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SOURCE: Aalborg University, Denmark
Multipath richness – a measure of MIMO capacity in an
environment
J. Bach Andersen, J.Ø.Nielsen
Dept. of Communication Technology
Aalborg University
Niels Jernes Vej 12
9220 Aalborg, Denmark
phone +45 9635 8641
email jba,jni@kom.aau.dk
Multipath richness – a measure of MIMO capacity in an environment .
J. Bach Andersen, J.Ø. Nielsen
Department of Communication Technology
Aalborg University, Denmark
Abstract
The capacity of a MIMO system in an environment depends on the environment, the antennas and
the SNR. For a given situation the normalised eigenvalues are determining the capacity, and by
expressing them as a cumulative sum of their log values, a measure of the intrinsic multipath
richness is obtained. For the case of waterfilling the capacity equals exactly the sum of the richness
and the log of water level. A set of indoor measurements for a 16 by 32 element MIMO channel
sounder is used to compare the richness of the environments.
Definition of richness
The capacity or spectral efficiency of an environment depends on the antenna structures at each end,
knowledge of channel information at the ends, correlations between antennas and between paths,
the distribution of scatterers and most importantly on the SNR level. It is most easily expressed in
the well-known formula for capacity [1]
C = log 2 (det( I +
P
HH ' ))
M
b/s/Hz
(1)
which may be calculated for many different cases once the channel matrix H has been measured. P
is the power normalised to the noise power, the SNR, and M the number of transmit antennas. It is
of interest to find a simple relationship or curve, which expresses the multipath richness of the
channel matrix without reference to the power or SNR. Of course, one could use the capacity for a
given SNR as such a measure, but that is just one number, which does not contain any additional
information. As suggested in [2] the EDOF (effective degrees of freedom) is essentially the slope of
capacity versus SNR at one value of SNR and gives an indication of the rank of the system. In [3]
the relative sum of the channel singular values of the channel matrix is used. Various measures are
also discussed in [4]. In this formulation the channel is first supposed to be unknown at the
transmitter, and the power divided equally between the M transmit antennas. For the moment we
assume N receive antennas, where N<M, which means that the maximum number of non-zero
eigenvalues equals N. The number of significant eigenvalues or singular values determines this
richness, and in this note we first explore this by letting SNR become so large that we can ignore the
identity matrix I in (1). As it has been emphasised in [5] we can think of the eigenvalues of HH’ as
gains of the independent, orthogonal channels. Expressing the determinant through the eigenvalues
we can obtain a convenient measure for the multipath richness, independent of SNR, and if
expressed in dB also a convenient measure of the gains. The channel matrices are normalised to
have mean gain of 1 (0 dB), which means that there is a constraint on the total gain,
N
∑λ
1
i
= NM
(2)
Using the arguments above to expand equation (1) we find
P
HH ' ))
M
N
P
= log 2 (∏ (1 +
λi )
M
i =1
C = log 2 (det( I +
N
≈ log 2 (∏ (
i =1
= N log 2 (
= 0.33 N (
P
P
λ i ) for
λi > 1
M
M
(3)
N
P
) + ∑ log 2 (λ i )
M
i =1
N
P
( dB)) + ∑ log 2 (λ i )
M
i =1
where the factor 0.33 stems from the transformation from log2 to dB. The richness curve (or vector)
is now defined as the cumulative sum of the log of the eigenvalues
k
R(k) = ∑ log 2 (λi )
(4)
i =1
As will be seen this measure has a significant amount of information concerning the multipath
richness, and apart from an easily calculated constant term depending on the SNR the capacity
equals the richness. It is the same richness no matter which end is the transmitter. The eigenvalues
are ordered in decreasing order.
The final richness (and capacity) using all eigenvalues and sufficiently high power equals R(N), but
if
P
λt < 1
(5)
M
for a particular value of i=t then the eigenvalues from t and above do not contribute to the richness,
and we can use R(t) as the measure of richness for that particular value of SNR.
Waterfilling
The formulation becomes even more appealing when we consider the case, when the channels are
known at the transmitter, so that waterfilling may be applied. The total power is now distributed
among the eigenvalues instead of among the antennas after the following scheme, which maximises
the capacity. The total power relative to the noise power is denoted P, and the powers allocated to
the individual channels or eigenvalues are Pi. The channels are filled up to a water level µ
1
1
+ P1 =
+ P2 = .... = µ
λ1
λ2
(6)
for the n channels with positive powers. The N-n channels are allocated zero power. Adding the
equations in (6) we find
µ=
1 n
P
?i +
∑
n 1
n
(7)
and the capacity
n
n
1
1
C = ∑ log 2 (1 + λ i Pi ) = ∑ log 2 (1 + λ i ( µ −
n
n
1
)) = ∑ log 2 (λ i µ ) = ∑ log 2 (λ i ) + nlog 2 ( µ )
λi
1
1
(8)
so the capacity simply equals the sum of the richness and the waterlevel expressed in log2 . with only
active channels included.
It is interesting to compare eqs. (8) and (3). In the case of uniform distribution of powers among
antennas µ=P/M compared with µ=P/n for large P. This explains why there is only little difference
between the two cases for symmetric systems where M=n=N for large P, but a significant
difference when M>>N.
Experimental results
The following cases are based on me asurements taken in various indoor environments at Aalborg
University, the measurement techniques and environments being described in [6]. The
measurements are performed with a 100 MHz sounder at 5.8 GHz. In this section we limit the
discussion to a single frequency (the narrowband richness) and later look at the richness as a
function of delay. The antennas are planar monopole arrays with 16 or 32 elements at the ends, a 4
by 4 and a 4 by 8 array. In Figure 1 we show the richness curves for the complete arrays with 512
(16*32) different paths for one particular position of the two antennas, bur the diversity order is so
high, that it is assumed to be representative. The richness is plotted as a curve, although it is only
defined at the integer values. The experimental environments are 1) a laboratory full of equipment
(lab), 2) a rather empty basement room (basement), 3) the link between two offices at two different
levels (levels), and 4) a link between two offices at the same floor (office-to-office). Apart from
these the theoretical uncorrelated Rayleigh case and the theoretical case of equal eigenvalues (called
max. capacity) are shown. All signals below –30 dB from the peak are ignored.
The general shape of the richness curve is parabolic, indicating an exponential decay of the
eigenvalues. The value of the first eigenvalue, the combined gain of the two antennas should be
around 10*log10 v(M2 +N2 ) [5] if uncorrelated, equal to 15.5 dB or 0.33*15.5=5.13 in richness. The
general order of the richness between the environments for sufficient power is lab, office-to-office,
basement, levels, and the differences in richness are also the differences in capacity. Note that the
turning point occurs where ?i=1 (0 dB) and the maximum richness occurs for SNR=M, cf. equation
5.
Figure 1. Richness for different environments for a 16*32 case, including the iid Rayleigh case,
and the maximum capacity case with equal eigenvalues.
There is a considerable difference between the laboratory richness (full of equipment) and the levels
case (offices at two different floors), in fact the difference in capacity is 30 b/s/Hz independent of
SNR as long as it is large. For M=32, N=16 and SNR = 20 dB, the lab capacity is 82 b/s/Hz and
levels capacity 52 b/s/Hz. The richness curves are independent of the environments for up to 6
eigenvalues of the total of 16.
A more realistic case with (N,M)=(4,4) is shown in figure 2 for the same environments. The 4
antennas are a random subset of the 16 and 32 antennas of the complete antenna.
Figure 2. The same environments as in Figure 1, but with (N,M)=(4,4).
The apparent richness and thus the capacity are now much less, which is natural with the smaller
number of elements. The ‘levels’ case is now no longer the worst case, indicating that the capacity
is a strong function of the antenna structure as well as the environment. It should be understood
though that this case is just one realisation and not a mean of many realisations. Roughly speaking
the richness for small eigenvalues is valid for low SNRs, about 0 dB, the middle range for around
10 dB and high range with all 4 eigenvalues for SNRs above 20 dB. Different environments have
the largest capacity for different SNRs. There is a cross-over such that the capacity is largest for
intermediate SNRs for basement relative to levels, while it is the opposite for large SNRs.
The gain of the first eigenvalue is close to 12 dB (3*3.8) valid for a single large eigenvalue for the
levels and office-to-office case, while the other cases have gains close to 10 dB, which is the true
mean value for a Rayleigh 4*4 case [5]. This is an indication of small angular spreads for the levels
and office-to-office case. In general the propagation aspects of the differences will be explored in
future work.
Richness in the delay domain
Since the matrix may be expressed as a function of delay it is possible to evaluate the richness as a
function of delay at a fixed position. This is not directly related to capacity, but illuminates the more
transient state of the impulse. Figure 3 a) and b) show the power delay profile, averaged over all
antennas, and the richness for the 16*32 case for ‘levels’ as a function of delay. At the peak of the
impulse the richness is dominated by a few dominant eigenvalues indicating that not all scatterers
are activated, and the richness saturates at5 a later time. Note that the maximum values are
considerably higher than the narrowband values in Figure 1 (levels, 30-40 b/s/Hz). It is interesting
that the maximum richness occurs later than the arrival of maximum power, but it is consistent with
the shape of the impulse response, where the exponential decay corresponds to the scatter, which
presumably has both a temporal and angular spread.
Figure 3 a) power delay profile for ‘levels’ and b) richness versus delay
Conclusion
The successful application of MIMO systems in indoor environments depends on a sufficiently high
richness of multipath components. The capacity depends strongly on sufficient power available, but
it is of interest to define the multipath richness independent of the power level. Such a measure is
the cumulative sum of the log of the eigenvalues of the channel matrix, approximating the capacity
apart from an additive term depending on the antennas and the power, in the case of unknown
channels at the transmitter. In the case of known channe ls waterfilling may be applied, and the
capacity equals the richness up to a certain eigenvalue plus the water level.
For the environments measured all eigenvalues are above a noise threshold, indicating full rank for
all environments. There is however considerable difference between the environments, the richest
being a laboratory full of equipment. It is possible to associate different parts of the richness curve
with various power levels. For very small power levels the environments are almost identical,
utilising only the first 5 to 6 eigenvalues. The ranking of the environments is also dependent on the
size of the arrays. The richness definition seems to be a useful tool for comparing environments.
The multipath richness may also be calculated as a function of delay, and for the specific
environments measured, the richness peaks somewhat later then the main impulse, which has a low
richness.
Acknowledgements
The work has been supported by DoCoMo – Eurolabs, Munich, Germany
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