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C H A P T E R
MIMO I: spatial multiplexing
and channel modeling
In this book, we have seen several different uses of multiple antennas in wireless communication. In Chapter 3, multiple antennas were used to provide diversity gain and increase the reliability of wireless links. Both receive and transmit diversity were considered. Moreover, receive antennas can also provide a power gain. In Chapter 5, we saw that with channel knowledge at the transmitter, multiple transmit antennas can also provide a power gain via transmit beamforming. In Chapter 6, multiple transmit antennas were used to induce channel variations, which can then be exploited by opportunistic communication techniques. The scheme can be interpreted as opportunistic beamforming and provides a power gain as well. In this and the next few chapters, we will study a new way to use multiple antennas. We will see that under suitable channel fading conditions, having both multiple transmit and multiple receive antennas (i.e., a MIMO channel) provides an additional spatial dimension for communication and yields a degree-of- freedom gain. These additional degrees of freedom can be exploited by spatially multiplexing several data streams onto the MIMO channel, and lead to an increase in the capacity: the capacity of such a MIMO channel with n transmit and receive antennas is proportional to n. Historically, it has been known for a while that a multiple access system with multiple antennas at the base-station allows several users to simultane- ously communicate with the base-station. The multiple antennas allow spatial separation of the signals from the different users. It was observed in the mid 1990s that a similar effect can occur for a point-to-point channel with multiple transmit and receive antennas, i.e., even when the transmit antennas are not geographically far apart. This holds provided that the scattering environment is rich enough to allow the receive antennas to separate out the signals from the different transmit antennas. We have already seen how channel fading can be exploited by opportunistic communication techniques. Here, we see yet another example where channel fading is beneficial to communication. It is insightful to compare and contrast the nature of the performance gains offered by opportunistic communication and by MIMO techniques.
290
291 7.1 Multiplexing capability of deterministic MIMO channels
Opportunistic communication techniques primarily provide a power gain. This power gain is very significant in the low SNR regime where systems are power-limited but less so in the high SNR regime where they are bandwidth- limited. As we will see, MIMO techniques can provide both a power gain and a degree-of-freedom gain. Thus, MIMO techniques become the primary tool to increase capacity significantly in the high SNR regime. MIMO communication is a rich subject, and its study will span the remain- ing chapters of the book. The focus of the present chapter is to investigate the properties of the physical environment which enable spatial multiplexing and show how these properties can be succinctly captured in a statistical MIMO channel model. We proceed as follows. Through a capacity analysis, we first identify key parameters that determine the multiplexing capability of a deterministic MIMO channel. We then go through a sequence of physical MIMO channels to assess their spatial multiplexing capabilities. Building on the insights from these examples, we argue that it is most natural to model the MIMO channel in the angular domain and discuss a statistical model based on that approach. Our approach here parallels that in Chapter 2, where we started with a few idealized examples of multipath wireless channels to gain insights into the underlying physical phenomena, and proceeded to statistical fading models, which are more appropriate for the design and performance analysis of communication schemes. We will in fact see a lot of parallelism in the specific channel modeling technique as well. Our focus throughout is on flat fading MIMO channels. The extensions to frequency-selective MIMO channels are straightforward and are developed in the exercises.
7.1 Multiplexing capability of deterministic MIMO channels
A narrowband time-invariant wireless channel with nt transmit and nr receive antennas is described by an nr by nt deterministic matrix H. What are the key properties 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 decomposition
The time-invariant channel is described by
y = Hx + w! (7.1)
where x ∈ nt^ , y ∈ nr^ and w ∼ " 0! N 0 Inr $ denote the transmitted sig- nal, received signal and white Gaussian noise respectively at a symbol time (the time index is dropped for simplicity). The channel matrix H ∈ nr^ ×nt
293 7.1 Multiplexing capability of deterministic MIMO channels
Figure 7.1 Converting the MIMO channel into a parallel channel through the SVD.
V x V^ U^ y U^ y
Pre-processing Post-processing Channel
λ 1
λ n min wn min
w 1
x ∼^ ∼
∼
~
×
×
where w˜ ∼ 0! N 0 Inr $ has the same distribution as w (cf. (A.22) in Appendix A), and "˜x"^2 = "x"^2. Thus, the energy is preserved and we have an equivalent representation as a parallel Gaussian channel:
˜yi = 'i x˜i + ˜wi! i = 1! 2! * * *! nmin+ (7.9)
The equivalence is summarized in Figure 7.1. The SVD decomposition can be interpreted as two coordinate transforma- tions: it says that if the input is expressed in terms of a coordinate system defined by the columns of V and the output is expressed in terms of a coordi- nate system defined by the columns of U, then the input/output relationship is 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 chan- nels and about time-varying fading channels with full CSI. The time-invariant MIMO channel is yet another example. Here, the spatial dimension plays the same role as the time and frequency dimensions in those other problems. The capacity is by now familiar:
C =
n∑min
i= 1
log
P i∗ '^2 i N 0
bits/s/Hz! (7.10)
where P∗ 1! * * *! P n∗min are the waterfilling power allocations:
P i∗ =
N 0
'^2 i
with / chosen to satisfy the total power constraint
i P i∗ =^ P. Each^ 'i corresponds to an eigenmode of the channel (also called an eigenchannel). Each non-zero eigenchannel can support a data stream; thus, the MIMO channel can support the spatial multiplexing of multiple streams. Figure 7. pictorially depicts the SVD-based architecture for reliable communication.
294 MIMO I: spatial multiplexing and channel modeling
AWGN coder
AWGN coder
{ x ~ 1 [ m ]} {~ y 1 [ m ]}
{ x ~ n^ min[ m ]} { y ~ n min[ m ]}
. . .
...... n (^) min information streams
{0}
{0}
{ w [ m ]}
V H U*
Decoder
Decoder
Figure 7.2 The SVD architecture There is a clear analogy between this architecture and the OFDM system for MIMO communication. (^) introduced 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 OFDM setting, 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 the input symbols. In fact, the decomposition C = Q−^1 Q in (3.143) is the SVD decomposition of a circulant matrix C, with U = Q−^1 and V∗^ = Q. The important 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 specific realization of the ISI channel, while for the latter, they do depend on the specific realization of the MIMO channel.
7.1.2 Rank and condition number
What are the key parameters that determine performance? It is simpler to focus separately on the high and the low SNR regimes. At high SNR, the water level is deep and the policy of allocating equal amounts of power on the non-zero eigenmodes is asymptotically optimal (cf. Figure 5.24(a)):
C ≈
∑^ k i= 1
log
P%^2 i kN 0
≈ k log SNR +
∑^ k i= 1
log
%^2 i k
bits/s/Hz( (7.12)
where k is the number of non-zero %^2 i , i.e., the rank of H, and SNR )= P/N 0. The parameter k is the number of spatial degrees of freedom per second per hertz. It represents the dimension of the transmitted signal as modified by the MIMO channel, i.e., the dimension of the image of H. This is equal to the rank of the matrix H and with full rank, we see that a MIMO channel provides nmin spatial degrees of freedom.
296 MIMO I: spatial multiplexing and channel modeling
rank and conditioning of their channel matrices. These deterministic examples will also suggest a natural approach to statistical modeling of MIMO chan- nels, which we discuss in Section 7.3. To be concrete, we restrict ourselves to uniform linear antenna arrays, where the antennas are evenly spaced on a straight line. The details of the analysis depend on the specific array structure but the concepts we want to convey do not.
7.2.1 Line-of-sight SIMO channel
The 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 a direct signal path between each antenna pair. The antenna separation is (^) r !c, where !c is the carrier wavelength and (^) r is the normalized receive antenna separation, normalized to the unit of the carrier wavelength. The dimension of the antenna array is much smaller than the distance between the transmitter and the receiver. The continuous-time impulse response hi$%& between the transmit antenna and the ith receive antenna is given by
hi$%& = a($% − di/c&, i = 1 , - - - , nr , (7.16)
Figure 7.3 (a) Line-of-sight channel with single transmit antenna and multiple receive antennas. The signals from the transmit antenna arrive almost in parallel at the receiving antennas. (b) Line-of-sight channel with multiple transmit antennas and single receive antenna.
Rx antenna i
∆rλc
d^ φ
( i −1)∆rλccosφ
(a)
∆tλc
φ
( i −1)∆tλccosφ
Tx antenna i
d
(b)
297 7.2 Physical modeling of MIMO channels
where 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 assume to be the same for all antenna pairs. Assuming di/c ≪ 1 /W , where W is the transmission bandwidth, the baseband channel gain is given by (2.34) and (2.27):
hi = a exp
j2'fcdi c
= a exp
j2'di )c
where fc is the carrier frequency. The SIMO channel can be written as
y = hx + w (7.18)
where x is the transmitted symbol, w ∼ , 0 * N 0 I. is the noise and y is the received vector. The vector of channel gains h = /h 1 * 0 0 0 * hnr 2 t^ is sometimes called the signal direction or the spatial signature induced on the receive antenna array by the transmitted signal. Since the distance between the transmitter and the receiver is much larger than the size of the receive antenna array, the paths from the transmit antenna to each of the receive antennas are, to a first-order, parallel and
di ≈ d + ,i − 1 .3r )c cos 4* i = 1 * 0 0 0 * nr * (7.19)
where d is the distance from the transmit antenna to the first receive antenna and 4 is the angle of incidence of the line-of-sight onto the receive antenna array. (You are asked to verify this in Exercise 7.1.) The quantity ,i− 1 .3r )c cos 4 is the displacement of receive antenna i from receive antenna 1 in the direction of the line-of-sight. The quantity
5 6= cos 4
is often called the directional cosine with respect to the receive antenna array. The spatial signature h = /h 1 * 0 0 0 * hnr 2 t^ is therefore given by
h = a exp
j2'd )c
exp,−j2'3r 5. exp,−j2' (^23) r 5. (^77) 7 exp,−j2',nr − 1 .3r 5.
299 7.2 Physical modeling of MIMO channels
The optimal transmission (transmit beamforming) is performed along the direction et !" of h, where
et !" #=
√n t
exp −j2%&t !" exp −j2% 2 &t !" '' ' exp −j2% nt − 1 "&t !"
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 that they add constructively at the receiver, yielding an nt -fold power gain. The capacity 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 path
Let us now consider a MIMO channel with only direct line-of-sight paths between the antennas. Both the transmit and the receive antennas are in linear arrays. Suppose the normalized transmit antenna separation is &t and the normalized receive antenna separation is &r. The channel gain between the kth transmit antenna and the ith receive antenna is
hik = a exp −j2%dik//c"( (7.26)
where dik is the distance between the antennas, and a is the attenuation along the line-of-sight path (assumed to be the same for all antenna pairs). Assuming again that the antenna array sizes are much smaller than the distance between the transmitter and the receiver, to a first-order:
dik = d + i − 1 "&r /c cos (^0) r − k − 1 "&t /c cos (^0) t ( (7.27)
where d is the distance between transmit antenna 1 and receive antenna 1, and (^0) t ( 0r are the angles of incidence of the line-of-sight path on the transmit and receive antenna arrays, respectively. Define !t #= cos (^0) t and !r #= cos (^0) r. Substituting (7.27) into (7.26), we get
hik = a exp
j2%d /c
· exp j2% k − 1 "&t !t " · exp −j2% i − 1 "&r !r " (7.28)
and we can write the channel matrix as
H = a√nt nr exp
− j2%d /c
er !r "et !t "∗( (7.29)
300 MIMO I: spatial multiplexing and channel modeling
where er ·! and et ·! are defined in (7.21) and (7.25), respectively. Thus, H is a rank-one matrix with a unique non-zero singular value " 1 = a√nt nr. The capacity of this channel follows from (7.10):
C = log
Pa^2 nt nr N 0
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 (the only non-zero eigenmode) and thus only one spatial degree of freedom is available. The receive spatial signatures at the receive antenna array from all the transmit antennas (i.e., the columns of H) are along the same direction, er *r !. Thus, the number of available spatial degrees of freedom does not increase even though there are multiple transmit and multiple receive antennas. The factor nt nr is the power gain of the MIMO channel. If nt = 1, the power gain is equal to the number of receive antennas and is obtained by maximal ratio combining at the receiver (receive beamforming). If nr = 1, the power gain is equal to the number of transmit antennas and is obtained by transmit beamforming. For general numbers of transmit and receive antennas, one gets benefits from both transmit and receive beamforming: the transmitted signals are constructively added in-phase at each receive antenna, and the signal at each receive antenna is further constructively combined with each other. In summary: in a line-of-sight only environment, a MIMO channel provides a power gain but no degree-of-freedom gain.
7.2.4 Geographically separated antennas
Geographically separated transmit antennas How do we get a degree-of-freedom gain? Consider the thought experiment where the transmit antennas can now be placed very far apart, with a separation of the order of the distance between the transmitter and the receiver. For concreteness, suppose there are two transmit antennas (Figure 7.4). Each
Figure 7.4 Two geographically separated transmit antennas each with line-of-sight to a receive antenna array.
Rx antenna array
φ r 2 φ r 1
Tx antenna 1
Tx antenna 2
302 MIMO I: spatial multiplexing and channel modeling
By direct computation (Exercise 7.3),
fr !"r # =
nr
exp!j%&r "r !nr − 1 ##
sin!%Lr "r # sin!%Lr "r /nr #
where Lr *= nr &r is the normalized length of the receive antenna array. Hence,
" cos +" =
sin!%Lr "r # nr sin!%Lr "r /nr #
∣ ,^ (7.36)
The conditioning of the matrix H depends directly on this parameter. For simplicity, consider the case when the gains a 1 = a 2 = a. The squared singular values of H are
.^21 = a^2 nr! 1 + " cos +"#).^22 = a^2 nr! 1 − " cos +"# (7.37)
and the condition number of the matrix is
. 1 . 2
1 + " cos +" 1 − " cos +"
The matrix is ill-conditioned whenever " cos +" ≈ 1, and is well-conditioned otherwise. In Figure 7.5, this quantity " cos +" = "fr !"r #" is plotted as a function of "r for a fixed array size and different values of nr. The function fr !·# has the following properties:
- (^) fr !"r # is periodic with period nr /Lr = 1 /&r ;
- (^) fr !"r # peaks at "r = 0; f! 0 # = 1;
- (^) fr !"r # = 0 at "r = k/Lr ) k = 1 ) 0 0 0 ) nr − 1.
The periodicity of fr !·# follows from the periodicity of the spatial signature er!·#. It has a main lobe of width 2/Lr centered around integer multiples of 1 /&r. All the other lobes have significantly lower peaks. This means that the signatures are close to being aligned and the channel matrix is ill conditioned whenever
""r −
m &r
Lr
for some integer m. Now, since "r ranges from −2 to 2, this condition reduces to
""r " ≪
Lr
whenever the antenna separation &r ≤ 1 /2.
303 7.2 Physical modeling of MIMO channels
Figure 7.5 The function |f( (^) r )| plotted as a function of (^) r for fixed Lr = 8 and different values of the number of receive antennas nr.
0
0.9^1
n r = 16
Ωr n r = 8 sinc function
Ωr
n r = 4
1
- 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2
0
1
- 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2 0
1
- 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2 Ωr
Ωr
| f (Ωr)| (^) | f (Ωr)|
| f (Ωr)| | f (Ωr)|
Increasing the number of antennas for a fixed antenna length Lr does not substantially change the qualitative picture above. In fact, as nr →! and "r → 0,
fr $%r & → ej'Lr^ %r^ sinc$Lr %r & (7.41)
and the dependency of fr $·& on nr vanishes. Equation (7.41) can be directly derived from (7.35), using the definition sinc$x& = sin$'x&/'x (cf. (2.30)). The parameter 1/Lr can be thought of as a measure of resolvability in the angular domain: if %r ≪ 1 /Lr , then the signals from the two transmit antennas cannot be resolved by the receive antenna array and there is effectively only one degree of freedom. Packing more and more antenna elements in a given amount of space does not increase the angular resolvability of the receive antenna array; it is intrinsically limited by the length of the array. A common pictorial representation of the angular resolvability of an antenna array is the (receive) beamforming pattern. If the signal arrives from a single direction * 0 , then the optimal receiver projects the received signal onto the vector er$cos * 0 &; recall that this is called the (receive) beamforming vector. A signal from any other direction * is attenuated by a factor of
%er$cos * 0 &∗er$cos *&% = %fr $cos * − cos * 0 &%+ (7.42)
The beamforming pattern associated with the vector er$cos *& is the polar plot
$*, %fr $cos * − cos * 0 &%& (7.43)
305 7.2 Physical modeling of MIMO channels
Figure 7.7 Beamforming patterns for different antenna array lengths. (Left) Lr = 4 and (right) Lr = 8. Antenna separation is fixed at half the carrier wavelength. The larger the length of the array, the narrower the beam.
1
30
210
60
240
90
270
120
300
150
330
180 0
1
30
210
60
240
90
270
120
300
150
330
180 0
L r = 4, n r = 8 L r = 8, n r = 16
resolvability of signals in the time domain: multipaths arriving at time sepa- ration much less than 1/W cannot be resolved by the receiver. The parameter 1 /Lr measures the resolvability of signals in the angular domain: signals that arrive within an angle much less than 1/Lr cannot be resolved by the receiver. Just as over-sampling cannot increase the time-domain resolvability beyond 1/W , adding more antenna elements cannot increase the angular- domain resolvability beyond 1/Lr. This analogy will be exploited in the statistical modeling of MIMO fading channels and explained more precisely in Section 7.3.
Geographically separated receive antennas We have increased the number of degrees of freedom by placing the transmit antennas far apart and keeping the receive antennas close together, but we can achieve the same goal by placing the receive antennas far apart and keeping the transmit antennas close together (see Figure 7.8). The channel matrix is given by
H =
[
h∗ 1 h∗ 2
]
Figure 7.8 Two geographically separated receive antennas each with line-of-sight from a transmit antenna array.
Tx antenna. array (^) φ t 1
φ t 2
Rx antenna 2
Rx antenna 1
306 MIMO I: spatial multiplexing and channel modeling
where
hi = ai exp
j2"di 1 $c
et %&ti'( (7.46)
and &ti is the directional cosine of departure of the path from the transmit antenna array to receive antenna i and di 1 is the distance between transmit antenna 1 and receive antenna i. As long as
&t )= &t2 − &t1 "= 0 mod
*t
the two rows of H are linearly independent and the channel has rank 2, yielding 2 degrees of freedom. The output of the channel spans a two-dimensional space as we vary the transmitted signal at the transmit antenna array. In order to make H well-conditioned, the angular separation &t of the two receive antennas should be of the order of or larger than 1/Lt , where Lt )= nt *t is the length of the transmit antenna array, normalized to the carrier wavelength. Analogous to the receive beamforming pattern, one can also define a trans- mit beamforming pattern. This measures the amount of energy dissipated in other directions when the transmitter attempts to focus its signal along a direc- tion. 0. The beam width is 2/Lt ; the longer the antenna array, the sharper the transmitter can focus the energy along a desired direction and the better it can spatially multiplex information to the multiple receive antennas.
7.2.5 Line-of-sight plus one reflected path
Can we get a similar effect to that of the example in Section 7.2.4, without putting either the transmit antennas or the receive antennas far apart? Consider again the transmit and receive antenna arrays in that example, but now suppose in 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 )= cos .ti) with the transmit antenna array and an angle of .ri%&ri )= cos .ri) with the receive antenna array. The channel H is given by the principle of superposition:
H = ab 1 er%&r1'et %&t1'∗^ + ab 2 er%&r2'er%&t2'∗^ (7.48)
where for i = 1 ( 2,
abi )= ai^ √nt nr exp
j2"d%i' $c
and d%i'^ is the distance between transmit antenna 1 and receive antenna 1 along path i. We see that as long as
&t1 "= &t2 mod
*t
308 MIMO I: spatial multiplexing and channel modeling
one can interpret the second matrix H′′^ as the matrix channel from two imaginary transmitters at A and B to the receive antenna array. This matrix has rank 2 as well; its conditioning depends on the parameter Lr !r. If both matrices are well-conditioned, then the overall channel matrix H is also well- conditioned. The MIMO channel with two multipaths is essentially a concatenation of the nt by 2 channel in Figure 7.8 and the 2 by nr channel in Figure 7.4. Although both the transmit antennas and the receive antennas are close together, mul- tipaths in effect provide virtual “relays”, which are geographically far apart. The channel from the transmit array to the relays as well as the channel from the relays to the receive array both have two degrees of freedom, and so does the overall channel. Spatial multiplexing is now possible. In this con- text, multipath fading can be viewed as providing an advantage that can be exploited. It is important to note in this example that significant angular separation of the two paths at both the transmit and the receive antenna arrays is crucial for 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 closer to the receiver than to the transmitter, then the angular separation !t at the transmitter is small. Similarly, if the reflector is local around the transmitter and is much closer to the transmitter than to the receiver, then the angular separation !r at the receiver is small. In either case H would not be very well-conditioned (Figure 7.10). In a cellular system this suggests that if the base-station is high on top of a tower with most of the scatterers and reflectors locally around the mobile, then the size of the antenna array at the base-station
Figure 7.10 (a) The reflectors and scatterers are in a ring locally around the receiver; their angular separation at the transmitter is small. (b) The reflectors and scatterers are in a ring locally around the transmitter; their angular separation at the receiver is small.
~~
~~
~~
~~
Tx antenna array
Tx antenna array
Rx antenna array
Rx antenna array
Very small angular separation
Large angular separation
(a)
(b)
309 7.3 Modeling of MIMO fading channels
will have to be many wavelengths to be able to exploit this spatial multiplexing effect.
Summary 7.1 Multiplexing capability of MIMO channels
SIMO and MISO channels provide a power gain but no degree-of-freedom gain. Line-of-sight MIMO channels with co-located transmit antennas and co-located receive antennas also provide no degree-of-freedom gain. MIMO channels with far-apart transmit antennas having angular separation greater than 1/Lr at the receive antenna array provide an effective degree- of-freedom gain. So do MIMO channels with far-apart receive antennas having angular separation greater than 1/Lt at the transmit antenna array. Multipath MIMO channels with co-located transmit antennas and co-located receive antennas but with scatterers/reflectors far away also provide a degree-of-freedom gain.
7.3 Modeling of MIMO fading channels
The examples in the previous section are deterministic channels. Building on the insights obtained, we migrate towards statistical MIMO models which capture the key properties that enable spatial multiplexing.
7.3.1 Basic approach
In the previous section, we assessed the capacity of physical MIMO channels by first looking at the rank of the physical channel matrix H and then its condition number. In the example in Section 7.2.4, for instance, the rank of H is 2 but the condition number depends on how the angle between the two spatial signatures compares to the spatial resolution of the antenna array. The two-step analysis process is conceptually somewhat awkward. It suggests that physical models of the MIMO channel in terms of individual multipaths may not be at the right level of abstraction from the point of view of the design and analysis of communication systems. Rather, one may want to abstract the physical model into a higher-level model in terms of spatially resolvable paths. We have in fact followed a similar strategy in the statistical modeling of frequency-selective fading channels in Chapter 2. There, the modeling is directly on the gains of the taps of the discrete-time sampled channel rather than on the gains of the individual physical paths. Each tap can be thought
