A well-known fact is that Cramer Rao Lower Bound (CRLB) in 2D Angle of Arrival estimation is inversely related to the (sin(AoA))^2 (or (cos(AoA))^2, depending on the reference of AoA). See [1,2] for example. Why is this the case? What is the physical explanation of this, that when the arriving signal is perpendicular to the array, CRLB tends toward infinity?
[1]. Shu F, Qin Y, Liu T, Gui L, Zhang Y, Li J, Han Z. Low-complexity and high-resolution DOA estimation for hybrid analog and digital massive MIMO receive array. IEEE Transactions on Communications. 2018 Feb 13;66(6):2487-501.
[2] Mailaender L. Bounds for 2-D angle-of-arrival estimation with separate and joint processing. EURASIP Journal on Advances in Signal Processing. 2011 Dec 1;2011(1):5.
Hi Masoud Arash
Because I did some work a long time before on this. Conference Paper Analysis of a joint space-time DOA/FOA estimator using MUSIC
please let me give a try to answer your question.
As you know, CRLB is given by the inverse of the Fisher information matrix (FIM) and is a function of the inverse of array correlation matrix. The array correlation matrix includes the information of the cross-correlation of different arrivals. And the cross-correlation between the different arrivals is a function of sin(AoA), isn't it? I have a feeling, this may be one of the reasons.
Cheers
Hi Masoud,
The CRLB is a measure of the sensitivity of your array to detect a particular AOA. Therefore it has a direct link with the derivative of the log likelihood function. It can be checked that the magnitude squared of this derivative scales with (cos(AoA))^2. Intuitively, this implies that for AoA = 0°, a small change of AoA implies a significant change of the likelihood function while at AoA = 90°, the change is negligible. You can easily see it physically. The response of one linear array element is exp (1j 2 pi r sin(AoA)). The function sin(AoA) is flat at AoA=90° which implies that a small change of AoA does not significantly affect the phase of the received signal. On the other hand at AoA = 0°, sin(AoA) has maximal slope (looks like a / ) -> a small change of AoA gives a large phase change, which implies more sensitivity, more robustness to additive noise...
I hope it helps... A picture of the incoming wave angle with respect to the array and the related phase shift might help as well.
Best regards,
François
Thank you very much for your answer Francois Rottenberg
Indeed, it is an ingenious way to explain it. My own thought was that in the end, what CRLB cares for (due to loglikelihood function form), is the difference of phase that is observed at the receiver side, and well the difference is (sort of) maximum at AoA=90°, so why we cannot say anything (literally anything) about the particular case of 90°?
By this explanation of yours, I get that in fact, CRLB returns this infinite answer, because the array cannot distinguish changes around AoA=90°. Great.
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