Synthesis Imaging -- from Eric Weisstein's World of Physics

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Synthesis Imaging

For a point source, let two receivers be a distance B apart, and the distances from each to the distant source be and with . Consider a harmonic Fourier component, or cosine wave

(1)

Each of the antennas sees the signal emitted a time

(2)

earlier. The signal is also attenuated by the distance, but because the waves can be considered planar, the signal strength received at each point will be the same, . If is delayed by a time

(3)

(4)

Using the identity

(5)

to determine the mixed (multiplied) signal

(6)

Running this through a low-pass filter removes the time-varying portion, so

(7)

This can be written using

(8)

(9)

where is the zenith (or "position") angle of the source measured from the center of the baseline and is the time light would take to travel the distance . The baseline remains fixed, but changes as the Earth rotates. The output will change by one fringe if is held constant when

(10)

(11)

(12)

Since is varying slowly, we are looking the solutions for which

(13)

Then

(14)

(15)

The fringe period is then the time it takes for to change by . Up until now, we have been dealing with only a single Fourier component. A real receiver, however, will accept frequencies of some finite bandwidth . If the receiver has zero response for frequencies outside this range and unity response for frequencies in this range,

(16)

where is the specific brightness. Since the brightness is roughly constant in this small interval,

(17)

and

(18)

But

(19)

(20)

so

(21)

where is chosen so that it will equal . Sources which are away from the point for which is chosen will therefore have a different , so they will have their signals attenuated by a factor , creating a limited field of view within which signals make an appreciable contribution. Notice that the bigger , the smaller can be and still make an appreciable contribution. Therefore, the wider the bandwidth, the narrower the field of view.

Amplitude Closure Relations, Complex Gain, Correlator, Interferometer, Multi-Frequency Synthesis, Phase Closure Relations, Recirculator, Redundancy Calibration, Redundancy Factor, Ring Lobes, Self Calibration, Sensitivity, Sensitivity Factor, Shadowing, Spectral Sensitivity Function, Synthesis Imaging Array, Transfer Function, van Cittert-Zernicke Theorem, Weighting Function (Synthesis Imaging)

References

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Thompson, A. R.; Moran, J. M.; and Swenson, G. W., Jr. Interferometry and Synthesis in Radio Astronomy. New York: Wiley, pp. 328-335, 1986.