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A novel ultra-fast ultra-simple adaptive blind beamforming algorithm for smart antenna arrays.

1. INTRODUCTION

Smart antenna arrays have become an essential part in modern wireless communication systems. The power of smart antenna comes from the fact that it can steer and reshape its radiation pattern to maximize Signal to Interference and Noise Ratio (SINR) [1-5]. This is done electronically using beamforming algorithms without the involvement of the mechanical parts to steer the array [6-10]. The beamforming algorithms are classified mainly into three types: (1) estimating the Angle Of Arrival (AOA) algorithms like MUSIC and ESPRIT [9,11-13], (2) non-blind algorithms where the transmitter and the receiver agree to certain code to detect the location of the transmitter to steer the main beam like Maximum Likelihood (ML), and Minimum Variance Distortionless Response (MVDR) [7], (3) blind beamforming algorithm. These algorithms are widely used and difficult to be built and developed. In the blind algorithms, the receiver has no idea or clue on how and where the transmitter is located. The receiver should estimate and steer blindly towards the transmitter. These algorithms are based on the maximization of SINR, or sometimes called training, by detecting the cyclostationary or modulus properties for a vector of stored transmitted symbols to predict AOA [14-17]. Least Mean Square LMS algorithm is the base for most of the known adaptive beamformers. This algorithm suffers from high computation load because LMS needs, theoretically, infinite vector length, high complexity, time consuming, and singularities. Modified algorithms were developed to reduce the computational load by reducing the required vector length such as the Recursive LMS (R-LMS). Nevertheless, this algorithm should wait until the required number of symbols is assembled to perform training [9,18]. Combing this algorithm with modulus detecting algorithms such as Constant Modulus Algorithm (CMA), Decision Direct Algorithm (DDA), or Conjugate Gradient Method (CGM) will produce the blind adaptive beamforming algorithms such as LMS-CMA, LMS-DDA, RLMS-CMA, ... [16-19].

In sensitive communication systems, such as military, these difficulties represent obstacles that should be solved by simplifying the beamforming algorithm. Hence, the need to develop new approaches becomes persistent. Researchers proposed new systems that are not based on LMS but still they need to store a vector of symbols to perform beamformation [20-27]. One of the interesting approaches uses the Particle Swarm Optimization (PSO) to accelerate the beamforming algorithm training by maximizing a suitable fitness function to find the required phase weights to steer the main beam [28-30]. Various versions of the PSO were developed to increase the speed, reduce the required memory, and maximize SINR to accommodate the requirements of real-time applications [31-33].

The algorithm presented in this research is completely independent of the LMS. The approach is simply based on the difference or the cost between the received and detected symbols at the receiver without the need to store a vector of symbols.

[FIGURE 1 OMITTED]

2. FORMULATING THE ALGORITHM

2.1. The Misalignment Effect

Assume a receiver that implements smart antenna to receive a transmitted signal as shown in Figure 1. The linear antenna array consists of M number of elements with inter-element distance d, and an incident wave at an angle of [phi].

The Array Factor AF for this array is [18-20]

AF = [M-1.aummation over (j = 0)][e.sup.i(M-j-1)kd cos[phi]] (1)

where k is the wavenumber = 2[pi] / [lambda].

If a transmitted message [alpha](t) is mapped into a complex symbol s = [s.sub.R] + [s.sub.I], where [s.sub.R] is the real part and [s.sub.I] the imaginary part, the complex symbol is received at an angle [phi], then the output of the above array is (neglecting the noise)

[PSI] = ([s.sub.R] + [s.sub.I]) * ([w.sub.H] * AF) (2)

The weighting factor or phase shifters [w.sub.j] are used to neutralize the effect of [phi] or in other words to steer the main lobe to the direction of arrival. These values are calculated as the Hermitian (complex conjugate referred to by [sup.H]) of an angle [[phi].sub.w] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

If [[phi].sub.w] = [phi] then ([[summation].sup.w] * AF = 1) and leaves only the complex symbol which is s = [s.sub.R] + [s.sub.I]. Assume that [[phi].sub.w] = [phi], then Equation (2) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [A.sub.R] and [A.sub.I] are the real and imaginary parts of the array residuals, and [s.sup.D.sub.R],[s.sup.D.sub.I] are the real and imaginary parts of the received symbol [s.sup.D] at the array output that will be the fed to the demodulator in the receiver.

2.2. Cost Calculation Using Demodulation Remodulation Technique

The algorithm is shown as a block diagram in Figure 2.

The demodulator output is the most probable message [[alpha].sup.D](t) corresponding to the array output [PSI] as shown in Figure 3.

The error added by the misalignment of the array with the transmitter angle will lead to an erroneous detection of the received symbol ([[alpha].sup.D] [not equal to] [alpha]). The procedure to calculate the cost is as follows: (1) calculate [PSI] to identify [[alpha].sup.D], (2) remodulate [[alpha].sup.D] using the same array at the receiver to generate [[PSI].sub.D], and then subtract [[PSI].sub.D] form [PSI] and take the absolute value to find the cost.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Converting the above block diagram into a mathematical model

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [s.sup.Dr.sub.R] and [s.sup.Dr.sub.I] are the real and imaginary parts of the remodulated message.

Multiplying Equation (5) by Equation (3) results in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

To calculate the cost C

C = [summation over (all elements)][absolute value of [PSI] - [[PSI].sub.D]] = [absolute value of ([s.sup.D.sub.R + [s.sup.D.sub.I]) - ([S.sup.Dr.sub.R] + [S.sup.Dr.sub.I])]

= [absolute value of (([s.sup.D.sub.R] - [S.sup.Dr].sub.R]) + ([s.sup.D.sub.I] - [S.sup.Dr.sub.I]))] (7)

If a proper alignment is acquired ([[phi].sub.w] = [phi]) then C = 0 otherwise C [not equal to] 0. The value of C will play a central role in switching the beam gradually towards the direction of transmission.

2.3. The Generation of [PHI] Set

In order to steer the beam iteratively in the direction of the transmitter, three new angles will be generated using the recent value of [[phi].sub.w] that are [PHI] = [[phi].sub.w], [[PHI].sup.+] = [[phi].sub.w] + C, and [[PHI].sup.-] = [[phi].sub.w] - C. After this step Equation (7) is used to calculate the cost C, [C.sup.+], and [C.sup.-] corresponding to [PHI], [[PHI].sup.+], and [[PHI].sup.-], respectively. Any angle of the calculated [PHI]s with the lowest cost indicates that it is the nearest angle to the transmitter direction. Take the value of the angle with the lowest cost and let ([[phi].sub.w] = [PHI]). Next, find a new set of three angles [PHI], [[PHI].sup.+], and [[PHI].sup.-]. Repeating this process iteratively will steer the beam towards the transmitter because the nearest angle will always be chosen. When total alignment occurs, then C will be zero, and all the angles will merge into one value [PHI] which is the angle of the transmitter [PHI] = [phi].

2.4. Dead Locks or Trap States

Although the above algorithm seems to be straightforward and simple to be implemented, it suffers from dead locks or trapping states. To explain a trapping state, assume that the current [PHI] has the lowest cost and [PHI] [not equal to] [phi]. At the same time, [[PHI].sup.+] and [[PHI].sup.-] having [C.sup.+] and [C.sup.-], respectively, are both greater than C. Thus the next iteration will select the value of [PHI] as the nearest angle to the transmitter, and the same values for [[PHI].sup.+] and [[PHI].sup.-] will be recalculated. This situation will continue indefinitely, which is called a trapping state or a dead lock.

To step out this trapping state, a simple modification in calculating the value of C is introduced to ensure that the algorithm will not fall in a dead lock. This modification is accomplished by generating a random variable [zeta] provided that (0 [less than or equal to] [zeta] [less than or equal to] [pi]) and adding it to the calculated value of C in the following manner

C' = C * [zeta] (8)

Equation (8) ensures that the next decision for the angles differs from the previous one. Also, the value of Z is related to C in such a way that as ([PHI] [right arriw] [phi]) then (C * [zeta] [right arrow] 0). The calculation of the angles set, [PHI] = [[phi].sub.w], [[PHI].sup.+] = [[phi].sub.w] + C', and [[PHI].sup.-] = [[phi].sub.w] - C', will also coincide with one angle that is [phi].

2.5. Number of Decisions Per Symbol

The algorithm should ensure a total alignment with the transmitter. If a single decision is made for the received symbol, the probability of having an erroneous decision will increase and affect the future received symbol because misalignment adds extra (real and imaginary) error to [s.sup.D] as in Equation (4). To avoid this case, the algorithm should make enough iterations per symbol which means repeating Equations (5) to (7) for N number of times such that N > 1. The repetition ensures that the main beam shifts from its current position to meet the transmitter angle as fast and accurate as possible. Approaching transmitter angle ensures a maximum SINR.

The value of N should be chosen carefully depending on the environment that the receiver is working within to maximize the probability of the last decision to be correct. If a faulty value of N is chosen (high or low) the algorithm may either works slowly, or the shift towards the transmitter is not enough to minimize the Symbol Error Rate SER.

3. THE MODEL OF THE ALGORITHM

The simulated system containing the algorithm is shown in Figure 4. It consists of three main parts: (1) a transmitter, (2) an antenna array, and (3) the receiver that hosts the proposed algorithm.

The transmitter section generates data messages [alpha](t), then encodes them using PSK modulation scheme, generates the complex symbol s = [s.sub.R] + [s.sub.I], and adds the AWGN noise to the transmitted symbol. The resultant transmitted symbol appears at (SymGenl) pin. The motion of the transmitter is fed to the next section, (Smart Antenna Array), using (MRA out) pin. The (Smart Antenna Array) generates the added phases corresponding to every element and feed them to the receiver using (PAA_Out).

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The model of the algorithm is shown in Figure 5, which consists of two sections: (1) cost calculation and (2) Phi calculation. The value of the iterations N is set externally as shown in Figure 4 (Iter No.). The switch is used to reset the system after each symbol period in order to clean any residuals from the previous calculations and starts a new estimation.

The inside of the cost section is shown in Figure 6.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The first three blocks (Signal AOA Array w) are identical and used for the demodulation. The next three blocks (Cost Calc) are used to calculate the cost for each generated [PHI] with its corresponding decoded symbol. Each one consists of a remodulater which remodulates the input symbol (Symb) using the generated set of [PHI]s (Phi) and applies Equations (7) and (8) to calculate the cost for the next iteration. All costs are multiplexed on a single bus in order to be fed to the next section which is shown in Figure 7.

The [PHI] calculation section is shown in Figure 7. The cost, Phi, and symbol buses are concatenated and sorted ascendingly according to the cost. The value of Phi corresponding to the lowest cost is fed to PHCST block to calculate the [PHI] set for the next iteration (Section 2.4). Here Phi = [PHI], Phip = [[PHI].sup.+], and Phim = [[PHI].sup.-].

[FIGURE 8 OMITTED]

4. SIMULATION SETTINGS AND RESULT

The settings for the simulated system are as follows: (1) the data source is a uniform random integer generator with 16 M-ary (0-15) used to generate [alpha](t), (2) a PSK modulator has 16 levels, (3) the channel is AWGN, (4) the array is linear with 16 elements with d = [lambda] / 2, (5) the angle of motion is restricted between 10[degrees] to 170[degrees], and (6) the value of [phi] (MRA) is set to take three motion types which are triangular, sinusoidal, and saw tooth as an extreme scenario.

[FIGURE 9 OMITTED]

The algorithm was tested under different levels of SNR (0-20dB). At each value of SNR, the number of iterations N is varied from 1-30. The average error (in degrees) and error variance (in [pi], 180[degrees]) were calculated using 10,000 samples for each point. The algorithm performance for the three motion types is shown in Figure 8.

Figure 8 shows that the algorithm performance is independent of SNR level because the average error and variance are almost constant along the SNR axis at the same N value. While the average error and variance drop significantly with the increase of N value. Also, we can see that the motion type has no effect on the algorithm performance because the performance planes (Figure 8) for each motion type are approximately the same. Table 1 gives a numerical comparison for the three motion types.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

The simulated tracking results for the algorithm are shown in Figures 9 to 20 with SER for each case. The total transmitted symbols are 100 with a symbol period 0.1.

4.1. Circular Motion

Circular motion is simulated at the transmitter using a periodic half sine wave. The results are shown in Figures 9 to 12.

4.2. Linear Motion

This motion is simulated using a simple triangular motion for the transmitter. The results are shown in Figures 13 to 16.

4.3. Saw Tooth Motion

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

This motion is a severe case that is used to simulate a very fast position changing objects. It is used to test the algorithm to track illusive objects. The results are shown in Figures 17 to 20.

In all types of motions and SNR values, the single decision (N = 1) has the worst tracking accuracy. The misalignment error bounds are [+ or -]170[degrees] while at (N = 30) the misalignment error bounds are at maximum of [+ or -]5[degrees] which confirms that N should be much higher than 1, as suggested in Section 2.5. Figure 12(a) shows an event of wrong decision. If this event occurs at the last iteration, then a wrong value for [PHI] will be assumed, leading to steer the beam in the wrong direction. But this wrong decision has happened during the estimation period, thus it did not affect the SER value because the next iteration has corrected this decision.

[FIGURE 17 OMITTED]

[FIGURE 18 OMITTED]

[FIGURE 19 OMITTED]

[FIGURE 20 OMITTED]

5. CONCLUSIONS

From the above, it can be concluded that the LMS is not necessary to steer the main beam blindly to track the transmitter movement. The cost is an alternative efficient method to be used for blind tracking. This algorithm does not suffer from the drawbacks of the LMS algorithm or its derivatives such as complexity, high computation load, and singularities that cripple the performance of the smart antenna because it does not store a symbol vector. This algorithm also decodes the incoming stream symbol by symbol unlike the LMS that needs to store enough symbols to perform maximization.

The increase in the number of array elements leads to a narrower beamwidth, thus the iterations should also be increased to ensure a precise alignment and minimum SER.

The algorithm is able to track objects that change their position abruptly (very high speed moving object) with high accuracy. For this type of motion, any symbol vector based algorithm might easily fail to track such objects because of the need for assembling the symbol vector.

The algorithm tracking ability is not affected by the high noise level. It can be seen that the algorithm can track the transmitter with high accuracy (with an average error 1[degrees]) whether SNR is high or low. This means that the algorithm can perform well in harsh environments without being affected by the high noise level. The tracking accuracy can be increased by increasing the number of iterations. The higher is the iterations the better is the tracking accuracy to a certain limit when higher iterations do not add significant improvement while increasing the computation load and decision delay.

The last decision in the iteration section has a major effect on the SER level. Thus setting the adequate iterations is vital to decreasing SER.

The type of motion of the transmitter has a minor effect on the tacking precision of the algorithm because the algorithm can make many decisions at each symbol and select the angle with least cost. Hence it can chase any sudden change in the object course.

Received 15 September 2011, Accepted 20 October 2011, Scheduled 2 November 2011

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A. N. Jabbar *

University of Babylon, Hilla, Babylon, Iraq

* Corresponding author: Ahmed N. Jabbar (Ahmed_Aljafari@Yahoo.com).
Table 1. The average misalignment error
and variance numerical value for the
three motion types at different SNR
levels and iterations values N.

 Motion     Misalignment average error in (deg)
  types

                SNR 0 dB          SNR 20 dB

            N = 1    N = 30    N = 1    N = 30

Circular    20.914    1.154    23.212    1.279
 Linear     14.378    1.037    19.972    1.044
Saw tooth   19.209    0.996    20.142    1.065

 Motion        Misalignment error variance
  types            in (180[degrees])

                SNR 0 dB          SNR 20 dB

            N = 1    N = 30    N = 1    N = 30

Circular    8.699    0.01453     11     0.05383
 Linear     4.374    0.01011   7.943    0.01004
Saw tooth    9.38    0.0085    9.009    0.0098
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Author:Jabbar, A.N.
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Date:Sep 1, 2011
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