Array pattern multiplication of phased array antennas

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Key focus: Array pattern multiplication: total radiation pattern of N identical antennas is product of single-antenna radiation vector and array factor.

Antenna arrays

Ferdinand Braun invented the Phased Array Antenna in 1905. He shared a Nobel Prize in physics in recognition of their contributions to the development of wireless telegraphy.

An antenna array is a collection of numerous linked antenna elements that operate together to broadcast or receive radio waves as if they were a single antenna. Phased array antennas are used to focus the radiated power towards a particular direction. The angular pattern of the phased array depends on the number of antenna elements, their geometrical arrangement in the array, and relative amplitudes and phases of the array elements.

Phased array antennas can be used to steer the radiated beam towards a particular direction by adjusting the relative phases of the array elements.

The basic property of antenna arrays is the translational phase-shift.

Time-shift property of Fourier transform

Let’s focus for a moment on the time-shifting property of Fourier transform. The time–shifting property implies that a shift in time corresponds to a phase rotation in the frequency domain.

\[F \left\{x(t−t_0) \right\}=e^{-j \omega t_0}X(\omega) \quad \quad (1)\]

Translational phase-shift property

Now, let’s turn our attention to antenna elements translated/shift in space. Figure 1 depicts a single antenna element having current density J(r) placed at the origin is moved in space to a new location that is l0 distant from the original position. The current density of the antenna element at the new position l0 is given by

\[J_{l_0}(r) = J(r – l_0) \quad \quad (2)\]
Translational phase-shift property of antenna elements
Figure 1: Current density of antenna element shifted in space

From the discussion on far-field retarded potentials, the radiation vector F(θ,ɸ) of an antenna element is given by the three dimensional spatial Fourier transform of current density J(z).

\[\mathbf{F} \left(\mathbf{k} \right) =\int_V J(r)e^{j \mathbf{k} \cdot r } d^3 r, \quad \quad \mathbf{k} = k\hat{r} \quad \quad (3) \]

Therefore, from equations (2) and (3), the radiation vector of the antenna element space-shifted to new position l0 is given by the space shift property (similar to time-shift property of Fourier transform in equation (1))

\[\begin{aligned} \mathbf{F}_{l_0} \left(\mathbf{k} \right) &=\int_V J_{l_0}(r)e^{j \mathbf{k} \cdot r } d^3 r \\ &= \int_V J(r-l_0)e^{j \mathbf{k} \cdot r } d^3 r \\ &= \int_V J(r)e^{j \mathbf{k} \cdot (r+l_0) } d^3 r \\ &= e^{j \mathbf{k} l_0}\int_V J(r)e^{j \mathbf{k} \cdot r } d^3 r \\ &= e^{j \mathbf{k} l_0} \mathbf{F} \left(\mathbf{k} \right) \end{aligned} \\ \Rightarrow \boxed{ \mathbf{F}_{l_0} \left(\mathbf{k} \right) = e^{j \mathbf{k} l_0} \mathbf{F}\left(\mathbf{k} \right) } \quad \quad (4) \]

Note: The sign of exponential in the Fourier transform does not matter (it just indicates phase rotation in opposite direction), as long as the same convention is used throughout the analysis.

From equation (4), we can conclude that the relative location of the antenna elements with regard to one another causes relative phase changes in the radiation vectors, which can then contribute constructively in certain directions or destructively in others.

Array factor and array pattern multiplication

Figure 2 depicts a more generic case of identical antenna elements placed in three dimensional space at various radial distances l0, l1, l2, l3, … and the antenna feed coefficients respectively are a0, a1, a2, a3,…

Figure 2: Current densities of antenna elements shifted in space - contributors to array factor of phased array antenna
Figure 2: Current densities of antenna elements shifted in space – contributors to array factor of phased array antenna

The current densities of the individual antenna elements are

\[\begin{aligned} J_{l_0}(r) &= a_0 J(r – l_0) \\ J_{l_1}(r) &= a_1 J(r – l_1) \\ J_{l_2}(r) &= a_2 J(r – l_2) \\ & \vdots \end {aligned} \quad \quad (5)\]

The total current density of the antenna array structure is

\[J_{total} = a_0 J(r – l_0) + a_1 J(r – l_1) + a_2 J(r – l_2) + \cdots \quad \quad (6)\]

Applying the translational phase-shift property in equation (4), the total radiation vector of an N element antenna array is given by

\[ \begin{aligned} \mathbf{F}_{total} \left(\mathbf{k} \right) &=\mathbf{F}_{l_0} \left(\mathbf{k} \right) + \mathbf{F}_{l_1} \left(\mathbf{k} \right) + \mathbf{F}_{l_2} \left(\mathbf{k} \right) + \cdots \\ &= a_0 e^{j \mathbf{k} l_0} \mathbf{F} \left(\mathbf{k} \right) + a_1 e^{j \mathbf{k} l_1} \mathbf{F} \left(\mathbf{k} \right) + a_2 e^{j \mathbf{k} l_2} \mathbf{F} \left(\mathbf{k} \right) + \cdots\\ &= \mathbf{F} \left(\mathbf{k} \right) \sum_{i=0}^{N} a_i e^{j \mathbf{k} l_i} \\ &= \mathbf{F} \left(\mathbf{k} \right) \mathbf{A} \left(\mathbf{k} \right)\end{aligned} \]
\[ \boxed{\mathbf{F}_{total} \left(\mathbf{k} \right) = \mathbf{F} \left(\mathbf{k} \right) \mathbf{A} \left(\mathbf{k} \right) \quad \quad (\text{array pattern multiplication}) } \quad \quad (7)\]

The quantity A(k) is called array factor which incorporates the relative translational phase shifts and the relative feed coefficients of the array elements.

\[\boxed{\mathbf{A} \left(\mathbf{k} \right) = \sum_{i=0}^{N} a_i e^{j \mathbf{k} l_i} \quad \quad (\text{array factor}) }\quad \quad ,\mathbf{k} = k\hat{r} \quad \quad (8)\]

or equivalently,

\[\boxed{\mathbf{AF} \left(\theta, \phi \right) = \sum_{i=0}^{N} a_i e^{j k \left( \hat{\theta} \times \hat{\phi} \right) l_i} \quad \quad (\text{array factor}) } \quad \quad (9)\]

The array pattern multiplication property states that the total radiation pattern of an antenna array constructed with N identical antennas is the product of radiation vector of a single individual antenna element (also called as element factor) and the array factor.

Effect of array factor on power gain and radiation intensity

Let U(θ,ɸ) and G(θ,ɸ) denote the radiation intensity and the power gain patterns of an antenna element. The total radiation intensity and the power gain of an antenna array, constructed with such identical antenna elements, will be modified by the array factor as follows.

\[\boxed{\begin{aligned} U_{total}\left( \theta, \phi \right) &= \mid A(\theta, \phi)\mid ^2 U\left( \theta, \phi \right)\\ G_{total}\left( \theta, \phi \right) &= \mid A(\theta, \phi)\mid ^2 G\left( \theta, \phi \right) \end{aligned}} \quad \quad (10)\]

The role of array factor is very similar to that of the transfer function of an linear time invariant system. Recall that if a wide sense stationary process x(t) is input to the LTI system defined by the transfer function H(f), then the power spectral density of the output is given by

\[S_y(f) = \mid H(f) \mid ^2 S_x(f)\]

Illustration using a linear array of half-wave dipole antennas

Linear antennas are electrically thin antennas whose conductor diameter is very small compared to the wavelength of the radiation λ.

A linear antenna oriented along the z-axis has radiation vector (field) whose components are along the directions of the radial distance and the polar angle. That is, the radiation intensity U(θ,ɸ) and the power gain G(θ,ɸ) depend only on the polar angle θ. In other words, the radiation intensity and the power gain are omnidirectional (independent of azimuthal angle ɸ).

Figure 3, illustrates an antenna array with linear half-wave dipoles placed on the x-axis at equidistant from each other.

A linear antenna array with half-wave dipole elements
Figure 3: A linear antenna array with half-wave dipole elements

We are interested in the power gain pattern G(θ,ɸ) of the antenna array shown in Figure 3.

The normalized power gain pattern of an individual antenna element (half-wave dipole) is given by

\[G_{dipole}(\theta, \phi) = \frac{cos^2 (\frac{\pi}{2} cos \theta)}{sin^2 \theta} \quad \quad (11)\]

From the Figure 4 given in this post, the maximum value for the normalized power gain occurs at θ =90°=π/2 radians, i.e, along the xy plane.

\[G_{dipole}(\theta = \pi/2, \phi) = 1 \quad \quad (12)\]

The array factor for the arrangement in Figure 3, computed at θ =90°=π/2 radians is given by

\[A(\theta = \pi/2, \phi) = a_0 + a_1 e^{(j \frac{2 \pi}{\lambda} \; l\; cos \phi)} + a_2 e^{(j \;2\; \frac{2 \pi}{\lambda} \; l \; cos \phi)} \quad \quad (13)\]

The total normalized power gain, along the xy plane (θ =90°=π/2 radians), of the array of dipole antennas arranged as given in Figure 3, is given by

\[G_{total}\left( \theta = \pi/2, \phi \right) = \mid A(\theta = \pi/2, \phi)\mid ^2 G_{dipole}\left( \theta = \pi/2, \phi \right) \quad \quad (14)\]

Dropping the θ for convenience in representation

\[G_{total}\left( \phi \right) = \mid A(\phi)\mid ^2 G_{dipole}\left( \phi \right) \quad \quad (15)\]

Simulation

Figure 4 illustrates equation (15) – the effect of array factor on normalized power gain of an array of half-wave dipole antennas. The plot is generated for separation distance between antenna elements l=λ and the feed coefficients for the antenna elements a = [1, -1, 1].

Check out my Google colab for the python code. The results are given below.

Illustrating the effect of array pattern multiplication on normalized power gain of antenna array
Figure 4: Illustrating the effect of array pattern multiplication on normalized power gain of antenna array

References

[1] Orfanidis, S.J. (2013) Electromagnetic Waves and Antennas, Rutgers University. https://www.ece.rutgers.edu/~orfanidi/ewa/

[2] Constantine A. Balanis, Antenna Theory: Analysis and Design, ISBN: 978-1118642061, Wiley; 4th edition (February 1, 2016)

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