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--- un:radio-inter [2024/09/08 22:34] – [Two elements in one dimension] asad
+++ un:radio-inter [2024/09/08 23:26] (current) – [1.2 Correlation interferometer] asad
@@ Line 2: / Line 2: @@
 In radio [[interferometry]], multiple radio antennas or apertures (elements) are used for observation in order to increase [[angular resolution]]. Without interferometry, radio telescopes would have resolutions comparable to the size of the sun and the moon. Let us begin our discussion by understanding a standard two-element interferometer in one dimension.
-===== Two elements in one dimension =====
+===== - Two elements in one dimension =====
 Consider a simple source in the far field generating plane waves at a single frequency $\nu$, two identical antennas receiving only one polarization at $\nu$ without any distorting effects either along the intervening media or in the receiving system. The geometric configuration is shown below.
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 ==== - Correlation interferometer ====
+According to the definition of [[cross-correlation]], the correlation between two voltage signals can be written as
+$$ R_{AB}(\tau) = \langle V_A \cos\omega t V_B \cos\omega(t-\tau) \rangle $$
+which can be reduced to only a fringe (interferential) term
+$$ R_{AB}(\tau) = \frac{1}{2} \langle V_AV_B\cos\omega\tau \rangle $$
+where the total-power terms are absent, unlike the equation for the adding interferometer, and this is plotted below.
+{{:un:correlation-interferometer.jpg?nolink&600|}}
+The dotted and dashed curves represent two signals (detected by two antennas) phase-shifted with respect to each other, the solid curve shows their multiplication and the solid line the corresponding time-average. As the direction toward the source changes (due to the rotation of the earth), $\tau$ changes and so does the relative phase. At the top phase shift is close to zero, in the middle it becomes $90^\circ$, and at the bottom $180^\circ$. The //fringe amplitude// varies from $+0.5$ to $-0.5$. One cycle of the RF phase shift corresponds to one //fringe cycle//.
+{{:un:fringe-cycles.jpg?nolink&650|}}
+Here you see the fringe oscillations (cycles) for a point source located on the celestial equator observed by a two-element interferometer lying along the terrestrial equator creating a 15-$\lambda$ long east-west baseline.
+The fringes oscillate around a zero mean and there are exactly 15 maxima on each side of the center, where $\theta=0^\circ$. The fringes near the center are more sinusoidal because there the small-angle approximation ($\sin\theta=\theta$) is more applicable. As $b_\lambda=15$, $\lambda/b=1/15 \sim 4^\circ$ which is the angular separation between successive lobes. The //fringe spacing// increases away from the center because the projected baseline $b\cos\theta$ decreases.