# Relationship of array factor and directivity for an antenna array

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Jonathan Williams on 20 Mar 2022
Edited: Honglei Chen on 17 Apr 2024
I have read every source that I can, including purchasing a few textbooks that did not answer the question, and I cannot find a satisfactory answer to this question. First, I am making the assumption that "array factor" (AF) differs from "radiation pattern," per IEEE Std 145-2013, in that the former assumes isotropic array elements. Therefore the latter is the product of the array factor and the antenna pattern of the individual elements, assuming they are equal, per the principle of pattern multiplication. Second, I am assuming that the "power pattern" is simply the square of magnitude of the radiation pattern.
Most authors develop the AF as the vector sum of the outputs of the individual elements. This results in a maximum value of N for the array factor, assuming an un-tapered array. Then, all of these authors magically normalize this AF to the maximum value and therefore the maximum becomes unity. In dB, the array factor is , and with a normalized AF this results in a proper power pattern, assuming that the elements are isotropic. If one instead does not normalize the AF, then the result of the conversion to a power pattern is a maximum of , or in dB, which is obviously wrong, as it exceeds the directivity, which cannot be. If the array is tapered, the plot thickens, as the normalized pattern maximum will still be unity, even though clearly the tapering causes a reduction in the directivity of the array. Therefore, if one wants to compare a tapered to an un-tapered array, one cannot simply plot the two patterns together and expect an accurate depiction.
So, the question is, how does one show a non-normalized power pattern for an array? The obvious solution is to compute the normalized AF in dB (as ), and assuming the elements are isotropic, compute the directivity in dB (as ) and add these results together. For isotropic elements, this seems like a reasonable answer. Van Trees, "Optimum Array Processing" does a very nice job describing how to compute the directivity of arbitrary array geometries for isotropic elements.
If the elements are not isotropic, one surmises that one should compute the element power pattern in dB (as ) and add this to the result as well. However, this often results in exceeding the maximum directivity of the array, i.e., there is an upper bound on the power pattern that is dictated by the universal relationship between effective area and directivity, with is . Instead, one must reduce the element pattern commensurate with the physical area occupied by each element.
Finally, there is the question of "array gain" as opposed to "gain." The former is not a sanctioned term by IEEE, but nonetheless appears in the literature often (such as in Van Trees). Array gain is the change in SNR caused by the array, which is not the same as the "gain" which is the directivity reduced by the array's radiation efficiency. For a planar array, the array gain is usually less than the gain by a factor of for an un-tapered array. It is unclear to me how the array gain factors into a link budget calculation as opposed to the gain, as clearly one cannot budget both of these.
Can anyone help explain these apparent contradictions to me?

Honglei Chen on 17 Apr 2024
Edited: Honglei Chen on 17 Apr 2024
I agree with your array factor definition but I don't quite follow your reasoning on normalization.
I don't think there should be a direct comparison between the power pattern and the directivity, normalized or not. Power pattern, like you said, is simply the multipication between the element pattern and the array factor. The directivity, on the other hand, is a gain compared to the same amount of power discipated through an isotropic antenna. Thus, the directivity is measured in dBi instead of dB. Therefore, the power pattern and the directivity are two different things and cannot be compared directly.
The array gain in Van Tree's book is, like you said, an SNR gain. It is the gain achieved by the array compared to the gain achieved by a single element. So it is a completely different dimension compared to the context of directivity or power pattern. Thus, I don't think there should be any relation between the two and I don't understand the pi relationship you mentioned in your original question. Could you clarify.
In my experience, for RF/antenna engineers, when they say gain, they mean directvity (or directivity scaled by the efficiency). But for signal processing engineers, when they say gain, they often refers to the SNR gain.
These are my thoughts. I would more than happy to discuss further.

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