Discussion
What mathematical relationship connects the aperture field and the far field?
Answer: D
The far field is the Fourier transform of the aperture distribution, because the path difference to a distant point varies linearly with position across the aperture and so contributes a linear phase. Every property of the pattern follows from this: a wider aperture gives a narrower beam, a sharp edge gives high side lobes, and a smooth taper suppresses them. It is not a scaling, a convolution or a derivative, and recognising the transform is what allows the whole vocabulary of signal processing windows to be carried across into antenna design unchanged.
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