《MIMO多进多出多天线》课程教学资源（文献资料）An Overview of MIMO Communications—A Key to Gigabit Wireless.pdf_大学文库

An Overview of MIMO Communications—A Key to Gigabit Wireless AROGYASWAMI J. PAULRAJ, FELLOW, IEEE, DHANANJAY A. GORE, ROHIT U. NABAR, MEMBER, IEEE, AND HELMUT BÖLCSKEI, SENIOR MEMBER, IEEE Invited Paper High data rate wireless communications, nearing 1-Gb/s transmission rates, is of interest in emerging wireless local area networks and home audio/visual networks. Designing very high speed wireless links that offer good quality-of-service and range capability in non-line-of-sight (NLOS) environments constitutes a significant research and engineering challenge. Ignoring fading in NLOS environments, we can, in principle, meet the 1-Gb/s data rate requirement with a single-transmit single-receive antenna wireless system if the product of bandwidth (measured in hertz) and spectral efficiency (measured in bits per second per hertz) is equal to 10 . As we shall outline in this paper, a variety of cost, technology and regulatory constraints make such a brute force solution unattractive if not impossible. The use of multiple antennas at transmitter and receiver, popularly known as multiple-input multiple-output (MIMO) wireless is an emerging cost-effective technology that offers substantial leverages in making 1-Gb/s wireless links a reality. This paper provides an overview of MIMO wireless technology covering channel models, performance limits, coding, and transceiver design. Keywords—Capacity, channel models, multiple-input multipleoutput (MIMO), MIMO orthogonal frequency division multiplexing (MIMO-OFDM), performance limits, receiver design, space–time coding, spatial multiplexing. I. INTRODUCTION High data rate wireless communications, nearing 1-Gb/s transmission rates, is of interest in emerging wireless local area networks (WLANs) and home audio/visual (A/V) networks. Currently, WLANs offer peak rates of 10 Mb/s, with 50–100 Mb/s becoming available soon. However, even 50 Mb/s is inadequate when faced with the Manuscript received October 16, 2002; revised November 6, 2003. A. J. Paulraj is with the Information Systems Laboratory, Stanford University, Stanford, CA 94305 USA (e-mail: apaulraj@stanford.edu). D. Gore is with Qualcomm Inc., San Diego, CA 92121 USA (e-mail: dgore@qualcomm.com). R. U. Nabar and H. Bölcskei are with the Communication Technology Laboratory, Swiss Federal Institute of Technology (ETH) Zürich, Zürich, Switzerland (e-mail: nabar@nari.ee.ethz.ch; boelcskei@nari.ee.ethz.ch). Digital Object Identifier 10.1109/JPROC.2003.821915 demand for higher access speeds due to the increase in rich media content and competition from 10-Gb/s wired LANs. Additionally, future home A/V networks will be required to support multiple high-speed high-definition television (HDTV) A/V streams, which again demand near 1-Gb/s data rates. Another challenge faced by WLANs and home A/V environments as well as outdoor wireless wide area network (WWAN) systems for fixed/nomadic access is non-line-of-sight (NLOS) propagation, which induces random fluctuations in signal level, known as fading. Designing very high speed wireless links that offer good quality-of-service (QoS) and range capability in NLOS environments constitutes a significant research and engineering challenge. Ignoring fading for the moment, we can, in principle, meet the 1-Gb/s data rate requirement if the product of bandwidth (measured in Hz) and spectral efficiency (measured in b/s/Hz) equals 10 . As we shall describe in the following, a variety of cost, technology, and regulatory constraints make such a brute force solution unattractive, if not impossible. In this paper, we provide an overview of an emerging technology, known as multiple-input multiple-output (MIMO) wireless, that offers significant promise in making 1-Gb/s wireless links in NLOS environments a reality. Several efforts are currently underway to build sub-Gb/s NLOS broadband wireless systems. In WWANs (corresponding standards are currently under development by IEEE 802.16), Iospan Wireless (founded by the first author of this paper and acquired by Intel Corp.) successfully developed a MIMO wireless system (physical layer and medium access control layer technology) using orthogonal frequency division multiplexing (OFDM) modulation for NLOS environments. The system is designed for a cellular plan with a reuse factor of two and delivers a peak spectral efficiency of 12 b/s/Hz. Current chipsets offer 13-Mb/s goodput in a 2-MHz channel. Future releases will support a goodput of 45 Mb/s in a 7-MHz channel. The system is 0018-9219/04$20.00 © 2004 IEEE 198 PROCEEDINGS OF THE IEEE, VOL. 92, NO. 2, FEBRUARY 2004 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply

aimed at fixed and nomadic/low mobility applications with cell sizes up to 4 mi. In mobile access, there is an effort under the International Telecommunications Union (ITU) working group to integrate MIMO techniques into the high-speed downlink packet access (HSDPA) channel, which is a part of the Universal Mobile Telecommunications System (UMTS) standard. Lucent Technologies recently announced a chip for MIMO enhancement of UMTS/HSDPA, but has released no further details. Preliminary efforts are also underway to define a MIMO overlay for the IEEE 802.11 standard for WLANs under the newly formed Wireless Next Generation (WNG) group. With the exception of Iospan’s product, the other efforts in MIMO technology are expected to take three to four years to reach deployment status. These efforts can serve as a good learning base for next-generation gigabit wireless systems. In this paper, we outline the value of MIMO technology in the development of viable gigabit wireless systems and provide an overview of this technology. A. Organization of the Paper The remainder of this paper is organized as follows. Section II discusses the design tradeoffs in building gigabit wireless systems and highlights the leverages of MIMO technology. Section III introduces a MIMO channel model for NLOS environments. In Section IV, we study the capacity gains resulting from the use of MIMO technology, while Sections V and VI review signaling and receiver design for MIMO systems, respectively. Section VII explores fundamental performance limits in communicating over MIMO channels. In Section VIII, we briefly review MIMO-OFDM, an increasingly popular modulation technique in broadband MIMO wireless channels. We present our conclusions in Section IX. B. Notation The superscripts , , and stand for transposition, conjugate transposition, and elementwise conjugation, respectively. denotes the expectation operator while is the convolution operator with . stands for the identity matrix, denotes the all zeros matrix of appropriate dimensions. , det , and Tr stand for the Frobenius norm, determinant, and trace, respectively, of the matrix . denotes the Euclidean norm of the vector . stands for the element in the th row and th column of . For an matrix , we define the 1 vector vec . A complex random variable is if and are independent identically distributed (i.i.d.) . II. BUILDING GIGABIT WIRELESS LINKS As noted in the preceding section, we can, in principle, reach 1-Gb/s link speed in a standard single-input singleoutput (SISO) wireless link by employing sufficiently high bandwidth along with coding and modulation that achieves the required spectral efficiency. However, there are several problems with such a simplistic approach. Let us start by discussing how transmit power and receive signal-to-noise ratio (SNR) constraints limit the maximum achievable spectral efficiency in SISO links. First, the transmit power in a terminal used by or located near human beings is limited to less than 1 W in indoor environments due to biohazard considerations. These limits are about a factor of ten higher in outdoor tower-based base stations. Second, the peak SNR limit in a wireless receiver rarely exceeds 30–35 dB because of the difficulty in building (at reasonable cost) highly linear receivers with low phase noise. More generally, the signal-to-interference-and-noise ratio (SINR) in cellular systems is capped due to the presence of cochannel interference. It is well known that aggressive cellular reuse with a low target SINR is advantageous for achieving high multicell spectral efficiency. Also, channel fading in the presence of imperfect power control and peak power limitations at the transmitter results in the peak achievable SINR being lower than the received SNR limit of 30–35 dB. The average SINR in a cellular reuse scheme lies in the range of 10–20 dB at best. This implies that increasing the spectral efficiency in a SISO NLOS cellular network beyond a peak value of 4–6 b/s/Hz (average value of 2–4 b/s/Hz) is not possible. In pure line-of-sight (LOS) links, practical SISO systems have reached spectral efficiencies of up to 9 b/s/Hz. However, such systems rely on fixed point-to-point links with very high gain directional antennas and Fresnel clearance to almost completely eliminate fading. The advantage of high-gain antennas in reducing the transmit power constraint is not available in NLOS environments, where large angle spread due to scattering can make such antennas highly inefficient. Let us next consider the implications of simply using the appropriate bandwidth and spectral efficiency product to achieve 1-Gb/s date rate. Consider a system that realizes a nominal spectral efficiency of 4 b/s/Hz over 250-MHz bandwidth, so that the data rate is 1 Gb/s. Two hundred fifty megahertz of bandwidth is scarce, if not impossible to obtain, particularly in frequency bands below 6 GHz, where NLOS networks are feasible. Two hundred fifty megahertz of bandwidth is easier to obtain in the 40-GHz frequency range. However, at frequencies higher than 6 GHz, the increased shadowing by obstructions in the propagation path render NLOS links unusable. Since transmit power and receive SNR are capped as pointed out above, a 250-MHz bandwidth will mean a reduction in range. Assuming a path (propagation) loss exponent of 3.0, the range reduces by a factor of two (or cell area by a factor of four) for every factor of eight increase in bandwidth. Therefore, compared to a 10-MHz bandwidth system used today, the range of a 250-MHz system will drop by a factor of 3 and the cell area by a factor of nine. On the positive side, a high bandwidth results in frequency diversity, which reduces the fade margin (excess transmit power required) in fading NLOS links. We should finally note that in a cellularized system, a total bandwidth of six to nine times the link bandwidth is needed in order to support a cellular reuse plan. This clearly places impossible bandwidth demands on SISO gigabit wireless systems. PAULRAJ et al.: AN OVERVIEW OF MIMO COMMUNICATIONS—A KEY TO GIGABIT WIRELESS 199 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply

Fig. 1. Bandwidth requirement and range of a 1-Gb/s link using MIMO technology. We summarize our discussion by noting that Gb/s wireless links in NLOS (and perhaps cellularized) networks using conventional approaches are in general not feasible due to peak and average SNR limits in practical receivers. Additionally, there is a serious range penalty to be paid for high bandwidth systems. MIMO wireless constitutes a technological breakthrough that will allow Gb/s speeds in NLOS wireless networks. The following example is designed to illustrate the performance gains delivered by MIMO. Consider a Rayleigh fading NLOS link with an average receive SNR of 20 dB and a constant total transmit power (independent of the number of transmit antennas). Let the coherence bandwidth be 20 MHz (typical value for indoor scenarios). The bandwidth needed to ensure 99% link reliability is obtained by computing the 1% outage capacity (see Section IV for details). Fig. 1 plots the bandwidth and range of symmetrical MIMO links (i.e., links with an equal number of transmit and receive antennas ) needed to support 1-Gb/s link speed. The range is normalized to unity with reference to a SISO system with 10-MHz bandwidth. For , we have a standard SISO link with a required bandwidth of 220 MHz, and a reduction in range to 35% of the reference system. On the other hand, a 10 10 MIMO system can deliver 1-Gb/s performance with only 20-MHz bandwidth and still support 80% of the reference range. Clearly, MIMO technology offers a substantial performance improvement. Note that a MIMO system does not require additional transmit power or receive SNR to deliver such performance gains. Furthermore, the spectral efficiency achieved over a 20-MHz bandwidth by the 10 10 MIMO channel is 50 b/s/Hz, which shows that high transmit power is not necessarily required to reach spectral efficiencies in excess of 10 b/s/Hz. We note that the downside of using a MIMO system is the increased transceiver complexity. The performance improvements resulting from the use of MIMO systems are due to array gain, diversity gain, spatial multiplexing gain, and interference reduction. We briefly review each of these leverages in the following considering a system with transmit and receive antennas. A. Array Gain Array gain can be made available through processing at the transmitter and the receiver and results in an increase in average receive SNR due to a coherent combining effect. Transmit/receive array gain requires channel knowledge in the transmitter and receiver, respectively, and depends on the number of transmit and receive antennas. Channel knowledge in the receiver is typically available whereas channel state information in the transmitter is in general more difficult to maintain. B. Diversity Gain Signal power in a wireless channel fluctuates randomly (or fades). Diversity is a powerful technique to mitigate fading in wireless links. Diversity techniques rely on transmitting the signal over multiple (ideally) independently fading paths (in time/frequency/space). Spatial (or antenna) diversity is preferred over time/frequency diversity as it does not incur an expenditure in transmission time or bandwidth. If the links composing the MIMO channel fade independently and the transmitted signal is suitably constructed, the receiver can combine the arriving signals such that the resultant signal exhibits considerably reduced amplitude variability in comparison to a SISO link and we get th-order diversity. Extracting spatial diversity gain in the absence of channel knowledge at the transmitter 200 PROCEEDINGS OF THE IEEE, VOL. 92, NO. 2, FEBRUARY 2004 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply

is possible using suitably designed transmit signals. The corresponding technique is known as space–time coding [1]–[4]. C. Spatial Multiplexing Gain MIMO channels offer a linear (in ) increase in capacity for no additional power or bandwidth expenditure [5]–[8]. This gain, referred to as spatial multiplexing gain, is realized by transmitting independent data signals from the individual antennas. Under conducive channel conditions, such as rich scattering, the receiver can separate the different streams, yielding a linear increase in capacity. D. Interference Reduction Cochannel interference arises due to frequency reuse in wireless channels. When multiple antennas are used, the differentiation between the spatial signatures of the desired signal and cochannel signals can be exploited to reduce interference. Interference reduction requires knowledge of the desired signal’s channel. Exact knowledge of the interferer’s channel may not be necessary. Interference reduction (or avoidance) can also be implemented at the transmitter, where the goal is to minimize the interference energy sent toward the cochannel users while delivering the signal to the desired user. Interference reduction allows aggressive frequency reuse and thereby increases multicell capacity. We note that in general it is not possible to exploit all the leverages of MIMO technology simultaneously due to conflicting demands on the spatial degrees of freedom (or number of antennas). The degree to which these conflicts are resolved depends upon the signaling scheme and transceiver design. III. MIMO CHANNEL MODEL We consider a MIMO channel with transmit and receive antennas. The time-varying channel impulse response between the th ( ) transmit antenna and the th ( ) receive antenna is denoted as . This is the response at time to an impulse applied at time . The composite MIMO channel response is given by the matrix with . . . . . . . . . . (1) The vector is referred to as the spatio-temporal signature induced by the th transmit antenna across the receive antenna array. Furthermore, given that the signal is launched from the th transmit antenna, the signal received at the th receive antenna is given by (2) where is additive noise in the receiver. Fig. 2. Schematic of wavefront impinging on an antenna array. Under the narrowband assumption the antenna outputs and are identical except for a phase shift. A. Construction of the MIMO Channel Through a Physical Scattering Model In the following, we derive a MIMO wireless channel model from a simplistic physical scattering description. For convenience, we suppress the time-varying nature of the channel and use the narrowband array assumption described in brief below. Consider a signal wavefront impinging at angle on an antenna array comprising two antennas spaced apart (see Fig. 2). We assume that the impinging wavefront has a bandwidth of and is represented as (3) where is the complex envelope of the signal (with bandwidth ) and is the carrier frequency in radians. Under the narrowband assumption, we take the bandwidth to be much smaller than the reciprocal of the transit time of the wavefront across the antenna array, i.e., . Denoting the signal received at the first antenna by , the signal received at the second antenna is then given by (4) where is the wavelength of the signal wavefront. It is clear from (4) that the signals received at the two antennas are identical, except for a phase shift that depends on the array geometry and the angle of arrival of the wavefront. This result can be extended to arrays with more than two antennas in a straightforward way. We emphasize that the narrowband assumption does not imply that the channel is frequency-flat fading. We shall next make use of the narrowband assumption in constructing the MIMO channel below. For the sake of simplicity we assume a single bounce based scattering model and consider a scatterer located at angle and delay with respect to the receive array and with complex amplitude (see Fig. 3). The same scatterer appears at angle with respect to the transmit antenna array. Thus, given the overall geometries of transmit and receive arrays, any two of the variables , , and define the third one. The PAULRAJ et al.: AN OVERVIEW OF MIMO COMMUNICATIONS—A KEY TO GIGABIT WIRELESS 201 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply

Fig. 3. Construction of the MIMO channel model from a physical scattering description. MIMO channel impulse response can now be constructed as ( is a function of and ) (5) where is the maximum delay spread in the channel, is the combined response of pulse shaping at the transmitter and matched-filtering at the receiver, and and are the 1 and 1 array response vectors at the receiver and transmitter, respectively. The single bounce based scattering model in (5) has a number of limitations and cannot adequately model all observed channel effects. A more general model is to assume multiple bounces, i.e., energy from the transmitter uses more than one scatterer to reach the receiver. If we use a double (or multiple) scattering model, the parameters , , and in (5) become independent of each other. B. Classical Frequency-Flat Rayleigh Fading i.i.d. MIMO Channel Model Assuming that the delay spread in the channel is small compared to the reciprocal of the signal bandwidth, i.e., , we can write (5) as (6) Furthermore, we take the combined response to be ideal, so that and henceforth focus on only. With suitable choices of antenna element patterns and array geometry, using a double scattering model, the elements of can be assumed to be independent zero mean unit variance circularly symmetric complex Gaussian random variables, i.e., i.i.d. . Summarizing, we get , the classical i.i.d. frequency-flat Rayleigh fading MIMO channel, which is known to be accurate in NLOS environments with rich scattering and sufficient antenna spacing at transmitter and receiver with all antenna elements identically polarized. C. Real-World MIMO Channels In the real world, the statistics of can deviate significantly from due to a variety of reasons including inadequate antenna spacing and/or inadequate scattering leading to spatial fading correlation, the presence of a fixed (possibly LOS) component in the channel resulting in Ricean fading, and gain imbalances between the channel elements through the use of polarized antennas. These effects have been modeled in [8]–[11] and have been shown to have a significant impact on the performance limits of MIMO channels. A number of MIMO channel measurements have been carried out across the globe [12]–[17]. Fig. 4 shows a measured time-frequency response of an MIMO channel for a fixed broadband wireless access system at 2.5 GHz. Parameters extracted from such measurements include path loss, Ricean -factor, fading signal correlation, delay spread, and Doppler spread. Clearly there is a tremendous variety in real channels. A set of six channels known as the Stanford University Interim (SUI) models [18], reflective of the three terrains (urban, suburban, and hilly) in the continental United States, have been developed and adopted by the IEEE 802.16 standards committee for fixed broadband wireless applications. D. Frequency-Flat Versus Frequency-Selective Fading If the bandwidth-delay spread product of the channel satisfies , the channel is generally said to be frequency selective [19]. Otherwise, the channel is said to be frequency flat. The variation of the matrix-valued transfer function (7) will depend on the delay spread and, hence, on the coherence bandwidth (approximated by the reciprocal of the delay spread). For frequencies and with , we have under Rayleigh fading conditions vec vec , i.e., the channel responses at two frequencies spaced sufficiently apart are uncorrelated. The spatial statistics of will depend on the scattering environment and the array geometry at both the transmitter and receiver. With rich scattering and sufficient antenna spacing, the channel matrix is i.i.d. for all frequencies, i.e., . We note, however, that the correlation between the for different frequencies depends on the power delay profile of the channel and the delay spread. IV. CAPACITY OF MIMO CHANNELS The Shannon capacity of a communication channel is the maximum asymptotically (in the block-length) error-free transmission rate supported by the channel. In the following, we will examine the capacity benefits of MIMO channels. We begin by introducing the discrete-time (sampled) MIMO input–output signal model. 202 PROCEEDINGS OF THE IEEE, VOL. 92, NO. 2, FEBRUARY 2004 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply

Fig. 4. Measured time-frequency response of an , MIMO channel. denotes the scalar subchannel between the th transmit and the th receive antenna. A. Discrete-Time Input–Output Relation For the sake of simplicity we assume that the channel is frequency-flat fading (the capacity of frequency-selective fading MIMO channels will be discussed later in this section). The input–output relation over a symbol period assuming single-carrier (SC) modulation is given by (8) where is the 1 received signal vector, with is the 1 transmitted signal vector, is the MIMO channel matrix, is additive temporally white complex Gaussian noise with , and is the total average energy available at the transmitter over a symbol period. We constrain the total average transmitted power over a symbol period by assuming that the covariance matrix of , , satisfies Tr . B. Capacity of a Deterministic MIMO Channel In the following, we assume that the channel is perfectly known to the receiver (channel knowledge at the receiver can be maintained via training and tracking). Although is random, we shall first study the capacity of a sample realization of the channel, i.e., we consider to be deterministic. It is well known that capacity is achieved with Gaussian code books, i.e., is a circularly symmetric complex Gaussian vector [7]. The corresponding mutual information for having covariance matrix is given by b/s Hz and the capacity of the MIMO channel follows as [7] det b/s Hz (9) where the maximization is performed over all possible input covariance matrices satisfying Tr . Furthermore, given a bandwidth of Hz, the maximum asymptotically (in the block-length) error-free data rate supported by the MIMO channel is simply b/s. Acquiring channel knowledge at the transmitter is in general very difficult in practical systems. In the absence of channel state information at the transmitter, it is reasonable to choose to be spatially white, i.e., . This implies that the signals transmitted from the individual antennas are independent and equi-powered. The mutual information achieved with this covariance matrix is given by [7] and [20] (10) which may be decomposed as (11) PAULRAJ et al.: AN OVERVIEW OF MIMO COMMUNICATIONS—A KEY TO GIGABIT WIRELESS 203 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply

Fig. 5. Ergodic capacity for different MIMO antenna configurations. Note that the SIMO channel has a higher ergodic capacity than the MISO channel. where is the rank of and denotes the positive eigenvalues of . Clearly, we have . Equation (11) expresses the spectral efficiency of the MIMO channel as the sum of the capacities of SISO channels with corresponding channel gains and transmit energy . It follows that multiple scalar spatial data pipes (also known as spatial modes) open up between transmitter and receiver resulting in significant performance gains over the SISO case. For example, increases by b/s/Hz for every 3-dB increase in transmit power (for high transmit power), as opposed to 1 b/s/Hz in conventional SISO channels. If the channel were known to the transmitter, the individual spatial channel modes can be accessed through linear processing at transmitter and receiver (modal decomposition), following which transmit energy can be allocated optimally across the different modes via the “waterfilling algorithm” [21], [7] so as to maximize the mutual information and achieve the capacity . C. Capacity of Fading MIMO Channels We now consider the capacity of fading MIMO channels. In particular, we shall assume with perfect channel knowledge at the receiver and no channel state information at the transmitter. Furthermore, we assume an ergodic block fading channel model where the channel remains constant over a block of consecutive symbols, and changes in an independent fashion across blocks. The average SNR at each of the receive antennas is given by , which can be demonstrated as follows. The signal at the th receive antenna is obtained as (12) where the 1 vector represents the th row of and is the th element of . Since and Tr , it follows that and, hence, the average SNR at the th receive antenna is given by . We shall see below that in a fading channel there are essentially two notions of capacity—ergodic capacity and outage capacity [7], [22], [23], which relate to the mean and tail behavior of , respectively. Ergodic Capacity: If the transmitted codewords span an infinite number of independently fading blocks, the Shannon capacity also known as ergodic capacity is achieved by choosing to be circularly symmetric complex Gaussian with resulting in [7], [24] (13) where the expectation is with respect to the random channel. It has been established that at high SNR [7], [25] (14) which clearly shows the linear increase in capacity in the minimum of the number of transmit and receive antennas. Fig. 5 depicts the ergodic capacity of several MIMO configurations as a function of SNR. As expected, the ergodic capacity increases with increasing and also with and . We note that the ergodic capacity of a SIMO ( 1) channel 204 PROCEEDINGS OF THE IEEE, VOL. 92, NO. 2, FEBRUARY 2004 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply

Fig. 6. 10% outage capacity for different MIMO configurations. MIMO yields significant improvements in terms of outage capacity. is greater than the ergodic capacity of a corresponding MISO (1 ) channel. This is due to the fact that in the absence of channel knowledge at the transmitter MISO channels do not offer array gain. We refer the interested reader to [24], [26], and [27] for analysis of the channel capacity when neither the transmitter nor the receiver knows the channel matrix . Outage Capacity: In applications where delay is an issue and the transmitted codewords span a single block only, the Shannon capacity is zero. This is due to the fact that no matter how small the rate at which we wish to communicate, there is always a nonzero probability that the given channel realization will not support this rate. We define the outage capacity as the information rate that is guaranteed for of the channel realizations [22], [23], i.e., (15) Fig. 6 shows the 10% outage capacity for several MIMO configurations as a function of SNR. As in the case of ergodic capacity, we can see that the outage capacity increases with SNR and that MIMO channels yield significant improvements in outage capacity. In fact the behavior of the 10% outage capacity as a function of SNR, and is almost identical to the behavior of ergodic capacity. The outage probability for a given transmission rate is the probability that the mutual information falls below that rate , i.e., , and can be interpreted as the packet error rate (PER). This interpretation will lead to an interesting tradeoff between transmission rate and outage probability, which we shall explore in greater detail in Section VII. Fig. 7. The capacity of a frequency-selective fading MIMO channel is the sum of (appropriately normalized) capacities of frequency-flat fading MIMO subchannels. D. Capacity of Frequency-Selective Fading MIMO Channels So far, we have restricted our discussion to frequency-flat fading MIMO channels. In the following, we shall briefly discuss frequency-selective fading MIMO channels. The capacity of a frequency-selective fading MIMO channel can be obtained by dividing the frequency band of interest into subchannels, each having bandwidth Hz. If is sufficiently large, each subchannel can be assumed frequency-flat fading (see Fig. 7). Denoting the th subchannel as and assuming that transmit power is allocated uniformly across space (transmit antennas) and frequency, the mutual information associated with a given realization of the frequency-selective MIMO channel is given by [8] det b/s Hz (16) where is the energy allocated to the th subchannel. PAULRAJ et al.: AN OVERVIEW OF MIMO COMMUNICATIONS—A KEY TO GIGABIT WIRELESS 205 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply

Fig. 8. CDF of the mutual information of an increasingly frequency-selective fading MIMO channel. Outage performance improves with frequency-selective fading, due to increased frequency diversity. Fig. 9. Generic coding architecture for MIMO channels. The ergodic capacity of the frequency-selective fading MIMO channel is given by (17) The outage capacity follows from the corresponding definition for the frequency-flat case. Note that the outage capacity (at low outage rates) of the frequency-selective fading channel will in general be higher than the outage capacity of a frequency-flat fading channel. This is due to frequency diversity which leads to increased tightening (the cumulative distribution function (CDF) becomes increasingly step-like) of the probability density function (PDF) of mutual information. Fig. 8 illustrates this effect by showing the CDF of the mutual information of a frequency-selective fading MIMO channel with , for increasing number of degrees of freedom1 (and, hence, increasing frequency diversity). The CDF of mutual information approaches a step function improving outage capacity at low outage rates. The influence of physical parameters such as delay spread, cluster angle spread, and total angle spread on ergodic and outage capacity of frequency-selective fading MIMO channels has been studied in detail in [8]. 1A uniform power delay profile was assumed in this example. V. MIMO SIGNALING In this section, we review some basic MIMO signaling techniques. We start by describing the framework employed in the remainder of this section. Consider the schematic in Fig. 9 where information bits are input to a block that performs the functions of forward-error-correction (temporal) encoding, symbol mapping and interleaving. In the process parity bits are added resulting in data symbols at the output with constellation size (for example, if 4-QAM modulation is employed). The resulting block of data symbols is then input to a space–time encoder that adds an additional parity data symbols and packs the resulting symbols into an matrix (or frame) of length . This frame is then transmitted over symbol periods and is referred to as the space–time codeword. The signaling (data) rate on the channel is b/s/Hz, which should not exceed the channel capacity if we wish to signal asymptotically error-free. Note that we can rewrite the signaling rate as (18) 206 PROCEEDINGS OF THE IEEE, VOL. 92, NO. 2, FEBRUARY 2004 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply

where is the (temporal) code rate of the outer encoder, while is the spatial code rate [3], defined as the number of independent data symbols in a space–time codeword divided by the frame length. Depending on the choice of the spatial signaling mode, the spatial rate varies between 0 and . For certain classes of space–time codes, discussed below, such as space–time trellis codes, the functions of the symbol mapper and space–time encoder are combined into a single block. In the following, we briefly discuss two space–time coding techniques—space–time diversity coding ( ) and spatial multiplexing ( ). Throughout this section we focus on the case where the transmitter does not have channel state information and the receiver knows the channel perfectly. For a discussion of the noncoherent case where neither the transmitter nor the receiver know the channel, the interested reader is referred to [24], [26], [28]. A. Space–Time Diversity Coding The objective of space–time diversity coding is to extract the total available spatial diversity in the MIMO channel through appropriate construction of the transmitted space–time codewords. As examples we consider two specific diversity coding techniques, the Alamouti scheme [2] and delay diversity [29], both of which realize full spatial diversity (without requiring channel knowledge at the transmitter). Alamouti Scheme: Consider a MIMO channel with two transmit antennas and any number of receive antennas. The Alamouti transmission technique is as follows: two different data symbols and are transmitted simultaneously from antennas 1 and 2, respectively, during the first symbol period, following which symbols and are launched from antennas 1 and 2, respectively (see Fig. 10). Note that (two independent data symbols are transmitted over two symbol periods) for the Alamouti scheme. We assume that the channel is i.i.d. frequency-flat fading with and remains constant over (at least) two consecutive symbol periods. Appropriate processing (details can be found in [2]) at the receiver collapses the vector channel into a scalar channel for either of the transmitted data symbols such that (19) where is the processed received signal corresponding to transmitted symbol and is scalar processed noise. Even though channel knowledge is not available to the transmitter, the Alamouti scheme extracts th-order diversity. We note, however, that (as shown in Fig. 11) array gain is realized only at the receiver (recall that the transmitter does not have channel state information). The Alamouti scheme may be extended to channels with more than two transmit antennas through orthogonal space–time block coding (OSTBC) [4] albeit at a loss in spatial rate (i.e., ). However, the low decoding complexity of Fig. 10. Schematic of the transmission strategy for the Alamouti scheme. The MISO channel is orthogonalized irrespectively of the channel realization. OSTBC renders this technique highly attractive for practical applications. Delay Diversity: The second simple scheme for space–time diversity coding we want to discuss is delay diversity [29] which converts spatial diversity into frequency diversity by transmitting the data signal from the first antenna and a delayed replica thereof from the second antenna (see Fig. 12). Retaining the assumption that and and assuming that the delay induced by the second antenna equals one symbol period, the effective channel seen by the data signal is a frequency-selective fading SISO channel with impulse response (20) where and are as defined above. We note that the effective channel in (20) looks exactly like a two-path (symbol spaced) SISO channel with independently fading paths and equal average path energy. A maximum-likelihood (ML) detector will, therefore, realize full second-order diversity at the receiver. General Space–Time Diversity Coding Techniques: The general case of space–time codeword construction for achieving full ( th-order) diversity gain has been studied in [3] and leads to the well-known rank and determinant criteria. Extensions of these design criteria to the frequency-selective fading case can be found in [30] and [31]. B. Spatial Multiplexing The objective of spatial multiplexing as opposed to space–time diversity coding is to maximize transmission rate. Accordingly, independent data symbols are transmitted per symbol period so that . In the following, we describe several encoding options that can be used in conjunction with spatial multiplexing. Horizontal Encoding (HE): The bit stream to be transmitted is first demultiplexed into separate data streams PAULRAJ et al.: AN OVERVIEW OF MIMO COMMUNICATIONS—A KEY TO GIGABIT WIRELESS 207 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply