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--- un:radio-inter [2024/09/08 06:22] – 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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 $$ R_{A+B}(\tau_g) \approx \langle [V_A^2 \cos^2 \omega t + V_B^2 \cos^2 \omega(t-\tau_g) + 2 V_AV_B \cos\omega t \cos\omega(t-\tau_g) + V_r^2] \rangle $$
-where the cross-terms between the signals and the receiver noise has been ignored because they do not have any long-term phase relationship. The third term on the right hand side can be written as \langle [\cos\omega\tau_g + \cos(2\omega t-\omega\tau_g)] \rangle using the relevant trigonometric identity (($\cos A \cos B = [\cos(A-B)+\cos(A+B)]/2$)).
+where the cross-terms between the signals and the receiver noise has been ignored because they do not have any long-term phase relationship. The third term on the right hand side can be written as $\langle [\cos\omega\tau_g + \cos(2\omega t-\omega\tau_g)] \rangle$ using the relevant trigonometric identity.(($\cos A \cos B = [\cos(A-B)+\cos(A+B)]/2$))
+The second term $\cos(2\omega t - \omega \tau_g)$ oscillates with RF frequency (as opposed to IF frequency) and, therefore, reduces to zero due to time-averaging. And we know that $\langle \cos^2\omega t \rangle=1/2$ which leads to
+$$ R_{A+B}(\tau_g) = \frac{1}{2} [(V_A^2+V_B^2)+V_r^2] + \langle V_AV_B\cos\omega\tau_g \rangle $$
+where the first part have the //total power// terms dominated by the system noise and the second 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.
+==== - 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.