6.962 Week 7 Summary:Capacity of Fading Channels with Multiple AntennasPresenter: Aaron CohenOctober 25, 20001 IntroductionWe now turn our attention to the study of wireless systems. We will use a mobile phone systemas the canonical example of such a system, which typically has many simultaneous users and ishighly time-varying. This talk will focus on the single user case, with earlier and later talks sheddingsome light onto the difficult multi-user case. Under appropriate conditions, the time-varying naturecan be modeled by a fading channel, where the received signal is simply a scaled version of thetransmitted signal plus some additive noise.In designing such a system, the most precious resources are bandwidth and power. We will showthat using multiple antennas at both the transmitter and the receiver gives a dramatic increase incapacity for a fixed amount of bandwidth and power [1, 2, 3]. We will be primarily interested herein capacity results and not in efficient coding and equalization methods used to obtain rates closeto capacity (see e.g. [4]).We will first discuss fading channels in general. We will next look at the case of a singletransmit and receive antenna. We will then examine the model proposed by Telatar [1] for themulti-antenna channel. We will finally look at the results obtained by Telatar. For more resultssee the very thorough review article [5]2 Fading channels overviewThis section is based on the presentation by Proakis [6, Chap. 14], and is intended to provide abrief introduction to the models and terminology widely used for wireless channels. We will assume1throughout that we are using a modulation scheme with bandwidth W and signaling rate TswhereW ≈ 1/Ts.The time-varying nature of a wireless channel is represented by a random process c(t; τ) whichgives the output of the channel at time t due to an impulse at time τ − t (plus some additivenoise). Our first assumptions are that this random process is wide sense stationary (in t) and thatresponses at different values of τ are uncorrelated. Thus,12E[c∗(τ1; t)c(τ2; t + ∆t)] =φc(τ1; ∆t) if τ1= τ20 otherwise,and we call φc(τ) ≡ φc(τ; 0) the delay power spectrum of the channel. The range of values of τwhere φc(τ) is non-zero is called the multipath spread of the channel is denoted by Tm. Invertingthis quantity gives us the coherence bandwidth,(∆f)c=1Tm,which provides a crude measure of how different frequencies are affected by the channel. If W (∆f)c, then we say that the channel is frequency-selective. This usually results in severe distortionsince different frequency bands are affected by different noise. Conversely, if W  (∆f)c, then wesay that the channel is frequency-nonselective or has flat fading. In this case, the entire signal isusually affected by a single multiplicative noise term.We can also consider the correlation function of the Fourier transform C(f; t) of c(τ ; t) (takenwith respect to τ with t fixed). In particular, letφC(∆f; ∆t) =12E[C∗(f; t)C(f + ∆f; t + ∆t)]and let SC(λ) be the Fourier transform of φC(0; ∆t). Then, the Doppler spread Bdis the range ofvalue of λ over which SC(λ) is non-zero. The inverse of the Doppler spread,(∆t)c=1Bdis the coherence time of the channel. This gives a crude measure of how long the impulse responseremains constant. For example, if Ts (∆t)c, then the noise remains constant over a signaling2interval.If (∆t)c(∆f)c 1, then we can design a system such that W  (∆f)cand Ts (∆t)c, whichwe call a slowly (flat) fading channel. We will focus exclusively on this case. In particular, weignore the problem of inter-symbol interference (ISI).3 Single antenna case3.1 ModelIn the single antenna case of a slowly fading channel, we can think of an equivalent baseband,discrete time system:yk= hkxk+ nk, k = 1, 2, . . .where {xk} is the (complex) input sequence designed by the transmitter; {hk} is a (complex)random sequence which we will call the fading coefficients; and {nk} is a (complex) additive noisesequence. Typically, these three sequences are jointly independent.The additive noise sequence is usually modeled as IID (complex) zero-mean Gaussian. Manydifferent models for fading coefficients have been proposed. One of the most common is Rayleighfading (other common distribution include Rician and Nakagami-m), which is appropriate when alarge number of scatterers contribute to the fading. In this case, one can think of hkwith inde-pendent real and complex components which are both zero-mean Gaussian with identical variance.Equivalently, hkhas uniform phase with magnitude given by the Rayleigh pdf,pR(r) =2rΩe−r2/Ω, r ≥ 0,where Ω = E[R2]. The fading coefficients are typically correlated but hkand hk+mare approxi-mately independent if m  (∆t)c/Ts.The power constraint is modeled by requiring that the input sequence {xk} satisfyN−1NXk=1|xk|2≤ Peither in expectation (average power constraint) or with probability one (peak power constraint),where N is the blocklength. Any number of other constraints have also been examined.3It is sufficient to consider E[|hk|2] = 1 and E[|nk|2] = 1 and to vary P (the power constraint)in order to study all possible configurations with SNR = P .3.2 Capacity resultsWe describe the capacity for three cases: fading coefficients known at receiver only, fading coeffi-cients known at transmitter and receiver, and fading coefficients known at neither transmitter norreceiver. These scenarios are extreme points for many realistic situations where the transmitterand/or receiver might have some estimate of the fading coefficients (see e.g. [7]). Perfect knowledgeof the fading coefficients is often called channel side information (CSI). In each of these case, weassume that {hk} form an IID sequence, but many of the same results apply if {hk} is stationaryand ergodic with the same marginal distribution.CSI at receiver only: For any distribution on the fading coefficients, the capacity achievingdistribution on the inputs is Gaussian with power P. The resulting capacity1is given by [8]CRCSI(1, 1, P ) = Eh[log(1 + |h|2P )], (1)where for consistent notation we will write C(r, t, P ) to denote the capacity with r receive antennas,t transmit antennas and SNR P . For Rayleigh fading, the capacity is given by [9]CRRCSI(1, 1, P ) = (log e)e1PZ∞1/Pe−ttdt.CSI at transmitter and receiver: The capacity achieving distribution is Gaussian with powergiven by the