Array Factor

In the image, ri means the location of the element in space(x,y,z). and k is alpha + pidcos(thetha)/Lambda and the R(theta,phi) is the radiation pattern of individual element.

In the array factor,

Please explain as to what is the weight wi, Is it the current that we give to the element? Because, the exp part takes care of the phased angle steering. So, what is the role of w, and what are its probable values ? Please elaborate.

  • \$\begingroup\$ This array factor is multiplied with the signal to cancel out the phase change due the propagation of wave. \$\endgroup\$
    – user134001
    Commented Dec 23, 2016 at 11:42

1 Answer 1


This notation is the usual notation for a complex oscillation:

$$f(x) = A*e^{-j2\pi f_0 t}$$

A is the amplitude, the rest is a complex pointer that rotates around zero. If there is no time dependend value, the exponential part introduces a phase shift.

In your case w_i is equivalent to the amplitude and - depending on what unit you want to have - you need to use the appropriate value there. The exponential part replaces the frequency from my example with a spatial location.

If wi is a complex factor it means it consists of amplitude and phase. The equation you showed in your image only takes the phaseshift that occurs by spatial placement off the center axis. Imagine one transmitter antenna and many receiver antennas. The RX antennas are seperated by the distance d which introduces a phaseshift of p. Each antenna has a small mixer+amplifier (which is why the factor wi has been introduced) whose output runs into a summing amplifier. Therefore you can use the multiplier (mathematically expressed by wi) to alter amplitude and phase of the signal. If you choose wi wisely you are able to do some beamforming (eg. by introducing a larger phaseshift to wm than to w1 in the picture)

enter image description here

I - and this is only my gut feeling - would use the power fed to the antenna. If all antenna elements are the same you can pull that factor out of the sum:

$$ \sum_{i=1}^n{(a_i*k)} = k*\sum_{i=1}^n{a_i}$$

  • \$\begingroup\$ Firstly, thank you. So, from what I understood, the wi in my image is the current fed into the individual antenna element. so, if all my elements are fed with a uniform current amplitude, I can remove this term from summation. But in case of antenna steering, where I consider the phase and amplitude of excitation, my wi will denote the current, and varying that will also vary my array factor? Am I right ? Or is it that, if the phase changes, the weight wi also gets affected. This wi is very important for me, because, I want to calculate the total powered radiated by the antenna. Please correct \$\endgroup\$ Commented Apr 4, 2014 at 10:54
  • \$\begingroup\$ But I remember reading in some books, that wi is called as complex weight of that element. What do exactly refer by saying complex weight ? Please elaborate. \$\endgroup\$ Commented Apr 4, 2014 at 10:57
  • \$\begingroup\$ Edited my question. Did that help? \$\endgroup\$
    – einball
    Commented Apr 4, 2014 at 11:52
  • \$\begingroup\$ Perfect ! This diagram got in me a good clarity. If you by any chance, you have to taken this diagram from an ebook, can you please let me know the name of the book. I am interested to read more. \$\endgroup\$ Commented Apr 4, 2014 at 14:47
  • \$\begingroup\$ I took this from a presentation a friend made about beamforming. I'm not allowed to share any more of this, sorry. If you do in fact have more unclear points on this topic please open another question. \$\endgroup\$
    – einball
    Commented Apr 4, 2014 at 14:54

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