Self-Interference Cancellation Models for
Full-duplex Wireless Communications
Andrew Thangaraj, Radha Krishna Ganti, and Srikrishna Bhashyam
Department of Electrical Engineering
Indian Institute of Technology Madras, Chennai 600036
Email: {andrew, rganti, skrishna}@ee.iitm.ac.in
Abstract—In this paper, we study two models for self or
loopback interference cancellation in full-duplex wireless com-
munications. Both models are based on an underlying Z-channel
with side information. We obtain achievable rate regions with
suitable coding schemes under both models. Under model 1,
where the self-interference channel gain is random, we employ
training to estimate the unknown gain, and optimize the required
training time. Under model 2, where the self-interference gain
is exactly known, we show that the capacity of an ideal full-
duplex node can be realized even when the side information is low
rate and quantized. Our results show that loopback interference,
rather than being treated as noise, can be effectively dealt with
by suitable coding.
I. INTRODUCTION
A wireless node is a device that can receive and transmit
radio frequency (RF) signals for communications purposes in a
fixed bandwidth. In most practical realizations, wireless nodes
are designed to be half-duplex i.e., the node can either transmit
or receive at any given time, but not both simultaneously. The
reason is as follows. Suppose a node has one receive antenna
and another transmit antenna. Since the receive antenna’s
input signals are several orders of magnitude lower than the
transmit antenna’s output signals, loopback interference from
the transmitting antenna to the receive antenna will drown out
the input signal.
Recently, work in [1][2][3] has shown practical imple-
mentations of full-duplex wireless nodes, where loopback
interference is cancelled by suitable signal processing in RF
and/or baseband level. A generic block diagram of such a full-
duplex wireless node is shown in Fig. 1. In Fig. 1, the node is
Rx
Rx message
Tx
Tx message
TxRx
Processing ProcessingSide information
Loopback
Interference
Input Output
signal signal
Fig. 1. Block diagram of a practical full-duplex wireless node.
internally shown to have two sub-blocks - one for transmitter
processing and the other for receiver processing. The two sub-
blocks can communicate to each other, and this is shown as
the side information exchanged between the two sub-blocks.
The output signal from the transmit antenna loops back as
interference to the receive antenna. In practical implementa-
tions, as reported in [2][3], exact cancellation of the loopback
interference may not be feasible at the receiver sub-block, even
with the use of side information from the transmitter sub-
block. This is because (a) the loopback interference signal
may not be exactly known to either the transmitter or the
receiver sub-blocks, and (b) constraints such as dynamic range,
resolution of sampling circuitry and phase noise in RF circuitry
might restrict the cancellation capability.
Another related area, where there has been recent interest
in full-duplex communications, is the installation of indoor
repeaters and relays to improve coverage [4][5]. For indoor
repeaters and relays, the transmitter and receiver processing
units operating in full-duplex mode could be located suf-
ficiently far apart within a building. The model of Fig. 1
still holds for such a scenario with suitable modifications to
incorporate the distance between the transmitter and receiver.
We will refer to such a transmit-receive pair situated within a
building as an indoor transceiver
1
to distinguish from a full-
duplex node. One effect of the increased distance is that the
loopback interference will be lower, but the capacity of side
information can also be lower.
The main goal of our work is to develop information-
theoretic models suitable for practical full-duplex commu-
nications in a wireless node and indoor transceivers with
possible limits on side information. In such models, there are
transmitter and receiver subnodes with a loopback link (see
Fig. 2). The transmitter and receiver subnodes are assumed
to be connected by side information links. In our first model
applicable to full-duplex nodes with co-located antennas, we
assume that the loopback gain is unknown and that the side
information links are of infinite capacity. We incorporate
training for the loopback channel gain, and provide achievable
rate regions for independent transmission rate to a destination
node and reception rate from another source node. In our
second model, we consider the scenario where the loopback
gain is known exactly and the side information channel has
a finite capacity. Such models can be used to study coding
methods and capacity of wireless networks with practically
realizable full-duplex nodes and indoor transceivers.
Full-duplex wireless communications with loopback in-
terference has been studied by several authors in recent
years. A lot of interest has been shown in the study of
1
We will not use the transceiver as a relay in this work. So, we have avoided
the term “relay”.
a relay channel with a full-duplex relay node with loop-
back interference, and comparison with half-duplex relaying
[6][7][5][8][9][10]. Signal processing methods for nulling the
loopback interference in MIMO relays has also received
attention [7][11][12][13][14][15]. While some of these works
derive capacity under specific assumptions on the processing
at the relay, information-theoretic models for the loopback
interference in wireless nodes seem to have emerged in the
literature only very recently [16]. In [16], the deterministic
approximate capacity method has been used for a full-duplex
relaying system with an unknown loopback gain. In contrast,
our study uses traditional random coding methods for a full-
duplex communication set-up, where a full duplex node or
indoor relay is transmitting information to a second destination
node and receiving independent information from a third
source node. Also, our second model with a known loopback
gain and limited side information capacity appears to be new
in the context of full-duplex communications.
The rest of this article is organized as follows. The models
for practical full-duplex communications are introduced in
Section II. In Section III, these models are applied and
schievable rate regions are derived in a simple setting where
a full-duplex node or indoor relay is receiving information
from one node, and simultaneously transmitting independent
information to another node.
II. MODELS FOR FULL-DUPLEX COMMUNICATIONS
In this section we introduce two models for full-duplex
communication capabilities with practical constraints on can-
cellation of loopback interference.
A base model for a full-duplex node or indoor transceiver
is shown in Fig. 2. A single full-duplex node or transceiver is
C
0
C
R
T
Fig. 2. Model of a practical full-duplex wireless node.
represented in Fig. 2 as a node R and a node T, representing
the receiving and the transmitting sub-blocks, respectively.
The solid directed link from T to R represents the loopback
interference link. We will typically assume that if X is
transmitted by T, a noisy version of δX is received on the
loopback link, where δ is a constant that may or may not be
known to both R and T.
The dotted lines in Fig. 2 represent the side information
links between R and T with capacities C and C
0
, as shown.
The incoming link carrying the input signal into R and the
outgoing link from T carrying the output signal are also
shown in the model. At R, the loopback signal and the
input signal will add to form the received signal. By the
broadcast nature of wireless nodes, the signal transmitted by T
is carried on both the solid edges out of T as the output signal
and the loopback interference. However, the side information
links are independent of all other links, and could carry any
independently encoded information. These capacities will be
assumed to be finite or infinite depending on the model. Since
R and T are part of the same node, encoding and decoding
operations can be performed at either node.
A. Full-duplex communications setting
We consider the setting where the full-duplex node or
indoor transceiver F receives information from a node A and
simultaneously transmits independent information to another
node B. Adding nodes A and B to Fig. 2, we obtain the model
in Fig. 3, which we immediately recognize as a Z-channel with
side information. In Fig. 3, X
1
represents the transmit symbol
Z
1
C
0
Z
2
T
A
C
R
Y
1
B
1
X
1
X
2
1
δ
Y
2
Fig. 3. The equivalent AWGN Z-channel model for practical full-duplex
communications. The main difference to the conventional Z-channel is the
availability of side-information. The capacity of the side-information channel
from the transmitter module T of F to its receive module R is denoted by C
0
,
while the capacity of the reverse side-information channel is denoted by C;
Z
1
and Z
2
represent the AWGN noises with variance σ
2
1
and σ
2
2
; δ is the
channel gain of the loopback interference path.
from A, X
2
represents the transmit symbol from F, Z
1
and Z
2
represent additive white Gaussian noise at the receivers R and
B, and δ represents the gain of the loopback interference.
The following equations are immediate from Fig. 3:
Y
1
= X
1
+ δX
2
+ Z
1
, (1)
Y
2
= X
2
+ Z
2
. (2)
We observe that (1) and (2) correspond to a Z-channel with
side-information. So, it appears that the capacity region can be
improved by viewing the system as a Z-channel and designing
appropriate codes, rather than viewing the transmissions as
independent and always treating X
2
as noise in (1).
B. Two models: Wireless node versus indoor transceiver
1) Full duplex wireless node: When the model of Fig.
3 is applied to a wireless node with co-located antennas,
T and R are located within the same block, we assume
that the side-information channel capacities are infinite, i.e.
C = C
0
= ∞. This is a practical assumption since the receiver
and the transmitter modules can be connected with a high
speed interconnect. The imparity in this network is that the
loopback interference channel gain δ is random and unknown
to the node (both the transmit module T and receive module
R). In the rest of this paper, this is referred to as Model 1.
2) Indoor transceiver: In contrast, when the model of Fig.
3 is applied to a wireless indoor transceiver with antennas
separated by some distance, we assume that side-information
channel capacities are finite. This is because the transceiver
and the receiver modules have to be interconnected by some
mechanism, for e.g., Lan or a cable. However, we will assume
that the loopback interference channel gain δ is assumed to be
known to both the transmit module T and receive module R.
We term such a model as Model 2.
III. ACHIEVABLE RATE REGIONS
In this section, we derive achievable rate regions for Models
1 and 2. More precisely, if R
1
denotes the rate of communi-
cation between F and A, and R
2
the rate of communication
between B and F, we are interested in characterizing the rate
pairs (R
1
, R
2
) such that error free communication is possible
in the standard information-theoretic sense. We denote this
rate region by R
F
. We then compare R
F
with the rate regions
obtained when F is either an ideal full-duplex node or a half-
duplex node. It is easy to see that R
H
⊂ R
F
⊂ R
I
, where
R
H
and R
I
are the achievable rate regions corresponding to
F being half duplex and ideal full-duplex respectively. Hence
Area(R
F
\ R
H
) provides a good measure of the gain obtained
because of the full-duplex capabilities of the node F, while
the quantity Area(R
I
\ R
F
) measures the performance loss
due to the imperfections in the full-duplex implementation.
We assume average power constraints on the inputs, i.e.,
E[X
2
1
] ≤ P
1
, E[X
2
2
] ≤ P
2
. Also define F (x) =
1
2
log
2
(1 + x).
A. Rate region of Model 1
Since δ is random and unknown, training is necessary to
learn the gain of the self-interference path. Hence, some of
the resources have to be allocated for training, which in turn
leads to a reduction in the overall throughput of the system .
Training is commonly used in wireless systems to estimate the
channel gains. The amount of training depends mainly on the
coherence time of the channel and the required precision of the
estimate usually specified in terms of the mean square error.
In a typical wireless system the transmitter and the receiver a
priori decide on a training sequence and use this sequence to
learn the channel.
In a full-duplex wireless node, the transmitter and the re-
ceiver are on the same physical device and the side information
channels have infinite capacity. Therefore, the transmit data
is available to the receiver instantaneously. Hence, during
the training period, while the node cannot receive external
information, it can transmit to an external receiver. It should
also be observed that there is additional feedback from the
receiver to the transmitter. During training, setting X
1
= 0 in
(1), we get that
Y
1
= δX
2
+ Z
1
, (3)
where δ is the unknown gain that has to be estimated.
Let X
2
[k], k = 1, . . . , T denote the training sequence.
The two sequences X
2
[k] and Y
2
[k] are known, and δ has
to be estimated. A sample average of
Y
1
[k]X
2
[k]
∗
|X
2
[k]|
can be
used to obtain an estimate of δ with a mean square error
of approximately T
−1
. Let us denote such an estimate as
ˆ
δ.
Subtracting
ˆ
δX
2
from the RHS of (3), we get
¯
Y
1
=
¯
δX
2
+ Z
1
, (4)
where the residual error
¯
δ tends to a Gaussian random variable
with variance T
−1
. We also assume a large coherence time
T
c
after which the channel might change necessitating a new
estimate.
In summary, after the training, the equations for Model 1
are as follows:
¯
Y
1
= X
1
+
¯
δX
2
+ Z
1
, (5)
Y
2
= X
2
+ Z
2
. (6)
An achievable rate region for Model 1, as represented in (5)
and (6), can be obtained by treating
¯
δX
2
as noise in (1).
Assuming a power constraint of P
1
to the input signal X
1
and
a power constraint of P
2
for X
2
, an achievable rate region
with Gaussian codebooks is given by
R
1
≤
1 −
T
T
c
E
"
F
P
1
1 +
X
2
2
T
!#
,
R
2
≤ F (P
2
). (7)
It is easy to observe that there is an optimal T
∗
(T
c
) which
maximizes the rate R
1
, a standard problem in non-coherent
communications. See Figure 4.
0 50 100 150
0
5
10
15
20
25
T
c
T
*
(T
c
)
Optimal training time and correpsonding rate R
1
for P
1
=200
0 50 100 150
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
R
1
: Rate of the first user for T=T
*
(T
c
)
Fig. 4. The curve without markers (in red) corresponds to T
∗
(T
c
) versus T
for P
1
= 200. The other curve is the corresponding rate R
1
for the optimal
threshold.
B. Rate region of Model 2
In Model 2, δ is constant and known to both the nodes T
and R. We provide an achievable region for this model by
neglecting the side-information from R to T. Currently, even
in practical full-duplex systems, this feedback is not used.
We provide the results for the discrete memoryless chan-
nel (DMC) corresponding to Fig. 3. This DMC is char-
acterized by the probability distribution p(y
1
, y
2
|x
1
, x
2
) =
p(y
1
|x
1
, x
2
)p(y
2
|x
2
). The capacity region of this channel even
without side-information is not known. Most of the known
achievable schemes are based on the Han-Kobayashi achiev-
able scheme [17] for the interference channel. We modify this
Han-Kobayashi scheme to incorporate the side-information
from T to R.
The transmitter T splits its codebook into two parts of
rates R
21
and R
22
, the common and the private information.
The intended receiver B decodes both parts targetting a rate
R
2
= R
21
+ R
22
. On the other hand, the receiver R decodes
information from its intended transmitter A and also the part
of information from T corresponding to rate R
21
akin to
a MAC channel. We use the side-information channel to
provide additional information to R about the public codeword
corresponding to rate R
21
.
Theorem 1. Let P be the set of distributions of the form
p(x
1
, w
2
, u
2
, q) = p(x
1
|q)p(w
2
|q)P(u
2
|q)p(q). Let R
p
denote
the set of all rate pairs (R
1
, R
2
) (R
2
= R
21
+ R
22
) satisfying
the following inequalities:
R
1
≤I(X
1
; Y
1
|U
2
, Q),
R
21
≤I(U
2
; Y
1
|X
1
, Q) + C
0
,
R
1
+ R
21
≤I(X
1
, U
2
; Y
1
|Q) + C
0
,
R
21
≤I(U
2
; Y
2
|W
2
, Q),
R
22
≤I(W
2
; Y
2
|U
2
, Q),
R
21
+ R
22
≤I(U
2
, W
2
; Y
2
|Q).
Then any rate pair in the closure of ∪
p∈P
R
p
is achievable
under Model 1.
Proof: When C
0
= 0, the above rate region corresponds
to the Han-Kobayashi region for the Z-channel. We do the
following to utilize the side-information:
1) Encoding: Encoding is exactly the same as in Han-
Kobayashi achievability scheme.
2) Random binning: Each of the 2
nR
21
codewords corre-
sponding to the public codebook is randomly assigned
an integer from the set {1, 2, . . . , 2
nC
0
}. Denote the set
of codewords which are assigned integer i by S
i
. The
sets S
i
are revealed apriori to R.
3) The transmitter sends the integer corresponding to the
public codebook using the side-information channel of
capacity C
0
.
4) As in Han-Kobayashi achievability scheme, the decoder
at R, does a joint typicality decoding of (Y
1
, X
1
, U
2
),
where U
2
corresponds to the public codebook. In our
case, the decoder takes the intersection of the sequences
(X
1
, U
2
) which are jointly typical with Y
1
and the set
S
k
, where k is the integer associated with the transmitted
public information.
At R, since the side information is used to remove the
ambiguity of public message it can easily shown that the rate
of the public codebook can be increased by C
0
compared to
the standard Han-Kobayashi scheme.
Using Fourier-Motzkin elimination [18], the region R
p
can
be simplified to
R
1
≤I(X
1
; Y
1
|U
2
, Q),
R
2
≤I(U
2
, W
2
; Y
2
, Q),
R
2
≤I(U
2
; Y
2
|W
2
, Q) + I(W
2
; Y
2
|U
2
, Q),
R
2
≤I(W
2
; Y
2
|U
2
, Q) + I(U
2
; Y
1
; X
1
, Q) + C
0
,
R
1
+ R
2
≤I(U
2
, X
1
; Y
1
|Q) + I(W
2
; Y
2
|U
2
, Q) + C
0
.
In the above inequalities, Q is the time sharing parameter.
We now evaluate the inequalities when the DMC is replaced
with an AWGN channel with power constraints and for a
time sharing variable that takes a single value Q = φ with
probability one. The AWGN channel model is given by
Y
1
= X
1
+ δ(U
2
+ W
2
) + Z
1
,
Y
2
= U
2
+ W
2
+ Z
2
,
where Z
1
and Z
2
are unit variance independent Gaussian
random variables. In particular, observe that we have assumed
X
2
= U
2
+ W
2
, although the rate splitting can be done in any
arbitrary way. We split the input power of X
2
as E[U
2
2
] ≤ λP
2
and E[W
2
2
] ≤ (1 − λ)P
2
, where 0 ≤ λ ≤ 1. Then the
pentagonal region given by inequalities in Theorem 1 equals
R
1
≤F
P
1
1 + δ
2
(1 − λ)P
2
,
R
2
≤F (P
2
) ,
R
2
≤F
δ
2
λP
2
1 + δ
2
(1 − λ)P
2
+ F (1 + (1 − λ)P
2
) + C
0
,
R
1
+ R
2
≤F
P
1
+ δ
2
λP
2
1 + δ
2
(1 − λ)P
2
+ F (1 + (1 − λ)P
2
) + C
0
.
It is well known that if δ > 1, then the Han-Kobayashi scheme
is optimal, i.e., achieves capacity. However, this region might
be less than the capacity region achieved without interference.
The next corollary characterizes the required rate of side
information for the node to act as an ideal-full duplex node.
Corollary 1. The capacity of the full-duplex node with limited
side-information equals the capacity of an ideal full duplex
node if the rate of the feedback channel from T to R
C
0
≥ C
∗
= F (P
1
) + F (P
2
) − F (P
1
+ δ
2
P
2
),
and is achieved by setting λ = 1.
Proof: We first choose λ = 1. The proof follows by
setting C
0
such that
F (P
1
) + F (P
2
) = F (P
1
+ δ
2
P
2
) + C
0
,
and observing that the region R
1
≤ F (P
1
) and R
2
≤ F (P
2
)
can be achieved.
Observe that C
∗
< F (P
2
), i.e., the side-information channel
can have a capacity that is less than the maximum transmission
rate of the T to B link. From a design perspective, it implies
that the capacity of an ideal-duplex node can be realized, even
when the side information is low rate and quantized (or non
ideal).
In Fig. 5, the achievable rate regions for the AWGN Z-
channel are presented for C
0
= 1.2 and C
0
= 3.6. Using
Corollary 1, we obtain C
∗
≈ 0.77 for this configuration. The
right figure in Fig. 5 correspond to C
0
= 3.6 > C
∗
and
hence the achievable region coincides with that of an ideal
full-duplex node. We also observe that the achievable regions
are always larger than those corresponding to a half-duplex
node.
Fig. 5. Illustration of achievable rate regions for the AWGN Z-channel
with side-information. The dark region corresponds to the rate region of the
full-duplex node with the modified Han-Kobayashi achievablity scheme. The
region within the dashed curve corresponds to the achievable rate region of
a half-duplex node. We have chosen P
1
= P
2
= 200, δ = 0.7. The left
figure corresponds to C
0
= 1.2 and the right figure corresponds to C
0
=
3.6 > C
∗
≈ 3.54. The jagged edges are because of numerical limitations in
computing unions of several pentagonal regions.
C. Relating Model 1 with Model 2
We now provide a heuristic relationship between both the
models. Observe that T
−1
in Model 1 denotes the ambiguity in
knowing the gain of the self interference loop. This ambiguity
results in interference which cannot be removed. Also for
this model, a capacity achieving scheme is to treat this self-
interference as noise. Hence, while not exact, we can consider
the following set of input-output equations for Model1.
Y
1
= X
1
+ X
2
+ N
2
,
with side-information Y
s
= X
2
+ N
s
where N
s
∼ N
0,
P
2
T
.
The receiver can now subtract Y
s
from Y
1
resulting in ”almost”
the same capacity given by (7).
Now it is easy to relate this model with Model 2 which has
limited rate side-information. Using rate-distortion theory, it is
easy to see that C
0
should be at least
1
2
log(P
2
/(P
2
/T
∗
(T
c
)))
so that the distortion in the side information is less than
P
2
. So Models 1 and 2 can be related by setting C
0
=
1
2
log(T
∗
(T
c
)). In Fig. 6, we plot the rate regions obtained
by Model 1 and Model 2.
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