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I am interested in understanding the proper terminology for the topic of Space Time Adaptive Processing (STAP) that extends beyond the example in all the text books and papers I could find that reference the Uniform Line Array. In the case of the ULA, the steering vector is easy for me to understand and is

\begin{equation*} \mathbf{a}(\phi) = A[e^{i \theta_1}, ...,e^{i \theta_n}]^T \end{equation*}

where $\theta_i$ is the phase response in the direction $\phi$. The optimal matched filter is thus:

\begin{equation*} \mathbf{h}(\phi) = \alpha\mathbf{R}_s^{-1} \mathbf{a}(\phi) \end{equation*}

where $\alpha$ grabs all of the constant terms and lumps them together

However, my expectation for the more generic case is that the array response on all channels is not equal (more like an array manifold) and is thus:

\begin{equation*} \mathbf{a}^\prime(\phi) = [A_1e^{i \theta_1}, ...,A_ne^{i \theta_n}]^T \end{equation*}

and now $A_i$ is the amplitude response in the direction $\phi$.

This leads me to my question. Is $\mathbf{a}^{\prime}(\phi)$ also a steering vector or is this something different (i.e. elements of the steering vector be other than unity)? And if the steering vector is still $\mathbf{a}(\phi)$, what is the convention for treating the amplitude response of the array. I am leaning towards the former.

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Yes, $a'(\phi)$ is a steering vector. In fact, the third equation (almost) describes a phased array that is capable of steering lobe patterns at different frequencies.

Each element of the array has a filter behind it which forms a weighted sum of (delayed) samples in Time. The same idea is applied one level above by now summing (delayed) samples of the received signal from the different elements that are in different points in Space.

In STAP, the adaptation in Time maintains the coherence of the probe signal (or carrier) against noise / doppler and other "structural" distortions and the adaptation in Space maintains the coherence of the signal along a specific direction by trying to minimise the covariance between different elements. Think of it as one convolution per element of the array(typical filtering in the Time domain) and one more ACROSS the same time instant (the output) of the element filters (Space).

BUT! With this configuration you can tune the Time filters at some sinusoid and then with the frequency of that sinusoid tune the Space filter to receive from a given bearing. To do the same thing simultaneously, you would need a Fourier Transform behind each element to decompose the received signal into a bank of sinusoids and then apply different space filters ($a'(\phi)$) to those outputs for specific bearings. In this way, you could have the array listening to some low frequency from the left and some high frequency from the right simultaneously. I feel that this is what is meant with "...the more generic case is that the array response on all channels is not equal (more like an array manifold) and is thus..."

Hope this helps.

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