# Measurement Equation

### From AstroBaki

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## Contents |

## 1 Measurement Equation

Let’s begin by restricting our discussion of interferometers to a single frequency ν, with corresponding wavelength λ. Then just as was the time delay between antennas *i*,*j*, we can also describe the number of wavelengths this is:

Knowing the number of wavelengths between the two antennas, we can now say that for a signal of a particular frequency emanating from a source in direction , the complex phase difference between that signal measured at antenna *i* and the signal at antenna *j* will be *e*^{ − iθ}, where θ is the angle swept out by the wave as it propagates from *i* to *j*. Using that θ is just 2π times the number of wavelengths, we know the phase difference Δφ is:

The phase difference Δφ is, of course, frequency dependent. And at a given frequency, it also varies with position on the sky. The pattern that this complex phase traces on the sky (see below for a graph of just the real component) is called the “fringe pattern” of an interferometer.

*A graph of the real component of at a fixed frequency, as a function of direction on the sky. The complex response of a baseline along the sky is called the “fringe pattern”, and it is suspiciously close to a sine wave.*

Now we will define a few variables that will help us extrapolate from a single baseline in a single direction to a picture of how a whole array might respond to the whole sky that falls within the primary beam of the correlated antennas. First, we will define coordinates representing the length of a baseline in units of wavelength:

where *u* is the east-west component of the baseline, *v* is the north-south component, and *w* is the vertical (up-down) component. We will also split the source direction vector into its components:

where *l* is the east-west direction on the sky, *m* is the north-south direction, and the third component comes from the fact that we restrict to have unit length (it’s a direction vector).

Using these components, we can now write down the response of a baseline (called the “visibility” *V*) as a function of the *u*,*v*,*w* separation of the antennas, integrating over all the source intensity *I* on the sky as a function of *l*,*m*:

The equation above is the full form of the “visibility equation”, otherwise known as the “measurement equation” of an interferometer. The only variable that we haven’t yet defined is *A*, which is the response of the primary beams of the antennas as a function of direction on the sky. In general, *A* and *I* are always grouped together, because the sky is always seen through the filter of the primary beam. The product is sometimes called the “perceived intensity”.

## 2 Understanding the Visibility Equation as a Fourier Transform

The equation we derived above can be much easier to understand if we make a simplifying assumption, known as the “flat-sky” approximation. This approximation is either that *w* = 0, or alternately, that the primary beam *A*(*l*,*m*) is sufficiently small that , making . In either case, we are asking that the response of a baseline not need to account for the fact that the sky is a curved surface of a sphere. Under this assumption, the term is no longer a function of *l*,*m*, and can be removed from the integral to give us:

This formulation of the Visibility Equation is much more illuminating. It says that when phased to a “phase center” via a choice of a corresponding *e*^{ − 2πiw}, with *w* being the baseline component along the direction toward the phase center in wavelengths, the visibility *V*(*u*,*v*) is just the Fourier Transform of the perceived sky.

So in addition to thinking about the fringe-pattern of a baseline on the sky, we can equivalently think of the following process. We take an image of the sky in *l*,*m* coordinates and Fourier transform it:

*The true image of the sky (left) and the true UV plane (right).*

The result is called the “uv-plane”, and its coordinates are inverse angles. An inverse angle is the same thing as a wavelength, so the uv-plane has coordinates (not surprisingly) of *u*,*v*.

Next, this uv-plane is sampled at particular *u*,*v*-coordinates by various baselines in an antenna array. The sampling pattern can be computed from the antenna configuration by choosing all of the antenna-to-antenna spacings. (Interestingly, this sampling pattern is the convolution of the antenna placement pattern with itself):

*The array sampling pattern in the uv-plane (right), and the associated synthesized beam pattern or “dirty beam” (left).*

Note that for each pair of antennas you get two samples: one at *u*,*v*, and one at − *u*, − *v*. Because the sky is real-valued (no complex fluxes), these two Fourier components are related by a complex conjugate. That is, if you measure *V*(*u*,*v*) at *u*,*v*, you will measure *V*^{ * }(*u*,*v*) at − *u*, − *v*.

Now, the sampling of the uv-plane is simply multiplying the true uv-plane by the sampling pattern you just computed. This is what you would get if you took visibilities recorded from an interferometer, and then placed each measured visibility *V*(*u*,*v*) at the corresponding *u*,*v* (and − *u*, − *v*) coordinates of a matrix. Finally, if you take the inverse Fourier Transform of this sampled uv-plane, you get an image:

*The “dirty image” (left) and the associated sampled uv-plane.*

As you may notice, this image is somewhat degraded from our original. In fact, it is usually called a “dirty image”. Why is it dirty? Because we lost information when we sampled the uv-plane. We multiplied the true uv-plane by our sampling function. This is equivalent to convolving the true sky by the Fourier Transform of our sampling function.