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Considering that the elevation angles of the rang bins that are adjacent to the rang cell under test change slowly, thus φ can be ignored in (5) for both the transmit and the following receive beamforming.
In the single snapshot case, the received noise-free echo of one ideal target can be represented as the following receive steering vector mathbf{b}(r, theta)=left[begin{array}{cccccc} e^{-jphi_{0}} & e^{-jphi_{1}} & ldots & e^{-jphi_{m}} & ldots & e^{-jphi_{M-1}} end{array}right]^{T}, (25).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com