Maximizing Degrees of Freedom in Wireless Networks
Shashibhushan Borade Lizhong Zheng
Robert Gallager
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
{spb,lizhong,gallager}@mit.edu
Abstract
We consider a wireless network with fading and a single source-destination pair.
The information reaches the destination through a sequence of layers of relays. A nonseparation based strategy is proposed which achieves a rate equal to the capacity of a
point-to-point multiantenna system [1, 2], in the high SNR limit. Our result can also
be thought of as an extension of [1, 2] for wireless networks. Later we derive formulas
for the tradeoffs between network size, rate, and diversity.
1
Introduction
In the seminal paper [3], it was essentially shown that the sum rate of an ad-hoc wireless
network can not grow linearly with the number of nodes in the network. So if every node
in the network communicates at the same data rate, that rate goes to zero as the number
of nodes in the network goes to infinity. The same result was proved in a more general
framework in [4, 5].
We also know that multiple antennas provide enormous performance gains in wireless systems. In a point-to-point multi-input multi-output (MIMO) channel with n transmit and
n receive antennas and i.i.d. flat fading, if the receiver knows the channel, the capacity is
approximately n log SNR at high SNR [1, 2]. This capacity is equal to the sum of capacities of
n parallel single antenna channels. Thus we say that this MIMO channel provides n degrees
of freedom to communicate and hence improves the performance significantly.
Applying this MIMO idea in a wireless network with fading has the potential to achieve
performance gains. An interesting example relating to a single source-destination pair was
given in [6]. There both source and destination use half the relay nodes. These relays are
so close to either the source or the destination that they act like the multiple antennas of a
single user (see Fig. 1). This network is like a point-to-point multiple transmit and receive
antenna system and its capacity grows linearly with the number of antennas. The idea of
making the relay nodes act like multiple antennas was also developed in [11].
Source
Destination
Virtual multi-antennas
Figure 1: Relays acting as multi-antenna
In this paper, we try to find whether more general wireless networks can have this MIMO
performance gain. Let the information be passed from one layer to the next till it reaches
the destination (Fig. 2). Every hop of the message from a layer to its next layer looks like
a MIMO system. Nevertheless it differs from a point-to-point MIMO system in that relay
nodes in a layer can not coordinate with each other. In this layered relay network having a
single source-destination pair, we assume that each layer has n single-antenna relay nodes
and both source and destination have n antennas each.
Figure 2: Message passed through sequence of relay-clusters
Assuming the destination knows the channel state, we demonstrate the following results in
this paper without proof:
• The full n degrees of freedom can be achieved at high SNR for a fixed network size.
This implies that lack of coordination between the relay nodes does not cost anything
in terms of degrees of freedom at high SNR.
• We formalize the meaning of high SNR above and compute the rate loss if the SNR is
not high enough.
• In an outage formulation (when codeword length is within one fading block), we derive
the tradeoff between increasing network size, outage probability, and achievable degrees
of freedom . It turns out that the penalty of increasing network size is much more severe
on outage capacity than on ergodic capacity.
2
Network Model
The network in consideration has a single information source s with n transmit antennas
and its destination d also has n receive antennas (see Fig. 3). There are k layers of relay
L1
L2
L3
Lk
L0
s
L k+1
H0
H1
H2
Hk
d
n Nodes (or antennas) in each layer
Figure 3: Network structure
nodes between s and d, with n relay nodes in each layer. Each relay node has only one
antenna. We denote the m’th layer of relay nodes by Lm and let L0 denote the layer of
transmit-antennas of s. Similarly let Lk+1 denote the layer of receive antennas at d. Each
layer can only receive transmissions from its previous layer. Interference from all other layers
is ignored. Now, transmission from every layer to next looks exactly like a MIMO system,
with the only difference being the lack of coordination between the relay nodes in a layer.
They can not coordinate with each other and act like multiantenna. This oversimplified
assumption allows us to study in isolation, the value of coordination between relay nodes.
Hence the complete network state is fully characterized by k + 1 channel matrices denoting
the channels between adjacent layers. Let matrix Hm denote the channel between layer m
and m + 1, i.e. Hm (i, j) is the value of the channel gain between the ith node (or antenna)
in Lm+1 and the jth relay node (or antenna) in Lm .
We assume an i.i.d. block fading model for each Hm . The block fading model means that
the channel state remains constant during a block of Tc symbols and changes to a new
independent realization after that. Each Hm is assumed to be known at the destination.
The source does not need to know the channel realizations. Each Hm is assumed to be
independent of all other Hj ’s. All entries of each Hm are i.i.d. complex Gaussian variables
with variance 1.
In our discrete-time model, xmi , ymi and wmi denote (respectively) the transmitted signal,
the received signal and the noise at the ith relay node (or antenna) in the mth layer. Let
xm = [xm1 , xm2 , ..xmN ]T denote the transmitted vector by mth layer, and let ym and wm be
defined similarly. For 0 ≤ m ≤ k we have,
ym+1 = Hm xm + wm+1
(1)
Each wmi is a proper complex Gaussian random variable of variance σ 2 and each is independent of the signal, the channels, and the other noise variables. Each of the relay nodes
or antennas1 has an average power constraint,
E[||xmi ||2 ] ≤ P
1 ≤ i ≤ n, 0 ≤ m ≤ k
(2)
The SNR is defined to be P/σ 2 and is assumed to be very large.
3
Single Layer Case
Although transmission from one relay layer to the next (in Fig. 2) ‘looks like’ a MIMO
system, it differs from a point-to-point MIMO system in an important way . The n transmit
antennas can fully coordinate in a multiantenna transmitter, and the multiantenna receiver
knows the signals received at all its n antennas. This is not the case in this wireless network
where the n relays in a layer can not coordinate fully. How crucial is this coordination for
achieving full degrees of freedom?
If we remove the coordination between receive antennas, i.e. if each node decodes only
according its own received data; but transmit antennas can fully coordinate, it becomes a
multiantenna broadcast channel. A sum rate of n log SNR can be achieved , which is the same
as with a MIMO channel [7, 8, 9]. This is achieved by beamforming independent data-streams
to different receive antennas, which do not interfere with each other (Fig. 4(a)). On the
other hand, if coordination between transmit antennas is removed, i.e. if transmit antennas
transmit mutually independent messages; but the receive antennas can fully coordinate, it
becomes multiple access of a multiantenna receiver (Fig. 4(b)). A sum rate of n log SNR
is still achievable. This is essentially like V-BLAST [10] in a point-to-point MIMO system
where the data-streams from all the transmit antennas are independent of each other. Thus
lack of coordination at any one side of a MIMO system does not reduce the total achievable
degrees of freedom.
Receivers
Transmitters
1
1
beam 1
Multiantenna
Transmitter
Multiantenna
Receiver
2
2
beam 2
d
s
beam 3
(a)
3
3
(b)
Figure 4: Value of coordination: (a)Multiantenna broadcast (b)Multi-access of multiantenna.
For the single layer case (k = 1), we simply combine the above two strategies. The source
splits its data into n equal rate sub-streams. These sub-streams are beamformed to individual
relay nodes simultaneously. Then each relay node decodes the sub-stream beamformed
1
This simplifying assumption means that the total available power at s is nP . Since we are assuming the
SNR is very high, the number of achievable degrees of freedom are not changed even if the total power at s
was limited to P .
towards it, and retransmits the message to the destination through the multiple access
channel. Hence all n data-streams can be decoded reliably at the destination.
Since beamforming ensures that there is no interference between the n data-streams, it
enables the source to transmit information reliably at a rate of log SNR to each of the relays.
On the other hand, we know from MIMO theory that the destination can reliably decode all
the data-streams coming from each relay at rate log SNR. Thus we get the following Lemma.
Lemma 1 The full n degrees of freedom can be achieved when only one relay layer exists
between multiantenna source and destination.
Note that multiple antennas at the source and destination, made it possible to have multiple
spatial channels between them. Only one such spatial channel is available if source and
destination had only one antenna on them (as in Gupta-Kumar and Xie-Kumar model [3, 6]).
This is one significant difference between our network model and other network models.
The previous result for the single layer case can be easily extended to a network such as Fig.
5. It has nodes with multiple antennas placed between any two layers of relay nodes. In this
network, each multiple antenna node decodes the full message and again separates it into n
data-streams which are beamformed to the nodes in the next relay layer. Here in every stage
of network operation, we have full coordination either at the transmitter or at the receiver
due to the multiantenna nodes placed between each relay layer.
Relay layer 1
Relay layer 2
Relay layer 3
Source
Relay layer 4
Destination
Figure 5: An extension of the single relay layer network
Returning to the general case in Fig. 2, neither the transmit antennas nor the receive
antennas at the relays can coordinate. Thus this scheme does not work in the case of more
than one relay layer (assuming no multiantenna nodes between the relay layers). This is
because the nodes in the first relay layer do not know the messages received by the other
nodes in that layer, so they can not perform beamforming to the next relay layer. Similarly,
the nodes in the second relay layer do not know the received symbols at other nodes in that
layer, so they can not decode the full message of rate n log SNR.
If we insist on using a separation based strategy where every relay node decodes the full
message before transmitting [3, 6], only one degree of freedom can be obtained. This is
because the single-antenna relay nodes can at most decode a rate of log SNR. It becomes
interesting to investigate if n degrees of freedom can still be achieved using some nonseparation based strategy if more than one relay layer is present. This would be a significant
performance gain over the separation based strategy. This motivates our next section.
4
General Case: Any Number of Layers
The optimal network operation is unknown for the network in Fig. 3. It is obvious that no
more than n degrees of freedom can be achieved since the source and destination have only
n antennas each. In this section, we propose a particular sub-optimal network operation.
We show that this strategy achieves all n possible degrees of freedom for any fixed number
of layers.
We first explain the network operation. We fix the functions of all the relay nodes and
convert the network to a point-to-point MIMO channel. Each relay node just re-transmits
the received symbol after scaling it down. Although
√ each relay node can perform a different
√
scaling, for simplicity
we
assume
a
fixed
scaling
n
at
every
relay
node,
i.e.
x
mi = ymi / n.
√
This scaling n can be shown to ensure that all relay nodes obey the average power constraint. Thus no relay does any kind of decoding (even partial) and all the decoding is done
only at the destination d, which employs an optimal decoding rule. This is a highly desirable
feature since it simplifies the network operation immensely and only s and d need to do the
computationally difficult tasks.
The source uses a distribution which is i.i.d. over space and time, and each symbol is a
complex Gaussian variable of variance P . Now conditioned on each channel realization,
the network looks like a point-to-point MIMO channel (which is a cascade of k + 1 MIMO
0
channels) with i.i.d. Gaussian input and correlated Gaussian noise wk+1
.
yk+1 =
Our choice of scaling
theorem2 ,
√
(Hk Hk−1 ...H0 )
0
x0 + wk+1
√ k
n
(3)
n is a finite number, independent of SNR. Hence we get the following
.
Theorem 2 For a fixed network size, a maximum rate of R = n log SNR can be achieved in
this network.
Thus the maximum number of achievable degrees of freedom in a point-to-point MIMO
system are achieved in this network. A simple simulation was done to check this theoretical
result. We compared the probability of error between a separation based strategy and our
non-separation based strategy for n = 2, k = 4, and SNR of 40dB. Both strategies had a
rate of 2bps/Hz. The separation based strategy used only one node in each layer, which
decoded the received message and then retransmitted it to the node in next layer. The nonseparation based strategy used the Alamouti space-time code [15]. The non-separation based
strategy achieved a probability of error 8.28 × 10−5 as compared to 6.57 × 10−4 achieved by
the separation based strategy.
2
.
Notation: f (SNR) = g(SNR) means
lim
SNR→∞
.
f (SNR)
=1
g(SNR)
Theorem 2 tells us that full degrees of freedom can be achieved for any fixed number of layers,
if SNR is high enough. However, the SNR required for achieving full degrees of freedom keeps
getting higher with increasing number of layers (i.e. increasing network size). On the other
hand, if SNR is fixed and the number of layers goes to infinity, no communication would be
possible due to noise accumulation. We want to study what is happening between these two
extremes. There is definitely some tradeoff between the number of layers and the achievable
degrees of freedom. The number of degrees of freedom achieved depends on the values of
SNR and k relative to each other. Although, the measure of degrees of freedom seems to be
hiding this detrimental effect of increasing network size by assuming very high SNR.
5
SNR vs. Network size
In this section, we study the interaction between SNR and network size. It is natural to
study the rate loss due to increasing the number of layers at any fixed finite SNR. It requires
to find the eigenvalue distribution of the product matrix of i.i.d. random matrices, which is
very difficult. Instead we study the asymptotic behavior of this scheme in the limit of large
SNR and large k. We let the SNR go to infinity and also let k grow to infinity with SNR.
We study for which functions k(SNR) of SNR, denoting the number of layers, full degrees
of freedom can be achieved. Taking both SNR and k to infinity enables us to formally find
out how fast SNR should grow with increasing number of layers to achieve full degrees of
freedom. Said differently, how fast can the number of layers grow with increasing SNR? We
prove that full degrees of freedom can be obtained for following functions k(SNR) of SNR
denoting number of layers:
Theorem 3 Full degrees of freedom will be obtained in this network if,
k(SNR)
=0
SNR→∞ log SNR
lim
If this condition is not satisfied, the maximum achievable degrees of freedom are reduced by
increasing number of layers. In other words, we can reach a larger network if we back off
from demanding full degrees of freedom. How large the network can be if one is a for r < n
degrees of freedom, is given by this sufficient condition.
Theorem 4 In this network, r degrees of freedom will be obtained if,
k(SNR)
≤ ϕ(n − r)
SNR→∞ log SNR
lim
(4)
where ϕ is a fixed positive parameter of the point-to-point MIMO channel. Its value is equal
−1
to (n log(n) − EHm log det(Hm HH
m )) .
We define the LHS of Eq. (4) as the network’s size penalty Sergodic (k) with respect to
ergodic capacity. This Sergodic (k) is a linear penalty on the achievable degrees of freedom in
the network. Theorem 4 gives us the following rule of thumb: To achieve the same number
of degrees of freedom when the number of layers are doubled, the SNR should be squared.
Thus the required SNR gets extremely large with increasing network size. It also says that
this non-separation based strategy outperforms any separation based strategy when the SNR
is large or network size is small. A separation based strategy guarantees at most one degree
of freedom.
A similar result was obtained in [12], which considered the case when n goes to infinity
instead of being a fixed finite number. It showed that negligible degrees of freedom are
obtained if k grows without any bound. We now know from Theorem 4 that even for the
finite n case, no degrees of freedom are achieved when k grows too fast. That is because the
size penalty for ergodic capacity becomes infinity when k grows without any bound.
6
Tradeoff between Rate, Network Size, and Diversity
We have discussed the ergodic capacity till now which averages out the mutual information
conditioned on the channel state, over all channel states. Achieving this ergodic capacity is
difficult because it requires the codelength to be long enough to span many fading blocks.
In practice, we are also interested in decoding error probability when codeword length is
finite, and within one fading block. If Pe (SNR) denotes probability of decoding error as a
function of SNR, diversity is defined as
d = lim −
SNR→∞
log (Pe (SNR))
log SNR
(5)
It is shown in [13] that for a moderate fading block length, outage probability is a tight lower
bound for decoding error probability. Hence the diversity is given by [13],
d = lim −
SNR→∞
log (Pout (SNR))
log SNR
where Pout (SNR) denotes the probability of an outage event. If r degrees of freedom are to
be achieved,
Ã
!
µ
¶
X
SNR
Pout (SNR) = P
log 1 + k µi < r log SNR
(6)
n
i=1:n
where µi represents the i’th eigenvalue of the effective channel matrix with i.i.d. white noise.
We obtain the following tradeoff between increasing network size, rate, and the diversity.
Theorem 5 If r degrees of freedom are to be achieved then the following diversity can be
achieved:
d(r) = n − r − Soutage (k)
where Soutage (k) is equal to limSNR→∞
respect to outage capacity.
nk(SNR) log log SNR
,
log SNR
(7)
which represents the size penalty with
Comparing Sergodic (k) in Eq. (4) to Soutage (k) in Eq. (7), we observe that the size penalty
on the outage capacity can be significantly higher than that on the ergodic capacity. For
example, if
∆
k(SNR) = k̄ =
log SNR
log log SNR
(8)
Soutage (k̄) = n, which implies that at any positive diversity gain, the supported outage
capacity yields 0 degrees of freedom. We can thus view k̄ as an upper limit on the supportable
network size, when coding over a single fading block. On the other hand, Sergodic (k̄) = 0,
which means that the ergodic capacity yields full n degrees of freedom. However, one has
to code over a large number of blocks to get close to the ergodic capacity. This behavior
is different from the point-to-point case. In a point-to-point MIMO channel, the outage
capacity approaches the ergodic capacity as the required diversity gain approaches 0. In
contrast, in a large network system (k comparable to k̄), the outage capacity, at any positive
diversity requirement, is in general much smaller than the ergodic capacity. Therefore, the
network throughput is mostly restricted by outage events.
Another important observation is that the maximum diversity gain for a network is n instead
n2 as in the point-to-point case. In fact, at r = 0, a typical outage event for the network is
that n out of the k + 1 channel matrices lose 1 degree of freedom each, rather than only one
channel matrix losing all n degrees of freedom. As shown in [13], the later has much smaller
probability than the former. In short, the dependence on many random matrices makes it
much easier to loose degrees of freedom in this network compared to the point-to-point case.
It is also worth noting that the network size in terms of Soutage (k) and diversity d can be
traded off. That means when r degrees of freedom are achieved, either Soutage (k) equal to
n − r can be achieved if no diversity is needed or vice versa. Of course any diversity and
Soutage (k) which add up to n − r can also be achieved.
7
Discussion
In this paper, we saw one more example (like [14]) where a very simple network operation
gives quite good performance. All the results in this paper were derived for SNR going to
infinity. Still we can also deduce some rules of thumb to be used in practice by substituting
actual values of parameters like SNR and k in the size penalty functions for ergodic or outage
capacity. For example, suppose we want to achieve the same diversity and degrees of freedom
after the number of layers are increased from k1 to k2 . The new SNR2 to ensure this is found
by equating the “empirical” value of the size penalty function,
log log SNR2
log log SNR1
= nk2 log
.
i.e. nk1 log
SNR1
SNR2
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