C H A P T E R
7 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 simultaneously 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 bandwidthlimited. 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 remaining 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 n
t 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 N0In
r
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×nt
292 MIMO I: spatial multiplexing and channel modeling
is deterministic and assumed to be constant at all times and known to both
the transmitter and the receiver. Here, hij is the channel gain from transmit
antenna j to receive antenna i. There is a total power constraint, P, on the
signals from the transmit antennas.
This is a vector Gaussian channel. The capacity can be computed by
decomposing the vector channel into a set of parallel, independent scalar
Gaussian sub-channels. From basic linear algebra, every linear transformation
can be represented as a composition of three operations: a rotation operation, a
scaling 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 real
numbers and whose off-diagonal elements are zero.2 The diagonal elements
1 ≥ 2 ≥ ··· ≥ nmin are the ordered singular values of the matrix H, where
n
min = minntnr. Since
HH∗ = UtU∗ (7.3)
the squared singular values 2 i are the eigenvalues of the matrix HH∗ and
also of H∗H. Note that there are nmin singular values. We can rewrite the
SVD as
| H = i=1 iuivi∗ |
(7.4) |
nmin
i.e., the sum of rank-one matrices iuivi∗. It can be seen that the rank of H is
precisely the number of non-zero singular values.
If we define
| x˜ = V∗x y˜ = U∗y w˜ = U∗w |
(7.5) (7.6) (7.7) |
then we can rewrite the channel (7.1) as
y˜ = 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 channels
Figure 7.1 Converting the
MIMO channel into a parallel
channel through the SVD.
| x | V | U |
| y | Channel λ1 λ nmin wnmin w1 + + ∼ ∼ … × × |
V*
| U* |
y
Pre-processing Post-processing
∼x
~
where w˜ ∼ 0 N0In
r
has the same distribution as w (cf. (A.22) in
Appendix A), and x˜2 = x2. Thus, the energy is preserved and we have
an 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 system
defined 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 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 channels 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 =
nmin
i=1
log1+ 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 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 7.2
pictorially depicts the SVD-based architecture for reliable communication.
294 MIMO I: spatial multiplexing and channel modeling
+
| AWGN coder |
~x1[m] |
| ]} … … 0 0 |
|
| { | V |
| H |
AWGN
coder
| ~y1 [m] | U* |
| {~ynmin[m] … |
~xnmin[mn min
information
streams
w[m]
| Decoder |
| Decoder |
| Figure 7.2 The SVD architecture for MIMO communication. |
There is a clear analogy between this architecture and the OFDM system 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-1Q 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 log1+ P ≈ klogSNR+klogbits/s/Hz |
(7.12) | ||
| i=1 | kN0 | i=1 | k |
2 i 2 i where k is the number of non-zero 2
i , i.e., the rank of H, and SNR = P/N0.
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.
295 7.2 Physical modeling of MIMO channels
The 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 singular
values themselves. By Jensen’s inequality,
1 k
k i=1
log1+ kN P 0 2 i ≤ log1+ kN P 0 k1 i=k1 2 i (7.13)
Now,
k i=1
2
i = Tr HH∗ =
ij
h
ij
2
(7.14)
which can be interpreted as the total power gain of the matrix channel if
one spreads the energy equally between all the transmit antennas. Then, the
above 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 values
equal. More generally, the less spread out the singular values, the larger the
capacity in the high SNR regime. In numerical analysis, maxi i/mini i is
defined to be the condition number of the matrix H. The matrix is said to be
well-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 high
SNR regime.
At low SNR, the optimal policy is to allocate power only to the strongest
eigenmode (the bottom of the vessel to waterfill, cf. Figure 5.24(b)). The
resulting capacity is
| C ≈ P max i log2 e bits/s/Hz |
(7.15) |
N0
2 i The MIMO channel provides a power gain of maxi 2 i . In this regime, the
rank or condition number of the channel matrix is less relevant. What matters
is how much energy gets transferred from the transmitter to the receiver.
7.2 Physical modeling of MIMO channels
In 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 the
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 channels, 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 rc,
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
h
i = 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 d
i 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):
h
i = aexp-j2f c cdi = aexp-j2d c i (7.17)
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 ∼ 0N0I is the noise and y is the
received vector. The vector of channel gains h = h1 hn
r
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
d
i ≈ d+i-1rc cos i = 1 nr (7.19)
where d is the distance from the transmit antenna to the first receive
antenna and 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-1rc cos is the displacement of receive antenna i from receive antenna
1 in the direction of the line-of-sight. The quantity
= cos
is often called the directional cosine with respect to the receive antenna array.
The spatial signature h = h1 hn
r
t is therefore given by
h = aexp-j2d c
1
exp-j2r
exp-j22 r
exp-j2nr -1r
(7.20)
298 MIMO I: spatial multiplexing and channel modeling
i.e., the signals received at consecutive antennas differ in phase by 2r
due to the relative delay. For notational convenience, we define
e
r = √1nr
1
exp-j2r
exp-j22 r
exp-j2nr -1r
(7.21)
as the unit spatial signature in the directional cosine .
The optimal receiver simply projects the noisy received signal onto the
signal direction, i.e., maximal ratio combining or receive beamforming
(cf. Section 5.3.1). It adjusts for the different delays so that the received
signals at the antennas can be combined constructively, yielding an nr-fold
power gain. The resulting capacity is
C = log1+ PNh02 = log1+ PaN20nr bits/s/Hz (7.22)
The SIMO channel thus provides a power gain but no degree-of-freedom
gain.
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 channel
The MISO channel with multiple transmit antennas and a single receive
antenna is reciprocal to the SIMO channel (Figure 7.3(b)). If the transmit
antennas are separated by tc and there is a single line-of-sight with angle
of departure of (directional cosine = cos), the MISO channel is
given by
y = h∗x+w (7.23)
where
h = aexpj2d c
1
exp-j2t
exp-j22 t
exp-j2nr -1t
(7.24)
299 7.2 Physical modeling of MIMO channels
The optimal transmission (transmit beamforming) is performed along the
direction e
t of h, where
e
t = √1nt
1
exp-j2t
exp-j22 t
exp-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 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
h
ik = aexp-j2dik/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-1rc cosr -k-1tc cost (7.27)
where d is the distance between transmit antenna 1 and receive antenna 1, and
t r are the angles of incidence of the line-of-sight path on the transmit and
receive antenna arrays, respectively. Define t = cost and r = cosr.
Substituting (7.27) into (7.26), we get
h
ik = aexp-j2d c ·expj2k-1tt·exp-j2i-1rr (7.28)
and we can write the channel matrix as
H = a√ntnr exp-j2d c errett∗ (7.29)
300 MIMO I: spatial multiplexing and channel modeling
where e
r· 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√ntnr. The
capacity 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 (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,
e
rr. Thus, the number of available spatial degrees of freedom does not
increase even though there are multiple transmit and multiple receive antennas.
The factor n
tnr 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
φr2 φr1
Tx antenna 1
Tx antenna 2
301 7.2 Physical modeling of MIMO channels
transmit 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 is
much 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 antenna
array is
h
k = 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 periodic
function 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 linearly
independent columns as long as the separation in the directional cosines
r = r2 -r1 = 0 mod
1
r
(7.32)
In this case, it has two non-zero singular values 2 1 and 2 2, yielding two
degrees of freedom. Intuitively, the transmitted signal can now be received
from two different directions that can be resolved by the receive antenna
array. Contrast this with the example in Section 7.2.3, where the antennas are
placed close together and the spatial signatures of the transmit antennas are
all aligned with each other.
Note that since
r1 r2, being directional cosines, lie in -11 and cannot
differ by more than 2, the condition (7.32) reduces to the simpler condition
r1 = r2 whenever the antenna spacing r ≤ 1/2.
Resolvability in the angular domain
The channel matrix H is full rank whenever the separation in the directional
cosines
r = 0 mod 1/r. However, it can still be very ill-conditioned. We
now give an order-of-magnitude estimate on how large the angular separation
has to be so that H is well-conditioned and the two degrees of freedom can
be effectively used to yield a high capacity.
The conditioning of H is determined by how aligned the spatial signatures
of the two transmit antennas are: the less aligned the spatial signatures are, the
better the conditioning of H. The angle between the two spatial signatures
satisfies
| cos = err1∗err2 | (7.33) |
| Note that e rr1∗err2 depends only on the difference r = r2 – r1. Define then |
|
| frr2 -r1 = err1∗err2 | (7.34) |
302 MIMO I: spatial multiplexing and channel modeling
By direct computation (Exercise 7.3),
frr = 1
n
r
expjrrnr -1 sinLrr
sinLrr/nr (7.35)
where L
r = nr r is the normalized length of the receive antenna array. Hence,
| cos = | sinLrr n r sinLrr/nr |
(7.36) |
The conditioning of the matrix H depends directly on this parameter. For
simplicity, consider the case when the gains a1 = a2 = a. The squared singular
values of H are
2
1 = a2nr1+ cos 2 2 = a2nr1- cos (7.37)
and the condition number of the matrix is
1
2
= 11+ – cos cos (7.38)
The matrix is ill-conditioned whenever cos ≈ 1, and is well-conditioned
otherwise. In Figure 7.5, this quantity cos = frr is plotted as a function
of
r for a fixed array size and different values of nr. The function fr· has
the 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 signature
e
r·. 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
1 L
r
(7.39)
for some integer m. Now, since r ranges from -2 to 2, this condition
reduces to
r
1 L
r
(7.40)
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 L
r = 8 and different
values of the number of
receive antennas n
r.
0
0.7
0.8
0.9
1
– 2 – 1.5 – 1
0.5
0.4
0.3
0.2
0.1
0.6
n
r = 16
| Ω r sinc function |
nr = 8 Ω r |
Ω
r
n
r = 4
– 0.5 0 0.5 1 1.5 2 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
– 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
– 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
– 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2
Ω
r
|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,
frr → ejLrrsincLrr (7.41)
and the dependency of fr· on nr vanishes. Equation (7.41) can be directly
derived 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 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 e
rcos0; recall that this is called the (receive) beamforming vector.
A signal from any other direction is attenuated by a factor of
e
rcos0∗ercos = frcos-cos0 (7.42)
The beamforming pattern associated with the vector ercos is the polar
plot
frcos-cos0 (7.43)
304 MIMO I: spatial multiplexing and channel modeling
Figure 7.6 Receive
beamforming patterns aimed
at 90, with antenna array
length Lr = 2 and different
numbers of receive antennas
n
r. Note that the beamforming
pattern is always symmetrical
about the 0 – 180 axis, so
lobes always appear in pairs.
For n
r = 4632, the antenna
separation r ≤ 1/2, and
there is a single main lobe
around 90 (together with its
mirror image). For nr = 2,
r = 1 > 1/2 and there is an
additional pair of main lobes.
0.2
0.4
| 0.6 |
0.8
1
30
210
60
240
90
270
120
300
150
330
180 0
L
r = 2, nr = 2
0.2
0.4
| 0.6 |
0.8
1
30
210
60
240
90
270
120
300
150
330
180 0
0.2
0.4
| 0.6 |
0.8
1
30
210
60
240
90
270
120
300
150
330
180 0
0.2
0.4
| 0.6 |
0.8
1
30
210
60
240
90
270
120
300
150
330
180 0
L
r = 2, nr = 4
L
r = 2, nr = 6 Lr = 2, nr = 32
(Figures 7.6 and 7.7). Two important points to note about the beamforming
pattern:
• It has main lobes around 0 and also around any angle for which
cos = cos0 mod 1
r
(7.44)
this follows from the periodicity of fr·. If the antenna separation r is
less than 1/2, then there is only one main lobe at , together with its mirror
image at -. If the separation is greater than 1/2, there can be several
more pairs of main lobes (Figure 7.6).
• The main lobe has a directional cosine width of 2/Lr; this is also called
the beam width. The larger the array length Lr, the narrower the beam
and the higher the angular resolution: the array filters out the signal from
all directions except for a narrow range around the direction of interest
(Figure 7.7). Signals that arrive along paths with angular seperation larger
than 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 and
the bandwidth W in a wireless channel. The parameter 1/W measures the
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.
| 0.5 30 60 |
120 |
| 300 |
1
210
240
90
270
150
330
180 0
| 0.5 30 60 |
120 |
| 300 |
1
210
240
90
270
150
330
180 0
L
r = 4, nr = 8 Lr = 8, nr = 16
resolvability of signals in the time domain: multipaths arriving at time separation 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 angulardomain 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 = hh1 2∗ ∗ (7.45)
Figure 7.8 Two geographically
separated receive antennas
each with line-of-sight from a
transmit antenna array.
.
| .. |
Tx antenna
array φt1
φt2
Rx antenna 2
Rx antenna 1
306 MIMO I: spatial multiplexing and channel modeling
where
h
i = ai expj2d c i1 etti (7.46)
and
ti is the directional cosine of departure of the path from the transmit
antenna array to receive antenna i and di1 is the distance between transmit
antenna 1 and receive antenna i. As long as
t = t2 -t1 = 0 mod
1
t
(7.47)
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 transmit beamforming pattern. This measures the amount of energy dissipated in
other 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 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 a
i, makes an angle of ti (ti = costi) with
the transmit antenna array and an angle of riri = cosri) with the receive
antenna array. The channel H is given by the principle of superposition:
| H = ab 1err1ett1∗ +ab 2err2ert2∗ where for i = 12, |
(7.48) |
ab
i = ai√ntnr exp-j2d c i (7.49)
and di is the distance between transmit antenna 1 and receive antenna 1
along path i. We see that as long as
t1 = t2 mod
1
t
(7.50)
307 7.2 Physical modeling of MIMO channels
Figure 7.9 (a) A MIMO
channel with a direct path and
a reflected path. (b) Channel is
viewed as a concatenation of
two channels H and H with
intermediate (virtual) relays
A and B.
Tx antenna
array
| Tx antenna array Tx antenna 1 |
~~ | ~~ | path 1 | φr | φt2 φt1 |
array
Rx antenna 1
…
(b)
(a)
A
B
~~
Rx antenna
array
path 2
| … |
| H″ |
H′ A B
2
φr
1
and
r1 = r2 mod
1
r
(7.51)
the matrix H is of rank 2. In order to make H well-conditioned, the angular
separation t of the two paths at the transmit array should be of the same
order or larger than 1/Lt and the angular separation r at the receive array
should be of the same order as or larger than 1/Lr, where
t = cos t2 -cos t1 Lt = nt t (7.52)
and
r = cos r2 -cos r1 Lr = nr r (7.53)
To see clearly what the role of the multipath is, it is helpful to rewrite H
as 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 as
the matrix for the channel from the transmit antenna array to two imaginary
receivers at point A and point B, as marked in Figure 7.9. Point A is the point
of incidence of the reflected path on the wall; point B is along the line-of-sight
path. Since points A and B are geographically widely separated, the matrix
H has rank 2; its conditioning depends on the parameter Ltt. Similarly,
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 Lrr. If both
matrices are well-conditioned, then the overall channel matrix H is also wellconditioned.
The MIMO channel with two multipaths is essentially a concatenation of the
n
t 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, 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 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 context, 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 degreeof-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
310 MIMO I: spatial multiplexing and channel modeling
of as a (time-)resolvable path, consisting of an aggregation of individual
physical paths. The bandwidth of the system dictates how finely or coarsely
the physical paths are grouped into resolvable paths. From the point of view
of communication, it is the behavior of the resolvable paths that matters,
not that of the individual paths. Modeling the taps directly rather than the
individual paths has the additional advantage that the aggregation makes
statistical modeling more reliable.
Using the analogy between the finite time-resolution of a band-limited
system and the finite angular-resolution of an array-size-limited system, we
can follow the approach of Section 2.2.3 in modeling MIMO channels. The
transmit and receive antenna array lengths Lt and Lr dictate the degree of
resolvability in the angular domain: paths whose transmit directional cosines
differ by less than 1/Lt and receive directional cosines by less than 1/Lr
are not resolvable by the arrays. This suggests that we should “sample” the
angular domain at fixed angular spacings of 1/Lt at the transmitter and at
fixed angular spacings of 1/Lr at the receiver, and represent the channel in
terms of these new input and output coordinates. The klth channel gain in
these angular coordinates is then roughly the aggregation of all paths whose
transmit directional cosine is within an angular window of width 1/Lt around
l/Lt and whose receive directional cosine is within an angular window of
width 1/Lr around k/Lr. See Figure 7.11 for an illustration of the linear
transmit and receive antenna array with the corresponding angular windows.
In the following subsections, we will develop this approach explicitly for
uniform linear arrays.
Figure 7.11 A representation
of the MIMO channel in the
angular domain. Due to the
limited resolvability of the
antenna arrays, the physical
paths are partitioned into
resolvable bins of angular
widths 1/Lr by 1/Lt. Here
there are four receive
antennas (Lr = 2) and six
transmit antennas (Lr = 3).
4
4
5
5 0
0
0
0
2
2
2
2
3
1
1 1
1
3
3
3
+1
| – r |
1 |
| 1 |
–1
path B
1/L
1/Lt
path A
path B
path A
Resolvable bins
Ω
t
Ω
r
311 7.3 Modeling of MIMO fading channels
7.3.2 MIMO multipath channel
Consider the narrowband MIMO channel:
y = Hx+w (7.55)
The n
t transmit and nr receive antennas are placed in uniform linear arrays
of normalized lengths Lt and Lr, respectively. The normalized separation
between the transmit antennas is
t = Lt/nt and the normalized separation
between the receive antennas is
r = Lr/nr. The normalization is by the
wavelength c of the passband transmitted signal. To simplify notation, we are
now thinking of the channel H as fixed and it is easy to add the time-variation
later 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 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 matrix H is
given by
| H = i ab i errietti∗ |
(7.56) |
where, as in Section 7.2,
ab
i = ai√ntnr exp-j2d c i
e
r = √1nr
1
exp-j2r
exp-j2nr -1r
(7.57)
e
t = √1nt
1
exp-j2t
exp-j2nt -1t
(7.58)
Also, di is the distance between transmit antenna 1 and receive antenna 1
along path i. The vectors et and er are, respectively, the transmitted
and received unit spatial signatures along the direction .
7.3.3 Angular domain representation of signals
The first step is to define precisely the angular domain representation of the
transmitted and received signals. The signal arriving at a directional cosine
312 MIMO I: spatial multiplexing and channel modeling
onto the receive antenna array is along the unit spatial signature er, given
by (7.57). Recall (cf. (7.35))
| fr = er0∗er | expjrnr -1 sinLr/nr |
(7.59) |
= 1
n
r
sinLr
analyzed in Section 7.2.4. In particular, we have
fr 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 basis
provides the representation of the received signals in the angular domain.
Why is this representation useful? Recall that associated with each vector e
r 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 small
side lobes. The different basis vectors e
rk/Lr have different main lobes.
This implies that the received signal along any physical direction will have
almost all of its energy along one particular erk/Lr vector and very little
along 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 resolution
of 1/Lr.
We can similarly define the angular domain representation of the transmitted signal. The signal transmitted at a direction is along the unit vector
e
t, 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 basis
provides the representation of the transmitted signals in the angular domain.
The transmitted signal along any physical direction will have almost all its
energy 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 channels
Figure 7.12 Receive
beamforming patterns of the
angular basis vectors.
Independent of the antenna
spacing, the beamforming
patterns all have the same
beam widths for the main
lobe, but the number of main
lobes 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 |
1
30
210
60
240
90
270
120
300
150
330
180 0
| 0.5 |
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
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
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
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
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
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
1
30
210
60
240
90
270
120
300
150
330
180 0
(a) Lr = 2, nr = 4
(b) Lr = 2, nr = 2
(c) Lr = 2, nr = 8
decomposition of the overall transmitted signal into the components transmitted along the different physical directions, up to a resolution of 1/Lt.
Examples of angular bases
Examples of angular bases, represented by their beamforming patterns, are
shown in Figure 7.12. Three cases are distinguished:
• Antennas are critically spaced at half the wavelength ( r = 1/2). In this
case, each basis vector erk/Lr has a single pair of main lobes around the
angles ± arccosk/Lr.
• Antennas are sparsely spaced ( r > 1/2). In this case, some of the basis
vectors have more than one pair of main lobes.
• Antennas are densely spaced ( r vectors have no main lobes.
314 MIMO I: spatial multiplexing and channel modeling
These statements can be understood from the fact that the function frr
is periodic with period 1/r. The beamforming pattern of the vector erk/Lr
is the polar plot
| fr cos- Lkr | (7.63) |
and the main lobes are at all angles for which
cos = k
L
r
mod
1
r
(7.64)
In the critically spaced case, 1/r = 2 and k/Lr is between 0 and 2; there is
a unique solution for cos in (7.64). In the sparsely spaced case, 1/r and for some values of k there are multiple solutions: cos = k/Lr +m/r
for integers m. In the densely spaced case, 1/r > 2, and for k satisfying
L
r do not correspond to any physical directions.
Only in the critically spaced antennas is there a one-to-one correspondence
between the angular windows and the angular basis vectors. This case is the
simplest and we will assume critically spaced antennas in the subsequent
discussions. The other cases are discussed further in Section 7.3.7.
Angular domain transformation as DFT
Actually the transformation between the spatial and angular domains is a
familiar one! Let U
t be the nt ×nt unitary matrix the columns of which are
the basis vectors in
t. If x and xa are the nt-dimensional vector of transmitted signals from the antenna array and its angular domain representation
respectively, then they are related by
| x = U txa |
x | a | = Ut∗x | (7.65) |
Now the klth entry of Ut is
1 √n
t
exp-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 that
the specific transformation for the angular domain representation is in fact
a DFT because of the use of uniform linear arrays. On the other hand, the
representation of signals in the angular domain is a more general concept and
can be applied to other antenna array structures. Exercise 7.8 gives another
example.
315 7.3 Modeling of MIMO fading channels
7.3.4 Angular domain representation of MIMO channels
We now represent the MIMO fading channel (7.55) in the angular domain.
U
t and Ur are respectively the nt ×nt and nr ×nr unitary matrices the columns
of which are the vectors in
t and r respectively (IDFT matrices). The
transformations
xa = U∗
t x (7.67)
ya = Ur∗y (7.68)
are the changes of coordinates of the transmitted and received signals into
the angular domain. (Superscript “a” denotes angular domain quantities.)
Substituting this into (7.55), we have an equivalent representation of the
channel in the angular domain:
ya = Ur∗HUtxa +Ur∗w
= Haxa +wa (7.69)
where
Ha = U∗
r HUt (7.70)
is the channel matrix expressed in angular coordinates and
wa = U∗
r w ∼ 0 N0Inr (7.71)
Now, recalling the representation of the channel matrix H in (7.56),
ha
kl = erk/Lr∗ Hetl/Lt
=
i
ab
i erk/Lr∗erri· etti∗etl/Lt (7.72)
Recall from Section 7.3.3 that the beamforming pattern of the basis vector
e
rk/Lr has a main lobe around k/Lr. The term erk/Lr∗erri is significant
for the ith path if
ri –
k Lr
1 L
r
(7.73)
Define then
k as the set of all paths whose receive directional cosine is
within a window of width 1/Lr around k/Lr (Figure 7.13). The bin k can be
interpreted as the set of all physical paths that have most of their energy along
the receive angular basis vector erk/Lr. Similarly, define l as the set of
all paths whose transmit directional cosine is within a window of width 1/Lt
316 MIMO I: spatial multiplexing and channel modeling
Figure 7.13 The bin k is the
set of all paths that arrive
roughly in the direction of the
main lobes of the
beamforming pattern of
e
rk/L. Here Lr = 2 and
n
r = 4.
1
| 30 60 0.8 0.6 0.4 0.2 |
120 |
| 240 | 300 |
210
90
270
150
330
180 0
k = 0
k = 1
k = 2
k = 3
around l/Lt. The bin l can be interpreted as the set of all physical paths that
have most of their energy along the transmit angular basis vector etl/Lt.
The entry ha kl is then mainly a function of the gains ab i of the physical paths
that fall in
l ∩ k, and can be interpreted as the channel gain from the lth
transmit angular bin to the kth receive angular bin.
The paths in l ∩ k are unresolvable in the angular domain. Due to
the finite antenna aperture sizes (Lt and Lr), multiple unresolvable physical
paths can be appropriately aggregated into one resolvable path with gain ha kl.
Note that
l ∩ kl = 01 nt – 1k = 01 nr – 1
forms a partition of the set of all physical paths. Hence, different physical paths
(approximately) contribute to different entries in the angular representation
Ha of the channel matrix.
The discussion in this section substantiates the intuitive picture in
Figure 7.11. Note the similarity between (7.72) and (2.34); the latter quantifies how the underlying continuous-time channel is smoothed by the limited
bandwidth of the system, while the former quantifies how the underlying
continuous-space channel is smoothed by the limited antenna aperture. In the
latter, the smoothing function is the sinc function, while in the former, the
smoothing functions are fr and ft.
To simplify notations, we focus on a fixed channel as above. But timevariation can be easily incorporated: at time m, the ith time-varying path
has attenuation a
i m, length di m, transmit angle ti m and receive angle
ri m. At time m, the resulting channel and its angular representation are
time-varying: H m and Ha m, respectively.
317 7.3 Modeling of MIMO fading channels
7.3.5 Statistical modeling in the angular domain
The basis for the statistical modeling of MIMO fading channels is the approximation that the physical paths are partitioned into angularly resolvable bins and
aggregated to form resolvable paths whose gains are ha kl m. Assuming that the
gains ab i m of the physical paths are independent, we can model the resolvable
path gainsha kl masindependent.Moreover,the anglesri mm andti mm
typically evolve at a much slower time-scale than the gains ab i mm; therefore, within the time-scale of interest it is reasonable to assume that paths do
not move from one angular bin to another, and the processes ha kl mm can be
modeled as independent across k and l (see Table 2.1 in Section 2.3for the analogous situation for frequency-selective channels). In an angular bin kl, where
there are many physical paths, one can invoke the Central Limit Theorem and
approximate the aggregate gain ha kl m as a complex circular symmetric Gaussian process. On the other hand, in an angular bin kl that contains no paths,
the entries ha
kl m can be approximated as 0. For a channel with limited angular
spread at thereceiver and/or the transmitter,many entries ofHa m may be zero.
Some examples are shown in Figures 7.14 and 7.15.
Figure 7.14 Some examples of
Ha. (a) Small angular spread at
the transmitter, such as the
channel in Figure 7.10(a). (b)
Small angular spread at the
receiver, such as the channel in
Figure 7.10(b). (c) Small
angular spreads at both the
transmitter and the receiver. (d)
Full angular spreads at both the
transmitter and the receiver.
5
10
15
20
25
30 5
10
15
20
25
30
5
10
15
20
25
30
k – Receiver bins
(a) 60° spread at transmitter, 360° spread at receiver
(c) 60° spread at transmitter, 60° spread at receiver
l – Transmitter bins
5
10
15
20
25
30 5
10
15
20
25
30
5
10
15
20
25
k – Receiver bins
(b) 360° spread at transmitter, 60° spread at receiver
(d) 360° spread at transmitter, 360° spread at receiver
l – Transmitter bins
5
10
15
20
25
30
5
10
15
20
25
30
10
20
30
40
50
l – Transmitter bins k – Receiver bins
5
10
15
20
25
30 5
10
15
20
25
30
5
10
15
l – Transmitter bins k – Receiver bins
|h
|hkl a | kl a |
|h
kl a | |hkl a |
318 MIMO I: spatial multiplexing and channel modeling
7.3.6 Degrees of freedom and diversity
Degrees of freedom
Given the statistical model, one can quantify the spatial multiplexing capability of a MIMO channel. With probability 1, the rank of the random matrix
Ha is given by
rankHa = minnumber of non-zero rows, number of non-zero columns
(7.74)
(Exercise 7.6). This yields the number of degrees of freedom available in the
MIMO channel.
The number of non-zero rows and columns depends in turn on two separate
factors:
• The amount of scattering and reflection in the multipath environment. The
Figure 7.15 Some examples of
Ha. (a) Two clusters of
scatterers, with all paths going
through a single bounce.
(b) Paths scattered via multiple
bounces.
more scatterers and reflectors there are, the larger the number of non-zero
entries in the random matrix Ha, and the larger the number of degrees of
freedom.
• The lengths Lt and Lr of the transmit and receive antenna arrays. With small
antenna array lengths, many distinct multipaths may all be lumped into a
single resolvable path. Increasing the array apertures allows the resolution
5
10
15
20
25
30
5
10
15
20
25
30
5
10
15
20
5
10
15
20
25
30
5
10
15
20
25
30
5
15
10
120°
–175°
–20°
Tx 40° Rx
10°
5°
15°
10°
70°
–175°
–120°
–60°
Tx
Rx
10°
5°
15°
10°
(a) (b)
|h
akl|
|h
akl|
l–Transmitter bins K–Receiver bins l–Transmitter bins K–Receiver bins
319 7.3 Modeling of MIMO fading channels
of more paths, resulting in more non-zero entries of Ha and an increased
number of degrees of freedom.
The number of degrees of freedom is explicitly calculated in terms of the
multipath environment and the array lengths in a clustered response model
in Example 7.1.
Example 7.1 Degrees of freedom in clustered response models
Clarke’s model
Let us start with Clarke’s model, which was considered in Example 2.2.
In this model, the signal arrives at the receiver along a continuum set
of paths, uniformly from all directions. With a receive antenna array of
length Lr, the number of receive angular bins is 2Lr and all of these
bins are non-empty. Hence all of the 2Lr rows of Ha are non-zero. If the
scatterers and reflectors are closer to the receiver than to the transmitter
(Figures 7.10(a) and 7.14(a)), then at the transmitter the angular spread t
(measured in terms of directional cosines) is less than the full span of 2.
The number of non-empty rows in Ha is therefore Ltt, such paths are
resolved into bins of angular width 1/Lt. Hence, the number of degrees
of freedom in the MIMO channel is
minLtt2Lr (7.75)
If the scatterers and reflectors are located at all directions from the transmitter as well, then t = 2 and the number of degrees of freedom in the
MIMO channel is
min2Lt2Lr (7.76)
the maximum possible given the antenna array lengths. Since the antenna
separation is assumed to be half the carrier wavelength, this formula can
also be expressed as
minntnr
the rank of the channel matrix H
General clustered response model
In a more general model, scatterers and reflectors are not located at all
directions from the transmitter or the receiver but are grouped into several
clusters (Figure 7.16). Each cluster bounces off a continuum of paths.
Table 7.1 summarizes several sets of indoor channel measurements that
support such a clustered response model. In an indoor environment, clustering can be the result of reflections from walls and ceilings, scattering from
furniture, diffraction from doorway openings and transmission through soft
partitions. It is a reasonable model when the size of the channel objects is
comparable to the distances from the transmitter and from the receiver.
320 MIMO I: spatial multiplexing and channel modeling
Table 7.1 Examples of some indoor channel measurements. The Intel
measurements span a very wide bandwidth and the number of clusters and
angular spread measured are frequency dependent. This set of data is further
elaborated in Figure 7.18.
Frequency (GHz) No. of clusters Total angular spread ()
USC UWB [27] 0–3 2–5 37
Intel UWB [91] 2–8 1–4 11–17
Spencer [112] 6.75–7.25 3–5 25.5
COST 259 [58] 24 3–5 18.5
Cluster of scatterers
Receive
array
Transmit
array
φ t Θ t,1 φ r
Θ t,2
Θ r,1
Θ r,2
Figure 7.16 The clustered response model for the multipath environment. Each cluster bounces
off a continuum of paths.
In such a model, the directional cosines r along which paths arrive
are partitioned into several disjoint intervals: r = ∪krk. Similarly, on
the transmit side, t = ∪ktk. The number of degrees of freedom in the
channel is
min k Lt tk k Lr tk (7.77)
For L
t and Lr large, the number of degrees of freedom is approximately
| minLtttotalLrrtotal | (7.78) |
| where | |
| ttotal =
tk and rtotal = rk |
(7.79) |
k
k
321 7.3 Modeling of MIMO fading channels
are the total angular spreads of the clusters at the transmitter and at the
receiver, respectively. This formula shows explicitly the separate effects
of the antenna array and of the multipath environment on the number of
degrees of freedom. The larger the angular spreads the more degrees of
freedom there are. For fixed angular spreads, increasing the antenna array
lengths allows zooming into and resolving the paths from each cluster,
thus increasing the available degrees of freedom (Figure 7.17).
One can draw an analogy between the formula (7.78) and the classic
fact that signals with bandwidth W and duration T have approximately
2WT degrees of freedom (cf. Discussion 2.1). Here, the antenna array
lengths Lt and Lr play the role of the bandwidth W, and the total angular
spreads ttotal and rtotal play the role of the signal duration T.
Effect of carrier frequency
As an application of the formula (7.78), consider the question of how
the available number of degrees of freedom in a MIMO channel depends
on the carrier frequency used. Recall that the array lengths Lt and Lr
are quantities normalized to the carrier wavelength. Hence, for a fixed
physical length of the antenna arrays, the normalized lengths Lt and Lr
increase with the carrier frequency. Viewed in isolation, this fact would
suggest an increase in the number of degrees of freedom with the carrier
frequency; this is consistent with the intuition that, at higher carrier frequencies, one can pack more antenna elements in a given amount of area
on the device. On the other hand, the angular spread of the environment
Cluster of scatterers
(a) Array length of L1
(b) Array length of L2 > L1
Cluster of scatterers
Receive
array
Receive
array
1/L1 1/L1
1/L2 1/L2
Transmit
array
Transmit
array
Figure 7.17 Increasing the antenna array apertures increases path resolvability in the angular
domain and the degrees of freedom.
322 MIMO I: spatial multiplexing and channel modeling
typically decreases with the carrier frequency. The reasons are
two-fold:
• signals at higher frequency attenuate more after passing through or
bouncing off channel objects, thus reducing the number of effective
clusters;
• at higher frequency the wavelength is small relative to the feature size
of typical channel objects, so scattering appears to be more specular in
nature and results in smaller angular spread.
These factors combine to reduce
ttotal and rtotal as the carrier frequency
increases. Thus the impact of carrier frequency on the overall degrees of
freedom is not necessarily monotonic. A set of indoor measurements is
shown in Figure 7.18. The number of degrees of freedom increases and
then decreases with the carrier frequency, and there is in fact an optimal
frequency at which the number of degrees of freedom is maximized. This
example shows the importance of taking into account both the physical
environment as well as the antenna arrays in determining the available
degrees of freedom in a MIMO channel.
0 2 3 4 5 6 7
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
02 3 4 5 6 7
7 6 5 4 3 2 1
Frequency (GHz) Frequency (GHz)
(a) (b)
Ω total in townhouse
Ω total
Ω total /λ c (m-1)
1/λ (m-1)
1/λ c
Ω total in office
Office
Townhouse
5 0
10
15
20
25
8 8
Figure 7.18 (a) The total angular spread total of the scattering environment (assumed equal at
the transmitter side and at the receiver side) decreases with the carrier frequency; the normalized
array length increases proportional to 1/c. (b) The number of degrees of freedom of the MIMO
channel, proportional to total/c, first increases and then decreases with the carrier frequency.
The data are taken from [91].
Diversity
In this chapter, we have focused on the phenomenon of spatial multiplexing
and the key parameter is the number of degrees of freedom. In a slow fading
environment, another important parameter is the amount of diversity in the
channel. This is the number of independent channel gains that have to be in
a deep fade for the entire channel to be in deep fade. In the angular domain
MIMO model, the amount of diversity is simply the number of non-zero
323 7.3 Modeling of MIMO fading channels
Figure 7.19 Angular domain
representation of three MIMO
channels. They all have four
degrees of freedom but they
have diversity 4, 8 and 16
respectively. They model
channels with increasing
amounts of bounces in the
paths (cf. Figure 7.15).
(a)
nt
n
r
n
r
n
r
nt nt
(b) (c)
entries in Ha. Some examples are shown in Figure 7.19. Note that channels
that have the same degrees of freedom can have very different amounts of
diversity. The number of degrees of freedom depends primarily on the angular
spreads of the scatters/reflectors at the transmitter and at the receiver, while
the amount of diversity depends also on the degree of connectivity between
the 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 with
single-bounced paths with signal sent along one transmit angle received at a
unique angle, even though the angular spreads may be the same.
7.3.7 Dependency on antenna spacing
So far we have been primarily focusing on the case of critically spaced
antennas (i.e., antenna separations t and r are half the carrier wavelength).
What is the impact of changing the antenna separation on the channel statistics
and 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 vary
the antenna separation, or equivalently the number of antenna elements. Let
us just focus on the receiver side; the transmitter side is analogous. Given the
antenna array length Lr, the beamforming patterns associated with the basis
vectors erk/Lrk all have beam widths of 2/Lr (Figure 7.12). This dictates
the maximum possible resolution of the antenna array: paths that arrive within
an angular window of width 1/Lr cannot be resolved no matter how many
antenna elements there are. There are 2L
r such angular windows, partitioning
all 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 physical
paths which have most of their energy along the basis vector etk/Lr. The
bins dictate the resolvability of the antenna array. In the critically spaced case
r = 1/2), the beamforming patterns of all the basis vectors have a single
main lobe (together with its mirror image). There is a one-to-one correspondence between the angular windows and the resolvable bins k, and paths
arriving in different windows can be resolved by the array (Figure 7.21). In
324 MIMO I: spatial multiplexing and channel modeling
Figure 7.20 An antenna array
of length Lr partitions the
receive directions into 2L
r
angular windows. Here, Lr = 3
and there are six angular
windows. Note that because of
symmetry across the 0 -180
axis, each angular window
comes as a mirror image pair,
and each pair is only counted
as one angular window.
3
4 2
5 1
5 1
4 2
3
0 0
Figure 7.21 Antennas are
critically spaced at half the
wavelength. Each resolvable
bin corresponds to exactly one
angular window. Here, there
are six angular windows and
six bins.
L
r = 3, nr = 6
4 2
| 0 | 1 | 2 | 3 | 4 | 5 |
k
5 1
5 1
4 2
3
0 0
3
Bins
the sparsely spaced case ( r > 1/2), the beamforming patterns of some of the
basis vectors have multiple main lobes. Thus, paths arriving in the different
angular windows corresponding to these lobes are all lumped into one bin
and cannot be resolved by the array (Figure 7.22). In the densely spaced case
( r main lobe; they can be used to resolve among the 2Lr angular windows. The
beamforming patterns of the remaining nr -2Lr basis vectors have no main
lobe and do not correspond to any angular window. There is little received
energy along these basis vectors and they do not participate significantly in
the communication process. See Figure 7.23.
The key conclusion from the above analysis is that, given the antenna
array lengths Lr and Lt, the maximum achievable angular resolution can
be achieved by placing antenna elements half a wavelength apart. Placing
antennas more sparsely reduces the resolution of the antenna array and can
325 7.3 Modeling of MIMO fading channels
(b)
Bins
0 0
1
0
1 1
0
1
0
1 1
0
k
| 0 | 1 |
L
r = 3, nr = 2
(a)
Bins
0 0
2
3
4 1
2
3
2
4 1
3
k
0 1 2 3 4
L
r = 3, nr = 5
Figure 7.22 (a) Antennas are reduce the number of degrees of freedom and the diversity of the channel.
sparsely spaced. Some of the
bins contain paths from
multiple angular windows.
(b) The antennas are very
sparsely spaced. All bins
contain several angular
windows of paths.
Placing the antennas more densely adds spurious basis vectors which do not
correspond to any physical directions, and does not add resolvability. In terms
of the angular channel matrix Ha, this has the effect of adding zero rows and
columns; in terms of the spatial channel matrix H, this has the effect of making
the entries more correlated. In fact, the angular domain representation makes
it apparent that one can reduce the densely spaced system to an equivalent
2L
t ×2Lr critically spaced system by just focusing on the basis vectors that
do correspond to physical directions (Figure 7.24).
Increasing the antenna separation within a given array length Lr does not
increase the number of degrees of freedom in the channel. What about increasing the antenna separation while keeping the number of antenna elements nr
the same? This question makes sense if the system is hardware-limited rather
than limited by the amount of space to put the antenna array in. Increasing
the antenna separation this way reduces the beam width of the nr angular
basis beamforming patterns but also increases the number of main lobes in
each (Figure 7.25). If the scattering environment is rich enough such that the
received signal arrives from all directions, the number of non-zero rows of
the channel matrix Ha is already nr, the largest possible, and increasing the
spacing 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 be
326 MIMO I: spatial multiplexing and channel modeling
Figure 7.23 Antennas are
densely spaced. Some bins
contain no physical paths.
0 0
7
8
9 1
2
3
2
9 1
8
k
| 0 | 1 | 2 | 3 | 4 5 6 | 7 8 9 |
Empty bins
L
r = 3, nr = 10
Figure 7.24 A typical Ha
when the antennas are
densely spaced.
10
20
30
40
50 5
10
15
20
25
30
35
40
45
50
5 4 3 2 1
L = 16, n = 50
|h
akl|
l–Transmitter bins K–Receiver bins
received 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 of
making the entries look more random and independent. At a base-station on
a high tower with few local scatterers, the angular spread of the multipaths is
small and therefore one has to put the antennas many wavelengths apart to
decorrelate the channel gains.
Sampling interpretation
One can give a sampling interpretation to the above results. First, think of
the discrete antenna array as a sampling of an underlying continuous array
-L
r/2Lr/2. On this array, the received signal xs is a function of the
327 7.3 Modeling of MIMO fading channels
Figure 7.25 An example of a
clustered response channel in
which increasing the
separation between a fixed
number of antennas increases
the number of degrees of
freedom from 2 to 3.
Cluster of scatterers
(a) Antenna separation of ∆1 = 1/2
(b) Antenna separation of ∆2 > ∆1
Cluster of scatterers
Receive
array
Receive
array
Transmit
array
Transmit
array
1 / (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 1
corresponds to directional cosines of actual physical directions, the angular
representation xa of the received signal is zero outside -1 1. Hence, the
spatial-domain signal xs is “bandlimited” to -W W, with “bandwidth”
W = 1. By the sampling theorem, the signal xs can be uniquely specified
by samples spaced at distance 1/2W = 1/2 apart, the Nyquist sampling
rate. This is precise when Lr → and approximate when Lr is finite. Hence,
placing the antenna elements at the critical separation is sufficient to describe
the received signal; a continuum of antenna elements is not needed. Antenna
spacing greater than 1/2 is not adequate: this is under-sampling and the loss
of resolution mentioned above is analogous to the aliasing effect when one
samples a bandlimited signal at below the Nyquist rate.
7.3.8 I.i.d. Rayleigh fading model
A very common MIMO fading model is the i.i.d. Rayleigh fading model:
the entries of the channel gain matrix H m are independent, identically
328 MIMO I: spatial multiplexing and channel modeling
distributed and circular symmetric complex Gaussian. Since the matrix H m
and its angular domain representation Ha m are related by
Ha m = Ur∗H mUt (7.80)
andU
r andUt arefixed unitarymatrices,thismeansthatHa should havethe same
i.i.d. Gaussian distribution as H. Thus, using the modeling approach described
here, we can see clearly the physical basis of the i.i.d Rayleigh fading model, in
terms of both the multipath environment and the antenna arrays. There should
be 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 some
of the angular directions, then the entries in H will be correlated. Moreover, the
antennasshould be either critically orsparselyspaced.Ifthe antennas are densely
spaced, then some entries ofHa are approximately zero and the entries inH itself
are highly correlated. However, by a simple transformation, the channel can be
reducedto an equivalent channelwithfewer antennaswhich are criticallyspaced.
Compared to the critically spaced case, having sparser spacing makes it
easier for the channel matrix to satisfy the i.i.d. Rayleigh assumption. This is
because each bin now spans more distinct angular windows and thus contains
more paths, from multiple transmit and receive directions. This substantiates
the intuition that putting the antennas further apart makes the entries of H
less dependent. On the other, if the physical environment already provides
scattering in all directions, then having critical spacing of the antennas is
enough to satisfy the i.i.d. Rayleigh assumption.
Due to the analytical tractability, we will use the i.i.d. Rayleigh fading
model quite often to evaluate performance of MIMO communication schemes,
but it is important to keep in mind the assumptions on both the physical
environment and the antenna arrays for the model to be valid.
Chapter 7 The main plot
The angular domain provides a natural representation of the MIMO channel, highlighting the interaction between the antenna arrays and the physical
environment.
The angular resolution of a linear antenna array is dictated by its length: an
array of length L provides a resolution of 1/L. Critical spacing of antenna
elements at half the carrier wavelength captures the full angular resolution
of 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 the
angular domain into 2Lt ×2Lr bins of unresolvable multipaths. Paths that
fall within the same bin are aggregated to form one entry of the angular
channel matrix Ha.
329 7.4 Bibliographical notes
A statistical model of Ha is obtained by assuming independent Gaussian
distributed 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 minimum
of the number of non-zero rows and the number of non-zero columns of
Ha. 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 with
the antenna array lengths. This number of degrees of freedom can be
achieved when the antennas are critically spaced at half the wavelength or
closer. With a maximum angular spread of 2, the number of degrees of
freedom is
min2Lt2Lr
and this equals
minnt nr
when 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 is
roughly equal amount of energy in each bin. The antenna elements should
be critically or sparsely spaced.
7.4 Bibliographical notes
The angular domain approach to MIMO channel modeling is based on works by
Sayeed [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 that
spatial multiplexability is limited not by the number of antenna elements but by the
size of the antenna array. We considered only linear arrays in this chapter, but [90] also
treated 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-like
model but with two rings of scatterers, one around the transmitter and one around the
receiver, to derive the MIMO channel statistics.
330 MIMO I: spatial multiplexing and channel modeling
7.5 Exercises
Exercise 7.1
1. For the SIMO channel with uniform linear array in Section 7.2.1, give an exact
expression 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 with
period 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 the
number of degrees of freedom in a MIMO channel with nt transmit, nr receive antennas
and K multipaths is given by
minntnrK (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 angular
representation of the MIMO channel.
1. Consider the critically spaced antenna array in Figure 7.21; there are six bins, each
one corresponding to a specific physical angular window. All of these angular
windows have the same width as measured in solid angle. Compute the angular
window width in radians for each of the bins l, with l = 0 5. Argue that the
width in radians increases as we move from the line perpendicular to the antenna
array to one that is parallel to it.
2. Now consider the sparsely spaced antenna arrays in Figure 7.22. Justify the depicted
mapping from the angular windows to the bins l and evaluate the angular window
width in radians for each of the bins l (for l = 01 nt – 1). (The angular
window width of a bin l is the sum of the widths of all the angular windows that
correspond to the bin l.)
3. Justify the depiction of the mapping from angular windows to the bins l in the
densely spaced antenna array of Figure 7.23. Also evaluate the angular width of
each 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 rank
of the matrix is given by the formula (7.74).
Exercise 7.7 In Chapter 2, we introduced Clarke’s flat fading model, where both the
transmitter and the receiver have a single antenna. Suppose now that the receiver has
n
r antennas, each spaced by half a wavelength. The transmitter still has one antenna
(a SIMO channel). At time m
y 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 Exercises
1. Consider first the case when the receiver is stationary. Compute approximately the
joint 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 and
the 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 the
difficulty 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 carrier
wavelength 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 1
and 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, that
is, they have non-uniform resolution along the -domain. Can you design a linear
array with uniform resolution along the -domain?
Exercise 7.9 (Spatial sampling) Consider a MIMO system with Lt = Lr = 2 in a
channel with M = 10 multipaths. The ith multipath makes an angle of i with the
transmit array and an angle of i with the receive array where = /M.
1. Assuming there are nt transmit and nr receive antennas, compute the channel
matrix.
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 of
frequency-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 = =0 Ha mxa m-+wa m |
(7.84) |
L-1
2. Consider the equivalent (except for the overhead in using the cyclic prefix) parallel
MIMO channel as in (8.113).
(a) Discuss the role played by the density of the scatterers and the delay spread in
the physical environment in arriving at an appropriate statistical model for H˜ n at
the different OFDM tones n.
(b) Argue that the (marginal) distribution of the MIMO channel H˜ n is the same for
each of the tones n = 0 N -1.
Exercise 7.11 A MIMO channel has a single cluster with the directional cosine ranges
as
t = r = 0 1. Compute the number of degrees of freedom of an n×n channel
as a function of the antenna separation t = r = .
