^{1}

^{1}

^{1}

^{*}

Directional antennas shape transmission patterns to provide greater coverage distance and reduced coverage angle. Use of adaptive directional antenna arrays can minimize interference while also being more energy efficient. When used in an ad-hoc network, this reduces interference among transmitting nodes and thereby increases throughput. Such “smart antennas” use digital beamforming based on signal processing algorithms to compute the appropriate weights to form effective antenna patterns. Smart antennas require the knowledge of the signal received at each antenna in the antenna array, thereby increasing the complexity of hardware and cost. Also, conventional smart antennas optimize results for each individual node, while it is preferable to have a global optimal solution. A problem that has not been addressed is how to compute individual beam patterns that maximize some measure of global network performance. Historically, the focus has been on finding node antenna patterns that give locally optimal performance. In this paper, we investigate a low hardware complexity beamforming approach aimed at improving global performance that uses average Noise-to-Signal ratio as the performance measure. Given a multi-hop route from source to destination, beam patterns are shaped to maximize average signal-to-noise ratio across all nodes on the route, which reduces bit-error rates and extends battery and network lifetime. The antenna weights are sequentially adjusted across all nodes in the route to achieve optimization across the network. By using phase-only weights, hardware costs are minimized. The performance of the algorithm using different path loss models is explored.

Ad hoc networks are wireless networks capable of autonomous communication independent of pre-established infrastructure. Energy efficiency is an important consideration as it determines node and network lifetime. Usually, communication takes place using omni-directional antennas that radiate signals in all directions. This not only wastes transmission power at the node but also acts as a source of interference to other nodes.

Power efficiency can be improved using smart antennas to direct the beam in the desired direction while minimizing gain in interference directions. Smart antennas use digital beamforming (DBF) methods, which require separate transceiver chains, A/D and D/A converters, and DSPs for each antenna in the array (

Analog beamforming (ABF) is a low complexity alternative to smart antennas. The system relies on a single transceiver and power splitter/combiner (

Therefore, the research described here focuses on techniques that use phase-only weights.

Since there is little established previous research on network-optimized antenna beamforming [

For a given route, we assume N nodes are active for relaying packets across a known route, and each node has an M-antenna array of omnidirectional antennas. All other nodes in the network are considered to be interference sources. The SNR at each node can be predicted using simple path loss models (such as the two-ray model or the Walfisch-Ikegami model [

We use SNR averaged over all network nodes as the measure of network performance. This is straightforward to compute and directly relates to network performance metrics such as bit error rate and battery life. To accomplish the optimization, we first find weights w j k , k = 1 , 2 , ⋯ , M for each node j that minimize average network noise-to-signal ratio (NSR). Once all N node weight vectors are known ( w j k , j = 1 , 2 , ⋯ , N ) weights are recomputed iteratively until the average NSR reaches a stable minimum. This provides a beamforming solution that intents to improve the performance of the network globally.

This approach can be simplified since each node in the route only communicates with previous and next hop neighbors and all other nodes are treated as interference. Thus, “average NSR” is taken to be the arithmetic mean over all the nodes along the route. Initially, only one route is considered; network nodes that are not in the current route are assumed to be interference nodes and their beam patterns remain omnidirectional or unchanged if assigned a pattern previously.

This paper is organized as follows: In Section 2, we discuss the related work on adaptive beamforming in ad hoc networks. Section 3 provides background theory on phased array antennas and the optimization technique used to obtain optimized beampattern in this paper. Section 4 explains the beamforming algorithm to provide improved global solution that increases the overall average Signal-to-Noise ratio (SNR) of the network. Section 5 outlines a brief discussion of noise in antenna systems. The simulation and analysis of results of the proposed beamforming algorithm is presented in Section 6. Section 7 concludes this paper with a discussion on performance issues and future work to be done on the developed algorithm.

The problem of finding a global optimal solution to minimize the power consumption or maximize the Signal-to-Noise ratio is challenging in ad hoc networks. Previous researchers [

Thornburg et al. [

The authors of [

Anbaran et al. have proposed a method using smart antennas that delivers beamforming performance close to that of phased array antennas without having any constraints on the antenna spacing, and compare it to the conventional Electrically Steerable Passive Array Radiator (ESPAR) [

The Kalman filter [

In this paper, we provide a low-hardware complexity phased array antenna beamforming technique that provides a network-wide optimized solution to deliver a global improvement in performance. We assume that the optimal route from source to destination is known a-priori, and that it can be obtained from any convenient routing protocol. All the antenna weights are calculated centrally (not in a distributed fashion at individual nodes) to minimize the signal-to-in- terference-plus-noise ratio (SINR) at each node. This makes our approach a separate layer that is independent of the routing protocol used and can be added on top of existing networks. Similar to cooperative techniques, beamforming is done to take into account terrain and node locations, but our method does not require internode communication and the associated overheads. Like smart antennas, we adjust beams to adapt to local terrain and other node signals, but antenna weights are computed off-line and prior to network setup, with periodic updates made as needed. Smart antennas perform local optimization, while our method seeks to optimize globally across the network. Also, hardware complexity of the proposed system is much lower than smart antenna-based radios.

The transmitted signal can be represented as r ( t ) = 2 s ( t ) cos ( ω 0 t ) = s ( t ) e j ω 0 t + s ( t ) e − j ω 0 t , where ω 0 = 2 π f 0 = angularfrequency . The block diagram of the receiver is represented in

A uniform linear array consisting of M antenna elements is shown in

B touches antenna #1 before it touches antenna #0. Let the wavefront at antenna #0 be s 0 ( t ) = s ( t ) e j ω 0 t , then the wavefront at the antenna #1 is s 1 ( t ) = s 0 ( t + Δ t ) . We assume a “low-pass narrow-band” signal s(t) with Bandwidth ≪ f 0 . There-

fore s ( t + Δ t ) ≈ s ( t ) , where Δ t = d cos θ c and c = λ f 0 . Therefore

Δ t = 2 π d cos θ ω 0 λ . The signal at the receiver for antenna m is given as

r m ( t ) ≃ s ( t ) e j ω 0 ( t + m Δ t ) = s ( t ) e j ω 0 ( t + 2 π m d cos θ ω 0 λ ) . (1)

For the entire array, the received signal will be

r ( t ) = s k ( t ) e j ω 0 t [ 1 , e j k d c o s θ , e j 2 k d c o s θ , ⋯ ] T , where k = 2 π λ = wavenumber .

The received signal is represented as r ( t ) = 2 s ( t ) cos ( ω 0 t ) + v ( t ) , where s ( t ) is a narrow band message signal and v ( t ) is white noise. The receiver down-converts the signal resulting in a complex base-band signal y ( t ) . For example, the result for a uniform linear array is:

y ( t ) = [ 1 e j k d cos θ ⋮ e j ( M − 1 ) k d cos θ ] s ( t ) + [ v 0 ( t ) ⋮ v M − 1 ( t ) ]

y ( t ) = h ( θ ) s ( t ) + V , (2)

where k, d, and θ are the wave number, antenna element separation distance, and direction of arrival (DOA), respectively, and h ( θ ) is called the steering vector. Note that h ( θ ) must be modified for each specific antenna array geometry to give proper delay characteristics in the direction θ. Now, it is possible to find a linear filter K that minimizes the effects of noise without distorting the signal.

s ^ = ∑ k i y i = K * y .

The filter output is unbiased as shown by

E [ s ^ ] = E [ K * y ] = E [ K * h ( θ ) s + K * V ] = K * h ( θ ) s = s .

The gain in direction θ is:

G ( θ ) = w T h ( θ ) , (3)

where w is the weight vector which applies to antenna elements and depends on the optimization method.

There are several ways to find the weight vector w . For example, it is possible to find w that minimizes output noise power while holding G ( θ s ) = 1 in signal direction θ_{s}. This is called a “Minimum Variance Distortion-less Response (MVDR)” filter [

For this research, we use the Nelder-Mead (NM) search algorithm to find phase-only weights minimizing a desired fitness function. NM is one of the most widely used methods for nonlinear unconstrained optimization. For example, to generate the optimal gain pattern G o p t ( θ ) , we enforce the phase-only constraint by fixing the weights as w k = e j ϕ k and then use NM to search the error surface | e ( ϕ ) | 2 = ∑ θ | G o p t ( θ ) − G ( θ , ϕ ) | 2 for a minimum. Here, G ( θ , ϕ ) is the gain pattern generated by phase-only weights w = e j ϕ , where ϕ = [ ϕ 1 , ϕ 2 , … , ϕ M ] T . However, in most cases the optimal gain pattern G o p t ( θ ) is not known a-priori so we use NSR as the fitness function. As this directly relates to global network performance and the resulting gain pattern G ( θ , ϕ m i n N S R ) is expected to be a reasonable approximation of G o p t ( θ ) .

The Nelder-Mead (NM) algorithm [_{i} denote the list of points in the current simplex, i = 1 , ⋯ , n + 1 . Because we seek to minimize the function f, x_{1} is referred to as the best point, and x_{n}_{+1} as the worst point. Four scalar parameters reflection (ρ), expansion (χ), contraction (γ), and shrinkage (σ) are specified for Nelder-Mead method.

The following indicates one iteration of the Nelder-Mead algorithm [

・ The n + 1 vertices are ordered such that f ( x 1 ) ≤ f ( x 2 ) ≤ ⋯ ≤ f ( x n + 1 ) .

・ The reflection point, x_{r}, is computed as

x r = x ¯ + ρ ( x ¯ − x n + 1 ) = ( 1 + ρ ) x ¯ − ρ x n + 1 , where x ¯ = ∑ i = 1 n x i n . Evaluate

・ f r = f ( x r ) . If the value f 1 ≤ f r < f n the reflected point x_{r} is accepted and the iteration terminates.

・ The expansion point is computed as x e = x ¯ + χ ( x r − x ¯ ) = x ¯ + ρ χ ( x ¯ − x n + 1 ) = ( 1 + ρ χ ) x ¯ − ρ χ x n + 1 if f_{r} < f_{1} and the value of the function f_{e} at x_{e} is evaluated. The iteration is terminated after retaining either x_{e} (f_{e} < f_{r}) or x_{r} (f_{e} > f_{r}).

・ Contraction is performed by computing the contracted point x c = x ¯ + γ ( x r − x ¯ ) . A new simplex is obtained by using the contracted point x c if it is better than the worst point.

・ The function is evaluated by replacing all the points by v i = x 1 + σ ( x i − x 1 ) , i = 2 , ⋯ , n + 1 , except for the best point. The new vertices x 1 , v 2 , ⋯ , v n + 1 are used for update in the next iteration.

Before discussing performance of NM solutions applied to an entire network, we first focus on individual antenna array performance by comparing NM-based array solutions to the popular ESPAR antenna array described earlier. We modeled an antenna array with 7 antennas arranged in a circular geometry that is similar to the ESPAR antenna used by [

In this section, we describe a technique for finding network-optimized beam patterns (and the associated complex phase-only antenna weights) for all nodes along a route. This is a joint solution, where individual beam patterns depend on the antenna patterns of adjacent nodes. The approach is iterative, such that the iteration proceeds in a sequential manner from node to node along the route. The average NSR of the nodes along the route is calculated at each iteration and NM is used to minimize the average NSR by trying different values of antennas weights. The scheme is repeated until convergence.

The algorithm is summarized in the following steps:

・ Given all node locations, compute the distance d_{ij} and the angle θ_{ij} between the nodes i and j.

・ For a given source and destination node, use a routing protocol to determine which nodes are members of the route. Nodes not included in the route are treated as interference sources.

・ Calculate path loss between the transmitting and receiving nodes using a suitable path loss model.

・ Compute received power at node i, P_{Rij} due to signal source j with transmit power P_{Tj}, where i and j represent nodes on the route. Then compute the total received signal power at node i using P R i S = ∑ j P R i j . Antenna gains are calculated using Equation (3).

・ Similarly, total interference power at node i can be calculated using P R i I = ∑ j P R i j where P_{Rij} represents received power from interference node j, i.e., node j is not a node on the route.

・ Ambient noise can be included by computing a suitable noise temperature and using it to calculate the noise power N i .

・ Assume an initial weight vector for the antennas at each node to compute an initial gain G i ( θ ) ; using this, calculate the received signal power, interference power, and the noise power. From these values calculate NSR at each of the M nodes in the route as well as the route-average NSR denoted by N S R ¯ :

・ N S R i = P R i I + N i P R i S

・ N S R ¯ = 1 M ∑ i = 1 M N S R i .

・ Apply NM to compute the weight vector at each node using N S R ¯ as the fitness function.

・ Using the obtained weight vector for the i-th node, calculate the gain in the direction of the j-th node G i ( θ i j ) = w i T h ( θ j ) , where w i = e j φ i is the weight vector at node i defined by antenna phase vector φ i T = [ φ i 1 , φ i 2 , ⋯ ] . NM minimizes NSR by trying different values of φ i to get the weights. Note that each candidate weight vector affects the beam pattern of the current node, thereby changing the node’s NSR as well as the average N S R ¯ for the route. The iteration proceeds in a sequence along the nodes in the route and is repeated until convergence. Each time the weight vector at a node is calculated, it considers the refined weight vector of its neighbors from the previous iteration. Convergence is reached when there is no longer significant reduction in N S R ¯ in a complete pass of the algorithm through the route.

Each node along the route must communicate with its previous and next hop neighbors. In each pass, the algorithm tries to refine the weight vector at each node, such that the average noise-to-signal ratio is minimized. As the algorithm tries to reduce the average NSR by minimizing the individual terms of the summation ∑ i = 1 M N S R i , it always tries to improve the SNR at each node. Also, as the iteration proceeds in a sequence, all the nodes in the route are equally favored.

Along with the desired and interference signals from various sources, the antenna system also receives noise from radiating sources of natural origin. These sources include cosmic noise, noise from the sun, noise from the ground, etc. Apart from these noises, the receiving system and amplifiers used with antennas also contribute to the system noise. Usually, the noise power received by an antenna is represented using antenna noise temperature and the noise from different sources can be combined in an additive manner.

The antenna noise temperature is the temperature of an equivalent fictitious resistor that would give rise to the same noise per unit bandwidth as that of the antenna output at a given frequency. The received noise power per unit bandwidth is given by S a = k T a , where k is the Boltzmann’s constant and T a is the noise temperature of the antenna and is computed as

T a = 1 4π ∬ G ( θ , ϕ ) T ( θ , ϕ ) d θ d ϕ ,

where G ( θ , ϕ ) is the antenna gain and T ( θ , ϕ ) is the sky brightness. Their product is integrated over the entire solid angle to compute the antenna noise temperature. There are empirical formulae available to calculate different factors that contribute to the sky brightness. For example, [

T c ( average ) = 290 λ 2 = 2.6 × 10 7 f 2 ,

where T c is the absolute temperature (in Kelvin), λ is the wavelength in meters and f is the frequency in MHz. The thermal noise in the receiving system will also have a noise temperature in addition to the noise from natural sources. The amplifier not only adds noise but also amplifies the noise at the input by a factor of the amplifier gain. Other factors like noise due to lossy elements also contribute to the system noise temperature. In general, the system noise temperature can be computed as

T s y s = T a + T r e c + T a m p + T f e e d + ⋯ .

Therefore, the noise power received by a receiver of bandwidth B would be k T s y s B .

The system noise temperatures of a typical directive antenna vary between 40 K to 3000 K depending on the frequency of operation and [

The simulations were performed using MatlabR2015a [

This shows that the proposed beamforming algorithm iteratively reduces the overall average NSR (therefore increases SNR) of the network, thereby, providing a globally improved solution by capturing all the changes in the network. The average NSR of the network using only omni-directional antennas is also shown in

Similarly, we fixed the SNR to be 10 dB (approximate lower limit for acceptable bit error rates) and calculated the transmitter power required at each node to maintain that SNR.

present results from experiments performed using the free-space propagation model and the Walfish-Ikegami model (WIM) [

In this approach, the improvement in performance is due to directional gain as well as nulling of interference. The gain pattern for the same topology as in

validated by the plot shown in

A low cost, low-complexity, and energy efficient solution for adaptive beam forming in ad hoc networks was proposed to increase the overall average SNR of the network. The approach uses the Nelder-Mead simplex method of unconstrained optimization to find antenna weights that provide a global solution for optimal beam patterns for a given network topology. This can provide lower bit error rate, increase throughput, and extend network life. The proposed method does not require transceivers and additional circuitry for each antenna in

the node’s antenna array as is the case for most smart antenna approaches. We have provided simulation results using the free-space model and the Walfisch- Ikegami model of radio propagation. Both these models show similar results of increased SNR (about 35 dB improvement using a 3 × 3 antenna array) and decreased transmission power (decrease of 25 - 40 dBW) for the Nelder-Mead optimized arrays.

One potential problem with the current implementation is that suboptimal solutions may occur when the algorithm settles for a local minimum. Ways to reduce this problem such as randomly visiting nodes during the iteration are being investigated. Another potential issue is that average NSR may not be the best fitness function to use since it can be affected by a few high or low NSR outlier values along the route. Alternative fitness function to be considered might include total route transmission energy, network lifetime, or average network throughput. Currently, the proposed algorithm is suitable only for stationary networks like wireless sensor networks. This is mainly because considerable processing time is required to compute new beam patterns for every node in the network, which would be required each time the network changes. For example, a network with 3 desired nodes and an interference node takes approximately 3.5 seconds to converge on an Intel Xeon Sandy Bridge CPU (2 GHz). The use of high performance computing and neural networks can potentially improve the convergence speed and make the beamforming algorithm suitable for mobile ad hoc networks.

Ramakrishnaiah, V.B., Kubichek, R.F. and Muknahallipatna, S.S. (2017) Nelder-Mead Based Iterative Algorithm for Optimal Antenna Beam Patterns in Ad Hoc Networks. Journal of Computer and Communications, 5, 117- 134. https://doi.org/10.4236/jcc.2017.57012