How to take the signals from multiple antenna pairs and turn them into an actual sky image using Fourier transforms
Good question — you’re asking about the heart of radio interferometric imaging: how to take the signals from multiple antenna pairs and turn them into an actual sky image using Fourier transforms. Here’s the step-by-step overview:
1. Signals from antenna pairs
- Each pair of antennas (a baseline) measures a complex correlation of the incoming signals at a given separation vector.
- This correlation gives you a visibility V(u,v)V(u,v), which is essentially one Fourier component of the sky brightness.
2. Mapping antenna separation to Fourier space
- Each baseline projects onto coordinates (u,v)(u,v) in units of wavelength.
u=bxλ,v=byλu = \frac{b_x}{\lambda}, \quad v = \frac{b_y}{\lambda}
where bx,byb_x, b_y are the east–west and north–south baseline components.
- Each (u,v)(u,v) sample corresponds to one Fourier coefficient of the sky image.
3. The Van Cittert–Zernike theorem
This is the central link:
V(u,v)=∬I(l,m) e−2πi(ul+vm) dl dmV(u,v) = \iint I(l,m)\, e^{-2\pi i(ul + vm)} \, dl \, dm
- I(l,m)I(l,m) = sky brightness as a function of angular coordinates.
- V(u,v)V(u,v) = measured visibilities (correlations).
So the visibilities are just the Fourier transform of the sky.
4. Collect visibilities
- As the Earth rotates, baselines trace out curves in the (u,v)(u,v)-plane (Earth rotation synthesis).
- With enough baselines and time, you build up coverage in Fourier space.
5. Fourier transform to make an image
- You take the set of visibilities and do an inverse Fourier transform:
I(l,m)=∬V(u,v) e+2πi(ul+vm) du dvI(l,m) = \iint V(u,v)\, e^{+2\pi i(ul + vm)} \, du \, dv
- This gives you the dirty image: the true sky brightness convolved with the point-spread function (the Fourier transform of your sampling pattern).
6. Deconvolution (CLEAN, MEM, etc.)
- Because the (u,v)(u,v) coverage is incomplete, the dirty image has sidelobes and artifacts.
- Algorithms like CLEAN iteratively remove the sidelobe pattern and reconstruct a better approximation of the true sky.
7. Calibration
- In practice, you also need phase and amplitude calibration (using known sources, or self-calibration) before imaging, or else the Fourier inversion won’t work properly.
✅ In summary:
- Each antenna pair gives you one Fourier component (visibility).
- Collect visibilities across many baselines/times → build a sampled Fourier plane.
- Do an inverse Fourier transform → dirty image.
- Apply deconvolution → final image.
Worked example with simulated antenna data (say, a small 4-antenna array), and generating the Fourier plane and dirty image step by step in Python.
Here’s the demonstration:
- Left: the true sky (two point sources).
- Middle: the sampled uv-plane from the four-antenna baselines.
- Right: the dirty image from the inverse Fourier transform.
Because of sparse sampling, the dirty image has artifacts (sidelobes), but you can already see the two sources emerge.