# Two-Way Transmission Capacity of Wireless Ad-hoc Networks

###### Abstract

The transmission capacity of an ad-hoc network is the maximum density of active transmitters per unit area, given an outage constraint at each receiver for a fixed rate of transmission. Most prior work on finding the transmission capacity of ad-hoc networks has focused only on one-way communication where a source communicates with a destination and no data is sent from the destination to the source. In practice, however, two-way or bidirectional data transmission is required to support control functions like packet acknowledgements and channel feedback. This paper extends the concept of transmission capacity to two-way wireless ad-hoc networks by incorporating the concept of a two-way outage with different rate requirements in both directions. Tight upper and lower bounds on the two-way transmission capacity are derived for frequency division duplexing. The derived bounds are used to derive the optimal solution for bidirectional bandwidth allocation that maximizes the two-way transmission capacity, which is shown to perform better than allocating bandwidth proportional to the desired rate in both directions. Using the proposed two-way transmission capacity framework, a lower bound on the two-way transmission capacity with transmit beamforming using limited feedback is derived as a function of bandwidth, and bits allocated for feedback.

## I Introduction

The transmission capacity of an ad-hoc wireless network is the maximum allowable spatial density of transmitting nodes, satisfying a per transmitter receiver rate, and outage probability constraint [1, 2, 3, 4]. Essentially, the transmission capacity characterizes the maximum number of transmissions per unit area that can be simultaneously supported in an ad-hoc network under a quality of service constraint. The transmission capacity framework allows the application of the rich tool set of stochastic geometry to derive closed-form bounds for the interference distribution in a spatial network when the locations of nodes form a Poisson point process (PPP) [5].

In prior work, the transmission capacity has been used successfully to characterize the effect of various physical and medium access (MAC) layer techniques on the ad-hoc network capacity, such as successive interference cancelation [6], multiple antennas [7, 8, 9, 10], and guard-zone based scheduling [11]. Most of the prior work on finding the transmission capacity has been limited to one-way communication (no data communication from the destination to the source), and precludes the possibility of two-way communication. In two-way (bidirectional) communication the destination also has data to send to its source, e.g. channel state feedback [12], packet acknowledgement [13], or route initiation and update requests [14].

In this paper we define the two-way transmission capacity, and derive tight upper and lower bounds on it when the transmitter location are distributed as a Poisson point process (PPP) distributed. The bounds are used to characterize the dependence of the two-way transmission capacity on the key system parameters, e.g. bandwidth allocation in two directions given a data rate requirement. We consider an ad-hoc network with two-way communication, where each source destination pair has data to exchange in both directions. We consider a general system where the data requirement in both directions can be different, and a frequency division duplex (FDD) communication model, where two separate frequency carriers are used for two directions, thereby forming a full-duplex link.

In a two-way communication model, where the transmitter locations are modeled as a PPP, the interference received in both directions is correlated, and hence the joint success probability in two directions is not equal to the product of the success probabilities in each direction. Therefore finding the exact expression for the joint success probability is complicated. To obtain meaningful insights on the two-way transmission capacity, we derive tight upper and lower bounds on the two-way transmission capacity with FDD, assuming that the channel coefficients on separate frequencies are independent and all the channel coefficients are Rayleigh distributed. The upper and lower bound only differ by a constant, i.e. the bounds have identical dependence on the parameters of interest (rate requirements, and bandwidths allocated in each direction). Thus, the derived bounds establish the two-way transmission capacity up to a constant.

The results of this paper in part have been presented in [15, 16]. The differences between [15, 16] and the present paper are as follows. For simplification of analysis, [15] assumed that the interference received in both directions is independent. The independence assumption was removed in [16], and upper and lower bounds on the two-way transmission capacity that derived which were shown to be tight. Compared to [16], the present paper extends the two-way transmission capacity framework to quantify the loss in transmission capacity with practical limited feedback [17] in comparison to genie-aided feedback (channel coefficients are known exactly, and without any cost at the transmitter), when the transmitter is equipped with multiple antenna and uses beamforming to transmit its signal to the receiver. In addition to this, the present paper offers more clarity of exposition, complete proof of Theorem , and added simulation results for more insights into the effects of two-way communication.

Using the derived bounds on the two-way transmission capacity, we find the optimal bandwidth allocation in two directions that maximizes the transmission capacity. The optimal bandwidth allocation problem is shown to be a convex program in a single variable which can be solved easily by finding the value where the function derivative is zero. Using the optimal bandwidth allocation solution, we show that an intuitive strategy that allocates the bandwidth in proportion to the desired rate in each direction is optimal only for symmetric traffic (same rate requirement in both directions) and performs poorly for asymmetric traffic in comparison to the optimal strategy. Examples of asymmetric traffic are channel feedback, and ack/nack messages, where there is huge disparity between the data rates required in two directions.

There is extensive related work on resource allocation in wireless ad hoc networks, but almost all of it focused on one-way communication. For instance, prior work studied the spectrum sharing between two one-way spatial networks in [18], between a spatial network and a cellular uplink network in [19], and one-way spatial networks where the total bandwidth is optimally split into sub-bands to maximize the transmission capacity [20]. Our bandwidth allocation, however, studies the bandwidth sharing between two directions within a single two-way spatial network.

As an application of the proposed two-way transmission capacity framework, we evaluate the performance degradation with practical limited channel feedback in comparison to genie aided channel feedback, when the transmitter has multiple antennas and uses beamforming for transmitting its signal to the receiver. We account for both the bandwidth used, and the bits required for feedback, to derive a lower bound on the two-way transmission capacity with transmit beamforming using limited feedback. We show that with practical limited channel feedback, the two-way transmission capacity is substantially reduction compared to the genie-aided case. The severe degradation results because with increasing the number of feedback bits, the transmission capacity increases sub-linearly due to improvement in signal strength, however, decreases exponentially because of the stringent requirement of feedback bits to be correctly decoded.

Notation: The expectation of function with respect to is denoted by . A circularly symmetric complex Gaussian random variable with zero mean and variance is denoted as . Let be a set and be a subset of . Then denotes the set of elements of that do not belong to . The integral is denoted by . We use the symbol to define a variable.

## Ii System Model

Consider an ad-hoc network with two sets of nodes and , where and want to exchange data between each other for each . We assume that each and have a single antenna. We consider a slotted Aloha random access protocol, where at any given time, the pair transmits data to each other with an access probability for each , independently of all other nodes. We assume that the distance between each and is . Let the location of be , and be . The set is modeled as a homogenous PPP on a two-dimensional plane with intensity , similar to [1, 2, 9]. Since is at a fixed distance in a random direction from the , the set is also a homogenous PPP on a two-dimensional plane with intensity , because of the random translation invariance property of PPP [21]. Because of the assumed Aloha random access protocol, at any given time, the active transmitter receiver location processes , and are homogenous PPPs on a two-dimensional plane with intensity . We consider a frequency division duplex system, where the total available bandwidth is , out of which is dedicated for communication to support a rate demand bits for all , and the rest for the communication to support a rate demand of bits for all .

In a time slot when the pair is active, the received signal at receiver is

(1) |

and the received signal at receiver is

(2) |

where is the transmit power, is the channel between and , and and is the channel from and , and is the channel between and , and and , respectively, and are the distances between and , and and , respectively, is the path loss exponent, are signals transmitted from and , respectively, and is the additive white Gaussian noise. The ad-hoc network is assumed to be interference limited [1], thus we drop the noise contribution from the received signal. We assume that , , and are independent and identically distributed with to model a Rayleigh fading channel.

With the received signal model (1) and (2), the signal to interference ratio (SIR) for the transmission from and from are

Assuming interference as noise, the mutual information [22] for the to communication using bandwidth , and for the to communication using bandwidth are

Recall that the rate requirement for the transmission is bits, and for the communication is bits. Thus, to account for the two-way or bidirectional nature of communication, we define the success probability (complement of the outage probability ) as the probability that communication in both directions is successful simultaneously, i.e.

(3) |

Let be maximum density of nodes per unit area that can support rate from , and bits from with success probability , using bandwidth .

###### Definition 1

The two-way transmission capacity is defined as

The problem to solve is to find the and consequently for a given rate , outage probability and bandwidth .

To compute the success probability we consider a typical transmitter receiver pair . Using the stationarity of the homogenous PPP and Slivnyak’s Theorem [19] (Page 121), it follows that the statistics of the signal received at the typical receiver are identical to that of any other receiver. Hence the outage probability is invariant with the choice of the receiver. Slivnyak’s Theorem also states that the locations of the interferers for the typical transmitter and receiver , i.e. and are also homogenous PPPs, each with intensity .

## Iii Computing the Two-Way Transmission Capacity

In this section we derive an upper and lower bound on the two-way transmission capacity. To derive a lower bound we use the Fortuin, Kastelyn, Ginibre (FKG) inequality [23], while for deriving an upper bound we make use of the Cauchy-Schwartz inequality. Before stating the FKG inequality, we need the following definitions.

###### Definition 2

A random variable defined on a probability space is called increasing if whenever , for some partial ordering on . is called decreasing if is increasing.

###### Example 1

and are decreasing random variables.

For the PPP under consideration, let where for ,

Then , if , i.e. configuration contains at least those interferers which are present in configuration . Recall our definition of . Clearly, if there are more interferers present, decreases, i.e. considering as a random variable , if . Thus is a decreasing random variable and so is .

###### Definition 3

Let be an event in , and be the indicator function of . Then the event is called increasing if , whenever . The event is called decreasing if its complement is increasing.

###### Example 2

The success event is a decreasing event, since if and , then .

###### Lemma 1

(FKG Inequality [23])

(a) If both and are increasing or decreasing random variables with , and , then
.

(b) If both are increasing or decreasing events then .

Now we are ready to derive bounds on the two-way transmission capacity. From (3), the success probability is

Let , , , and . Then,

(4) | |||||

where follows since , and and are independent, and follows by taking the expectation with respect to , and , and noting that , and are independent and exponentially distributed.

The difficulty in evaluating the expectation with respect to and in the success probability (4) lies in the fact that and are not independent. To visualize this, consider a network where there are only two active pairs of nodes, , and as depicted in Figure 1. For the receiver receiving over bandwidth , the transmission from is interference. As defined before, the distance between and be . Thus, the interference power at is . Similarly, for receiving over bandwidth , the transmission from is interference. The distance between and be . Thus, the interference power at is . For the case when is very small , , and thus distances and are not independent. Moreover, explicitly computing the correlation between and is also a hard problem. Thus, to get a meaningful insight into the two-way transmission capacity we derive a lower and upper bound.

Lower Bound: Similar to Example 1, and are decreasing random variables, since each term in the product is less than , and with the increasing the number of terms (number of interferers) in the product the total value of each expression decreases. Thus, using Lemma 1, from (4)

(6) |

where follows from the probability generating functional of the Poisson point process [24, Example 4.2], and is a constant, where is co-secant.

Upper Bound: Using the Cauchy-Schwartz inequality, from (4)

(7) | |||||

where follows from the probability generating functional of the Poisson point process [24, Example 4.2], and is a constant, different from the constant of the lower bound.

###### Theorem 1

The two-way transmission capacity is upper and lower bounded by

where and are constants, and .

Discussion: In this section we derived an upper and lower bound on the two-way transmission capacity. The upper and lower bound only differ by a constant, and, most importantly, both have identical dependence on the parameters of interest in the two-way communication, and . Thus, the derived bounds establish the two-way transmission capacity up to a constant. The derived upper and lower bounds for the two-way transmission capacity are in a fairly simple form and can be used to calculate the two-way transmission capacity for given rates , , success probability , and . Since the upper and lower bound are identical functions of and , an added advantage of our bounds on the two-way transmission capacity expression is that they can be used to find the optimal value of for given rates , , success probability , and . The optimal bandwidth allocation that maximizes the two-way transmission capacity is derived next in the Section IV.

## Iv Two-Way Bandwidth Allocation

In Section III, we derived the two-way transmission capacity of ad-hoc networks within a constant as a function of bandwidth allocated to the and connections. Since the total bandwidth is finite, an important question to answer is: what is the optimal bandwidth allocation between that maximizes the transmission capacity? For the special case of equal rate requirement in both directions, i.e. , equal bandwidth allocation is optimal. For the non-symmetric case, however, the answer is not that obvious and is derived in the following theorem.

###### Theorem 2

The optimum bidirectional bandwidth allocation that maximizes the transmission capacity with two-way communication is and where is the unique positive solution to the following equation:

(8) |

where for .

Proof: Neglecting the constant, the two-way transmission capacity is

To derive the optimal bandwidth partitioning, i.e. the optimal that maximizes , we need to minimize .

Let . Let . Thus, the problem we need to solve is

The first-order derivative of is , where for . The second-order derivative of is . Since is monotonically increasing in over , then we have for all . Therefore, for all . This means that is a convex function of over and its minimum corresponds to that is the unique positive solution of the following equation , or equivalently, . \QED

Discussion: In Theorem 2 we derived the optimal bandwidth allocation for two-way communication in ad-hoc networks that maximizes the transmission capacity. The result is derived by showing that the optimization problem is convex in one variable, hence the optimal solution corresponds to the value for which the function derivative is zero.

Using Theorem 2, if the traffic is symmetric, i.e., , the optimal strategy is naturally allocate equal bandwidths for two directions with . This result is intuitive since the counterpart parameters in two directions are equal. For asymmetric traffic , however, allocating bandwidths proportional to the desired rate in each direction does not satisfy (8). Thus the proportional bandwidth allocation policy is not optimal for asymmetric traffic for maximizing the transmission capacity, and (8) must be satisfied to find the optimal policy.

## V Effect of Limited Feedback on Two-Way Transmission Capacity with Beamforming

In this section we consider an ad-hoc network where each transmitter is equipped with antennas while each receiver has a single antenna. All other system parameters and assumptions remain the same as defined in Section II. With multiple transmit antennas, and channel state information CSI at each transmitter, transmission rate can be increased by transmitting the signal along the strongest eigenmode of the channel (called beamforming). Beamforming, however, requires that the transmitter know the channel coefficients, which in general is a challenging problem. In a FDD system, the transmitter can learn the channel coefficients, or equivalently the optimum beamformer, through the use of a finite rate feedback channel from the receiver. Assuming a genie aided feedback (channel coefficients are exactly known at the transmitter, and without accounting for the feedback bandwidth, and SIR required for the feedback), [7] derived the transmission capacity with beamforming, and showed that the transmission capacity increases as with increasing . In reality, however, feedback requires sufficient bandwidth, and the channel coefficients can be fed back only up to a certain precision.

Limited feedback techniques [25] are commonly used in practical systems to exploit finite rate feedback channels. With limited feedback, a beamforming codebook is assumed known to both the receiver and the transmitter. The receiver computes the best beamforming vector from the beamforming codebook and sends the index of this vector back to the transmitter. The larger the codebook size, the better is the quality of feedback, and consequently better is the data rate from the transmitter to the receiver with beamforming. With a codebook size of , each codeword requires bits of feedback. Thus, the use of a large codebook increases the required bandwidth for the feedback channel, thereby restricting the bandwidth allocated for transmitter to receiver communication. Thus, there is a three-fold tradeoff between the bandwidth allocated in forward channel, the feedback channel, and the size of the codebook. In this section, we quantify this tradeoff and evaluate its effect on the two-way transmission capacity.

The received signal at receiver over bandwidth is

where is the transmit power of each transmitter, are the beamformers used by , is the channel between and , is the channel between and , is the distance between and , and are the data symbols transmitted from and , respectively. For simplicity we assume that each receiver computes the beamforming vectors only depending on , independent of the interferers’ channels.

The received signal at transmitter corresponding to the feedback by receiver over bandwidth is

(9) |

where is the channel between and , is the channel between and , and are the feedback signals transmitted by and , respectively.

With genie-aided feedback, the optimal beamforming vector is known to be . In practice, however, only a finite number of bits are available for feedback, and hence can be modeled as , where is the additive error term which represents the uncertainty due to limited feedback. The quantization error degrades the signal power compared to genie aided feedback. With bits of feedback bits, the signal power [26] is ( is a constant), compared to for genie aided feedback (). Thus, the SIR for to communication with bits of feedback is

(10) |

and the corresponding mutual information from to using bandwidth is

Similarly, the SIR for the feedback link is , and thus with bandwidth , the mutual information of the feedback link is

Similar to (3), we define the success probability as the probability that communication in both directions is successful simultaneously, i.e.

Consequently, with the two-way transmission capacity is defined as

As stated before, in a two-way communication model, where the transmitter locations are modeled as a PPP, the interference received in both directions is correlated. Therefore, computing the success probability in closed form is a hard problem. To derive a meaningful insight into the dependence of bandwidth allocation, and feedback bits on two-way transmission capacity, we derive a lower bound on the success probability using the FKG inequality as follows.

###### Theorem 3

Accounting for feedback bandwidth, the two-way transmission capacity with beamforming is lower bounded by

where , and

with .

Proof:

Similar to Example 3, the success events in two directions , and , respectively, are decreasing events. Thus, invoking Lemma 1,

By definition,

where follows from the definition of , follows by substituting for (10), follows by defining , and follows from Theorem [7].

Directly applying Theorem [7], , where

Thus, . Then,

Discussion: In this section we derived a lower bound on the two-way transmission capacity when the transmitter uses beamforming with limited feedback, as a function of the bandwidth allocated in two directions, and the number of feedback bits. Note that as (the number of feedback bits) increases, the two-way transmission capacity increases as