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2.5 Source and noise model

2.5.1 Narrowband signals

A signal is narrowband if its bandwidth is much smaller than its center frequency, that is:

$W_{B}/f_0< 1/10$

Among them, $W_B$ is the signal bandwidth, $f_0$ and is the center frequency. Generally, the sine signal and the cosine signal are collectively referred to as the sine signal, and the sine signal is a typical narrowband signal. Unless otherwise specified, the narrowband signals mentioned in this book are expressed as:

$s(t)=a(t)e^{j[w_0t+\theta (t)]}$

In the formula, $a(t)$ is the slowly varying amplitude modulation function (or real envelope), $\theta(t)$ is the slowly varying phase modulation function, and $w_0=2\pi f_0$ is the carrier frequency. In general, $a(t)$ and $\theta (t)$ contain all useful information.

2.5.2 Correlation coefficient

For multiple received signals, the correlation coefficient (or cross-correlation coefficient) can generally be used to measure the degree of correlation between signals. $s_i(t)$ For the sum of two stationary signals $s_j(t)$ , the correlation coefficient is defined as:

$\rho _{ij}=\frac{E[(s_i(t)-E[s_i(t)])(s_j(t)-E[s_j(t)])]}{\sqrt{E[s_i(t)-E[s_i(t)]]^2E[s_j(t)-E[s_j(t)]]^2}}$

At that time $\rho _{ij}=0$ , it was called $s_i(t)$ irrelevant $s_j(t)$ (or irrelevant); $0<\left | \rho _{ij} \right |<1$ at that time , it was $s_i(t)$ called $s_j(t)$ partially related; $\rho _{ij}=1$ at that time , it $s_i(t)$ was called $s_j(t)$ completely related (or coherent).

2.5.3 Noise model

In this book, without special instructions, the noise received by the array element is assumed to be a stationary zero-mean Gaussian white noise with a variance of $\sigma^2$ . The noise between each array element is not correlated with each other, and is not correlated with the target source. Thus, $\thing{n}(t)$ the second moment of the noise vector satisfies:

$E\begin{Bmatrix} \vec n(t_1),\vec n^H(t_2) \end{Bmatrix}=\sigma ^2I \delta_{t_1,t_2}$

$E\begin{Bmatrix} \vec n(t_1),\vec n^T(t_2) \end{Bmatrix}=0$

2.6 Statistical Model of Array Antenna

2.6.1 Premises and assumptions

Assumptions about the receiving antenna array: the receiving array is arranged in a certain form by passive array elements located at known coordinates in space. It is assumed that the receiving characteristic of the array element is only related to the position and has nothing to do with the size (considered as a point), and the array elements are all omnidirectional array elements, the gains are equal, and the mutual coupling between each other is negligible. When the array element receives the signal, noise will be generated. Assuming it is additive Gaussian white noise, the noise on each array element is statistically independent from each other, and the noise and signal are statistically independent.

Assumptions about the space source signal: Assume that the propagation medium of the space signal is uniform and isotropic, which means that the space signal propagates in a straight line in the medium; at the same time, it is assumed that the array is in the far field of space signal radiation, so the space source signal reaches the array When can be regarded as parallel plane waves, the time delay of the space source signal arriving at each element of the array can be determined by the geometric structure of the array and the direction of the space wave. The direction of the space wave is usually characterized by elevation angle and azimuth angle in three-dimensional space.

2.6.2 Basic concepts of arrays

Let the signal carrier be , and propagate $e^{jwt}$ in space along the direction of the wave number vector in the form of a plane wave , let the signal at the reference point be , then the received signal is: $\thing to$ $s(t)e^{jwt}$

$s_r(t)=s(t-\frac{1}{c}\vec r ^T \vec \alpha)exp[j(wt-\vec r^T\vec k)]$

In the formula, $\thing to$ is the wave number vector; $\vec{\alpha}=\vec{k}/ \left | \thing k\right |$ is the electromagnetic wave propagation direction unit vector; $\left | \vec k \right |=w/c=2\pi /\lambda$ is the wave number, where c is the speed of light, $\lambda$ is the wavelength of the electromagnetic wave; $\frac{1}{c}\vec r ^T \vec \alpha$ is the delay time of the signal relative to the reference point $\thing r^T\thing k$ ; The phase lag of the electromagnetic wave propagating to the reference point. $\theta$ is the electromagnetic wave propagation direction angle, which is defined by the counterclockwise rotation direction relative to the x-axis. Obviously, the wave number vector can be expressed as:

$\vec k = k[cos\theta,sin\theta]^T$

Electromagnetic waves propagate outward from point radiation sources as spherical waves. As long as the distance is far enough, the spherical waves can be approximated as plane waves in the local area of reception.

It is assumed that there are M array elements in space to form an array, and the array elements are numbered from 1 to M , and the array element 1 (other array elements can also be selected) is used as a datum or reference point. Assuming that each array element has no directionality (omnidirectional), the position vectors relative to the reference point are respectively $r_i(i=1,...,M;r_1=0)$ . If the received signal at the reference point is $s(t)e^{jwt}$ , then the received signals on each array element are:

$s_r(t)=s(t-\frac{1}{c}\vec r_i ^T \vec \alpha)exp[j(wt-\vec r_i^T\vec k)]$

In communication, the frequency band B of the signal is much smaller than the carrier value w, so the change of s(t) is relatively slow and delayed, so there is: , that is, the $\frac{1}{c}\vec r_i ^T \vec \alpha<<(1/B)$ difference $s(t-\frac{1}{c}\vec r_i ^T \vec \alpha)\approx s(t)$ of the signal envelope on each array element can be ignored , which is called narrowband signal .

In addition, the array signal is always converted to baseband and then processed, so the array signal can be expressed in vector form as:

$s(t)=[s_1(t),s_2(t),...,s_M(t)]^T=s(t)[e^{-j\vec r_1^T},e^{-j\vec r_2^T},...,e^{-j\vec r_M^T}]^T$

The vector part in the above formula is called the direction vector , because when the wavelength and the geometric structure of the array are determined, the vector is only related to the space angle vector of the arrival wave , and the direction vector is $\theta$ denoted as $\vec a(\theta)$ For example, if the first array element is selected as the reference point, the direction vector is:

$\vec a(\theta)=[1,e^{-j\bar r_2\vec k},...,e^{-j\bar r_M\vec k}]$

formula: $\bar{r_i}=\vec r_i-\vec r_1(i=2,...,M)$

The actual array structure requires that the direction vector $\vec a(\theta)$ must be $\theta$ in one-to-one correspondence with the space angle vector, and no ambiguity can occur. When there are multiple (for example, K) sources, the direction vectors of the arriving waves can be $\vec a(\theta)$ represented by . The matrix composed of these K direction vectors $A=[\vec a(\theta_1),\vec a (\theta_2),...,\vec a(\theta_K)]$ is called the direction matrix or response matrix of the array, which represents the direction of all signal sources .

Change the space angle $\theta$ to make the direction vector $\vec a (\theta)$ scan in the M-dimensional space, and the formed surface is called array flow .

Arrays are popularly represented by the symbol A , that is:

$A=\begin{Bmatrix} \vec a(\theta|\theta \in \Theta ) \end{Bmatrix}$

Among them, $\Theta=[0.2\pi)$ is $\theta$ the set of all possible values of direction of arrival. So the array pop A is a collection of array direction vectors. Array popularity A includes the effects of array geometry, array element mode, coupling between array elements, and frequency.

2.6.3 Antenna array model

Assume that there is an antenna array, which is composed of M array elements with any directionality arranged in any order. At the same time, K spatial narrowband plane waves (M>K) with the same center frequency $w_0$ and wavelength $\lambda$ are respectively $\Theta_1,\Theta_2,...,\Theta_K$ incident on the array at a direction angle. As shown below.

Among them $\Theta_i=(\theta_i,\phi_i),i=1,2,...K$ . $\theta_i$ and $\phi_i$ are the elevation angle and azimuth angle of the i-th incident signal, respectively, $0\leq \theta_i <90^{\circ},0\leq \phi_i<360^{\circ}$

Then the output of the mth element of the array can be expressed as:

$x_m(t)=\sum_{i=1}^{K}s_i(t)e^{jw_0\tau_m(\Theta_i) }+n_m(t)$

Among them, $s_i(t)$ is the i-th source signal incident to the array, $n_m(t)$ is the additive noise of the m-th array element, and is the time delay relative to the selected reference point when the source signal from the direction is projected to the m-th array element $\tau_m(\Theta_i)$ . $\Theta_i$ And remember:

$X(t)=[x_1(t),x_2(t),...,x_M(t)]^T$

$N(t)=[n_1(t),n_2(t),...,n_M(t)]^T$

In addition, S(t) is a Kx1-dimensional column vector

$S(t)=[s_1(t),s_2(t),...,s_K(t)]$

$A(\Theta)$ is the direction matrix of the MxK matrix: $A(\Theta)=[\vec a(\Theta_1),\vec a(\Theta_2),...,\vec a(\Theta_K)]$

$A(\Theta)$ Any column vector in the matrix $\vec a(\Theta_i)$ is a direction vector of the array in the space source signal $\Theta_i$ , and it is an Mx1-dimensional column vector: $\vec a(\Theta)=[e^{jw_0\tau_1(\Theta_1)},e^{jw_0\tau_2(\Theta_2)},...,e^{jw_0\tau_M(\Theta_M)}]^T$

Therefore, if described by a matrix, even in the most generalized case, the array signal model can be simply expressed as:

$X(t)=A(\Theta)S(t)+N(t)$

Obviously, the direction matrix $A(\Theta)$ is related to the shape of the matrix and the direction of the signal source. Generally, in practical applications, the shape of the antenna array will not change once it is fixed. Therefore, $A(\Theta)$ any column in the matrix is always closely related to the direction of a certain spatial source signal.

2.6.4 Orientation diagram of the array

The relationship between the absolute value of the array output and the incoming wave direction is called the pattern of the antenna. There are generally two types of pattern, one is the direct addition of the array output (regardless of the signal and its direction), that is, the static pattern; the other is the pattern with pointing (considering the pointing of the signal), of course, the pointing of the signal It is achieved by controlling the weighted phase. It can be seen from the previous signal model that, for a certain $M$ element space array, under the condition of ignoring the noise, $l$ the complex amplitude of the first array element is:

$x_l=g_0e^{-jw\tau_l},l=1,2,...,M$

where, $g_0$ is the complex amplitude of the incoming wave, $\tau_l$ and is $l$ the delay between the th array element and the reference point. Assuming that $l$ the weight of the first array element is $w_l$ , then the weighted output of all array elements is:

$Y_0=\sum_{l=1}^{M}w_lg_0e^{-jw\tau_l},l=1,2,...,M$

After taking the absolute value of the above formula and normalizing it, $G(\theta)$ the pattern :

$G(\theta)=\frac{\begin{vmatrix} Y_0 \end{vmatrix}}{ max(\begin{vmatrix} Y_0 \end{vmatrix}) }$

If the above $w_l=1,l=1,2,...,M$ formula becomes a static pattern $G(\theta)$

Next consider the pattern of a uniform line array . Assume that the spacing of the uniform linear array is $d$ , and the leftmost array element is used as the reference point; and the azimuth of the signal incident is assumed to be $\theta$ , where the azimuth represents the angle between the incident direction of the signal and the normal direction of the line array. Then $l$ the wave path difference (time delay) between the th array element and the reference point is $\tau_l=(x_xin\theta)/c=(l-1)dsin\theta/c$ , then the output of the array is:

$Y_0=\sum_{l=1}^{M}w_lg_0e^{-jw\tau_l}=\sum_{l=1}^Mw_lg_0e^{-jw(\frac{(l-1)dsin\theta}{c })}=\sum_{l=1}^Mw_lg_0e^{-j\frac{w}{c}(l-1)dsin\theta}=\sum_{l=1}^Mw_lg_0e^{-j\frac {2\pi}{\lambda}(l-1)dsin\theta}=\sum_{l=1}^Mw_lg_0e^{-j(l-1)\beta}$

In the above formula, $\beta=2\pi d sin\theta/\lambda$ , $w=2\pi f$ , $\lambda$ is the wavelength of the incident signal.

When in the above formula $w_l=1,l=1,2,...,M$ , the formula can be further simplified as:

$Y_0=Mg_0e^{j(M-1)\beta /2}\frac{sin(M\beta/2)}{Msin(\beta/2)}$

The static pattern of the uniform linear array can be obtained as (Formula 2.6.17):

$G_0(\theta)= |\frac{sin(M\beta/2)}{Msin(\beta/2)}|$

When in the above formula $w_l=e^{j(l-1)\beta_d},\beta_d = 2\pi d sin\theta_d/\lambda,l=1,2,...,M$ , the formula can be simplified as:

$Y_0=Mg_0e^{j(M-1)(\beta-\beta_d)/2}\frac{sin[M(\beta-\beta_d)/2]}{Msin[(\beta-\beta_d)/2 ]}$

$\theta_d$ Then the array pattern of pointing to can be obtained as (Formula 2.6.19):

$G(\theta)=| \frac{sin[M(\beta-\beta_d)/2]}{Msin[(\beta-\beta_d)/2]}|$

2.6.5 Beamwidth

The direction finding range of the linear array is [-90°, 90°], while the direction finding range of the general area array (such as the circular array) is [-180°, 180°]. To illustrate beamwidth, only linear arrays are considered below.

It can be seen from formula 2.6.17 that the static pattern of the uniform linear array with M array elements is:

$G_0(\theta)= |\frac{sin(M\beta/2)}{Msin(\beta/2)}|$

In the formula, the spatial frequency $\beta = (2\pi d sin\theta)/\lambda$

For the null point of the main lobe of the static pattern of the antenna, $|G_0(\theta)|^2=0$ the beam width of the null point can be obtained $BW_0$ as:

$BW_0 = 2$arcsin$(\lambda/Md)$

And $|G_0(\theta)|^2=1/2$ the half-power point beam width can be obtained $BW_{0.5}$ under $Md\gg \lambda$ the condition of:

$BW_{0.5}\approx 0.886\lambda/Md$

In this book, the half-power point beamwidth of the static pattern is generally considered, that is, for a uniform linear array, its beamwidth is:

$BW_{0.5}\approx \frac{51^{\circ}}{D/\lambda}=\frac{0.89}{D\lambda}rad$

In the formula, D is the effective aperture of the antenna, $\lambda$ is the wavelength of the signal, and rad is the unit of radian. For an equidistant uniform linear array with M array elements, the array element spacing is $\lambda/2$ , and the effective aperture of the antenna is $D=(M-1)\lambda/2$ , so for the ULA array, the approximate calculation formula for the array beam width is:

$BW\approx 102^{\circ}/M$

Regarding the beam width, the following points need to be noted.

(1) The beam width is inversely proportional to the antenna aperture. In general, the relationship between the half-power point beam width of the antenna and the antenna aperture is:

$BW_{0.5}=(40\sim 60)\frac{\lambda}{D}$

(2) For some arrays (such as linear arrays), the beam width of the antenna is related to the beam pointing. For example, when the beam pointing $\theta_d$ is beam width of a uniform linear array is:

$BW_0=2$arcsin$(\frac{\lambda}{Md}+sin\theta_d)$

$BW_{0.5}\approx 0.886\frac{\lambda}{Md}\frac{1}{cos\theta_d}$

(3) The narrower the beam width, the better the directivity of the array, which means the stronger the ability of the array to distinguish space signals.

2.6.6 Resolution

In array direction finding, the resolution of the signal source in a certain direction is directly related to the change rate of the array direction vector near this direction. Near the direction where the direction vector changes quickly, the array snapshot data changes more with the change of the source angle, and the corresponding resolution is also higher. Define a representation resolution $D(\theta)= ||\frac{d\vec a(\theta)}{d\theta}||\propto ||\frac{d\tau}{d\theta} ||$

$D(\theta)$ The larger the value, the higher the resolution in that direction.

For a uniform linear array, yes $D(\theta)\proto cos\theta$ , indicating that the resolution of the signal is the highest in the 0° direction, and the resolution in the 60° direction has been reduced by half, so the direction finding range of the general linear array is -60° to 60°.

"Array Signal Processing and MATLAB Realization" Source and Noise Model, Array Antenna Statistical Model