# Radio astronomy - integration time vs. sample rate

Sooo I wasn't sure if I should put this here or in the astrophysics section, but I figured electrical engineering was more relevant for my question.

I'm involved in a project at my university to build a (very) simple radio telescope, but I'm having a bit of trouble with the radiometer equation:

$\sigma_T \approx \dfrac{T_{sys}}{\sqrt{\Delta \nu_{RF} \cdot \tau}}$

Where $\sigma_T$ is the "sensitivity" or the minimum detectable temperature, $T_{sys}$ is the system temperature, $\Delta \nu_{RF}$ is the bandwidth, and $\tau$ is the integration time.

I'm mostly having trouble with "integration time". This seems to imply that one has to use an integrating analog to digital converter (something like a dual-slope ADC I guess...), but this seems to prevent the application of Fourier analysis (am I wrong in thinking this?). However, I have seen some examples of radiometers that appear to take discrete samples, and apply some sort of "integration time".

Since the product $\nu \cdot \tau$ is effectively the number of samples that would have been taken over the integration time, I was wondering if this product could equated to some function of sample rate of a non-integrating ADC?

Rereading my above questions, they aren't the most coherent sentences in the world... But I'm not quite sure how else to pose them. I have a feeling I'm overthinking this, or missing something simple. If anyone needs anything clarified, please let me know!

Thanks so much for the help!

Last edited:

f95toli
Gold Member
I think the answer is that it depends a bit on the application. One way to get an "integration time" is to simply use a very fast sample rate and then integrate numerically.
However, one thing to keep in mind is that the circuit before the ADC can ALSO give you an "effective" integration time (just think of it as a low-pass filter), and as long as you sample fast enough the dead-time between sample will presumably not matter (since there isn't enough time for successive samples to change anyway). Most of the formulas of these types assume a negligible dead-time between samples, which is certainly not the the case of you use a fast ADC with a low(ish) rate.
Ideally, these two "methods" should of course agree.

So again, it depends. The main things to keep an eye on is the BW of your circuit and the dead-time of the ADC.

Thanks for the response!

We are taking a software defined radio approach to radio astronomy, which is something that hasn't really been done, or at least documented to a great extent, so it's difficult to find information. As a result of using SDR, we aren't using any sort of frequency-to-voltage converter, and instead are just going to run the discrete Fourier transform on a sample of data. If our sample rate is fast enough (I read somewhere where the sample rate should be comparable to the bandwidth? I don't really see why this should be true though.) would it then be acceptable to use the total time over which we collected data as our "effective" integration time?

Thanks again for the help!

As you know, any radiometer have the Low Pass Filter (LPF) at it's output. The term 'integration time' relates to the device, named an ideal integrator, used as LPF. This is not good LPF, since it power response looks like (sin f)^2 / f^2. However, the such responce corresponds to the 'moving average' filter, which can be performed programmatically (after the sampling) but not in continuous time, as physical device. Instead this kind of LPF you can use any you prefer, but take into account the effective integration time, which defined in the Kraus book, Radioastronomy, Chapter 7. This effective integration time defined via the Effective Noise Bandwidth (ENB) of your real LPF. No matter at this point, are you using the ADC or not.

If you use an ADC, the sampling frequency should be equal or greater the Fcutoff*2, othervise the aliasing phenomena (sampling theorem, Shannon, Nyquist, Kotelnikov). Fcutoff - the cutoff frequency your LPF (before the ADC).

You don't want to know what I read instead of "Fcutoff" :D

Yes! I'm do not