7 MIMO I: spatial multiplexing | My Assignment Tutor

C H A P T E R7 MIMO I: spatial multiplexingand channel modelingIn this book, we have seen several different uses of multiple antennas inwireless communication. In Chapter 3, multiple antennas were used to providediversity gain and increase the reliability of wireless links. Both receiveand transmit diversity were considered. Moreover, receive antennas can alsoprovide a power gain. In Chapter 5, we saw that with channel knowledge atthe transmitter, multiple transmit antennas can also provide a power gain viatransmit beamforming. In Chapter 6, multiple transmit antennas were usedto induce channel variations, which can then be exploited by opportunisticcommunication techniques. The scheme can be interpreted as opportunisticbeamforming and provides a power gain as well.In this and the next few chapters, we will study a new way to use multipleantennas. We will see that under suitable channel fading conditions, havingboth multiple transmit and multiple receive antennas (i.e., a MIMO channel)provides an additional spatial dimension for communication and yields adegree-of- freedom gain. These additional degrees of freedom can be exploitedby spatially multiplexing several data streams onto the MIMO channel, andlead to an increase in the capacity: the capacity of such a MIMO channelwith n transmit and receive antennas is proportional to n.Historically, it has been known for a while that a multiple access systemwith multiple antennas at the base-station allows several users to simultaneously communicate with the base-station. The multiple antennas allow spatialseparation of the signals from the different users. It was observed in the mid1990s that a similar effect can occur for a point-to-point channel with multipletransmit and receive antennas, i.e., even when the transmit antennas are notgeographically far apart. This holds provided that the scattering environmentis rich enough to allow the receive antennas to separate out the signals fromthe different transmit antennas. We have already seen how channel fadingcan be exploited by opportunistic communication techniques. Here, we seeyet another example where channel fading is beneficial to communication.It is insightful to compare and contrast the nature of the performancegains offered by opportunistic communication and by MIMO techniques.290291 7.1 Multiplexing capability of deterministic MIMO channelsOpportunistic communication techniques primarily provide a power gain.This power gain is very significant in the low SNR regime where systems arepower-limited but less so in the high SNR regime where they are bandwidthlimited. As we will see, MIMO techniques can provide both a power gainand a degree-of-freedom gain. Thus, MIMO techniques become the primarytool to increase capacity significantly in the high SNR regime.MIMO communication is a rich subject, and its study will span the remaining chapters of the book. The focus of the present chapter is to investigatethe properties of the physical environment which enable spatial multiplexingand show how these properties can be succinctly captured in a statisticalMIMO channel model. We proceed as follows. Through a capacity analysis,we first identify key parameters that determine the multiplexing capability ofa deterministic MIMO channel. We then go through a sequence of physicalMIMO channels to assess their spatial multiplexing capabilities. Building onthe insights from these examples, we argue that it is most natural to model theMIMO channel in the angular domain and discuss a statistical model basedon that approach. Our approach here parallels that in Chapter 2, where westarted with a few idealized examples of multipath wireless channels to gaininsights into the underlying physical phenomena, and proceeded to statisticalfading models, which are more appropriate for the design and performanceanalysis of communication schemes. We will in fact see a lot of parallelismin the specific channel modeling technique as well.Our focus throughout is on flat fading MIMO channels. The extensions tofrequency-selective MIMO channels are straightforward and are developed inthe exercises.7.1 Multiplexing capability of deterministic MIMO channelsA narrowband time-invariant wireless channel with nt transmit and nr receiveantennas is described by an nr by nt deterministic matrix H. What are the keyproperties of H that determine how much spatial multiplexing it can support?We answer this question by looking at the capacity of the channel.7.1.1 Capacity via singular value decompositionThe time-invariant channel is described byy = Hx+w (7.1)where x ∈ nt, y ∈ nr and w ∼ 0 N0Inr denote the transmitted signal, received signal and white Gaussian noise respectively at a symbol time(the time index is dropped for simplicity). The channel matrix H ∈ nr×nt292 MIMO I: spatial multiplexing and channel modelingis deterministic and assumed to be constant at all times and known to boththe transmitter and the receiver. Here, hij is the channel gain from transmitantenna j to receive antenna i. There is a total power constraint, P, on thesignals from the transmit antennas.This is a vector Gaussian channel. The capacity can be computed bydecomposing the vector channel into a set of parallel, independent scalarGaussian sub-channels. From basic linear algebra, every linear transformationcan be represented as a composition of three operations: a rotation operation, ascaling operation, and another rotation operation. In the notation of matrices,the matrix H has a singular value decomposition (SVD):H = UV∗ (7.2)where U ∈ nr×nr and V ∈ nt×nt are (rotation) unitary matrices1 and ∈nr×nt is a rectangular matrix whose diagonal elements are non-negative realnumbers and whose off-diagonal elements are zero.2 The diagonal elements1 ≥ 2 ≥ ··· ≥ nmin are the ordered singular values of the matrix H, wherenmin = minntnr. SinceHH∗ = UtU∗ (7.3)the squared singular values 2 i are the eigenvalues of the matrix HH∗ andalso of H∗H. Note that there are nmin singular values. We can rewrite theSVD as H = i=1iuivi∗(7.4) nmini.e., the sum of rank-one matrices iuivi∗. It can be seen that the rank of H isprecisely the number of non-zero singular values.If we define x˜ = V∗xy˜ = U∗yw˜ = U∗w(7.5)(7.6)(7.7) then we can rewrite the channel (7.1) asy˜ = x˜ +w˜ (7.8)1 Recall that a unitary matrix U satisfies U∗U = UU∗ = I.2 We will call this matrix diagonal even though it may not be square.293 7.1 Multiplexing capability of deterministic MIMO channelsFigure 7.1 Converting theMIMO channel into a parallelchannel through the SVD. xVUyChannelλ1λnmin wnminw1+ +∼ ∼…× × V* U* yPre-processing Post-processing∼x~where w˜ ∼ 0 N0Inr has the same distribution as w (cf. (A.22) inAppendix A), and x˜2 = x2. Thus, the energy is preserved and we havean equivalent representation as a parallel Gaussian channel:y˜i = ix˜ i + ˜ wi i = 12 nmin (7.9)The equivalence is summarized in Figure 7.1.The SVD decomposition can be interpreted as two coordinate transformations: it says that if the input is expressed in terms of a coordinate systemdefined by the columns of V and the output is expressed in terms of a coordinate system defined by the columns of U, then the input/output relationshipis very simple. Equation (7.8) is a representation of the original channel (7.1)with the input and output expressed in terms of these new coordinates.We have already seen examples of Gaussian parallel channels in Chapter 5,when we talked about capacities of time-invariant frequency-selective channels and about time-varying fading channels with full CSI. The time-invariantMIMO channel is yet another example. Here, the spatial dimension plays thesame role as the time and frequency dimensions in those other problems. Thecapacity is by now familiar:C =nmin i=1log1+ PNi∗0 2 i bits/s/Hz (7.10)where P∗1 Pn∗min are the waterfilling power allocations:P∗i = – N2 i0 + (7.11)with chosen to satisfy the total power constraint i Pi∗ = P. Each icorresponds to an eigenmode of the channel (also called an eigenchannel).Each non-zero eigenchannel can support a data stream; thus, the MIMOchannel can support the spatial multiplexing of multiple streams. Figure 7.2pictorially depicts the SVD-based architecture for reliable communication.294 MIMO I: spatial multiplexing and channel modeling+ AWGNcoder{~x1[m]}]}……{0}{0}{VH AWGNcoder {~y1 [m]}U*{~ynmin[m]… {~xnmin[mn mininformationstreamsw[m]} Decoder Decoder Figure 7.2 The SVD architecturefor MIMO communication.There is a clear analogy between this architecture and the OFDM systemintroduced in Chapter 3. In both cases, a transformation is applied to convert a matrix channel into a set of parallel independent sub-channels. In the OFDMsetting, the matrix channel is given by the circulant matrix C in (3.139),defined by the ISI channel together with the cyclic prefix added onto theinput symbols. In fact, the decomposition C = Q-1Q in (3.143) is the SVDdecomposition of a circulant matrix C, with U = Q-1 and V∗ = Q. Theimportant difference between the ISI channel and the MIMO channel is that,for the former, the U and V matrices (DFTs) do not depend on the specificrealization of the ISI channel, while for the latter, they do depend on thespecific realization of the MIMO channel.7.1.2 Rank and condition numberWhat are the key parameters that determine performance? It is simpler tofocus separately on the high and the low SNR regimes. At high SNR, thewater level is deep and the policy of allocating equal amounts of power onthe non-zero eigenmodes is asymptotically optimal (cf. Figure 5.24(a)): C ≈klog1+ P ≈ klogSNR+klogbits/s/Hz(7.12)i=1kN0i=1k 2 i 2 i where k is the number of non-zero 2i , i.e., the rank of H, and SNR = P/N0.The parameter k is the number of spatial degrees of freedom per second perhertz. It represents the dimension of the transmitted signal as modified bythe MIMO channel, i.e., the dimension of the image of H. This is equal tothe rank of the matrix H and with full rank, we see that a MIMO channelprovides nmin spatial degrees of freedom.295 7.2 Physical modeling of MIMO channelsThe rank is a first-order but crude measure of the capacity of the channel.To get a more refined picture, one needs to look at the non-zero singularvalues themselves. By Jensen’s inequality,1 kk i=1log1+ kN P 0 2 i ≤ log1+ kN P 0 k1 i=k1 2 i (7.13)Now,k i=12i = Tr HH∗ = ijhij2 (7.14)which can be interpreted as the total power gain of the matrix channel ifone spreads the energy equally between all the transmit antennas. Then, theabove result says that among the channels with the same total power gain,the one that has the highest capacity is the one with all the singular valuesequal. More generally, the less spread out the singular values, the larger thecapacity in the high SNR regime. In numerical analysis, maxi i/mini i isdefined to be the condition number of the matrix H. The matrix is said to bewell-conditioned if the condition number is close to 1. From the above result,an important conclusion is:Well-conditioned channel matrices facilitate communication in the highSNR regime.At low SNR, the optimal policy is to allocate power only to the strongesteigenmode (the bottom of the vessel to waterfill, cf. Figure 5.24(b)). Theresulting capacity is C ≈Pmax i log2 e bits/s/Hz(7.15) N02 i The MIMO channel provides a power gain of maxi 2 i . In this regime, therank or condition number of the channel matrix is less relevant. What mattersis how much energy gets transferred from the transmitter to the receiver.7.2 Physical modeling of MIMO channelsIn this section, we would like to gain some insight on how the spatial multiplexing capability of MIMO channels depends on the physical environment.We do so by looking at a sequence of idealized examples and analyzing the296 MIMO I: spatial multiplexing and channel modelingrank and conditioning of their channel matrices. These deterministic exampleswill also suggest a natural approach to statistical modeling of MIMO channels, which we discuss in Section 7.3. To be concrete, we restrict ourselvesto uniform linear antenna arrays, where the antennas are evenly spaced on astraight line. The details of the analysis depend on the specific array structurebut the concepts we want to convey do not.7.2.1 Line-of-sight SIMO channelThe simplest SIMO channel has a single line-of-sight (Figure 7.3(a)). Here,there is only free space without any reflectors or scatterers, and only adirect signal path between each antenna pair. The antenna separation is rc,where c is the carrier wavelength and r is the normalized receive antennaseparation, normalized to the unit of the carrier wavelength. The dimensionof the antenna array is much smaller than the distance between the transmitterand the receiver.The continuous-time impulse response hi between the transmit antennaand the ith receive antenna is given byhi = a -di/c i = 1 nr (7.16)Figure 7.3 (a) Line-of-sightchannel with single transmitantenna and multiple receiveantennas. The signals from thetransmit antenna arrive almostin parallel at the receivingantennas. (b) Line-of-sightchannel with multiple transmitantennas and single receiveantenna.… …Rx antenna i∆rλcd φ(i -1)∆rλccosφ(a)… …∆tλcφ(i -1)∆tλccosφTx antenna id(b)297 7.2 Physical modeling of MIMO channelswhere di is the distance between the transmit antenna and ith receive antenna,c is the speed of light and a is the attenuation of the path, which we assumeto be the same for all antenna pairs. Assuming di/c 1/W, where W isthe transmission bandwidth, the baseband channel gain is given by (2.34)and (2.27):hi = aexp-j2f c cdi = aexp-j2d c i (7.17)where fc is the carrier frequency. The SIMO channel can be written asy = hx+w (7.18)where x is the transmitted symbol, w ∼ 0N0I is the noise and y is thereceived vector. The vector of channel gains h = h1 hnrt is sometimescalled the signal direction or the spatial signature induced on the receiveantenna array by the transmitted signal.Since the distance between the transmitter and the receiver is much largerthan the size of the receive antenna array, the paths from the transmit antennato each of the receive antennas are, to a first-order, parallel anddi ≈ d+i-1rc cos i = 1 nr (7.19)where d is the distance from the transmit antenna to the first receiveantenna and is the angle of incidence of the line-of-sight onto the receiveantenna array. (You are asked to verify this in Exercise 7.1.) The quantityi-1rc cos is the displacement of receive antenna i from receive antenna1 in the direction of the line-of-sight. The quantity = cosis often called the directional cosine with respect to the receive antenna array.The spatial signature h = h1 hnrt is therefore given byh = aexp-j2d c 1exp-j2rexp-j22 rexp-j2nr -1r (7.20)298 MIMO I: spatial multiplexing and channel modelingi.e., the signals received at consecutive antennas differ in phase by 2rdue to the relative delay. For notational convenience, we defineer = √1nr1exp-j2rexp-j22 rexp-j2nr -1r (7.21)as the unit spatial signature in the directional cosine .The optimal receiver simply projects the noisy received signal onto thesignal direction, i.e., maximal ratio combining or receive beamforming(cf. Section 5.3.1). It adjusts for the different delays so that the receivedsignals at the antennas can be combined constructively, yielding an nr-foldpower gain. The resulting capacity isC = log1+ PNh02 = log1+ PaN20nr bits/s/Hz (7.22)The SIMO channel thus provides a power gain but no degree-of-freedomgain.In the context of a line-of-sight channel, the receive antenna array is sometimes called a phased-array antenna.7.2.2 Line-of-sight MISO channelThe MISO channel with multiple transmit antennas and a single receiveantenna is reciprocal to the SIMO channel (Figure 7.3(b)). If the transmitantennas are separated by tc and there is a single line-of-sight with angleof departure of (directional cosine = cos), the MISO channel isgiven byy = h∗x+w (7.23)whereh = aexpj2d c 1exp-j2texp-j22 texp-j2nr -1t (7.24)299 7.2 Physical modeling of MIMO channelsThe optimal transmission (transmit beamforming) is performed along thedirection et of h, whereet = √1nt1exp-j2texp-j22 texp-j2nt -1t (7.25)is the unit spatial signature in the transmit direction of (cf. Section 5.3.2).The phase of the signal from each of the transmit antennas is adjusted so thatthey add constructively at the receiver, yielding an nt-fold power gain. Thecapacity is the same as (7.22). Again there is no degree-of-freedom gain.7.2.3 Antenna arrays with only a line-of-sight pathLet us now consider a MIMO channel with only direct line-of-sight pathsbetween the antennas. Both the transmit and the receive antennas are in lineararrays. Suppose the normalized transmit antenna separation is t and thenormalized receive antenna separation is r. The channel gain between thekth transmit antenna and the ith receive antenna ishik = aexp-j2dik/c (7.26)where dik is the distance between the antennas, and a is the attenuation alongthe line-of-sight path (assumed to be the same for all antenna pairs). Assumingagain that the antenna array sizes are much smaller than the distance betweenthe transmitter and the receiver, to a first-order:dik = d+i-1rc cosr -k-1tc cost (7.27)where d is the distance between transmit antenna 1 and receive antenna 1, andt r are the angles of incidence of the line-of-sight path on the transmit andreceive antenna arrays, respectively. Define t = cost and r = cosr.Substituting (7.27) into (7.26), we gethik = aexp-j2d c ·expj2k-1tt·exp-j2i-1rr (7.28)and we can write the channel matrix asH = a√ntnr exp-j2d c errett∗ (7.29)300 MIMO I: spatial multiplexing and channel modelingwhere er· and et· are defined in (7.21) and (7.25), respectively. Thus, His a rank-one matrix with a unique non-zero singular value 1 = a√ntnr. Thecapacity of this channel follows from (7.10):C = log 1+ PaN2n0tnr bits/s/Hz(7.30) Note that although there are multiple transmit and multiple receive antennas,the transmitted signals are all projected onto a single-dimensional space (theonly non-zero eigenmode) and thus only one spatial degree of freedom isavailable. The receive spatial signatures at the receive antenna array from allthe transmit antennas (i.e., the columns of H) are along the same direction,err. Thus, the number of available spatial degrees of freedom does notincrease even though there are multiple transmit and multiple receive antennas.The factor ntnr is the power gain of the MIMO channel. If nt = 1, the powergain is equal to the number of receive antennas and is obtained by maximalratio combining at the receiver (receive beamforming). If nr = 1, the powergain is equal to the number of transmit antennas and is obtained by transmitbeamforming. For general numbers of transmit and receive antennas, one getsbenefits from both transmit and receive beamforming: the transmitted signalsare constructively added in-phase at each receive antenna, and the signal ateach receive antenna is further constructively combined with each other.In summary: in a line-of-sight only environment, a MIMO channel providesa power gain but no degree-of-freedom gain.7.2.4 Geographically separated antennasGeographically separated transmit antennasHow do we get a degree-of-freedom gain? Consider the thought experimentwhere the transmit antennas can now be placed very far apart, with a separationof the order of the distance between the transmitter and the receiver. Forconcreteness, suppose there are two transmit antennas (Figure 7.4). EachFigure 7.4 Two geographicallyseparated transmit antennaseach with line-of-sight to areceive antenna array.…Rx antennaarrayφr2 φr1Tx antenna 1Tx antenna 2301 7.2 Physical modeling of MIMO channelstransmit antenna has only a line-of-sight path to the receive antenna array,with attenuations a1 and a2 and angles of incidence r1 and r2, respectively.Assume that the delay spread of the signals from the transmit antennas ismuch smaller than 1/W so that we can continue with the single-tap model.The spatial signature that transmit antenna k impinges on the receive antennaarray ishk = ak√nr exp-j2d c 1k errk k = 12 (7.31)where d1k is the distance between transmit antenna k and receive antenna 1,rk = cosrk and er· is defined in (7.21).It can be directly verified that the spatial signature er is a periodicfunction of with period 1/r, and within one period it never repeats itself(Exercise 7.2). Thus, the channel matrix H = h1h2 has distinct and linearlyindependent columns as long as the separation in the directional cosinesr = r2 -r1 = 0 mod1r (7.32)In this case, it has two non-zero singular values 2 1 and 2 2, yielding twodegrees of freedom. Intuitively, the transmitted signal can now be receivedfrom two different directions that can be resolved by the receive antennaarray. Contrast this with the example in Section 7.2.3, where the antennas areplaced close together and the spatial signatures of the transmit antennas areall aligned with each other.Note that since r1 r2, being directional cosines, lie in -11 and cannotdiffer by more than 2, the condition (7.32) reduces to the simpler conditionr1 = r2 whenever the antenna spacing r ≤ 1/2.Resolvability in the angular domainThe channel matrix H is full rank whenever the separation in the directionalcosines r = 0 mod 1/r. However, it can still be very ill-conditioned. Wenow give an order-of-magnitude estimate on how large the angular separationhas to be so that H is well-conditioned and the two degrees of freedom canbe effectively used to yield a high capacity.The conditioning of H is determined by how aligned the spatial signaturesof the two transmit antennas are: the less aligned the spatial signatures are, thebetter the conditioning of H. The angle between the two spatial signaturessatisfies cos = err1∗err2 (7.33)Note that err1∗err2 depends only on the difference r = r2 – r1.Define thenfrr2 -r1 = err1∗err2(7.34) 302 MIMO I: spatial multiplexing and channel modelingBy direct computation (Exercise 7.3),frr = 1nrexpjrrnr -1 sinLrrsinLrr/nr (7.35)where Lr = nr r is the normalized length of the receive antenna array. Hence, cos =sinLrrnr sinLrr/nr(7.36) The conditioning of the matrix H depends directly on this parameter. Forsimplicity, consider the case when the gains a1 = a2 = a. The squared singularvalues of H are21 = a2nr1+ cos 2 2 = a2nr1- cos (7.37)and the condition number of the matrix is12= 11+ – cos cos (7.38)The matrix is ill-conditioned whenever cos ≈ 1, and is well-conditionedotherwise. In Figure 7.5, this quantity cos = frr is plotted as a functionof r for a fixed array size and different values of nr. The function fr· hasthe following properties:• frr is periodic with period nr/Lr = 1/r;• frr peaks at r = 0; f0 = 1;• frr = 0 at r = k/Lr k = 1 nr -1.The periodicity of fr· follows from the periodicity of the spatial signatureer·. It has a main lobe of width 2/Lr centered around integer multiples of1/r. All the other lobes have significantly lower peaks. This means that thesignatures are close to being aligned and the channel matrix is ill conditionedwheneverr –m r1 Lr(7.39)for some integer m. Now, since r ranges from -2 to 2, this conditionreduces tor1 Lr(7.40)whenever the antenna separation r ≤ 1/2.303 7.2 Physical modeling of MIMO channelsFigure 7.5 The function |f(r)|plotted as a function of r forfixed Lr = 8 and differentvalues of the number ofreceive antennas nr.00.70.80.91– 2 – 1.5 – 10.50.40.30.20.10.6nr = 16 Ωrsinc functionnr = 8Ωr Ωrnr = 4– 0.5 0 0.5 1 1.5 2 00.10.20.30.40.50.60.70.80.91– 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 200.10.20.30.40.50.60.70.80.91– 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2 00.10.20.30.40.50.60.70.80.91– 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2Ωr|f(Ωr)| |f(Ωr)||f(Ωr)| |f(Ωr)|Increasing the number of antennas for a fixed antenna length Lr does notsubstantially change the qualitative picture above. In fact, as nr → andr → 0,frr → ejLrrsincLrr (7.41)and the dependency of fr· on nr vanishes. Equation (7.41) can be directlyderived from (7.35), using the definition sincx = sinx/x (cf. (2.30)).The parameter 1/Lr can be thought of as a measure of resolvability in theangular domain: if r 1/Lr, then the signals from the two transmit antennascannot be resolved by the receive antenna array and there is effectively onlyone degree of freedom. Packing more and more antenna elements in a givenamount of space does not increase the angular resolvability of the receiveantenna array; it is intrinsically limited by the length of the array.A common pictorial representation of the angular resolvability of an antennaarray is the (receive) beamforming pattern. If the signal arrives from a singledirection 0, then the optimal receiver projects the received signal onto thevector ercos0; recall that this is called the (receive) beamforming vector.A signal from any other direction is attenuated by a factor ofercos0∗ercos = frcos-cos0 (7.42)The beamforming pattern associated with the vector ercos is the polarplot frcos-cos0 (7.43)304 MIMO I: spatial multiplexing and channel modelingFigure 7.6 Receivebeamforming patterns aimedat 90, with antenna arraylength Lr = 2 and differentnumbers of receive antennasnr. Note that the beamformingpattern is always symmetricalabout the 0 – 180 axis, solobes always appear in pairs.For nr = 4632, the antennaseparation r ≤ 1/2, andthere is a single main lobearound 90 (together with itsmirror image). For nr = 2,r = 1 > 1/2 and there is anadditional pair of main lobes.0.20.4 0.6 0.81302106024090270120300150330180 0Lr = 2, nr = 20.20.4 0.6 0.81302106024090270120300150330180 00.20.4 0.6 0.81302106024090270120300150330180 00.20.4 0.6 0.81302106024090270120300150330180 0Lr = 2, nr = 4Lr = 2, nr = 6 Lr = 2, nr = 32(Figures 7.6 and 7.7). Two important points to note about the beamformingpattern:• It has main lobes around 0 and also around any angle for whichcos = cos0 mod 1r (7.44)this follows from the periodicity of fr·. If the antenna separation r isless than 1/2, then there is only one main lobe at , together with its mirrorimage at -. If the separation is greater than 1/2, there can be severalmore pairs of main lobes (Figure 7.6).• The main lobe has a directional cosine width of 2/Lr; this is also calledthe beam width. The larger the array length Lr, the narrower the beamand the higher the angular resolution: the array filters out the signal fromall directions except for a narrow range around the direction of interest(Figure 7.7). Signals that arrive along paths with angular seperation largerthan 1/Lr can be discriminated by focusing different beams at them.There is a clear analogy between the roles of the antenna array size Lr andthe bandwidth W in a wireless channel. The parameter 1/W measures the305 7.2 Physical modeling of MIMO channelsFigure 7.7 Beamformingpatterns for different antennaarray lengths. (Left) Lr = 4 and(right) Lr = 8. Antennaseparation is fixed at half thecarrier wavelength. The largerthe length of the array, thenarrower the beam. 0.53060120300 121024090270150330180 0 0.53060120300 121024090270150330180 0Lr = 4, nr = 8 Lr = 8, nr = 16resolvability of signals in the time domain: multipaths arriving at time separation much less than 1/W cannot be resolved by the receiver. The parameter1/Lr measures the resolvability of signals in the angular domain: signalsthat arrive within an angle much less than 1/Lr cannot be resolved by thereceiver. Just as over-sampling cannot increase the time-domain resolvabilitybeyond 1/W, adding more antenna elements cannot increase the angulardomain resolvability beyond 1/Lr. This analogy will be exploited in thestatistical modeling of MIMO fading channels and explained more preciselyin Section 7.3.Geographically separated receive antennasWe have increased the number of degrees of freedom by placing the transmitantennas far apart and keeping the receive antennas close together, but we canachieve the same goal by placing the receive antennas far apart and keepingthe transmit antennas close together (see Figure 7.8). The channel matrix isgiven byH = hh1 2∗ ∗ (7.45)Figure 7.8 Two geographicallyseparated receive antennaseach with line-of-sight from atransmit antenna array.. .. Tx antennaarray φt1φt2Rx antenna 2Rx antenna 1306 MIMO I: spatial multiplexing and channel modelingwherehi = ai expj2d c i1 etti (7.46)and ti is the directional cosine of departure of the path from the transmitantenna array to receive antenna i and di1 is the distance between transmitantenna 1 and receive antenna i. As long ast = t2 -t1 = 0 mod1t (7.47)the two rows of H are linearly independent and the channel has rank 2, yielding2 degrees of freedom. The output of the channel spans a two-dimensionalspace as we vary the transmitted signal at the transmit antenna array. In orderto make H well-conditioned, the angular separation t of the two receiveantennas should be of the order of or larger than 1/Lt, where Lt = nt t is thelength of the transmit antenna array, normalized to the carrier wavelength.Analogous to the receive beamforming pattern, one can also define a transmit beamforming pattern. This measures the amount of energy dissipated inother directions when the transmitter attempts to focus its signal along a direction 0. The beam width is 2/Lt; the longer the antenna array, the sharperthe transmitter can focus the energy along a desired direction and the betterit can spatially multiplex information to the multiple receive antennas.7.2.5 Line-of-sight plus one reflected pathCan we get a similar effect to that of the example in Section 7.2.4, withoutputting either the transmit antennas or the receive antennas far apart? Consideragain the transmit and receive antenna arrays in that example, but now supposein addition to a line-of-sight path there is another path reflected off a wall(see Figure 7.9(a)). Call the direct path, path 1 and the reflected path, path 2.Path i has an attenuation of ai, makes an angle of ti (ti = costi) withthe transmit antenna array and an angle of riri = cosri) with the receiveantenna array. The channel H is given by the principle of superposition: H = ab1err1ett1∗ +ab 2err2ert2∗where for i = 12,(7.48) abi = ai√ntnr exp-j2d c i (7.49)and di is the distance between transmit antenna 1 and receive antenna 1along path i. We see that as long ast1 = t2 mod1t(7.50)307 7.2 Physical modeling of MIMO channelsFigure 7.9 (a) A MIMOchannel with a direct path anda reflected path. (b) Channel isviewed as a concatenation oftwo channels H and H withintermediate (virtual) relaysA and B.Tx antennaarray Tx antennaarrayTx antenna 1~~~~path 1φrφt2φt1 arrayRx antenna 1…(b)(a)AB~~Rx antennaarraypath 2 …H″ H′ A B2φr1andr1 = r2 mod1r (7.51)the matrix H is of rank 2. In order to make H well-conditioned, the angularseparation t of the two paths at the transmit array should be of the sameorder or larger than 1/Lt and the angular separation r at the receive arrayshould be of the same order as or larger than 1/Lr, wheret = cos t2 -cos t1 Lt = nt t (7.52)andr = cos r2 -cos r1 Lr = nr r (7.53)To see clearly what the role of the multipath is, it is helpful to rewrite Has H = HH, where H = ab 1err1 ab 2err2 H =(7.54) e et t∗ ∗ t1 t2H is a 2 by nt matrix while H is an nr by 2 matrix. One can interpret H asthe matrix for the channel from the transmit antenna array to two imaginaryreceivers at point A and point B, as marked in Figure 7.9. Point A is the pointof incidence of the reflected path on the wall; point B is along the line-of-sightpath. Since points A and B are geographically widely separated, the matrixH has rank 2; its conditioning depends on the parameter Ltt. Similarly,308 MIMO I: spatial multiplexing and channel modelingone can interpret the second matrix H as the matrix channel from twoimaginary transmitters at A and B to the receive antenna array. This matrixhas rank 2 as well; its conditioning depends on the parameter Lrr. If bothmatrices are well-conditioned, then the overall channel matrix H is also wellconditioned.The MIMO channel with two multipaths is essentially a concatenation of thent by 2 channel in Figure 7.8 and the 2 by nr channel in Figure 7.4. Althoughboth the transmit antennas and the receive antennas are close together, multipaths in effect provide virtual “relays”, which are geographically far apart.The channel from the transmit array to the relays as well as the channel fromthe relays to the receive array both have two degrees of freedom, and sodoes the overall channel. Spatial multiplexing is now possible. In this context, multipath fading can be viewed as providing an advantage that can beexploited.It is important to note in this example that significant angular separationof the two paths at both the transmit and the receive antenna arrays is crucialfor the well-conditionedness of H. This may not hold in some environments.For example, if the reflector is local around the receiver and is much closerto the receiver than to the transmitter, then the angular separation t at thetransmitter is small. Similarly, if the reflector is local around the transmitterand is much closer to the transmitter than to the receiver, then the angularseparation r at the receiver is small. In either case H would not be verywell-conditioned (Figure 7.10). In a cellular system this suggests that if thebase-station is high on top of a tower with most of the scatterers and reflectorslocally around the mobile, then the size of the antenna array at the base-stationFigure 7.10 (a) The reflectorsand scatterers are in a ringlocally around the receiver;their angular separation at thetransmitter is small. (b) Thereflectors and scatterers are ina ring locally around thetransmitter; their angularseparation at the receiver issmall.~~~~~~~~Tx antenna arrayTx antennaarrayRx antennaarrayRx antennaarrayVery smallangular separationLarge angularseparation(a)(b)309 7.3 Modeling of MIMO fading channelswill have to be many wavelengths to be able to exploit this spatial multiplexingeffect.Summary 7.1 Multiplexing capability of MIMO channelsSIMO and MISO channels provide a power gain but no degree-of-freedomgain.Line-of-sight MIMO channels with co-located transmit antennas andco-located receive antennas also provide no degree-of-freedom gain.MIMO channels with far-apart transmit antennas having angular separationgreater than 1/Lr at the receive antenna array provide an effective degreeof-freedom gain. So do MIMO channels with far-apart receive antennashaving angular separation greater than 1/Lt at the transmit antenna array.Multipath MIMO channels with co-located transmit antennas andco-located receive antennas but with scatterers/reflectors far away alsoprovide a degree-of-freedom gain.7.3 Modeling of MIMO fading channelsThe examples in the previous section are deterministic channels. Building onthe insights obtained, we migrate towards statistical MIMO models whichcapture the key properties that enable spatial multiplexing.7.3.1 Basic approachIn the previous section, we assessed the capacity of physical MIMO channelsby first looking at the rank of the physical channel matrix H and then itscondition number. In the example in Section 7.2.4, for instance, the rankof H is 2 but the condition number depends on how the angle between thetwo spatial signatures compares to the spatial resolution of the antenna array.The two-step analysis process is conceptually somewhat awkward. It suggeststhat physical models of the MIMO channel in terms of individual multipathsmay not be at the right level of abstraction from the point of view of thedesign and analysis of communication systems. Rather, one may want toabstract the physical model into a higher-level model in terms of spatiallyresolvable paths.We have in fact followed a similar strategy in the statistical modelingof frequency-selective fading channels in Chapter 2. There, the modeling isdirectly on the gains of the taps of the discrete-time sampled channel ratherthan on the gains of the individual physical paths. Each tap can be thought310 MIMO I: spatial multiplexing and channel modelingof as a (time-)resolvable path, consisting of an aggregation of individualphysical paths. The bandwidth of the system dictates how finely or coarselythe physical paths are grouped into resolvable paths. From the point of viewof communication, it is the behavior of the resolvable paths that matters,not that of the individual paths. Modeling the taps directly rather than theindividual paths has the additional advantage that the aggregation makesstatistical modeling more reliable.Using the analogy between the finite time-resolution of a band-limitedsystem and the finite angular-resolution of an array-size-limited system, wecan follow the approach of Section 2.2.3 in modeling MIMO channels. Thetransmit and receive antenna array lengths Lt and Lr dictate the degree ofresolvability in the angular domain: paths whose transmit directional cosinesdiffer by less than 1/Lt and receive directional cosines by less than 1/Lrare not resolvable by the arrays. This suggests that we should “sample” theangular domain at fixed angular spacings of 1/Lt at the transmitter and atfixed angular spacings of 1/Lr at the receiver, and represent the channel interms of these new input and output coordinates. The klth channel gain inthese angular coordinates is then roughly the aggregation of all paths whosetransmit directional cosine is within an angular window of width 1/Lt aroundl/Lt and whose receive directional cosine is within an angular window ofwidth 1/Lr around k/Lr. See Figure 7.11 for an illustration of the lineartransmit and receive antenna array with the corresponding angular windows.In the following subsections, we will develop this approach explicitly foruniform linear arrays.Figure 7.11 A representationof the MIMO channel in theangular domain. Due to thelimited resolvability of theantenna arrays, the physicalpaths are partitioned intoresolvable bins of angularwidths 1/Lr by 1/Lt. Herethere are four receiveantennas (Lr = 2) and sixtransmit antennas (Lr = 3).4455 00002222311 11333+1 –r11 –1path B1/L1/Ltpath Apath Bpath AResolvable binsΩtΩr311 7.3 Modeling of MIMO fading channels7.3.2 MIMO multipath channelConsider the narrowband MIMO channel:y = Hx+w (7.55)The nt transmit and nr receive antennas are placed in uniform linear arraysof normalized lengths Lt and Lr, respectively. The normalized separationbetween the transmit antennas ist = Lt/nt and the normalized separationbetween the receive antennas isr = Lr/nr. The normalization is by thewavelength c of the passband transmitted signal. To simplify notation, we arenow thinking of the channel H as fixed and it is easy to add the time-variationlater on.Suppose there is an arbitrary number of physical paths between the transmitter and the receiver; the ith path has an attenuation of ai, makes an angleof ti (ti = cos ti) with the transmit antenna array and an angle of ri(ri = cos ri) with the receive antenna array. The channel matrix H isgiven by H = iabi errietti∗(7.56) where, as in Section 7.2,abi = ai√ntnr exp-j2d c i er = √1nr1exp-j2rexp-j2nr -1r (7.57)et = √1nt1exp-j2texp-j2nt -1t (7.58)Also, di is the distance between transmit antenna 1 and receive antenna 1along path i. The vectors et and er are, respectively, the transmittedand received unit spatial signatures along the direction .7.3.3 Angular domain representation of signalsThe first step is to define precisely the angular domain representation of thetransmitted and received signals. The signal arriving at a directional cosine 312 MIMO I: spatial multiplexing and channel modelingonto the receive antenna array is along the unit spatial signature er, givenby (7.57). Recall (cf. (7.35)) fr = er0∗erexpjrnr -1sinLr/nr(7.59) = 1nrsinLranalyzed in Section 7.2.4. In particular, we havefr Lkr = 0 andfr -Lkr = fr nrL-r k k = 1 nr -1 (7.60)(Figure 7.5). Hence, the nr fixed vectors:r = er0 er L1r er nrL-r 1 (7.61)form an orthonormal basis for the received signal space nr. This basisprovides the representation of the received signals in the angular domain.Why is this representation useful? Recall that associated with each vector er is its beamforming pattern (see Figures 7.6 and 7.7 for examples). It has one or more pairs of main lobes of width 2/Lr and smallside lobes. The different basis vectors erk/Lr have different main lobes.This implies that the received signal along any physical direction will havealmost all of its energy along one particular erk/Lr vector and very littlealong all the others. Thus, this orthonormal basis provides a very simple(but approximate) decomposition of the total received signal into the multipaths received along the different physical directions, up to a resolutionof 1/Lr.We can similarly define the angular domain representation of the transmitted signal. The signal transmitted at a direction is along the unit vectoret, defined in (7.58). The nt fixed vectors:t = et0 et L1t et ntL-t 1 (7.62)form an orthonormal basis for the transmitted signal space nt. This basisprovides the representation of the transmitted signals in the angular domain.The transmitted signal along any physical direction will have almost all itsenergy along one particular etk/Lt vector and very little along all the others. Thus, this orthonormal basis provides a very simple (again, approximate)313 7.3 Modeling of MIMO fading channelsFigure 7.12 Receivebeamforming patterns of theangular basis vectors.Independent of the antennaspacing, the beamformingpatterns all have the samebeam widths for the mainlobe, but the number of mainlobes depends on the spacing.(a) Critically spaced case; (b)Sparsely spaced case. (c)Densely spaced case. 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1302106024090270120300150330180 0 0.5 1302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 01302106024090270120300150330180 0(a) Lr = 2, nr = 4(b) Lr = 2, nr = 2(c) Lr = 2, nr = 8decomposition of the overall transmitted signal into the components transmitted along the different physical directions, up to a resolution of 1/Lt.Examples of angular basesExamples of angular bases, represented by their beamforming patterns, areshown in Figure 7.12. Three cases are distinguished:• Antennas are critically spaced at half the wavelength ( r = 1/2). In thiscase, each basis vector erk/Lr has a single pair of main lobes around theangles ± arccosk/Lr.• Antennas are sparsely spaced ( r > 1/2). In this case, some of the basisvectors have more than one pair of main lobes.• Antennas are densely spaced ( r < 1/2). In this case, some of the basisvectors have no main lobes.314 MIMO I: spatial multiplexing and channel modelingThese statements can be understood from the fact that the function frris periodic with period 1/r. The beamforming pattern of the vector erk/Lris the polar plot fr cos- Lkr (7.63) and the main lobes are at all angles for whichcos = kLrmod1r(7.64)In the critically spaced case, 1/r = 2 and k/Lr is between 0 and 2; there isa unique solution for cos in (7.64). In the sparsely spaced case, 1/r < 2and for some values of k there are multiple solutions: cos = k/Lr +m/rfor integers m. In the densely spaced case, 1/r > 2, and for k satisfyingLr < k < nr -Lr, there is no solution to (7.64). These angular basis vectorsdo not correspond to any physical directions.Only in the critically spaced antennas is there a one-to-one correspondencebetween the angular windows and the angular basis vectors. This case is thesimplest and we will assume critically spaced antennas in the subsequentdiscussions. The other cases are discussed further in Section 7.3.7.Angular domain transformation as DFTActually the transformation between the spatial and angular domains is afamiliar one! Let Ut be the nt ×nt unitary matrix the columns of which arethe basis vectors in t. If x and xa are the nt-dimensional vector of transmitted signals from the antenna array and its angular domain representationrespectively, then they are related by x = Utxaxa= Ut∗x(7.65) Now the klth entry of Ut is1 √ntexp-j2nkl t kl = 0 nr -1 (7.66)Hence, the angular domain representation xa is nothing but the inverse discrete Fourier transform of x (cf. (3.142)). One should however note thatthe specific transformation for the angular domain representation is in facta DFT because of the use of uniform linear arrays. On the other hand, therepresentation of signals in the angular domain is a more general concept andcan be applied to other antenna array structures. Exercise 7.8 gives anotherexample.315 7.3 Modeling of MIMO fading channels7.3.4 Angular domain representation of MIMO channelsWe now represent the MIMO fading channel (7.55) in the angular domain.Ut and Ur are respectively the nt ×nt and nr ×nr unitary matrices the columnsof which are the vectors in t and r respectively (IDFT matrices). Thetransformationsxa = U∗t x (7.67)ya = Ur∗y (7.68)are the changes of coordinates of the transmitted and received signals intothe angular domain. (Superscript “a” denotes angular domain quantities.)Substituting this into (7.55), we have an equivalent representation of thechannel in the angular domain:ya = Ur∗HUtxa +Ur∗w= Haxa +wa (7.69)whereHa = U∗r HUt (7.70)is the channel matrix expressed in angular coordinates andwa = U∗r w ∼ 0 N0Inr (7.71)Now, recalling the representation of the channel matrix H in (7.56),hakl = erk/Lr∗ Hetl/Lt= iabi erk/Lr∗erri· etti∗etl/Lt (7.72)Recall from Section 7.3.3 that the beamforming pattern of the basis vectorerk/Lr has a main lobe around k/Lr. The term erk/Lr∗erri is significantfor the ith path ifri –k Lr L1Cluster of scatterersReceivearrayReceivearray1/L1 1/L11/L2 1/L2TransmitarrayTransmitarrayFigure 7.17 Increasing the antenna array apertures increases path resolvability in the angulardomain and the degrees of freedom.322 MIMO I: spatial multiplexing and channel modelingtypically decreases with the carrier frequency. The reasons aretwo-fold:• signals at higher frequency attenuate more after passing through orbouncing off channel objects, thus reducing the number of effectiveclusters;• at higher frequency the wavelength is small relative to the feature sizeof typical channel objects, so scattering appears to be more specular innature and results in smaller angular spread.These factors combine to reduce ttotal and rtotal as the carrier frequencyincreases. Thus the impact of carrier frequency on the overall degrees offreedom is not necessarily monotonic. A set of indoor measurements isshown in Figure 7.18. The number of degrees of freedom increases andthen decreases with the carrier frequency, and there is in fact an optimalfrequency at which the number of degrees of freedom is maximized. Thisexample shows the importance of taking into account both the physicalenvironment as well as the antenna arrays in determining the availabledegrees of freedom in a MIMO channel.0 2 3 4 5 6 70.050.10.150.20.250.30.350.40.4502 3 4 5 6 77 6 5 4 3 2 1Frequency (GHz) Frequency (GHz)(a) (b)Ω total in townhouseΩ totalΩ total /λ c (m-1)1/λ (m-1)1/λ cΩ total in officeOfficeTownhouse5 0101520258 8Figure 7.18 (a) The total angular spread total of the scattering environment (assumed equal atthe transmitter side and at the receiver side) decreases with the carrier frequency; the normalizedarray length increases proportional to 1/c. (b) The number of degrees of freedom of the MIMOchannel, proportional to total/c, first increases and then decreases with the carrier frequency.The data are taken from [91].DiversityIn this chapter, we have focused on the phenomenon of spatial multiplexingand the key parameter is the number of degrees of freedom. In a slow fadingenvironment, another important parameter is the amount of diversity in thechannel. This is the number of independent channel gains that have to be ina deep fade for the entire channel to be in deep fade. In the angular domainMIMO model, the amount of diversity is simply the number of non-zero323 7.3 Modeling of MIMO fading channelsFigure 7.19 Angular domainrepresentation of three MIMOchannels. They all have fourdegrees of freedom but theyhave diversity 4, 8 and 16respectively. They modelchannels with increasingamounts of bounces in thepaths (cf. Figure 7.15).(a)ntnrnrnrnt nt(b) (c)entries in Ha. Some examples are shown in Figure 7.19. Note that channelsthat have the same degrees of freedom can have very different amounts ofdiversity. The number of degrees of freedom depends primarily on the angularspreads of the scatters/reflectors at the transmitter and at the receiver, whilethe amount of diversity depends also on the degree of connectivity betweenthe transmit and receive angles. In a channel with multiple-bounced paths,signals sent along one transmit angle can arrive at several receive angles(see Figure 7.15). Such a channel would have more diversity than one withsingle-bounced paths with signal sent along one transmit angle received at aunique angle, even though the angular spreads may be the same.7.3.7 Dependency on antenna spacingSo far we have been primarily focusing on the case of critically spacedantennas (i.e., antenna separations t and r are half the carrier wavelength).What is the impact of changing the antenna separation on the channel statisticsand the key channel parameters such as the number of degrees of freedom?To answer this question, we fix the antenna array lengths Lt and Lr and varythe antenna separation, or equivalently the number of antenna elements. Letus just focus on the receiver side; the transmitter side is analogous. Given theantenna array length Lr, the beamforming patterns associated with the basisvectors erk/Lrk all have beam widths of 2/Lr (Figure 7.12). This dictatesthe maximum possible resolution of the antenna array: paths that arrive withinan angular window of width 1/Lr cannot be resolved no matter how manyantenna elements there are. There are 2Lr such angular windows, partitioningall the receive directions (Figure 7.20). Whether or not this maximum resolution can actually be achieved depends on the number of antenna elements.Recall that the bins k can be interpreted as the set of all physicalpaths which have most of their energy along the basis vector etk/Lr. Thebins dictate the resolvability of the antenna array. In the critically spaced caser = 1/2), the beamforming patterns of all the basis vectors have a singlemain lobe (together with its mirror image). There is a one-to-one correspondence between the angular windows and the resolvable bins k, and pathsarriving in different windows can be resolved by the array (Figure 7.21). In324 MIMO I: spatial multiplexing and channel modelingFigure 7.20 An antenna arrayof length Lr partitions thereceive directions into 2Lrangular windows. Here, Lr = 3and there are six angularwindows. Note that because ofsymmetry across the 0 -180axis, each angular windowcomes as a mirror image pair,and each pair is only countedas one angular window.34 25 15 14 230 0Figure 7.21 Antennas arecritically spaced at half thewavelength. Each resolvablebin corresponds to exactly oneangular window. Here, thereare six angular windows andsix bins.Lr = 3, nr = 64 2 012345 k5 15 14 230 03Binsthe sparsely spaced case ( r > 1/2), the beamforming patterns of some of thebasis vectors have multiple main lobes. Thus, paths arriving in the differentangular windows corresponding to these lobes are all lumped into one binand cannot be resolved by the array (Figure 7.22). In the densely spaced case( r < 1/2), the beamforming patterns of 2Lr of the basis vectors have a singlemain lobe; they can be used to resolve among the 2Lr angular windows. Thebeamforming patterns of the remaining nr -2Lr basis vectors have no mainlobe and do not correspond to any angular window. There is little receivedenergy along these basis vectors and they do not participate significantly inthe communication process. See Figure 7.23.The key conclusion from the above analysis is that, given the antennaarray lengths Lr and Lt, the maximum achievable angular resolution canbe achieved by placing antenna elements half a wavelength apart. Placingantennas more sparsely reduces the resolution of the antenna array and can325 7.3 Modeling of MIMO fading channels(b)Bins0 0101 10101 10k 01 Lr = 3, nr = 2(a)Bins0 0234 12324 13k0 1 2 3 4Lr = 3, nr = 5Figure 7.22 (a) Antennas are reduce the number of degrees of freedom and the diversity of the channel.sparsely spaced. Some of thebins contain paths frommultiple angular windows.(b) The antennas are verysparsely spaced. All binscontain several angularwindows of paths.Placing the antennas more densely adds spurious basis vectors which do notcorrespond to any physical directions, and does not add resolvability. In termsof the angular channel matrix Ha, this has the effect of adding zero rows andcolumns; in terms of the spatial channel matrix H, this has the effect of makingthe entries more correlated. In fact, the angular domain representation makesit apparent that one can reduce the densely spaced system to an equivalent2Lt ×2Lr critically spaced system by just focusing on the basis vectors thatdo correspond to physical directions (Figure 7.24).Increasing the antenna separation within a given array length Lr does notincrease the number of degrees of freedom in the channel. What about increasing the antenna separation while keeping the number of antenna elements nrthe same? This question makes sense if the system is hardware-limited ratherthan limited by the amount of space to put the antenna array in. Increasingthe antenna separation this way reduces the beam width of the nr angularbasis beamforming patterns but also increases the number of main lobes ineach (Figure 7.25). If the scattering environment is rich enough such that thereceived signal arrives from all directions, the number of non-zero rows ofthe channel matrix Ha is already nr, the largest possible, and increasing thespacing does not increase the number of degrees of freedom in the channel.On the other hand, if the scattering is clustered to within certain directions,increasing the separation makes it possible for the scattered signal to be326 MIMO I: spatial multiplexing and channel modelingFigure 7.23 Antennas aredensely spaced. Some binscontain no physical paths.0 0789 12329 18k 01234 5 67 8 9 Empty binsLr = 3, nr = 10Figure 7.24 A typical Hawhen the antennas aredensely spaced.1020304050 51015202530354045505 4 3 2 1L = 16, n = 50|hakl|l–Transmitter bins K–Receiver binsreceived in more bins, thus increasing the number of degrees of freedom(Figure 7.25). In terms of the spatial channel matrix H, this has the effect ofmaking the entries look more random and independent. At a base-station ona high tower with few local scatterers, the angular spread of the multipaths issmall and therefore one has to put the antennas many wavelengths apart todecorrelate the channel gains.Sampling interpretationOne can give a sampling interpretation to the above results. First, think ofthe discrete antenna array as a sampling of an underlying continuous array-Lr/2Lr/2. On this array, the received signal xs is a function of the327 7.3 Modeling of MIMO fading channelsFigure 7.25 An example of aclustered response channel inwhich increasing theseparation between a fixednumber of antennas increasesthe number of degrees offreedom from 2 to 3.Cluster of scatterers(a) Antenna separation of ∆1 = 1/2(b) Antenna separation of ∆2 > ∆1Cluster of scatterersReceivearrayReceivearrayTransmitarrayTransmitarray1 / (nt∆1) 1 / (nr∆1)1 / (nt∆2)1 / (nr∆2)continuous spatial location s ∈ -Lr/2 Lr/2. Just like in the discrete case(cf. Section 7.3.3), the spatial-domain signal xs and its angular representation xa form a Fourier transform pair. However, since only ∈ -1 1corresponds to directional cosines of actual physical directions, the angularrepresentation xa of the received signal is zero outside -1 1. Hence, thespatial-domain signal xs is “bandlimited” to -W W, with “bandwidth”W = 1. By the sampling theorem, the signal xs can be uniquely specifiedby samples spaced at distance 1/2W = 1/2 apart, the Nyquist samplingrate. This is precise when Lr → and approximate when Lr is finite. Hence,placing the antenna elements at the critical separation is sufficient to describethe received signal; a continuum of antenna elements is not needed. Antennaspacing greater than 1/2 is not adequate: this is under-sampling and the lossof resolution mentioned above is analogous to the aliasing effect when onesamples a bandlimited signal at below the Nyquist rate.7.3.8 I.i.d. Rayleigh fading modelA very common MIMO fading model is the i.i.d. Rayleigh fading model:the entries of the channel gain matrix H m are independent, identically328 MIMO I: spatial multiplexing and channel modelingdistributed and circular symmetric complex Gaussian. Since the matrix H mand its angular domain representation Ha m are related byHa m = Ur∗H mUt (7.80)andUr andUt arefixed unitarymatrices,thismeansthatHa should havethe samei.i.d. Gaussian distribution as H. Thus, using the modeling approach describedhere, we can see clearly the physical basis of the i.i.d Rayleigh fading model, interms of both the multipath environment and the antenna arrays. There shouldbe a significant number of multipaths in each of the resolvable angular bins,and the energy should be equally spread out across these bins. This is the socalled richly scattered environment. If there are very few or no paths in someof the angular directions, then the entries in H will be correlated. Moreover, theantennasshould be either critically orsparselyspaced.Ifthe antennas are denselyspaced, then some entries ofHa are approximately zero and the entries inH itselfare highly correlated. However, by a simple transformation, the channel can bereducedto an equivalent channelwithfewer antennaswhich are criticallyspaced.Compared to the critically spaced case, having sparser spacing makes iteasier for the channel matrix to satisfy the i.i.d. Rayleigh assumption. This isbecause each bin now spans more distinct angular windows and thus containsmore paths, from multiple transmit and receive directions. This substantiatesthe intuition that putting the antennas further apart makes the entries of Hless dependent. On the other, if the physical environment already providesscattering in all directions, then having critical spacing of the antennas isenough to satisfy the i.i.d. Rayleigh assumption.Due to the analytical tractability, we will use the i.i.d. Rayleigh fadingmodel quite often to evaluate performance of MIMO communication schemes,but it is important to keep in mind the assumptions on both the physicalenvironment and the antenna arrays for the model to be valid.Chapter 7 The main plotThe angular domain provides a natural representation of the MIMO channel, highlighting the interaction between the antenna arrays and the physicalenvironment.The angular resolution of a linear antenna array is dictated by its length: anarray of length L provides a resolution of 1/L. Critical spacing of antennaelements at half the carrier wavelength captures the full angular resolutionof 1/L. Sparser spacing reduces the angular resolution due to aliasing.Denser spacing does not increase the resolution beyond 1/L.Transmit and receive antenna arrays of length Lt and Lr partition theangular domain into 2Lt ×2Lr bins of unresolvable multipaths. Paths thatfall within the same bin are aggregated to form one entry of the angularchannel matrix Ha.329 7.4 Bibliographical notesA statistical model of Ha is obtained by assuming independent Gaussiandistributed entries, of possibly different variances. Angular bins that contain no paths correspond to zero entries.The number of degrees of freedom in the MIMO channel is the minimumof the number of non-zero rows and the number of non-zero columns ofHa. The amount of diversity is the number of non-zero entries.In a clustered-response model, the number of degrees of freedom is approximately:minLtttotal Lrrtotal (7.81)The multiplexing capability of a MIMO channel increases with the angular spreads ttotal rtotal of the scatterers/reflectors as well as withthe antenna array lengths. This number of degrees of freedom can beachieved when the antennas are critically spaced at half the wavelength orcloser. With a maximum angular spread of 2, the number of degrees offreedom ismin2Lt2Lrand this equalsminnt nrwhen the antennas are critically spaced.The i.i.d. Rayleigh fading model is reasonable in a richly scattering environment where the angular bins are fully populated with paths and there isroughly equal amount of energy in each bin. The antenna elements shouldbe critically or sparsely spaced.7.4 Bibliographical notesThe angular domain approach to MIMO channel modeling is based on works bySayeed [105] and Poon et al. [90, 92]. [105] considered an array of discrete antenna elements, while [90, 92] considered a continuum of antenna elements to emphasize thatspatial multiplexability is limited not by the number of antenna elements but by thesize of the antenna array. We considered only linear arrays in this chapter, but [90] alsotreated other antenna array configurations such as circular rings and spherical surfaces.The degree-of-freedomformula(7.78) is derived in[90]for the clusteredresponsemodel.Other related approaches to MIMO channel modeling are by Raleigh and Cioffi[97], by Gesbert et al. [47] and by Shiu et al. [111]. The latter work used a Clarke-likemodel but with two rings of scatterers, one around the transmitter and one around thereceiver, to derive the MIMO channel statistics.330 MIMO I: spatial multiplexing and channel modeling7.5 ExercisesExercise 7.11. For the SIMO channel with uniform linear array in Section 7.2.1, give an exactexpression for the distance between the transmit antenna and the ith receive antenna.Make precise in what sense is (7.19) an approximation.2. Repeat the analysis for the approximation (7.27) in the MIMO case.Exercise 7.2 Verify that the unit vector err, defined in (7.21), is periodic withperiod r and within one period never repeats itself.Exercise 7.3 Verify (7.35).Exercise 7.4 In an earlier work on MIMO communication [97], it is stated that thenumber of degrees of freedom in a MIMO channel with nt transmit, nr receive antennasand K multipaths is given byminntnrK (7.82)and this is the key parameter that determines the multiplexing capability of the channel.What are the problems with this statement?Exercise 7.5 In this question we study the role of antenna spacing in the angularrepresentation of the MIMO channel.1. Consider the critically spaced antenna array in Figure 7.21; there are six bins, eachone corresponding to a specific physical angular window. All of these angularwindows have the same width as measured in solid angle. Compute the angularwindow width in radians for each of the bins l, with l = 0 5. Argue that thewidth in radians increases as we move from the line perpendicular to the antennaarray to one that is parallel to it.2. Now consider the sparsely spaced antenna arrays in Figure 7.22. Justify the depictedmapping from the angular windows to the bins l and evaluate the angular windowwidth in radians for each of the bins l (for l = 01 nt – 1). (The angularwindow width of a bin l is the sum of the widths of all the angular windows thatcorrespond to the bin l.)3. Justify the depiction of the mapping from angular windows to the bins l in thedensely spaced antenna array of Figure 7.23. Also evaluate the angular width ofeach bin in radians.Exercise 7.6 The non-zero entries of the angular matrix Ha are distributed as independent complex Gaussian random variables. Show that with probability 1, the rankof the matrix is given by the formula (7.74).Exercise 7.7 In Chapter 2, we introduced Clarke’s flat fading model, where both thetransmitter and the receiver have a single antenna. Suppose now that the receiver hasnr antennas, each spaced by half a wavelength. The transmitter still has one antenna(a SIMO channel). At time my m = h mx m+w m (7.83)where y mh m are the nr-dimensional received vector and receive spatial signature(induced by the channel), respectively.331 7.5 Exercises1. Consider first the case when the receiver is stationary. Compute approximately thejoint statistics of the coefficients of h in the angular domain.2. Now suppose the receiver is moving at a speed v. Compute the Doppler spread andthe Doppler spectrum of each of the angular domain coefficients of the channel.3. What happens to the Doppler spread as nr → ? What can you say about thedifficulty of estimating and tracking the process h m as n grows? Easier, harder,or the same? Explain.Exercise 7.8 [90] Consider a circular array of radius R normalized by the carrierwavelength with n elements uniformly spaced.1. Compute the spatial signature in the direction .2. Find the angle, f1 2, between the two spatial signatures in the direction 1and 2.3. Does f1 2 only depend on the difference 1 -2? If not, explain why.4. Plot f1 0 for R = 2 and different values of n, from n equal to R/2, R,2R, to 4R. Observe the plot and describe your deductions.5. Deduce the angular resolution.6. Linear arrays of length L have a resolution of 1/L along the cos -domain, thatis, they have non-uniform resolution along the -domain. Can you design a lineararray with uniform resolution along the -domain?Exercise 7.9 (Spatial sampling) Consider a MIMO system with Lt = Lr = 2 in achannel with M = 10 multipaths. The ith multipath makes an angle of i with thetransmit array and an angle of i with the receive array where = /M.1. Assuming there are nt transmit and nr receive antennas, compute the channelmatrix.2. Compute the channel eigenvalues for nt = nr varying from 4 to 8.3. Describe the distribution of the eigenvalues and contrast it with the binning interpretation in Section 7.3.4.Exercise 7.10 In this exercise, we study the angular domain representation offrequency-selective MIMO channels.1. Starting with the representation of the frequency-selective MIMO channel in time(cf. (8.112)) describe how you would arrive at the angular domain equivalent(cf. (7.69)): ya m = =0Ha mxa m-+wa m(7.84) L-12. Consider the equivalent (except for the overhead in using the cyclic prefix) parallelMIMO channel as in (8.113).(a) Discuss the role played by the density of the scatterers and the delay spread inthe physical environment in arriving at an appropriate statistical model for H˜ n atthe different OFDM tones n.(b) Argue that the (marginal) distribution of the MIMO channel H˜ n is the same foreach of the tones n = 0 N -1.Exercise 7.11 A MIMO channel has a single cluster with the directional cosine rangesas t = r = 0 1. Compute the number of degrees of freedom of an n×n channelas a function of the antenna separation t = r = .

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