Wireless Networks Security Over Smart Antennas Computer Science Essay

Published Date: 02 Nov 2017

Franklin George Jobin#, Dr. M. Rajaram*

#Research Scholar, Anna University, #[email protected]

*Professor & Head, PG Studies, Department of Electrical Engineering

Government College of Technology, Coimbatore

*[email protected]

Abstract— Smart antennas consist of an antenna array, combined with signal processing in both space and time. They offer a broad range of ways to improve wireless system performance. These antenna systems include a large number of techniques that attempt to enhance the receiver signal, suppress all interfering signals and increase capacity. This chapter begins with the review of types of smart antennas. This is followed by the signal modelling for smart antenna system. Finally beamforming schemes of smart antennas are reviewed, and the conventional beamforming and recursive least square error algorithms are presented in detail.

Keywords— Smart Antennas, Signal Processing, Beamforming, Adaptive Algorithm, RLS Algorithm

Introduction

Smart antennas consist of an antenna array, combined with signal processing in both space and time. Spatial processing leads to more degrees of freedom in the system design, which can help to improve the overall performance of the system. Smart antenna, when used appropriated, help in improving the system performance by increasing channel capacity and spectrum efficiency, extending range coverage by steering multiple beams to track many mobiles [1]. Smart antenna systems are usually categorized as switched beamer, and adaptive array systems [2-4], [8].

A switched beam antenna system consists of several highly directive, fixed, pre-defined beams, and choose the beam which gives the best SNR. Fig. 1 shows a switched beam antenna system. A switch is used to select the best beam to receive a particular signal. Because no Direction of Arrival (DOA) information of the desired user is assumed, the desired user may not fall on the maximum of the chosen beam. The system is not able to provide any protection from multipath components which arrive with DOAs near that of desired components. They are unable to take advantage of path diversity by combining coherent multipath components. Switched beam antennas are effective in low to moderate to co-channel interfering environments owing to their lack of ability to distinguish a desired user from an interferer.

Fig 1. Switched Beam for Antenna Pattern

In adaptive array systems, the weights are adjusted by some means using available information derived from the array output. Fig.2 shows an adaptive beam form antenna pattern. They maximize the signal-to-interference-plus-noise ratio (SINR) and provide the maximum discrimination against interfering signals. The exact structure of the signal processor is dependent on the amount of information that is available or can be estimated at the receiver. This information includes the type of modulation and signalling format, the number of resolvable signal paths that received at the receiver, the DOA and time delay of each path signal, availability of reference to training signals, and the complexity of the propagation environment.

Fig 2. Adaptive beamform antenna pattern

II. SIGNAL GENERATION AND MODELLING

Modeling signal is an important step in the design of array signal processing. Consider a linear antenna array of L isotropic elements, shown in Fig. 3. Under these assumptions, if antennas are identical in the array and all antennas have the same response in any given direction. The signals induced at different receiving antennas only have phase difference caused by time delay [4], [9-11]

τl (φi ,θi ) = (1)

Where is the position vector of the nth element, is the unit vector in direction (φi ,θi ), and c is the speed of propagation of the plane wavefront. For the simple case, a Uniform Linear Array (ULA) of identical antennas, aligned with the x-axis such that the first element is situated at the origin (Fig. 3), this becomes

τl (θi )= (n-1) cos θi n =1,2,...B (2)

Fig 3. The coordinate system for the signal model

Assume the signal on the 1st element due to the ith source is xi(t)

xi(t) = mi (t) exp ( j 2 π f0 t ) (3)

Where mi (t) denotes the complex modulating function. The received signal vector on ULA can be expressed as

r(t)==+n(t) =+n(t) = s+n(t) (4)

Where n(t) is the white noise with zero mean and variance equal to σ2, and s is the steering vector.

From (2) and (4), the channel propagation function vector for Single Input Multiple Output (SIMO) channels that have Uniform Linear Multiple Receiver Antennas (ULA) can be shown

H=h(t,τ)=V(θ) h(t, τ) (5)

Where h(t, τ) is the channel response function for given reference point, and V(θ) is the array response vector which is with respect to a given reference point and is functions of the array geometry and direction of arrival (DOA).

We can extend the results above to the case of M multipaths, each multipath component here can be considered to be a planar wave, arriving from a unique direction at a unique time delay. The general results for multipath case can be simply obtained by the superposition principle

Hm = (6)

Where hm (t) has been derived and expressed as h (t, , θ) = βj (t)(  - j ) ( θ - θj)

For the discrete baseband model in matrix notation with M multipath signal and ULA used, the received signal is given by

r = + n=SCd+n.

Where

=

: the ith data symbol transmitted by user k

: the zero-padded spreading sequence for symbols generated by user k

: the channel coefficients over the mth multipath for the ith symbol generated by user k.

L : the total amount of the data symbol sent by every active user

K : the total amount of active users sending data to the BTS

: the maximum delay spread normalised to sampling interval

M : the amount of resolvable multipath components per user data sequence

C : channel matrix, it also needs to be modified to reflect the multipath profile, the matrix is LKM x KL block diagonal:

C =

S: the matrix of received spreading waveform with the column = , j = iKM+(k-1)M+m. The dimension of S is (LNQ++P) x KLM, consequently, the length of r is LNQ++P.

Consider a broad-band beamformer, shown in Fig. 4, which has a digital filter in each antenna element, can also control their own frequency response. The pattern of an array is easily controlled by adjusting the amplitude and phase of the signal from each element before combining the signals.

Fig 4. Broad-band beam-former structure

For a received signal xi (t) in Fig.3, the array output from Fig. 4 can be given by

y(t) = wn rn (t) (7)

Where w = [wn1, wn2, ..., wnM ], the weight vector, and r is given in (4)

III Beamforming Schemes

3.1 Conventional beamforming

The simplest beamformer has all the weights of equal magnitudes and is called a conventional beamformer or a delay-and-sum beamformer. This array has unity response in the look direction, which means that the mean output power of the processor duo to a source in the look direction is the same as the source power. To steer the array in a particular direction, the phases are selected appropriately [12]. This can be explained as follows.

Assume that there is a source of power pi in the look direction, it is referred to as the signal source. According to (3) for a particular direction (φi, θi),, the signal rin (t) induced on the nth element due to this source is given

rin (t) = mi (t) exp ( j 2 π f0 ( t +τ (φi ,θi ))) (8)

Array signal vector due to the look direction signal becomes

ri (t) = mi (t) exp ( j 2 π f0 t ) s (9)

Where s is the steering vector in the look direction, it has been given in (4).

For the conventional beamforming, the array weights are given by

wc = (1 /B ) s (10)

Where B is the number of elements. The output of the array with weight vector wc becomes

y(t)= wcH ri (t) = mi (t) exp ( j 2 π f0 t ) (11)

yielding the mean output power of the processor

P(wc )=E[y(t) y*(t)]= pi (12)

Thus, the mean output power of the conventional bean former is steered in the look direction. The process is similar to steering the array mechanically in the look direction except that it is done electronically by adjusting the phases. This beamformer provides the maximum output SNR for the case that no directional jammer operating at the same frequency exists, but it is not effective in the presence of directional jammers, intentional or unintentional [9].

Optimum beamforming

In optimum beamforming techniques, a weight vector is determined which minimizes a cost function. Such beamforming techniques are also known as statistically optimal techniques as they determine a weight vector which is optimum in at the array output. Two of the most popular techniques which have been applied extensively in communication systems are the Minimum Mean Square Error (MMSE) and Least some statistical sense. Typically, this cost function is inversely associated with the quality of the signal squares (LS) criteria. In both of these techniques, the square of the difference between the array output, z(t)=wHk ri (t), and dk(t), a locally generate estimation of the desired signal for the kth subscriber, is minimized by finding an appropriated weight vector, wk. MMSE solutions are posed in terms of ensemble averages and produce a single weight vector, wk , which is optimal over the ensemble of possible realizations of the stationary environment [16], [4], [12-13].

Table 1 Statistically optimum beamforming techniques [4]

MMSE

Max SNR

LCMV

criteria

Minimize the difference between the output of the array and some desired response

Maximize the ratio of the power in the desired signal component to the power in the noise component at the array output

Minimize the variance at the output of the array subject to linear constraints. For a single constraint, this corresponds to forcing the beam pattern to be a constant in a particular direction

Cost Function

J(w)= E[ | wH r(t)-d(t)|2 ]

Where r(t) is the array input and d(t) is the desired response

J(w)=

Where Rn is the covariance matrix of the noise component of r(t) and RS is the covariance matrix of the signal component.

J(w)=wH Rw

Subject to the linear constraint wHa(φ) = g. when g=1, this is called the Minimum Variance Distortionless Response (MVRD) beamformer

Optimal solution

w = R-1 p

where R = E[r(t) rH(t)]

and p = E[r(t) d*(t)]

Rn-1Rs w = λmax w

Where λmax is the maximum eigenvalue of Rs

w = R-1c[cHR-1c]-1g

where c = a(φ) is the steering vector in the direction of constraint

Advantages

Knowledge of the DOA is not required

True maximization of SNR

Generalized constraint technique

Disadvantages

Generation of reference signal

Must know statistics of noise and DOA of desired signal

Must know DOA of desired component

There are other techniques which can be used to form statistically optimal beam patterns based on data received by the array. The statistically optimal beamforming techniques are shown in Table 1. The Max SNR approach which maximizes the actual SNR at the array output and the Linearly Constrained Minimum Variance (LCMV) approach, require knowledge of Direction of Arrival (DOA) of the desired signal which is not typically known in mobile and portable wireless systems [4], [12-13].

3.3 Adaptive beamforming

Since in practice situations, the R used in optimal beamformer above are not available to calcULAte the optimal weights of the array. Adaptive techniques are often used with iterative approach which provides an updated weight vector w, after each computation. Typically, these algorithms have a per-step complexity which is much lower than the direct solution in optimal beamformer above, and can track non-stationary channels. The adaptive algorithm operates either in block mode or recursive mode. In block estimation the input data stream is arranged in the form of blocks of equal length (duration), and the current weight vector w(i) is adjusted on a block-by-block basis. In recursive estimation, on the other hand, the tap weights of spatial processor are updated on a sample-by-sample basis [16], [4], [11-12]. An adaptive solution which minimizes the cost function is

wi+1 = wi –J(wi) (13)

Where μ is the convergence factor which controls the rate of adaptation, and J(wi) is the gradient of a function J(k), which is given

J(k)= E[ | ζ(k)|2 ] (14)

Where ζ(k) is the a priori estimation error defined by

ζ(k) = d(k) - wH (k-1) r(k) (15)

The function J(w) is the cost function which defines an error performance surface.

There are a large number of adaptive beamforming algorithms. One of them, RLS (Recursive Least Squares) algorithms, is well known to pursue fast convergence even when the eigenvalue spread of the input signal correlation matrix is large. In RLS algorithms, we minimize the cost function

J(k) = (| ζ(k)|2 ) (16)

Where 1<<λ≤ 1 is the exponential weighting factor or forgetting factor, i=1,2,...,n. When λ equals 1, we have the ordinary methods of least squares. The inverse of 1- λ is, roughly speaking, a measure of the memory of the algorithms. The special case λ=1 corresponds to infinite memory.

The RLS family of linear adaptive filtering algorithms is classified 3 into distinct categories [14]:

Standard RLS algorithm, which assumes the use of a transversal filter as the structural basis of the linear adaptive filter.

Square-root RLS algorithms, which are based on QR-decomposition of the incoming data matrix.

Fast RLS algorithms, which combine the desirable characteristics of recursive linear least squares estimation by the use of linear least-squares predictionin both the forward and backward directions. Two types fast RLS algorithms may be identified

Order-recursive adaptive filters, which are based on a lattice like structure for making linear forward and backward predictions.

Fast transversal filters, in which the linear forward and backward predictions are performed using separate transversal filters.

In a nonstationary environment, the algorithm is required to continuously track the statistical variations of the input, the occurrence of which is assumed to be "slow" enough for tracking to be feasible. Tracking is a steady-state phenomenon. This is to be contrasted with convergence, which is transient phenomenon. An adaptive filter must first pass through the transient mode to the steady-state mode of operation to exercise its tracking capability. And there must be provision for continuous adjustment of the free parameters of the filter.

For RLS algorithms, how to optimally choose the forgetting factor λ when details of the underlying physical model of the system and its variability with time are not known? In this work the RLS algorithm with adaptive memory will be reviewed to automatically tune the forgetting factor λ [14].

IV Simulation and Practical Realization

The convergence speed of the LMS algorithm depends on the Eigen values of the array correlation matrix. In an environment yielding an array correlation matrix with large eigen value spread the algorithm converges with a slow speed This problem is solved with the RLS algorithm by replacing The gradient step size µ with a gain matrix at the nth iteration, producing the weight update gradient equation

Define gradient

= (17)

Differentiating the cost function J(k) with respect to λ yields

E[] (18)

The updated weight vector for time k is given by gain vector G(k) = P(k) r(k)

w(k)= w(k-1) + G(k) ζ *(k) (19)

Where G(k) is gain vector

G(k) = (20)

and

P(k) = λ-1 P(k-1) - λ-1 G(k) rH (k) P(k-1) (21)

Let S(k) denote the derivative of the inverse correlation matrix P(k) with respect to λ:

S(k) = (22)

Differentiating Eq. (21) with respect to λ

S(k) = λ-1[I - G(k) rH (k)] S(k-1) [I - r (k)GH(k)]+ λ-1 G(k) GH(k)- λ-1 P(k) (23)

Then using Eqs. (15), (19), and (22) in Eq. (17) yields

= [I - G(k) rH (k)] + S(k) r(k) ζ *(k) (24)

According to Eq. (18) , the forgetting factor λ(k) is adaptively computed

λ(k)= λ(k-1)+α Re[r(k) ζ *(k)] (25)

Where α is a small, positive learning – rate parameter.

The applicability of the RLS algorithm requires that we initialize the recursion of the Eq. (21) by choosing a starting value P(0) that assures the nonsingULArity of the correlation matrix [14]. To meet this requirement we may choose the initial value of P(k) with

P(0)= δ-1 I (26)

Where δ is the small positive constant.

And the initial value of weight vector is set to

w(0) = 0 (27)

Where 0 is the N-by-1 null vector.

The initialization procedure incorporating Equation (26) and (27) is referred to as a soft constrained initialization. The positive δ is that it should be small compared to 0.01 σ2, where σ2 is the variance of a data sample r(k). Such a choice is based on practical experience with the RLS algorithm supported by a statistical analysis of the soft constrained initialization of the algorithm [14]. Now we may summarize the RLS algorithm with adaptive memory as follows:

Starting with the initial values , P(0), λ(0), S(0), and , compute for n>0 :

G(k) =

ζ(k) = d(k) - H (k-1) r(k)

w(k)= (k-1) + G(k) ζ *(k)

P(k) = λ-1 P(k-1) - λ-1 G(k) rH (k) P(k-1)

λ(k)= λ(k-1)+α Re[r(k) ζ *(k)]]

S(k) = λ-1[I - G(k) rH (k)] S(k-1) [I - r (k)GH(k)]+ λ-1 G(k) GH(k)- λ-1 P(k)

= [I - G(k) rH (k)] + S(k) r(k) ζ *(k)

The convergence behaviour for are illustrated through the learning curves depicted in Fig. 5, where in this case α= 0.001, δ = 0.01 λ+=1, λ-=0.82, MS is moving at the speed 8.1 Km/h, processing gain equal 16. The channel data is updated symbol by symbol, this means the channel data is updated once every 16 chips. The estimated MSE was plotted only for the first 300 iterations in chip level, enough to display the convergence behaviour. The fast convergence of this algorithm is traded off high computational complexity.

Fig 5. The learning curve for the RLS algorithms ith adaptive memeory with α= 0.001, δ = 0.01 λ+=1, λ-=0.82,

V. Conclusion

In this chapter, different types of smart antennas are classified and reviewed. The signal modeling for smart antenna system is developed in order to analyze the beamforming algorithms. In beamforming techniques, bemforming techniques are classified and their characteristics are compared. The conventional beamforming and RLS algorithm with adaptive memory are studied in detailed. RLS algorithm with adaptive memory can continuously track the time-varying channel and obtain fast convergence by the trading off high computation complexity.

VI. Reference

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