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Antenna Array Background

In the image above, there is a formula for the delay of a signal at a given sensor 'i' in an antenna array. I don't understand how they arrived at that formula.

Each antenna array element is separated by a distance of 'd' and we know the signal hits the array at an angle 'theta'. Dividing by 'c' gives us the time taken for light to travel the additional distance, but I don't see how the quantity 'dsin(theta)' represents the extra distance. Would appreciate any clarification.

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but I don't see how the quantity 'dsin(theta)' represents the extra distance

Let analyse just sensors 1 and 2 also put $\theta$ 90 and 0 degrees and see what happens.

Let $\theta$ = 0 degrees, which means that transmitter of the signal is situated in front and both sensors should receive the signal at the same time (isn't it?).

$$ \Delta_i = { {d \cdot \sin {\theta} } \over {c} } = { {d \cdot 0 } \over {c} } = 0 $$

Let $\theta$ = 90 degrees, which means that transmitter of the signal is situated at the right hand and first sensors should receive the signal later. And delay should be time which need for signal travel the distance d with velocity c.

$$ \Delta_i = { {d \cdot \sin {\theta} } \over {c} } = { {d \cdot 1 } \over {c} } = {d \over c} $$

But anyway in my opinion the indexes of sensors should be in the opposite direction.

I hope it helps you.

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