Determination of Array Gain of Single Hop to Achieve the Performance of a 2-Hop Wireless Link

In adaptive beamforming system adaptive algorithm of digital filter is applied to update the weighting vector of the antenna elements to get antenna gain along the desired direction and attenuation along the jammer. The objective of the paper is to evaluate the threshold gain of the adaptive beam former along the line of sight (LOS) between the transmitter and the receiver (including jammer suppression) to make the single hop link comparable with 2-hop link. The single hop and 2-hop communication systems are compared in context of symbol error rate (SER) under fading condition theoretically and verified by simulation. Finally we evaluate the numerical value of threshold gain of adaptive beamformer of two antenna elements under Rayleigh and Nakagami-m fading conditions.


Introduction
To combat the fading affect of long wireless link, 2-hop model is widely used where a relay is placed between transmitter and receiver. The concept is applicable to enhance the performance of point to point link. The combined SNR of two links becomes approximately parallel combination of two SNRs like equivalent parallel resistor of electrical circuit. Therefore the combined SNR is less than the individual SNR (between transmitter and repeater or repeater and receiver) but the SNR between the transmitter and receiver (without the repeater station) is Journal of Computer and Communications concept of AAS, overall SINR (signal to interference and noise ratio) at receiving end and algorithms of adaptive beamforming; Section 4 provides the results based on theoretical analysis of Section 3; and Section 5 concludes the entire analysis.

Dual-Hop Wireless Link
First of all we consider the 2-hop wireless link of Figure 1 where the SNR between transmitter and relay is Γ SR and that of relay to destination is Γ RD .
The received signal vector at the relay is expressed as: where E is the transmitted power, SR H is the channel matrix from the source to relay, x is the signal vector of source, and SR n is the noise vector of source to relay link. Above expression is applicable for MIMO system but for single antenna system instead of signal vector we consider individual symbol (for example symbols of 16-QAM) of transmitter and instead of channel matrix we consider simply channel gain may be complex number. When the signal is detected at relay, it is processed and transmitted with some delay expressed as: RD ′ y hence the received signal vector at the destination is, The noise vector on both the links are uncorrelated and possesses the proper- This overall SNR will be compared with overall SNR of AAS with adaptive beamformer under single hop of same path length.
For single antenna case Equations (1) and (2) is written conventionally like: where each variable indicates a real or complex number instead of vector.

Adaptive Beamforming
In array antenna system the number of antenna elements is a vital factor to achieve a wide variation of directivity and gain visualized from The weighting factors of reference antenna elements are updated using adaptive algorithms of [21].
To match with multiple jammers of different direction at the same time, it is necessary to have more than one reference omni as explained in [22] [23]. Radiation pattern of two reference adaptive beamforming is shown in Figure   4 taking the weighting factors as: 11 0.48 w = − , 12 0.871 w = − , 21 1.7 w = and 22 1 w = based on [24]. Here the null forms along 30˚, 150˚, 60˚ and 120˚; where the main beam is directed along 270˚.
The overall SINR at receiving end is, 0 where P r is the received signal power along desired direction, G d is gain of the AAS along the link between T x and R x , A j is the gain along the jth jammer, I j is the interference/jammer along jth direction and N 0 is the awgn noise. Overall SINR is approximated as G Γ ⋅ ; where Γ is the SNR of conventional single antenna system and G is the ratio of G d and A j .   In free space path loss model the received signal at a distance d is expressed as:

Algorithm-1
In this algorithm we use the concept of Wiener filter theory in minimization of mean square error of cost function which actually represents the difference between output of the adaptive filter and desired output. For input signal u(n) and kth weighting factor W k the output signal from a linear combiner is given as, Output power, If the beamformer is subjected to a linear constraint, where g is the complex-valued gain, θ 0 desired electrical angle of arrival. Taking gradient of J and putting the result equal to zero like Wiener filter theorem, we get the optimum weighting vector like,

Algorithm-2
This algorithm is an extension of previous one where the desired and undesired signals are analyzed separately. A widely used method of adaptive beamforing is LCMV (linearly constraint minimum variance) algorithm where the weighting vector of adaptive filter is divided into two portion W q and W a . Here the first one is dependent on desired signal i.e. signal along the desired AOA (Angle of Arrival) and the second one is related to sidelobe and un-correlated with the desired signal. A constraint matrix C a correspond to a band rejection filter is used to segregate the sidelobe from the desired signal shown in Figure 6. Finally   Winner filter theorem is applied to in getting weighting vector for minimum output y(n). Let us introduce multiple linear constraints by the constraint matrix C and gain vector g like, (14) where the overall weight vector is, The desired signal of the beamformer, If C a is a stop band FIR filter with zero response in desired direction θ 0 then the signal with sidelobe is

Results
In this paper the matric used against the performance of wireless link is SER or BER for single and 2-hop cases. Figure 8 shows the profile of SER against average SNR for both 2-hop link and single hop link of same path length incorporating AAS of two antenna elements. Here our proposal becomes successful when overall gain (enhancement of desired signal and attenuation of interferences) G is greater than or equal to 3.44 dB for Nakagami-m fading with m = 4.
In Figure 8 we found that SER of single hop is better than the case of 2-hop for G = 4.77 and 6.02 dB but worse than it for G = 3.42 dB. In case of Rayleigh fading the threshold gain G becomes 1.86 visualized from Figure 9 where the SER is found better than 2-hop for G = 1.76, 4.77 and 6.02 dB but worse than it for G = 1.76 dB.   First the simulation is done for 2000 random symbols shown in Figure 10(a) then for 200,000 symbols shown in Figure 10(b). The simulation curves become smothering for more symbols validates the concept Monte Carlo simulation. The similar result is shown in logarithmic scale in Figure 10(c). The difference between simulation and theoretical result is less than 5% hence ensure 95% confidence level. All the curves reveal the success of our proposed scheme when we use G = 1.86 under the simulation as well. Similar job is done for Nakagami-m fading case shown in Figures 11(a)-(c).
The relative performance between Rayleigh and Nakagami-m is found subtle for SNR in the range of 0 dB to 20 dB. To observe the variation we plot BER  against SNR in the range of −10 dB to 0 dB shown in Figure 12(a) and Figure   12(b); where the performance under Nakagami-m fading is found better than that of Rayleigh case because of strong link between transmitter and receiver for former case. Finally the performance of AAS is found better than conventional 2-hop wireless link for SNR in the range of 0 dB to 20 dB visualized from Figure   10 and Figure 11. Situation is found reverse for SNR in the range of −10 dB to 0 dB shown in Figure 12(a) and Figure 12(b) taking the same overall array gain.
To overcome the above cumbersome situation the array gain has to be changed at low SNR condition even we have to consider the environment of fading.

Conclusion
The weak radiation of the antenna is strengthened and made more directional