**3. Problem formulation**

**Figure 3** shows a linear *N*-isotropic element adaptive array with inter-element separation distances *<sup>y</sup>*1, *<sup>y</sup>*2, *<sup>y</sup>*3, … , *yN*�<sup>1</sup>, which, in general, may be nonuniform. Here, we consider the case of one desired signal and *M* interference signals, lying in the same 2-D plane containing the linear array.

The received signal from the *i*th element (which is assumed to be a complex random process) is multiplied by a complex weight *wi* and summed with the other *N* � 1 output signals to produce the array output *So*ð Þ*t* . Using the feedback loop

*Steered Beam Adaptive Antenna Arrays DOI: http://dx.doi.org/10.5772/intechopen.100168*

**Figure 2.** *Feedback loop of the steering vector adaptive array [3].*

**Figure 3.** N*-elements adaptive array with nonuniform spacings.*

configuration of **Figure 2**, the steady-state weight vector that maximizes the output *SINR* is given by [2, 3]:

$$\boldsymbol{w} = \begin{bmatrix} \boldsymbol{w}\_1, \boldsymbol{w}\_2, \boldsymbol{w}\_3, \dots, \boldsymbol{w}\_N \end{bmatrix}^T = \begin{bmatrix} \boldsymbol{I} + k\boldsymbol{\Phi} \end{bmatrix}^T \overline{\boldsymbol{w}}\_0 \tag{8}$$

where, <sup>Φ</sup> <sup>¼</sup> *E X*<sup>∗</sup>*X<sup>T</sup>* � � is the covariance matrix of the received signal, and *<sup>w</sup>*<sup>0</sup> <sup>¼</sup> ½ � *w*10, *w*20, *w*3, … , *wN*<sup>0</sup> *<sup>T</sup>* is the steering vector of the array [1], *I* is the identity matrix, *k* is the feedback loop gain,*T* denotes transpose, \* denotes complex conjugate, *E* {.} denotes the expectation, and *X* is the received signal vector as defined in Eq. (1). The received signal of the *j*th element is given by the following:

$$\overline{\mathbf{x}}\_{j}(t) = \overline{\mathbf{S}}\_{d\_{j}}(t) + \sum\_{i=1}^{M} \overline{\mathbf{S}}\_{l\_{\overline{i}}^{i}}(t) + \overline{\mathbf{S}}\_{n\_{j}}(t), \mathbf{j} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{N} \tag{9}$$

where *Sd <sup>j</sup>*ð Þ*t* is the signal induced on the *j*th element due to the desired signal, *SIij*ð Þ*t* is the signal induced on the *j*th element due to the *i*th interference signal, *M* is the number of interference signals, and *Sn <sup>j</sup>*ð Þ*t* is the white thermal noise at the *j*th element, which has zero mean and is uncorrelated with other thermal noise signals such that

$$E\left[\overline{\mathbf{S}}\_{ni}(t)\overline{\mathbf{S}}\_{n\,j}(t)\right] = \sigma^2 \delta\_{\vec{\eta}}\tag{10}$$

where *σ*<sup>2</sup> is the variance of the thermal white noise, and *δij* is the Kronecker delta function. In this array, uniform plane waves are assumed to be incident on the sensors of the array, and the medium is assumed to be linear and its only effect on the signals is a time delay. Hence, the desired signals *Sd*1ð Þ*t* , *Sd*2ð Þ*t* , … *:*, *SdN*ð Þ*t* differ by an inter-element propagation time

$$T\_{dp} = \left(\frac{\mathcal{V}\_p}{c}\right) \sin \theta\_d \tag{11}$$

where *c* denotes the velocity of light, *θ<sup>d</sup>* is the arrival angle of the desired signal, *yp* is the inter-element spacing between the *p*th element and the ð Þ *p* þ 1 th element, and *Tdp* denotes the propagation time between the *p*th element and the ð Þ *p* þ 1 th element. Hence

$$\overline{S}\_{d\_j}(t) = \overline{S}\_{d\_1} \left( t - \sum\_{p=1}^{j-1} T\_{d\_p} \right) . \gamma\_d, \mathbf{j} = \mathbf{1}, 2, \dots, \mathbf{N} \tag{12}$$

where *γ<sup>d</sup>* ¼ exp �*jω<sup>c</sup>* P *<sup>j</sup>*�<sup>1</sup> *<sup>p</sup>*¼<sup>1</sup>*Tdp* � �, and *<sup>ω</sup><sup>c</sup>* is the center frequency of the power spectral density of the desired signal. Similarly, *SIij*ð Þ*t* is given by the following:

$$\overline{\mathbf{S}}\_{I\dot{\boldsymbol{\eta}}}(t) = \overline{\mathbf{S}}\_{I\dot{\boldsymbol{\alpha}}1} \left( t - \sum\_{p=1}^{j-1} T\_{I\dot{\boldsymbol{\alpha}}p} \right) . \boldsymbol{\eta}\_{I}, \mathbf{j} = \mathbf{1}, 2, \dots, \text{N} \tag{13}$$

where *γ<sup>I</sup>* ¼ exp �*jω<sup>c</sup>* P *<sup>j</sup>*�<sup>1</sup> *<sup>p</sup>*¼<sup>1</sup>*TIip* � �, *<sup>i</sup>* denotes the *<sup>i</sup>*th interference signal, *<sup>j</sup>* denotes the *j*th sensor element of the array, and

$$T\_{Ipi} = \left(\frac{\mathcal{V}\_p}{c}\right) \sin \theta\_{Ii} \tag{14}$$

where *θIi* is the arrival angle of the *i*th interference signal. The desired and the interference signals are assumed to have zero mean, and are stationary and statistically independent from each other and the thermal noise. This applies in many cases, especially in narrowband-faded signals, with different Directions of Arrival (DOA's).

Using these results in Eq. (8), together with the initial steering vector *w*0, a signal component-based expression for the steady-state weight vector of the array can be determined.

With the aim of obtaining a complete signal component-based expression for the steady-state weight vector *w* of Eq. (8), we now focus on the steering vector *w*0. The steering vector *w*<sup>0</sup> is chosen to provide a beam maximum of the quiescent pattern in a given direction *θmax*. It can be noted that a CW signal from an angle *θmax* will produce a signal vector:

$$X = \left[ \mathbf{1}, e^{-j\beta l\_1 \sin \theta\_{\text{max}}}, e^{-j\beta (l\_1 + l\_1) \sin \theta\_{\text{max}}}, \dots, e^{-j\beta \sum\_{i=1}^{N-1} l\_i \sin \theta\_{\text{max}}} \right] . \mathbf{e}^{j\alpha\_i t} \tag{15}$$

*Steered Beam Adaptive Antenna Arrays DOI: http://dx.doi.org/10.5772/intechopen.100168*

where *ω<sup>c</sup>* is the center frequency of the power spectral density of the desired signal and *<sup>β</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>λ</sup>* is the wave number of the desired signal. The output of the array for such a signal would be

$$
\overline{\mathbf{S}}\_o(t) = \mathbf{X}^T \boldsymbol{w} \tag{16}
$$

so that

$$\overline{\mathbf{S}}\_o(t) = \left[ w\_1, w\_2 e^{-j\mu\_1} + w\_3 e^{-j\mu\_2} + \dots + w\_N e^{-j\mu\_{N-1}} \right] e^{j\alpha\_i t} \tag{17}$$

where *<sup>μ</sup><sup>i</sup>* <sup>¼</sup> <sup>P</sup> *i j*¼1 *l j* !*β* sin ð Þ *θmax* .

The quiescent pattern of the array will have a maximum on this signal if

$$w\_1 = w\_2 e^{-j\mu\_1} = w\_3 e^{-j\mu\_2} = \dots = w\_N e^{-j\mu\_{N-1}} \tag{18}$$

Therefore, for a given *θmax* and according to the phased array theory, *w*<sup>0</sup> should be chosen as

$$\overline{w}\_0 = \begin{bmatrix} e^{-j\mu\_{N-1}}, \dots, e^{-j\mu\_2}, e^{-j\mu\_1}, \mathbf{1} \end{bmatrix}^T \tag{19}$$

Using this *w*0, the steady-state weight vector *w* may then be calculated from Eq. (8), that is, *<sup>w</sup>* <sup>¼</sup> ½ � *<sup>I</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>Φ</sup> *<sup>T</sup> w*0. Here, it should be noted that the difference between *θmax* and the actual direction of the desired signal *θ<sup>d</sup>* is called the pointing error (*θper*) of the main beam of the array.

The signals *Sd*ð Þ*t* and *SI*1ð Þ*t* , *SI*2ð Þ*t* , *SI*3ð Þ*t* , … *:*, *SIN*�<sup>1</sup>ð Þ*t* are defined such that the normalized autocorrelations in Eqs. (20) and (21) could be found; thus, they are each assumed to be a wide sense stationary random process with flat band-limited power spectral density centered at *ωc*.

Using Eq. (7), the steady-state weight vector *w* may then be calculated from Eq. (1). The output power of the desired signal can be written as

$$P\_d = \mathbb{S}\_d \left[ \sum\_{n=1}^{N} |w\_N| \right]^2 + \sum\_{i=1}^{N-1} \sum\_{m=2}^{N} 2 \operatorname{Re} \left\{ w\_N w\_m^\* \rho\_{d\_{ij}} \right\} |, for \, m > n \tag{20}$$

where *ρdij* is the normalized autocorrelation between the desired signals coming from the *i*th and *j*th antenna elements. The output power of the *i th* interference signal is [5]:

$$P\_{I\_i} = \mathbb{S}\_{I\_i} \left[ \sum\_{n=1}^{N} |w\_N| \right]^2 + \sum\_{i=1}^{N-1} \sum\_{m=2}^{N} 2 \operatorname{Re} \left\{ w\_N w\_m^\* \rho\_{I\_{[ij]}} \right\} |, \text{for } m > n \tag{21}$$

and the output power of the thermal noise is given by the following:

$$P\_n = \sigma^2 \sum\_{n=1}^{N} |w\_N|^2 \tag{22}$$

From *Pd,* P*<sup>M</sup> <sup>i</sup>*¼<sup>1</sup>*PIi* , and *Pn*, the output *SINR* can be computed as

$$\text{SINR} = \mathbf{10} \ast \log \left( P\_d / \left( \left( \sum\_{i=1}^{M} P\_{I\_i} \right) + P\_n \right) \right)$$

which can be used as a measure of the performance of the adaptive array in the sequel. The radiation/sensitivity pattern of the array can be calculated from

$$E(\theta) = 20 \log \left( w\_1 + w\_2 e^{-j\mu\_1} + w\_3 e^{-j\mu\_2} + \dots + w\_N e^{-j\mu\_{N-1}} \right) \tag{24}$$

where *<sup>μ</sup><sup>i</sup>* <sup>¼</sup> <sup>P</sup>*<sup>i</sup> <sup>j</sup>*¼1*<sup>y</sup> <sup>j</sup>* � �*βsinθ:* Now, all of the equations needed to compute the output SINR as a function of the input signal-to-noise ratio (SNR), input interference-to-noise ratios, arrival angles of the desired and interference signals, and bandwidths of the desired and interference signals are readily available. MATLAB application is used to simulate the operation of the array.
