Performance Analysis of Analytical Approaches to Smart … · 2020. 1. 23. · Page Number: 320-331 . Performance Analysis of Analytical Approaches to Smart Antennas Modeling . Emmanuel

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Volume-5, Issue-4, August-2015

International Journal of Engineering and Management Research

Page Number: 320-331

Performance Analysis of Analytical Approaches to Smart Antennas Modeling

Emmanuel Tonye1, Simon Kepchabe2

1Department of Electrical and Telecommunications Engineering, National Advanced School Polytechnic, University of Yaoundé I, CAMEROON

2

Department of Physics, Faculty of Science, University of Yaoundé I, CAMEROON

ABSTRACT

This article presents the results of MATLAB based implementation of various analytical approaches to the modeling of planar smart antennas. These latter, which are found in the form of linear networks, square or circular, are often implemented in two stages: determining the directions of arrivals, followed by the beamforming over these directions. In the literature, many laboratories have implemented one or the other of these steps relying on either of these configurations. For our part, we present the results of both the first step and the second, over all these three types of configurations. In addition, we implement a host of methods to determine the one that best suits each antenna array. This has led us to develop a performance simulation tool of smart antennas by analytical approaches. Keywords—Adaptive planar antennas, analytical approaches, beamforming, directions of arrivals, smart antennas.

I. INTRODUCTION A smart antenna is a system consisting of a linear, planar, circular or volume array of elementary antennas and a digital signal processor wherein implemented adaptive algorithms are used to control in real time their radiation patterns. This is achieved in two phases: the determination of the directions of arrival, followed by the focusing of the radiation in these directions. The implementation of these algorithms remains a major challenge

. Analytical approaches, also known as

constructive, which group estimation and optimization without constraints based on an established formalism [1],

are often used primarily to simulate the developed mathematical model. A thorough study of the results and the influence of different characteristic parameters so will validate the utility of the model [2], which could be used for a possible optimization, prediction or characterization. For the estimation of the directions of arrival, we implement algorithms Bartlett, Prony, MVDR, MEM, MMSE, MUSIC and MIN-NORM. For beamforming, we implement the conventional shaper, the null-steering, MVDR, DMI, LMS, RLS and CMA. This article is organized as follows: in Section 2, we model the three main configurations of adaptive planar antennas, in Section 3, we review the major conventional implementation methods, in Section 4, we present the simulation tool and the results obtained; we validate results with those of articles [3], [4] and [5] before extrapolating.

II. MODELING For each configuration, we make the following

assumptions:

All incoming signals to the antenna array consist only of plane waves

.

The transmitter and objects causing multi-path are all located in the far-field

region.

The mutual coupling between the various antenna elements

is negligible.

The inter-element distance is very small so that the amplitudes of the received signals

remain constant. This distance is set to λ/2 but changeable on the simulation tool

.

Each antenna element has the same radiation pattern and the same polarization.



2.1. Theoretical model of the linear array Consider M uniformly spaced antennas on which arrive L signals (

W1 W2 Wm WM

∑

Y(t)

dx

y

-(m-1)dcosφ

φφ

Oz.

Fig.1).

Figure 1: Uniform linear array of M antennas

The origin O is the first network element. The coordinates of the rank m antenna is (xm, ym, zm). The phase difference between the originally received signal and that received by the element of rank m is, according to [6]: Δγm = γm(t) – γ1(t) = − kxmcosϕsinθ − kymsinϕsinθ − kzmcosϕ (1) where k = 2π/λ is the wave number and ϕ the angle between the direction of the incident signals and network axis (Ox). We have xm = (m−1)d and ym = zm = 0. On the other hand, θ = π/2 because for planar smart antenna, all elements are aligned along (Ox). Since then: Δγm = − kd(m−1)cos ϕ (2) Note that Δγ1 = 0. The incoming signal on the first element from a source ℓ in the absence of noise is: Sℓ(t) = mℓ(t)ej2πf0t (3) where mℓ(t) is the modulation function of the ℓth source and f0 the used frequency. That signal at the mth element, in noisy condition becomes: xm(t) = mℓ(t) ej(2πf0t+Δγm) + bm(t) = Sℓ(t)am(ϕℓ) + bm(t) (4) where am(ϕℓ) = ejΔγm = e-jkd(m-1)cos(ϕℓ) (5) and bm(t), the mth component of random noise which consists of the background noise and electronic noise generated in the mth channel. It is assumed to be temporally white with zero mean and variance equal to σn

2. Therefore, the steering vector from a source ℓ on the network elements is: a(ϕℓ) = [1, a2(ϕℓ), …, am(ϕℓ), …, aM(ϕℓ)]T (6) where T is the transpose operator. If we consider all sources simultaneously, the signal at the mth

]1

( ) ( ) ( ) ( )L

m m mX t S t a b tφ=

= +∑

element will be:

(7)

We can then define the array signal vector by:

X(t) = [X1(t), X2(t), …, Xm(t), …, XM(t)]T

and the incoming signal vector by :

(8)

S(t) = [S1(t), S2(t), …, Sℓ(t), …, SL(t)]T

The noise vector is : b(t) = [b

(9)

1(t), b2(t),…, bm(t),…, bM(t)]T. (10) The steering matrix (dimension MxL) is given by: A = [a(ϕ1), a(ϕ2), …, a(ϕℓ), …, a(ϕL)]. (11) We can now write in matrix notation: X(t) = AS(t) + b(t) (12) Let’s define the weights of the beam former as: W = [w1, w2, …, wm, …, wM]T

*

1( ) ( )

M

m mm

Y t w x t=

= ∑

(13) The total array output is then:

= WHX(t) (14)

Where H denotes the complex conjugate transpose. If the components of X(t) can be modeled as zero mean stationary processes, then for a given W, the mean output power of the process is: P = E[Y(t)Y*(t)] = WHRxxW (15) where E[.] denotes the expectation operator and Rxx is the array correlation matrix defined by Rxx = E[X(t)XH(t)] (16) By replacing X(t) with (12) and developing, we obtain : Rxx = ASAH + σn

2I (17) where I is the identity matrix of dimension M. 2.2. Theoretical model of the planar array Consider a square lattice (by matrix inversion need), comprising MxM uniformly spaced antennas of d = λ/2, assumed punctual, on which arrive L signals from L sources (Fig. 2). The reference element is one of the peaks, origin of the landmark (O, x, y, z). The coordinates of an element respectively of rank m along x and p along y will be denoted (xmp, ymp, zmp). Returning to equation (1), and considering that zmp = 0, xmp = (m-1)d, ymp = (p-1)d and θ = π/2, we obtain: ∆γmp = γmp(t) − γ1(t) = − kxmpcosϕsinθ − kympsinϕsinθ − kzmpcosϕ = − kd[(m−1)cos(ϕ) + (p−1)sin(ϕ)] (18)



Figure 2: Uniform planar array of MxM antennas

The received signal on the sensor at (m, p) coming from a source ℓ is: xmp(t) = mℓ(t)ej(2πf0t+∆γmp) +bmp(t) (19) where Sℓ(t) = mℓ(t)ej(2πf0t) is the incoming signal received on the reference element and bmp(t) the additive noise on the sensor. Since then: Xmp(t) = Sℓ(t)ej∆γmp + bmp(t) = Sℓ(t)amp(ϕℓ) + bmp(t) (20) Where amp(ϕℓ) = ej∆γmp = e-jkd[(m-1)cos(ϕℓ) + (p-1)sin(ϕℓ)]

1( ) ( ) ( ) ( )

L

mp mp mpX t S t a b tφ=

= +∑

(21) Considering all sources together, the incoming signal on the sensor at (m, p) is:

(22)

The array signal vector becomes a matrix:

11 12 1 1

21 22 2 2

1 2

1 2

... ...

... ...( )

... ...

... ...

p M

p M

m m mp mM

M M Mp MM

x x x xx x x x

X tx x x xx x x x

=

(23)

We can also define the steering matrix of the signal coming from a source ℓ over all the elements by:

12 1 1

21 22 2 2

1 2

1 2

1 ... ...

... ...( ( ))

... ...

...

p M

p Ml mp l

m m mp mM

M M Mp MM

a a aa a a a

A aa a a aa a a a

φ

= =

(24)

The steering matrix produced by the L sources will then be of M lines and MxL columns:

( )( ) ( )

12 1

21 22 22 3

1 2

1 2

1 ......

.........

M

ML

m m mM

M M MM

a aa a a

A A A Aa a aa a a

=

(25)

Like in the case of linear array, the incoming signal vector is: S(t) = [S1(t), S2(t), …, Sℓ(t), …, SL(t)]T

The noise over the array elements becomes a matrix B(t). we can then rewrite in matrix notation:

.

X(t) = AS(t) + B(t) (26) On the other hand, the weight vector for beamforming becomes a MxM matrix

11 12 1 1

21 22 2 2

1 2

1 2

... ...

... ...

... ...

... ...

p M

p M

m m mp mM

M M Mp MM

w w w ww w w w

Ww w w ww w w w

=

:

(27)

The total array output becomes:

Y(t) = *

1 1

M M

mp mpm p

W X= =

∑ ∑ = WH X (t) (28)

Like for the linear array, equations (15), (16) and (17) will permit to calculate the mean output power and evaluate the correlation matrix. 2.3. Theoretical model of the circular array Consider a circular array of radius a (Fig.3), comprising M uniformly spaced elements, contained in the plane (x, O, y).

Figure 3: Uniform circular array of M elements

The coordinates of an element of rank m is (xm, ym, zm). We still have θ = π/2 ; xm = acos(φm) ; ym = asin(φm) and zm = 0. By replacing them in equation (1), we obtain: ∆γm = γm(t) − γ1(t) = − kxmcos(ϕ) − kymsin(ϕ) = − ka[cos( φm)cos(ϕ) − sin( φm)sin(ϕ)] (29) The signal received at the mth element is given by: xm(t) = mℓ(t)ej(2πf0t+Δγm) + bm(t) = Sℓ(t)am(φm,ϕℓ) + bm(t) (30) where: am(φm,ϕℓ) = ejΔγm = e−jka[cos(φm)cos(ϕℓ) − sin(φm)sin(ϕℓ)] (31) and bm(t), the mth component of random noise. The steering vector of the signal coming from a source ℓ over all the elements is:



a(φm,ϕℓ) = [1, a2(φm,ϕℓ), …, am(φm,ϕℓ), …, aM(φm,ϕℓ)]T (32) If we consider all the sources simultaneously, the signal received at the mth

0(2 )

1( ) ( ) ( )m

Lj f t

m mX t m t e b tπ γ+∆

=

= +∑

element is:

]1

( ) ( , ) ( )L

m m mS t a b tϕ φ=

= +∑

(33)

Equations (8), (9) and (10) remain the same. The steering matrix (which is of dimension MxL) is given by: A = [a(φm,ϕ1), a(φm,ϕ2), …, a(φm,ϕℓ), …, a(φm,ϕL)] (34) Equations (12) to (17) remain the same.

III. METHODOLOGY

Smart antenna implementation involves two stages: determining the directions of arrival, followed by the subsequent beamforming. 3.1. Conventional algorithms for estimating directions of

arrival (DOA) We group them into two categories (Fig. 4)

:

The spectral estimation methods

;

3.1.1. The spectral estimation methods [7],[8]. The Eigenstructure methods.

3.1.1.1. The basic method : the spatial Fourier transform

Developed by Bartlett, it is one of the first methods used to detect arrivals angles. Its principle is to perform the Fourier transform in the space of received signals. If we plot this function for a given wavelength, we obtain an energy peak to the direction in which the source is located. That energy’s pseudo-spectrum

2

1 HxxP A R A

M=

is given by:

(35)

3.1.1.2. The linear prediction method Developed by Prony in 1795, its principle is to minimize the prediction error on the response of any element of the array. Search for weights that will minimize the mean value of this error leads to the pseudo-spectrum:

1

1 2| |

Hm xx mHm xx

u R uP

u R A

−

−=

(36)

where um is the mth column of the identity matrix IM

3.1.1.3. Maximum Likelihood Methods (MLM)

.

They are developed according to several criteria: Maximizing Signal to Interference plus Noise

Ratio: developed by Capon in 1969, it is based on an unbiased estimate and minimum variance, that’s why it’s also called MVDR

1

1H

xx

PA R A−=

(Minimum Variance Distorsionless Response). Its pseudo-spectrum is given by:

(37)

Minimizing the square error: the problem is considered as an inverse problem, minimizing a least square criterion: 2|| ||mcr X AS= − (38) The goal is to find for what value of ϕ, mcr is minimal.

( ) 2 2 0H Hd mcr A X A ASdS

= − + =

For this, we do:

(39)

We find : 1( )H HS A A A X−= (40) (40) in (38) gives:

1 2|| ( ) ||H Hmcr X A A A A X−= − (41) Then we minimize mcr, which contains as single the unknown vector ϕ desired, contained in

The minimization of a cost function f(ϕ) :

A.

For example in [8],

( ) ln det( )HMf APA qIφ = + (42)

{ }1 1 1 11( ) ( ) ( ) ( )H H H H H Hxx M xxP A A A R A A A tra ce I A A A A R A A

M L− − − − = − − −

(43)

3.1.1.4. The Maximum Entropy Method (MEM) Developed by Burg, we look for directions that maximize

the pseudo-spectrum: 1H H

m m

PA C C A

= (44)

where Cm is the mth column of the inverse of the correlation matrix Rxx3.1.2. The eigenvalues or sub-spaces methods [7],[8].

.

They follow historically, the Capon method and are based on a decomposition of the space into a signal space (Es) and a noise space (En) by searching eigenvalues

3.1.2.1. Pisarenko harmonic decomposition (Minimum Mean Square Error: MMSE)

.

Its purpose is to minimize the mean square error at the output of the array under the constraint that the norm of the weight vector shall be equal to unity. The eigenvector of the corresponding correlation matrix is that associated with the smallest eigenvalue. Its pseudo-spectrum is:

1

1| |HPA e

= (45)

where e1 is the eigenvector corresponding to λ13.1.2.2. The minimum norm method

.

Developed by Reddi, Kumaresan and Tufs, this method optimize the weight vector by resolving the following system equation: min( )HW W ; 0H

SE W = ; 1.e 1W = . (46) The solution leads to the pseudo-spectrum:



21

1| |H H

b b

PA E E e

= (47)

3.1.2.3. MUSIC (Multiple Signal Classification) [6],[9]. The basic version was proposed by Schmidt: generally assess Rxx by expression (17) is not easy. Instead, using its estimated also denoted Rxx

1

1 ( ) ( )N

Hxx

tR X t X t

N =

= ∑ given by:

(48)

where N is the number of samples. By calculating the eigenvalues of the estimated Rxx (in descending order), we obtain M eigenvectors, which the first L correspond to the signal space Es = [e1 e2 … eL], and the (M-L) last to the noise subspace En = [eL+1 eL+2 … eM]. Is then plot the cost function, which performs a projection of the noise subspace on the signal subspace; thus making it seeks for what ϕ values the noise subspace is orthogonal to the signal subspace, which corresponds to the directions of arrivals sought. This is the pseudo MUSIC spectrum

1| |H H

b b

PA E E A

=

:

(49)

Variants have been developed to deal with shortcomings in order to meet certain specific conditions [6],[7],[10].

3.1.2.4. ESPRIT (Estimation of Signal Parameters via Rotationnal Invariance Techniques) [7]

The basic version breaks the antenna array into two sub-networks X and Y offset Δ. The signal received on the second sub-network is then phase-shifted relative to the first. Instead of calculating the eigenvalues of a matrix alone then browse a spectrum as is the case with MUSIC, eigenvalues are calculated for both the autocorrelation matrices Rxx and Ryy respectively and then searching for the matrix Ψ for moving from to each other. Then deduced the angles of arrivals: θn = arcos[arg(λ1) / 2π∆] (50) where ℓ= 1 … L and λℓ the eigenvalues of Ψ. The same improvements brought to MUSIC, are often applied to ESPRIT. We summarize in Fig. 4, the main classical algorithms for estimating DOA.

Figure 4: Classical algorithms for estimating

3.2. Beamforming algorithms

DOA

The arrays developed models in paragraph 2 allow to obtain in the absence of noise: X(t) = A(θ, ϕ)S. by replacing in Y = WHX(t), we obtain Y = WHA(θ, ϕ)S. we set AF = WHA(θ,ϕ) (51) AF represents the array factor. It calculates the radiation pattern when the weights of individual antenna elements are known. Conform the radiation pattern to privileged directions returns then to adjust the various weight. Its normalized

( , )max( )AFNAF

AFθ φ

=

value is:

(52)

We identify two main types of beamformers: fixed and adaptive

3.2.1. Fixed beamformers

(Fig. 5).

They are user data independent. 3.2.1.1. The conventionnal shaper It is the most simple to make and is also known as the "sum-and-delay beamformer." Generally, it is used in combination with MUSIC to point the main beam in a desired fixed direction. All its weights have the same value

0 01 ( , )W AM

θ φ= (53)

For more information have a look in [6]

3.2.1.2. The null-steering beamformer [6] It is used to cancel a plane wave from a known direction by canceling the radiation pattern in that direction. The weights are determined as: WHA=e1

T where e1 = (1,0,0…0)T (54) If all sub-matrices a(θℓ,ϕℓ) of A are linearly independent, and A a square matrix, then A can be inverted and the weight matrix W calculated by:



W = WHA = e1T.A-1 (55)

Although canceling the interfering signals, the shaper does not take into account the useful signals3.2.2. Adaptive beamformers

.

They depend on the user data and are therefore indicated for adaptive antennas

. The optimal beamformer [6],[11] It overcomes two of the major limitations of the "Null-steering beamformer":

It does not require information about the directions of interfering

;

It maximizes the signal to interference plus noise ratio at the output of the array (

The weights are given by : W=µ

MVDR).

0 Rxx-1A(θ0,ϕ0)

(56) where µ0 is a constant given by: μ0 = 1/[AH(θ0,ϕ0)RxxA(θ0,ϕ0

10 0

0 0 0 0

( , )( , ) ( , )

xxH

xx

R AW

A R Aθ φ

θ φ θ φ

−

=

)] (57) By replacing (57) in (56), we obtain:

(58)

These weights minimize the output average power while maintaining a response unit in the direction of the user. Thus the process minimizes the total noise including interference and correlation noises. As the value of Rxx is not available, we use its estimated

[ ] [ ] [ ] [ ]1 11

1

Hxx

xx

nR n X n X nR n

n+ + +

+ =+

:

(59)

3.2.2.1. Training sequence beamformers For more information have a look in [6],[8].

The training sequence is a sent information part known by the receiver, allowing it to deduce from the bits arrival state, the channel transfer function. Thus, there is a reference signal with which the antenna compares the output of the array signal. The most commonly used algorithms are

• DMI (Direct Matrix Inversion, also called SMI : Sampled Matrix Inversion)

:

• LMS (Least Mean Square) • RLS (Recursive Least Square)

The Least Mean Square algorithm (LMS) It is said stochastic gradient and is the recurrent version of the Wiener filter [8]. This algorithm calculates the weights according to the equation: W(n+1) = W(n) + µX(n)[d*(n) - XH (n)W(n)] (60) where W(n+1) represents W at the (n+1)th iteration and µ the constant gain that controls the degree of adaptation, that is, how fast and how well, the estimated weight are close to optimal weight. The Recursive Least Square algorithm (RLS)

In our code, we call it mu_cst.

The LMS convergence depends on the eigenvalues of Rxx. If Rxx has a very broad spectrum, LMS

becomes quite slow. This problem can be solved by replacing the previous gain µ by a matrix Rxx

-1(n) at the nth

W(n) = W(n-1) – Riteration. We then have according to [12]:

xx-1(n)X(n) ε*

Where: ε

(W(n-1)) (61)

*(W(n-1)) = [d*(n-1) - XH (n-1)W(n-1)] (62) and: Rxx(n) = δo Rxx(n-1) + X(n)XH

00

( ) ( )n

n k H

kX k X kδ −

=∑

(n)

= (63)

with δo

1

0

(0) Mxx

IRδ

− =

=(1-1/N) less than but close to 1. N is the number of iterations. In [12], we have:

(64)

where IM is the identity matrix of order M and M the number of elements of the antenna array. Although the algorithm converges about 10 times faster than LMS, the number of iterations N must be kept high in order to get δo close to 1, which is its main Direct Matrix Inversion algorithm (DMI or SMI)

limitation.

The weightings are chosen to minimize the quadratic error between the array output signal and the reference signal. According to [8], this error is given by: E [{r(t)-WHX(t)}²] = E [{r²(t)] - 2WHRr+WHRmW (65) where X(t) is the array output signal at time t and r(t) the reference signal. Rm = E[X(t)XH(t)] is the autocorrelation matrix of the signal or covariance matrix. Rr = E [r(t)XH(t)] is the cross-correlation matrix between the reference signal and the array output signal. The weight vector for which the equation (65) has a minimum is obtained by canceling its gradient vector with respect to W. This is to say

w∇

: {E[{r(t)- WHX(t)}²]} = - 2 Rr + 2RmW = 0. We draw :

Wopt = Rm-1Rr

(66) Thus, the optimum weight can be easily obtained by direct inversion of the 3.2.2.2. Constant Modulus Amplitude algorithm (CMA)

covariance matrix.

Its configuration is the same as the DMI system with the sole difference that here a reference signal is not required [4]. This is a gradient algorithm which operates on the theory that the existence of interference generally leads to variations in amplitude of the transmitted signal, which nevertheless has a constant envelope. The updating of the weight is obtained by minimizing the mean of the positive cost function Jn defined by: Jn = (½) E [(|y(n)|² - yo

where y(n)

²)²] (67) The weights are given by: W(n+1) = W(n) – µ.g(W(n)) (68)

is the array output after the nth iteration; yo the amplitude of the envelope of the desired signal in the



absence of interference; g(w(n)) an estimate of the cost function and µ the adjustment coefficient. We summarize in Fig. 5, the main conventional beamforming algorithms.

Fig. 5: Classical beamforming algorithms

IV. RESULTS AND DISCUSSION

We have developed a tool in MATLAB 7.0 with an HP laptop (300Go, 1.8GHz, 2GB, Windows7). 4.1. Results of DOA estimation Fig.6 shows the home page of DOA detector.

Figure 6: Home page for DOA detection

By clicking on "User guide" in blue at the bottom left of the home page, there is a recall on smart antennas and simulation process. By varying the different parameters of antennas or signals, one can study their influence on the accuracy of the results

. 4.1.1. Radiation Patterns Figs.7 and 8 show the detection spectra for a direction ϕ=180° using a circular array of M = 16 elements, N = 1000 and SNR = 0 (case a) or 15dB (case b) respectively obtained in [14] and with our tool.

Figure 7: Results of DOA detection obtained in [14]

Figure 8: Results of DOA detection obtained with our tool. Figs. 9 to 11 allow to generalize.

Figure 9: Various spectra of DOA estimators over linear array of M=8 elements, f=1.8GHz, d=0.5λ, SNR=30dB, 3

directions to find (50°,80°,120°)



Figure 10: Various spectra of DOA estimators over planar array of M = 9x9 elements, N=100, d=0.5λ, f=1.8GHz,

SNR=30dB, 3 directions to find (50°,80°,120°)

Figure 11: Various spectra of DOA estimators over circular array of M = 16 elements, N=100, d=0.5λ, f=1.8GHz, SNR=30dB, 3 static directions to find

(50°, 80°, 120°)

Figure 12: MUSIC power spectrum on square lattice

4.1.2. Elapsed times

of M = 9x9 elements, N=100, d=0.5λ, f=1.8GHz, SNR=30dB, 3

static directions to find (50°, 80°, 120°)

The other parameter to which we are interested is the computation time of the algorithms. We assess with "Tic Toc" function of MATLAB. The results are shown in Tables I, II and III.

4.2. Beamforming results The home page for the beamforming is shown in Fig. 13.

Figure 13: Home page for the beamforming



4.2.1. Beamforming spectra Fig. 14 compares for validation, the results of RLS (a) obtained in [3] and (b) obtained with our tool.

(a) (b)

Figure 14: Results of RLS in logarithmic coordinates on a linear array of M = 8 elements, N=100, δ=0.9, user to

cover at 60°, interferer at 40°, f=1.8GHz, SNR= 30dB for different values of d, (a) obtained in [3], (b) with our tool

Figs. 15 and 16 compare the results of LMS and CMA to those of Article [4] published in June 2015.

(a) (b)

Figure 15: Results of LMS on a linear array in cartesian coordinates with N=200, μ=0.008, d = 0.5λ, user to cover at 30°, interferer at 60°, f=0.4GHz and SNR= 30dB for

different values of M, (a) in [4], (b) with our tool

Figure 16: Results of CMA in logarithmic coordinates on a linear array of M=21 elements,

N=100, μ=0.008, d = 0.5λ, user to cover at 60°, f=0.4GHz, SNR= 30dB, (a) in [4], (b)

with our tool.

Fig. 17 compares our results of DMI to those of article [5].

(a) (b)

Figure 17: Results of DMI in cartesian coordinates on a linear array with N=200, d = 0.5λ, f=1.8GHz, SNR= 30dB,

user to cover at 30°, interferer to cancel at -60°, for different values of M, (a) in [5], (b) with our tool.

Figs. 18 to 20 show various beamformers we implemented

respectively on linear networks, planar and circular.

Figure 18: Various adaptive shapers on linear network of M = 8 elements, N=200, d=0.5λ, SNR = 30dB, f=1.8GHz,

user to cover at 80° and interferer to cancel at 120°



Figure 19: Various adaptive beamformers on planar array of M = 9x9 elements, SNR=30dB, N = 100, d=0.5λ, f=1.8GHz, two static users to cover at (80°, 120°)

Figure 20: Various adaptive beamformers on circular array of M=16 elements, SNR = 30dB, N = 100, d=0.5λ, a=0.25,

f=1.8GHz, 2 users to cover at (80°, 120°)

4.2.2. Output errors We present in Fig. 21 to 23, the error curves of various conventional beamformers on linear networks, planar and circular respectively.

Figure 21: Various error curves of adaptive beamformers on the linear array of Fig.18

Figure 22: Various error curves of adaptive beamformers

on the planar array of Fig.19

Figure 23: Various error curves of adaptive beamformers on the circular array of Fig. 20

4.2.3. Elapsed times Tables IV to VI summarize the various elapsed times of classical beamformers. The simulations were conducted with N = 100 for 1 or 2 lobes and N = 200 for cases with interferers (1L1Z, 1L3Z). The other settings are the default on the tool.



4.3. Discussions We see a bad interference cancellation by CMA on linear network (Fig.18) and a poor resolution of DMI on planar array (Fig. 19). The algorithms that have retained our attention are MUSIC and Capon for estimating DOA, LMS and MVDR for beamforming. Nevertheless all these algorithms admit variants and acceptable results according to study parameters such as spacing between the radiating elements d, the number of radiating elements M, the number of iterations N, the signal to noise ratio SNR, a much lesser extent the frequency f, the number of sources to be detected or the number of lobes to form in the presence or not of interferers, mobility and particularly the configuration of the antenna array. We summarize in Table VII, a comparative study of the three main configurations of planar adaptive antennas.

For the computing times, those for DOA estimators revolve around 0.2s; which is satisfactory because for real-time applications, the maximum permissible value is 0.3s. The elapsed times of beamformers by cons, are of several seconds (5 to over 20s for some) and will therefore

V. CONCLUSION

be improved.

In this article, we modeled three common configurations of adaptive planar antennas (linear, square and circular). We also presented the main analytical approaches generally used to implement them. Subsequently a simulation tool was developed and conducted a performance study. Under this work, we can confirm that smart antennas allow a significant increase of quality of service in mobile radio networks (including scope, capacity, speed, etc.). In fact, the cancellation of some directions eliminates spurious emissions that could disrupt communications or decrease the transmission rate. This avoids interacting with other systems or damaging some equipment, "preserving" not only the battery but also and especially the environment. However, these antennas have some drawbacks, such as their heavy structure and the more material need. On the other hand, the computation time of the algorithms of "beamforming" sometimes exceed 20s, well above the permissible limit of 0.3s. We must therefore seek more rapid techniques of lobes formation. Particular attention today focuses on heuristic approaches which include genetic algorithms and neural networks.

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