Every measurement a radio telescope makes is a correlation of some kind. A single horn correlates a signal with itself; two horns correlate one against the other. This lesson explains convolution, autocorrelation, and cross-correlation. It then uses these operations to explain what a two-element interferometer measures.
It was written by Pranav Sanghavi at this lab in 2018. Everything here can
be run yourself: the notebook is at
code/interferometry/
and needs nothing but numpy, scipy and matplotlib.
- Convolution
- Cross-correlation and autocorrelation
- Why interferometry
- The two-element interferometer
- The effect of bandwidth
- Where to go next
- Credits
Convolution
\[(f * g)(x) \,=\, \int_{-\infty}^{+\infty} f(x-t)\,g(t)\,dt\]“In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated”
for multiple dimensions
\[\begin{align} (f * g)(x_1,\ldots,x_n ) \,&=\, (f * g)({\bf x})\\ \,&=\, \int_{-\infty}^{+\infty} \ldots \int_{-\infty}^{+\infty} f(x_1-t_1, \ldots , x_n-t_n)\,g(t_1, \ldots, t_n) \,d^nt\\ \,&=\, \int_{-\infty}^{+\infty} f({\bf x}-{\bf t})\,g({\bf t}) \,d^nt\end{align}\]Properties of convolution
\[\begin{align} f * g \,&=\, g * f&\qquad (\text{commutativity})\\ (f * g)* h \,&=\, f * (g * h)&\qquad (\text{associativity})\\ f * (g + h) \,&=\, (f * g) + (f * h) &\qquad (\text{distributivity})\\ (a\, g) * h \,&=\, a \, (g * h)&\qquad (\text{associativity with scalar multiplication})\\ \end{align}\]Convolution smooths. Convolving a rectangle with itself produces a triangle. Convolving that result with the rectangle again approaches a Gaussian. That is the central limit theorem happening in front of you — repeated convolution drives almost anything towards a bell curve.
The reason this matters for a telescope is the next picture. Point sources on the sky are spikes. Your telescope has a beam — a response pattern roughly 15° across for a DSPIRA horn. What you record is the sky convolved with the beam.
Two sources closer together than the beam merge into one bump. No amount of integration time separates them; the information is gone at the moment of measurement. That single fact is the whole motivation for interferometry.
Cross-correlation and autocorrelation
Cross-correlation asks how much two signals resemble each other when one is slid past the other:
\[\begin{split} (f\star g)(x) \,&=\, ({f_-}^*\circ g)(x)\\ &=\, \int_{-\infty}^{+\infty} f^*(t-x)\,g(t)\,dt\\ &\underset{t^\prime = t-x}{=}\, \int_{-\infty}^{+\infty} f^*(t^\prime)\,g(t^\prime+x)\,dt^\prime\\ \end{split}\qquad \text{,}\] \[\begin{align} (f\star g)(x_1,\ldots,x_n ) \,&=\, (f\star g)({\bf x})\\ &=\, ({f_-}^*\circ g)(x)\\ &=\, \int_{-\infty}^{+\infty} \ldots \int_{-\infty}^{+\infty} f^*(t_1-x_1, \ldots , t_n-x_n)\,g(t_1, \ldots, t_n) \,d^nt\\ \end{align}\]Autocorrelation is the same operation with one signal played against itself:
\[\begin{split} R\{f\}(x) \,&=\, (f\star f)(x)\\ &=\, (f_-^* * f)(x)\\ &=\, \int_{-\infty}^{+\infty} f^*(t-x)\,f(t)\,dt\\ &\underset{t^\prime = t-x}{=}\, \int_{-\infty}^{+\infty} f^*(t^\prime)\,f(t^\prime+x)\,dt^\prime\\ \end{split}\qquad \text{.}\]Unlike convolution, cross-correlation does not flip a function. Thus, \( f \star g \neq g \star f \), while \( f * g = g * f \).
Autocorrelation is how a radio telescope finds signal underneath noise. Below is a sine wave three times smaller than the noise it sits in — invisible in the raw trace. Its autocorrelation still reveals the period. The noise is uncorrelated at nonzero lags, while the sine wave remains correlated.
This is not an analogy for what the spectrometer does — it is what the
spectrometer does. spectrometer_w_cal.grc displays a power spectrum. This is the Fourier transform of the horn voltage’s autocorrelation. The Wiener–Khinchin theorem establishes this equivalence. It explains how the hydrogen line emerges from a much larger noise floor.
Why interferometry
The resolution of a single dish is set by diffraction:
\[\theta \,\approx\, 1.22 \frac{\lambda}{D}\]At 21 cm that is a brutal constraint. Rearranged for the diameter you would need:
| You want | At 21 cm you need a dish |
|---|---|
| 1 degree | 15 m |
| 1 arcminute | 881 m |
| 1 arcsecond | 53 km |
| 15 milliarcseconds | 3,523 km |
The last row is the resolution the VLA reaches routinely. A single dish that did the same would be a quarter of the diameter of the Earth. A dish of that size is impractical, regardless of budget.
The way out is to stop trying to fill the aperture and instead sample it at a few points. Two horns separated by \( b \) resolve detail on the scale \( \lambda / b \). They match that large dish’s resolution, but not its collecting area.
The two-element interferometer
Take two antennas and combine their signals. There are two ways to do it:
- by addition, to form a sum interferometer, or \( \sum \)-interferometer
- by multiplication, to form a product interferometer, or \( \prod \)-interferometer
The sum interferometer
\[P ={2 V_0^2(1+ \cos\Delta \Phi)}\]The product interferometer
\[R_c= \langle V_1 V_2 \rangle_t\] \[R_c =\frac{V_0^2}{2}\cos{\omega \tau}=\frac{V_0^2}{2}\cos \Delta \Phi\]where \( \tau \) is the time delay in the signal path. A source at angle \( \theta \) from the zenith reaches one antenna first. The geometric delay is \( \tau = b\cos\theta / c \). As the Earth turns, that delay sweeps, and the output oscillates — the fringe.
This is a cosine correlator. Add a phase delay of \( \pi/2 \) to one signal path to make a sine correlator:
\[V_1=V_{01} \cos (\omega (t + \tau)) \quad V_2 = V_{02} \cos (\omega t + \frac{\pi}{2} )\] \[\begin{align} R&=\langle V_{01} V_{02} \cos{(\omega t + \frac{\pi}{2})} \cos{[\omega (t + \tau) ]} \rangle_t\\ R&= V_0^2 \frac{\langle \cos(2 \omega t + \tau + \frac{\pi}{2})+\cos (\omega \tau - \frac{\pi}{2})\rangle_t}{2} \end{align}\] \[R_s =\frac{V_0^2}{2}\sin{\omega \tau}\]Run both at once and you have a complex correlator, whose output is the visibility:
\[R = \int_{\Omega} I_\nu(\mathbf{s}) e^{-\imath 2\pi \frac{\mathbf{b}\cdot\mathbf{s}}{\lambda}} d\Omega = V\]That integral is a Fourier transform of the sky brightness. An interferometer measures the Fourier transform of the sky, one spatial frequency per baseline. Everything in aperture synthesis follows from it.
Here is the fringe pattern of a two-element interferometer with a baseline three wavelengths long — the left panel. The lobes are the directions where the two paths arrive in phase.
The effect of bandwidth
The right panel above is the same interferometer observing across a wide band rather than at one frequency. Across the band, the baseline length in wavelengths changes. Each frequency places its lobes differently, so averaging washes out the pattern. Near \( \theta = 90^\circ \), the geometric delay is zero and all frequencies agree.
This is bandwidth decorrelation. A broadband interferometer sees clearly only near the direction where its delays match. Arrays compensate with a delay that tracks the source. The zero-delay direction follows the target instead of remaining at the zenith.
DSPIRA’s two-horn setup produces its sharpest fringes for sources transiting near the phase center. Narrowing the band preserves fringes across a wider region, at the cost of sensitivity.
Where to go next
Two elements give you one spatial frequency at a time. Earth’s rotation sweeps the baseline through the Fourier plane. More antennas fill that plane faster. This process is aperture synthesis.
See Fundamentals of Radio Interferometry for the standard treatment. Rhodes University and the National Astrophysics and Space Science Programme produced this course book. It is free to read and uses runnable notebooks. Chapter 4, Visibility Space, continues from this page. Start there.
Closer to home, and worth doing in this order:
- Additive Interferometry Using Two DSPIRA Radio Horn Telescopes — LightWork Memo 31. This is a measured version of the fringe pattern above. A solar transit on a 5.0 m baseline produced 0.043 rad spacing, compared with 0.042 rad predicted. This is what the theory on this page looks like when it comes out of a real horn.
- Setting up a 2 Horn Interferometer — building one and running it
- Interferometry — the rest of the interferometry material on this site
- Fourier Analysis and Expert Mode — the transforms this page leans on
Credits
Pranav Sanghavi wrote this page’s text, equations, and code at WVU’s Radio Astronomy Instrumentation Laboratory on May 21, 2018. They are published under this site’s MIT license. The notebook was ported from Python 2 to Python 3 in 2026. The autocorrelation figure was added then; the remaining material is his.
The definition of convolution is quoted from Wikipedia under CC BY-SA 4.0.
Fundamentals of Radio Interferometry is a separate work licensed under GPL v2. Its authors are the Rhodes University Centre for Radio Astronomy Techniques & Technologies and the NASSP community. It is linked above rather than reproduced.