# Physical Network Coding in Two–Way Wireless Relay Channels

###### Abstract

It has recently been recognized that the wireless networks represent a fertile ground for devising communication modes based on network coding. A particularly suitable application of the network coding arises for the two–way relay channels, where two nodes communicate with each other assisted by using a third, relay node. Such a scenario enables application of physical network coding, where the network coding is either done (a) jointly with the channel coding or (b) through physical combining of the communication flows over the multiple access channel. In this paper we first group the existing schemes for physical network coding into two generic schemes, termed 3–step and 2–step scheme, respectively. We investigate the conditions for maximization of the two–way rate for each individual scheme: (1) the Decode–and–Forward (DF) 3–step schemes (2) three different schemes with two steps: Amplify–and–Forward (AF), JDF and Denoise–and–Forward (DNF). While the DNF scheme has a potential to offer the best two–way rate, the most interesting result of the paper is that, for some SNR configurations of the source—relay links, JDF yields identical maximal two–way rate as the upper bound on the rate for DNF.

## I Introduction

It has been recently noted [1] that broadcast and unreliable nature of the wireless medium sets a fertile ground for developing network–coding [2] solutions. The network coding can offer performance improvement in the wireless networks for two–way (or multi–way) communication flows [3] [4] [5] [6] [7] [8] [9]. In general, there are two generic schemes for two–way wireless relay (Fig. 1): (a) 3–step scheme (b) 2–step scheme.

The node has packets for the node and vice versa. In Step
1 of the 3–step scheme, transmits the packet
, in Step 2 transmits the packet
. Here decodes both packets, such that the
3–step schemes are Decode–and–Forward (DF) schemes. In
the simpler DF schemes [3]
[4] [5], the direct link
between and is ignored by the receivers in Steps 1
and 2, such that in Step 3 broadcasts the packet
, where is `XOR`

operation, after
which the node () is able to decode the packet
. While it is hard to
characterize such a simple DF scheme as “physical” network
coding, such an attribute can be attached to the 3–step DF
scheme [7], where the direct link is not
ignored in the Steps 1 and 2 and a joint network–channel coding
is needed. In that case, the packet is a many–to–one function of the packet
, since () already has
some information from the Step 2 (1). In the 2–step schemes the
communication flows are combined through a simultaneous
transmission over a multiple access channel. In Step 1
receives a noisy signal that consists of interference between the
signals of and . Due to the half–duplex operation, the
direct link is naturally ignored in the 2–step schemes. The
signal that is broadcasted in Step 2 depends on the
applied 2–step scheme. In Amplify–and–Forward
(AF) [5], is simply an amplified
version of the signal received by in step 1. After receiving
, the node () subtracts its own signal and
decodes the signal sent by () in Step 1. The 2–step scheme
termed Denoise–and–Forward (DNF) has been introduced
in [6]. A related scheme appeared
in [10]. In DNF, the node again does not
decode the packets sent by and in Step 1, but it maps the
received signal to a codeword from a discrete set. Hence, the
signal carries now the information about the set of
codeword pairs which are
considered by the node as likely to have been sent in the Step
1. In general, this set can consist of several codeword pairs,
such that has an ambiguity which information has been sent.
Nevertheless, since () knows , after receiving , it will
extract exactly one codeword as a likely one to have been sent by
() in Step 1. The final considered 2–step scheme is
Joint Decode–and Forward (JDF), recently considered
in [9]. In JDF, the transmission rates in Step 1 of
Fig. 1(b) are selected such that can jointly
decode both and , and then use
`XOR`

to obtain the signal for broadcast in Step 2.

In this paper we investigate the strategies that can maximize the overall two–way rate for several 2– and 3–step schemes for physical network coding. We show that the key to maximizing the two–way rate in the system for the 3–step schemes is the relation between the durations of Step 1 and Step 2. On the other hand, we show that the key factor for maximizing the two–way rate in the 2–step schemes is the choice of the rates at which and transmit in Step 1. Note that we are not providing the absolute capacities of the two–way relay channel, since we are putting some operational restrictions to the applied schemes. Nevertheless, the results give an excellent overview of what can be achieved by each scheme for physical network coding.

## Ii Notations and Definitions

We assume that there are only two communication flows, and , respectively. The relay is neither a source nor a sink of any data in the system. All the nodes are half–duplex, such that a node can either transmit or receive at a given time. We use to denote the th complex baseband transmitted symbol from node . A complex–valued vector is denoted by . A packet of bits is denoted by D, and the number of bits in the packet is . If only one node is transmitting, then the th received symbol at the node is given by:

(1) |

where is the complex channel coefficient between and . is the complex additive white Gaussian noise . The transmitted symbols have and a normalized power . Each node uses the same transmission power, which makes the links symmetric:

(2) |

We consider time–invariant channels and are perfectly known by all nodes. This assumption allows us to find the two–way rates at which a reliable communication is possible. The bandwidth is normalized, such that we consider the following signal–to–noise ratios (SNRs):

(3) |

The bandwidth is normalized to 1 Hz, such that a link with SNR of can reliably transfer up to:

(4) |

The time is measured in number of symbols, such that when a packet of symbols is sent at the data rate , the packet contains bits. The packet lengths are sufficiently large, such that we can use codebooks that offer zero errors if the rate is chosen to be below the channel capacity.

Without loss of generality, we assume that

(5) |

The source–to–relay links are assumed better than the direct link [11]:

(6) |

If and transmit simultaneously, then receives:

(7) |

In this paper we will be interested in the two–way rate:

###### Definition 1

Let, during a time of symbols, receive reliably bits from and receive reliably bits from . Then the two–way rate is given by:

(8) |

We seek to maximize the two–way rate under the following two operational restrictions. First, in each round and transmit only fresh data, which is independent of any information exchange that took part in the previous rounds. Second, is applying potentially suboptimal broadcast strategy, as we have not explicitly considered the broadcast strategies that achieve the full capacity region of the Gaussian broadcast channel [12]. Hence, the obtained two–way rates are lower bounds on the achievable rates in the two–way relay systems.

## Iii 3–Step Scheme

A single round in a 3–step scheme is (Fig. 1(a)): Step 1: Node transmits, nodes and receive. Step 2: Node transmits, nodes and receive. Step 3: Node transmits, nodes and receive. In this scheme, should decode the data transmitted by node (node ) in Step 1 (Step 2). The data transmitted by in Step 2 is independent of the data received from in Step 1. The data transmitted by the node in Step 3 is a function of the data that was transmitted by and in Step 1 and 2, respectively, from the same round.

We first determine the size of the data broadcasted by . If is transmitting symbols at a data rate , then receives reliably the packet of bits. At the same time, the total amount of information received at the node is bits, where , due to (6). Hence, in the next step the relay needs to transmit at least:

(9) |

bits to in order to completely remove the uncertainty at about the message transmitted by . It is crucial to note that the node knows the content of the packet . The argument to show this is that, after receives , both and have the same information and no information what has been received at . Even then, the random binning technique [12] can be used to create the packet , such that is uniquely and in advance determined for each .

Let the node in Step 1 transmit a packet of symbols at a rate , where . Upon successfully decoding , the relay node prepares that needs to be forwarded to , with a packet size of:

(10) |

During the next symbols, in Step 2, the node transmits at a rate , out of which creates with:

(11) |

It follows from above that knows and knows . In addition, the node does not know , but it knows a priori the size of the packet . The same is valid for and the packet size . This is reasonable for the assumed time–invariant systems with fixed .

###### Theorem 1

The maximal two–way rate for DF is

(12) |

where .

In Step 3, the node first compares the packet sizes and . Two cases can occur:

#### Iii-1 Case 1:

Using (10) and (11), we can translate this condition into inequality for :

(13) |

The relay partitions the packet into and :

(14) |

consists of the first bits from and consists of the rest of the bits from . Now creates:

(15) |

where is bitwise `XOR`

. Due to the
condition (5) and the fact that both
and need to receive it, the packet is
transmitted at the lower rate . After receiving
, the node extracts the packet
as . This packet is then used together with the
information that has received from node in Step 2 to
decode the packet . On the other hand, after
receiving , the node extracts
. Now
transmits the packet to the node
at a higher rate of , as does not need to receive
this information. With and
, the node creates ,
which is further on used jointly with the information that has
received in Step 1 to decode the packet . The
total duration of the three steps is ,
resulting in a two–way rate of:

(16) |

where and are functions of and are given by (10) and (11), respectively. It can be proved that is monotonically increasing function of , such that achieves its maximal value for the upper limiting value of , given in (13). By applying into the terms of (16), we obtain the two–way rate given by (12).

#### Iii-2 Case 2:

This is the region:

(17) |

The packet is padded with zeros to obtain the packet such that . Since and know the size of , they also know how many zeros are used for padding. The node creates the packet . In Step 3 only the packet is broadcasted at a transmission rate . The node extracts as and uses the information received in Step 2 to decode . Similarly, obtains from , removes the padding zeros and obtains , which is then used jointly with the information from Step 1 to decode the packet .

## Iv 2–Step Schemes

In this section we deal with three schemes: Amplify–and Forward (AF), Joint Decode–and–Forward (JDF) and Denoise–and–Forward (DNF). The two steps are: Step 1: Nodes and transmit, node receives. Step 2: Node transmits, nodes and receive.

The transmission rates for and in Step 1 are denoted by and , respectively. As we will see, the choice of and is a feature of each transmission scheme AF, JDF or DNF. Except for the selection of the rate pair rates, the Step 1 is identical for all three schemes, where its duration is fixed to symbols and the th received symbol at node is given by (7).

### Iv-a Amplify–and–Forward (AF)

After Step 1, the node amplifies the received signal for a factor and broadcasts to and . As also consists of symbols, the total duration of the two steps is . The amplification factor is chosen as:

(19) |

to make the the average per–symbol transmitted energy at equal to 1 ( is the noise variance). The th symbol received by in Step 2 is:

Since knows and , it can subtract from and obtain:

(20) |

which is a Gaussian channel for receiving with SNR:

(21) |

This notation denotes that is the SNR that determines the rate at which can communicate to . Similarly, we can find the SNR which determines the rate :

(22) |

Hence, the rate pair used in Step 1 should be:

(23) |

Finally, the two–way rate achieved by the AF scheme is:

(24) |

### Iv-B Joint Decode–and–Forward (JDF)

Here the at rates and are chosen such that the node is able to decode both packets in Step 1. The rate pairs with such a property should lie inside the convex region [12] on Fig. 2. The sum–rate is maximized if the rate pair lies on the segment :

(25) |

while in all other points of the region of achievable rates. The points and are determined as:

(26) |

For the rate pair at , the packet is decoded first, it is then subtracted from the received signal and then is decoded. At the point , these operations are reversed. Any other point on the line has rates

(27) | |||

(28) |

where can be the time–sharing parameter, see [12].

###### Theorem 2

The maximal two–way rate for the joint decode–and–forward (JDF) scheme is

(29) |

The starting point is the fact that the line segment contains at least one rate pair that maximizes the two–way rate. We omit this proof as it can be done in a similar way as the part of the proof that follows. We consider two different cases, one for each region of .

#### Iv-B1 Case

In this region of values for there is a value , such that:

(30) |

i. e. the dashed line on Fig. 2 intersects with the segment . The value of is determined as:

(31) |

There are two subcases:

Subcase . Here and the packet sent by node contains more bits than the packet . After decoding both packets, the node pads the packet with zeros to obtain with and creates:

(32) |

Note again that the nodes and know a priori how many padding zeros are used. Since , in Step 2 of the JDF scheme the node broadcasts at a rate . After receiving , the node obtains and the node obtains and hence obtains . The total number of symbols used in the two steps is , such that the two–way rate is:

(33) |

since (25) holds for each . As decreases with , the value is maximized for , where is given by (31), such that .

Subcase . Here
and hence
. The proof uses similar line
of argument as in case 1 of the proof of theorem 1
and therefore we briefly sketch it. The first part of the packet
is `XOR`

–ed with the packet
and the resulting packet is broadcasted at rate
. Then, the rest of the packet is
broadcasted at a higher rate . The total number of
symbols in the two steps is:

(34) |

This leads to two–way rate of

(35) |

It can be shown that is monotonically decreasing with , while , which proves that the maximal rate is achieved at .

#### Iv-B2 Case

. In this case for any , it holds that . Hence, we can use the transmission method for the subcase , discussed above. The obtained two–way rate is again given by (33), which is monotonically increasing with and attains the maximum for . Hence, the maximal two–way rate is:

(36) |

It can be shown that there are other pairs that achieve the maximal two–way rate. Those pairs lie on the segment , where is the point where . Note that when .

### Iv-C Denoise–and–forward (DNF)

In the first step of this scheme, the nodes and transmit the packets and at rates and but we do not require that the node is able to decode the packets and . During the symbols of Step1, receives the dimensional complex vector , where the th symbol of is given by (7). If the selected rate pair is not achievable for the multiple access channel (i. e. lies outside the convex region on Fig. 2), then cannot find unique pair of codewords , such that the triplet is jointly typical. The concept of joint typicality is rather a standard one in information theory and the reader is referred to [12] for precise definition. For our discussion it is sufficient to say that is jointly typical when the codeword is likely to produce at . When the pair is not achievable over the multiple–access channel, then, upon observing , the node has a set of codeword pairs such that:

(37) |

###### Lemma 1

Let be a typical sequence. Let and be two distinct codeword pairs in . If and , then and can always select the codebooks such that

(38) |

If knows packet of , then can transmit to reliably up to the rate . We prove the lemma by contradiction. Let us assume that the contrary is true: and . Now, assume that, after receiving , the node is told by a genie–helper which is the codeword . Then, would still have ambiguity whether has sent or . But that contradicts the fact that can communicate reliably to at a rate if is known a priori to .

From this lemma it follows that, if in Step 2 manages to send the exact value (with no additional noise) to and , then () will be able to retrieve the packet sent by () in Step 1. In the DNF scheme the node maps to a discrete set of codewords and, in Step 2 it broadcasts the codeword to which is mapped. Such a mapping to discrete codewords is referred to as denoising. Let denote the set of typical sequences , each of size . Let be a set of denoising codewords , where is the cardinality of the set. The denoising is defined through the following mapping:

(39) |

The codewords in are random i. e. selected in a manner that achieves the capacity of the associated Gaussian channel. Upon observing in Step 1, in Step 2 the node broadcasts the codeword . The mapping should have the following property:

###### Property 1

Given the codeword and with known codeword (), the other codeword () can be retrieved unambiguously.

Such a property enables and to successfully decode each other’s packets after Step 2. The important question is: For given from Step 1, what should be the minimal size , such that Property 1 is satisfied? Assume that , then there are possible codewords that can send in Step 1 vs. sent by . Clearly, the cardinality should be at least , because otherwise it is impossible for to reconstruct the codeword sent by . In this paper we conjecture, without proof, that it is always possible to design the denoising by using a set of minimal possible cardinality that can satisfy the Property 1:

(40) |

Such a choice is guaranteed to offer an upper bound on the two–way rate of DNF and is equal to the achievable rate of DNF if the conjecture is valid.

###### Theorem 3

The upper bound on the two–way rate for denoise–and–forward (DNF) is

(41) |

where is the SNR of the weaker link to the relay.

The rate is maximal possible, while the rate , where . After the Step 1, the node maps the received sequence according to the denoising to . As there are denoising codewords, each one is represented by bits. Since both and need to receive it, the codeword needs to be sent at a rate . The total duration of the two steps is which makes the two–way rate:

(42) |

This result implies that the node does not need to “fully load” the channel by setting and any value of will result in the maximal two–way rate. Hence, the higher transmission rate does not improve the two–way rate, as it accumulates more data at which needs to be broadcasted at a low rate in Step 2. Finally, while the JDF scheme achieves a two–way rate of only when , the DNF scheme achieves it even for .

## V Numerical Illustration

Fig. 3 and Fig. 4 depict the two–way rate vs. the SNR . In both figures, the scheme is evaluated for two different values of the SNR on the direct link, and . Fig. 3 shows the results when the SNR of the link is . As expected, the upper bound is always highest for all . While is lower than for low SNRs, at high SNR the noise amplification loses significance and thus AF achieves higher two–way rate than JDF. Also, note that the improvement of the direct link , brings significant increase of the two–way rate in the DF scheme. Fig. 4 shows the results when , the lowest value for at which the rate of JDF becomes equal to teh upper bound for DNF. Clearly, the curve for DNF remains the same as in Fig. 3, while the increased is reflected in improved two–way rates for AF and DF. The improvement is larger for AF, which now slightly outperforms DF with at higher SNRs.

## Vi Conclusion

We have investigated several methods that implement physical network coding for two–way relay channel. We have grouped the physical network coding schemes into two generic groups of 3–step and 2–step schemes, respectively. The 3–step scheme is Decode–and–Forward (DF), while we consider are three 2–step schemes Amplify–and Forward (AF), Joint Decode–and–Forward (JDF) and Denoise–and–Forward (DNF). We have derived the achievable rates for DF, AF, and JDF, as well as an upper bound on the achievable rate of DNF. The numerical results confirm that no scheme can achieve higher two–way rate than the upper bound of DNF. Nevertheless, there are certain SNR configurations of the source–relay links under which the maximal two–way rate of JDF is identical with the uppper bound of DNF. As a future work, we are first going to provide a proof that the upper bound for DNF is achievable. Another important aspect is investigation of the impact that the efficient broadcasting schemes [12] can have on the DF and JDF scheme. It is interesting to investigate how to design a 3–step scheme when the direct link is better than one of the source–relay links. Although some practical DNF methods have been outlined in [6], it is important to investigate how to perform DNF when different modulation/coding methods are applied. Finally, a longer–term goal is to investigate how the physical network coding can be generalized to the scenarios with multiple communicating nodes and multiple relays.

## References

- [1] C. Fragouli, J. Y. Boudec, and J. Widmer, “Network coding: An instant primer,” ACM SIGCOMM Computer Communication Review, vol. 36, no. 1, pp. 63–68, 2006.
- [2] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inf. Theory, vol. IT-46, pp. 1204–1216, 2000.
- [3] Y. Wu, P. A. Chou, and S.-Y. Kung, “Information exchange in wireless networks with network coding and physical-layer broadcast,” in Proc. 39th Annual Conference on Information Sciences and Systems (CISS), Mar. 2005.
- [4] P. Larsson, N. Johansson, and K.-E. Sunell, “Coded bi–directional relaying,” in 5th Scandinavian Workshop on Ad Hoc Networks (ADHOC’05), Stockholm, Sweden, May 2005.
- [5] P. Popovski and H. Yomo, “Bi-directional amplification of throughput in a wireless multi-hop network,” in IEEE 63rd Vehicular Technology Conference (VTC), Melbourne, Australia, May 2006.
- [6] ——, “The anti-packets can increase the achievable throughput of a wireless multi-hop network,” in Proc. IEEE International Conference on Communication (ICC 2006), Istanbul, Turkey, Jun. 2006.
- [7] C. Hausl and J. Hagenauer, “Iterative network and channel decoding for the two-way relay channel,” in Proc. IEEE International Conference on Communication (ICC 2006), Istanbul, Turkey, Jun. 2006.
- [8] S. Katti, H. Rahul., W. Hu, D. Katabi, M. Médard, and J. Crowcroft, “XORs in the Air: Practical Wireless Network Coding,” in Proc. of ACM SIGCOMM 2006 Conference, Sep. 2006.
- [9] B. Rankov and A. Wittneben, “Achievable rate regions for the two-way relay channel,” in Proc. IEEE Int. Symposium on Information Theory (ISIT), Jul.
- [10] L. Xiao, T. E. Fuja, J. Kliewer, and D. J. Costello, Jr., “Nested codes with multiple interpretations,” in Proc. 40th Conference on Information Sciences and Systems (CISS), Princeton, NJ, Mar. 2006.
- [11] T. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inf. Theory, vol. IT-25, pp. 572–584, 1979.
- [12] T. M. Cover and J. A. Thomas, Elements of Information Theory. John Wiley & Sons Inc., 1991.