How Array gain is calculated in Phased array?

2014 10 8
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更新时间：2014 10 10
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In text books on phased array there are standard equations for array gains of phased arrays (for linear , circular etc)
But I am unable to match the array gains same with Matlab's array gain obtained in Phased Array toolbox
The formula given in http://www.mathworks.in/help/phased/ref/phased.arraygain-class.html but i am observing same aray gain with or without steering.

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Honglei Chen
Honglei Chen 2014-10-8

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The ArrayGain in Phased Array System Toolbox calculates the SNR improvement due to the array. The computation is outlined in the reference page as the ratio of SNR at the output of the array over SNR at the input of the array. This is more relevant for array signal processing but may not be what you are looking for.
Since you are asking this question, if I'm not mistaken, you are looking for the gain in terms of directivity instead? If that's the case, you can plot the directivity of the array by setting the unit as dbi, e.g.,
% Compute the directivity of a 10-element ULA with quarter
% wavelength spacing. The element is assumed to be isotropic.
c = 3e8; fc = 3e8; lambda = c/fc;
myArray = phased.ULA(10,lambda/4);
plotResponse(myArray,fc,c,'Type','dbi')
The above code should work since R2014a and if you have the latest R2014b, you can also compute the directivity for a given direction directly, such as
d = directivity(myArray,fc,0,'PropagationSpeed',c)
HTH

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Kotresh
Kotresh 2014-10-10
编辑：Kotresh 2014-10-10
Thanks Chen for detailed answer.
Yes. I am looking for DIRECTIVITY INDEX value.
Following picture shows design of URA using Sensor Array Analyzer app. Url: http://www.mathworks.in/help/phased/ref/sensorarrayanalyzer.html
The array directivity for this URA is 17.2 dB. With tapering, the array directivity loses 1 dB to yield 16.0 dB.
If I calculate the DI for rectangular array using formula as given in
http://ocw.mit.edu/courses/mechanical-engineering/2-011-introduction-to-ocean-science-and-engineering-spring-2006/readings/hw5_sonar_leonar.pdf
DI = 10 log(4πLxLy/λ*λ)
From this formula DI = 18.95.
This is the source of confusion.Either the forumala I am using is not correct or Matlab calculates in different way.
Honglei Chen
Honglei Chen 2014-10-10
I'm not an acoustics expert but my understanding is that those formulas are approximate ones. The one you quote is specifically for rectangular array and it is based on the 3dB beamwidth. Basically it says the gain can roughly be considered as the product of two 3dB beamwidth (lambda/L) and compare that to 4*pi beamwidth for an isotropic source.
On the other hand, MATLAB computes it using numerical integration of field, so it's a more accurate approximation. Thus I won't be surprised that they differ a bit.
Honglei Chen
Honglei Chen 2014-10-10
BTW if you want to take into consideration of Taylor taper using the approximation formula, you need to take into consideration of beambroadening. For a Taylor window of 30dB suppression, the factor is 1.14. So the approximation will give a result of 17.81 dB, approximately a 1 dB loss, which is consistent to MATLAB result's difference.
Kotresh
Kotresh 2014-10-10
Thank you Honglei Chen. You cleared all my doubts.

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