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--- un:radio-inter [2024/09/08 22:32] – [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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 The signals from the two antennas are combined in the **combiner**: either added or multiplied.
+==== - Adding interferometer ====
 {{:un:adding-inter.jpg?nolink&600|}}
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  $R_{A+B}(\tau_g) = \frac{1}{2} [(V_A^2+V_B^2)+V_r^2] + \langle V_AV_B\cos\omega\tau_g \rangle$
-where the first part have the //total power// terms dominated by the system noise and the second interference varying cosinusoidally with time delay represents the interference effects.
+where the first part have the //total power// terms dominated by the system noise and the second part varying cosinusoidally with time delay represents the interference effects.
-Adding interferometers are dominated by gain fluctuations in the total-power terms which might be okay for strong sources, but very bad for weak, i. e. most, sources. In a //correlation interferometer//, the total-power terms vanish.
+//Adding interferometers// are dominated by gain fluctuations in the total-power terms which might be okay for strong sources, but very bad for weak, i. e. most, sources. In a //correlation interferometer//, the total-power terms vanish.
+==== - 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.