributions, and is not treated here.
`A. Fundamental Results
`For a memoryless 1
`by
`
`1 (SISO) system the capacity is given
`
`(1)
`b/s/Hz
`is the normalized complex gain of a fixed wireless
`where
`channel or that of a particular realization of a random channel.
`In (1) and subsequently,
`is the SNR at any RX antenna. As we
`deploy more RX antennas the statistics of capacity improve and
`3The goodput can be defined as the error-free fraction of the conventional
`physical layer throughput.
`
`RX antennas, we have a SIMO system with capacity
`
`with
`given by
`
`b/s/Hz
`
`(2)
`
`is the gain for RX antenna . Note the crucial fea-
`where
`ture of (2) in that increasing the value of
`only results in a
`logarithmic increase in average capacity. Similarly, if we opt
`for transmit diversity, in the common case, where the trans-
`mitter does not have channel knowledge, we have a multiple-
`input–single-output (MISO) system with
`TX antennas and
`the capacity is given by [1]
`
`b/s/Hz
`
`(3)
`
`ensures a fixed total transmitter
`where the normalization by
`power and shows the absence of array gain in that case (com-
`pared to the case in (2), where the channel energy can be com-
`bined coherently). Again, note that capacity has a logarithmic
`relationship with
`. Now, we consider the use of diversity at
`both transmitter and receiver giving rise to a MIMO system. For
`TX and
`RX antennas, we have the now famous capacity
`equation [1], [3], [21]
`
`(4)
`
`b/s/Hz
`where ( ) means transpose-conjugate and
`is the
`channel matrix. Note that both (3) and (4) are based on
`equal power (EP) uncorrelated sources, hence, the subscript
`in (4). Foschini [1] and Telatar [3] both demonstrated that
`the capacity in (4) grows linearly with
`rather than logarithmically [as in (3)[. This result can be
`intuited as follows: the determinant operator yields a product
`of
`nonzero eigenvalues of its (channel-dependent)
`matrix argument, each eigenvalue characterizing the SNR over
`a so-called channel eigenmode. An eigenmode corresponds to
`the transmission using a pair of right and left singular vectors
`of the channel matrix as transmit antenna and receive antenna
`weights, respectivelly. Thanks to the properties of the
`, the
`overall capacity is the sum of capacities of each of these modes,
`hence the effect of capacity multiplication. Note that the linear
`growth predicted by the theory coincides with the transmission
`example of Section II. Clearly, this growth is dependent on
`properties of the eigenvalues. If they decayed away rapidly then
`linear growth would not occur. However (for simple channels),
`the eigenvalues have a known limiting distribution [22] and
`tend to be spaced out along the range of this distribution.
`Hence, it is unlikely that most eigenvalues are very small and
`the linear growth is indeed achieved.
`With the capacity defined by (4) as a random variable, the
`issue arises as to how best to characterize it. Two simple sum-
`maries are commonly used: the mean (or ergodic) capacity [3],
`[21], [23] and capacity outage [1], [24]–[26]. Capacity outage
`measures (usually based on simulation) are often denoted
`or
`, i.e., those capacity values supported 90% or 99% of
`the time, and indicate the system reliability. A full description
`of the capacity would require the probability density function
`
`Authorized licensed use limited to: University of Canterbury. Downloaded on November 18, 2008 at 20:46 from IEEE Xplore. Restrictions apply.
`
`IPR2018-01476
`Apple Inc. EX1016 Page 4
`
`

`

`GESBERT et al.: FROM THEORY TO PRACTICE: AN OVERVIEW OF MIMO SPACE–TIME CODED WIRELESS SYSTEMS
`
`285
`
`or equivalent. Some results are available here [27] but they are
`limited.
`Some caution is necessary in interpreting the above equa-
`tions. Capacity, as discussed here and in most MIMO work
`[1], [3], is based on a “quasi-static” analysis where the channel
`varies randomly from burst to burst. Within a burst the channel
`is assumed fixed and it is also assumed that sufficient bits are
`transmitted for the standard infinite time horizon of information
`theory to be meaningful. A second note is that our discussion
`will concentrate on single user MIMO systems but many results
`also apply to multiuser systems with receive diversity. Finally,
`the linear capacity growth is only valid under certain channel
`conditions. It was originally derived for the independent and
`identically distributed (i.i.d.) flat Rayleigh fading channel and
`does not hold true for all cases. For example, if large numbers
`of antennas are packed into small volumes, then the gains in
`may become highly correlated and the linear relationship will
`plateau out due to the effects of antenna correlation [28]–[30].
`In contrast, other propagation effects not captured in (4) may
`serve to reinforce the capacity gains of MIMO such as multi-
`path delay spread. This was shown in particular in the case when
`the transmit channel is known [4] but also in the case when it is
`unknown [5].
`More generally, the effect of the channel model is critical.
`Environments can easily be chosen which give channels where
`the MIMO capacities do not increase linearly with the numbers
`of antennas. However, most measurements and models available
`to date do give rise to channel capacities which are of the same
`order of magnitude as the promised theory (see Section V). Also
`the linear growth is usually a reasonable model for moderate
`numbers of antennas which are not extremely close-packed.
`B. Information Theoretic MIMO Capacity
`1) Background: Since feedback is an important component
`of wireless design (although not a necessary one), it is useful to
`generalize the capacity discussion to cases that can encompass
`transmitters having some a priori knowledge of channel. To this
`end, we now define some central concepts, beginning with the
`MIMO signal model
`
`(5)
`
`is the
`received signal vector,
`is the
`In (5),
`transmitted signal vector and
`is an
`vector of additive
`noise terms, assumed i.i.d. complex Gaussian with each element
`having a variance equal to
`. For convenience we normalize the
`noise power so that
`in the remainder of this section. Note
`that the system equation represents a single MIMO user com-
`municating over a fading channel with additive white Gaussian
`noise (AWGN). The only interference present is self-interfer-
`ence between the input streams to the MIMO system. Some au-
`thors have considered more general systems but most informa-
`tion theoretic results can be discussed in this simple context, so
`we use (5) as the basic system equation.
`Let
`denote the covariance matrix of
`, then the capacity of
`the system described by (5) is given by [3], [21]
`b/s/Hz
`
`(6)
`
`where
`holds to provide a global power constraint.
`Note that for equal power uncorrelated sources
`and (6) collapses to (4). This is optimal when
`is unknown at
`the transmitter and the input distribution maximizing the mutual
`information is the Gaussian distribution [3], [21]. With channel
`feedback may be known at the transmitter and the optimal
`is not proportional to the identity matrix but is constructed from
`a waterfilling argument as discussed later.
`The form of equation (6) gives rise to two practical questions
`of key importance. First, what is the effect of
`? If we compare
`the capacity achieved by
`(equal power transmis-
`sion or no feedback) and the optimal
`based on perfect channel
`estimation and feedback, then we can evaluate a maximum ca-
`pacity gain due to feedback. The second question concerns the
`effect of the
`matrix. For the i.i.d. Rayleigh fading case we
`have the impressive linear capacity growth discussed above. For
`a wider range of channel models including, for example, corre-
`lated fading and specular components, we must ask whether this
`behavior still holds. Below we report a variety of work on the
`effects of feedback and different channel models.
`It is important to note that (4) can be rewritten as [3]
`
`b/s/Hz
`
`where
`
`, and
`
`are the nonzero eigenvalues of
`
`,
`
`(7)
`
`(8)
`
`This formulation can be easily obtained from the direct use
`of eigenvalue properties. Alternatively, we can decompose the
`MIMO channel into m equivalent parallel SISO channels by
`performing a singular value decomposition (SVD) of
`[3],
`[21]. Let the SVD be given by
`, then
`and
`are unitary and
`is diagonal with entries specified by
`. Hence (5) can
`be rewritten as
`
`(9)
`where
`. Equation (9) repre-
`and
`,
`sents the system as m equivalent parallel SISO eigen-channels
`with signal powers given by the eigenvalues
`.
`Hence, the capacity can be rewritten in terms of the eigen-
`values of the sample covariance matrix
`. In the i.i.d. Rayleigh
`fading case,
`is also called a Wishart matrix. Wishart matrices
`have been studied since the 1920s and a considerable amount is
`known about them. For general
`matrices a wide range of
`limiting results are known [22], [31]–[34] as
`or
`or both
`tend to infinity. In the particular case of Wishart matrices, many
`exact results are also available [31], [35]. There is not a great
`deal of information about intermediate results (neither limiting
`nor Wishart), but we are helped by the remarkable accuracy of
`some asymptotic results even for small values of
`,
`[36].
`We now give a brief overview of exact capacity results,
`broken down into the two main scenarios, where the channel is
`either known or unknown at the transmitter. We focus on the
`
`Authorized licensed use limited to: University of Canterbury. Downloaded on November 18, 2008 at 20:46 from IEEE Xplore. Restrictions apply.
`
`IPR2018-01476
`Apple Inc. EX1016 Page 5
`
`

`

`286
`
`IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 3, APRIL 2003
`
`two key questions posed above; what is the effect of feedback
`and what is the impact of the channel?
`2) Channel Known at the Transmitter (Waterfilling): When
`the channel is known at the transmitter (and at the receiver), then
`is known in (6) and we optimize the capacity over
`subject
`to the power constraint
`. Fortunately, the optimal
`in this case is well known [3], [4], [21], [26], [37]–[39] and is
`called a waterfilling (WF) solution. There is a simple algorithm
`to find the solution [3], [21], [26], [37], [39] and the resulting
`capacity is given by
`
`where
`
`is chosen to satisfy
`
`b/s/Hz
`
`(10)
`
`(11)
`
`and “ ” denotes taking only those terms which are positive.
`Since
`is a complicated nonlinear function of
`,
`the distribution of
`appears intractable, even in the Wishart
`case when the joint distribution of
`is known.
`Nevertheless,
`can be simulated using (10) and (11) for any
`given
`so that the optimal capacity can be computed numeri-
`cally for any channel.
`The effect on
`of various channel conditions has been
`studied to a certain extent. For example in Ricean channels in-
`creasing the LOS strength at fixed SNR reduces capacity [23],
`[40]. This can be explained in terms of the channel matrix rank
`[25] or via various eigenvalue properties. The issue of correlated
`fading is of considerable importance for implementations where
`the antennas are required to be closely spaced (see Section VI).
`Here, certain correlation patterns are being standardized as suit-
`able test cases [41]. A wide range of results in this area is given
`in [26].
`In termsof theimpact offeedback (channelinformation being
`supplied to the transmitter), it is interesting to note that the WF
`gains over EP are significant at low SNR but converge to zero as
`the SNR increases [39], [40], [42]. The gains provided by WF
`appear to be due to the correlations in
`rather than any unequal
`power allocation along the diagonal in
`. This was shown in
`[40], where the gains due to unequal power uncorrelated sources
`were shown to be small compared to waterfilling. Over a wide
`range of antenna numbers and channel models the gains due to
`feedback are usually less than 30% for SNR above 10 dB. From
`zero to 10 dB the gains are usually less than 60%. For SNR
`values below 0 dB, large gains are possible, with values around
`200% being reported at
`10 dB. These results are available in
`the literature, see for example [39], but some simulations are
`also given in Fig. 3 for completeness. The fact that feedback
`gain reduces at higher SNR levels can be intuitively explained
`by the following fact. Knowledge of the transmit channel mainly
`provides transmit array gain. In contrast, gains such as diversity
`gain and multiplexing gain do not require this knowledge as
`these gains can be captured by “blind” transmit schemes such as
`STCs and V-BLAST (see later). Since the relative importance
`of transmit array gain in boosting average SNR decreases in the
`high SNR region, the benefit of feedback also reduces.
`
`Fig. 3. Shows the percentage relative gains in capacity due to feedback at
`various SNR values, channel models (
`is the Ricean factor), and array sizes.
`
`3) Channel Unknown at the Transmitter: Here, the capacity
`is given by
`in (4). This was derived by Foschini [1] and
`Telatar [3], [21] from two viewpoints. Telatar [3], [21] started
`from (6) and showed that
`is optimal for i.i.d.
`Rayleigh fading. Foschini derived (4) starting from an equal
`power assumption. The variable,
`, is considerably more
`amenable to analysis than
`. For example, the mean capacity
`is derived in [3], [21] and the variance in [36] for i.i.d. Rayleigh
`fading, as well as [43]. In addition, the full moment generating
`function (MGF) for
`is given in [27] although this is rather
`complicated being in determinant form. Similar results include
`[44].
`For more complex channels, results are rapidly becoming
`available. Again, capacity is reduced in Ricean channels as the
`relative LOS strength increases [25], [37] . The impact of cor-
`relation is important and various physical models and measure-
`ments of correlations have been used to assess its impact [26],
`[45]–[47]. For example,
`is shown to plateau out as the
`number of antennas increases in either sparse scattering envi-
`ronments [48] or dense/compact MIMO arrays [29], [30].
`
`C. Limiting Capacity Results
`The exact results of Section III-B above are virtually all de-
`pendent on the i.i.d. Rayleigh fading (Wishart) case. For other
`scenarios exact results are few and far between. Hence, it is
`
`Authorized licensed use limited to: University of Canterbury. Downloaded on November 18, 2008 at 20:46 from IEEE Xplore. Restrictions apply.
`
`IPR2018-01476
`Apple Inc. EX1016 Page 6
`
`

`

`GESBERT et al.: FROM THEORY TO PRACTICE: AN OVERVIEW OF MIMO SPACE–TIME CODED WIRELESS SYSTEMS
`
`287
`
`useful to pursue limiting results not only to cover a broader
`range of cases but also to give simpler and more intuitive re-
`sults and to study the potential of very large scale systems.
`The surprising thing about limiting capacity results