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I am given a recording of an ionosonde sounding. The signal $S$ is represented as a series of I/Q pairs at a sample rate $F_{sr} = 10\,\text{MHz}$. I am told that the center frequency of the recording is $F_{center} = 7\,\text{MHz}$.

I have a $T=200\,\text s$ recording as $I=[I[0],..,I[T\times F_{sr}-1]]$, $Q=[Q[0],..,Q[T\times F_{sr}-1]]$.

The actual signal $R[t]$ being recorded can vary from $2\,\text{MHz}$ to $12\,\text{MHz}$, that is from ${F_{sr}\over 2} -F_{center}$ to ${F_{sr}\over 2}+F_{center}$.

Let $S=I+Q j$ be the complex representation of the recording.

How do I recover the original signal $R[t]$ at a sample rate of $F_{up}=24\,\text{MHz}$, which is high enough to represent a $12\,\text{MHz}$ signal directly, as a function $R(S,F_{center},F_{sr})$?

That is, $F_{up}=2F_{sr}+ 2 F_{center}$, so $R=R[0],...,R[T\times F_{up}-1]$.

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  • $\begingroup$ I'm not quite sure what you mean with "$R$ can vary from 2 to 12 MHz", because that's actually exactly the bandwidth represented by complex sampling of a baseband signal representing the spectrum around 7 MHz at a rate of 10 MS/s. What is "the original $R$"? To me, the original signal is your $S[n] = I[n] + jQ[n]$. What'd be "more" original than that? $\endgroup$ – Marcus Müller Jun 18 '19 at 7:12
  • $\begingroup$ The "original R" is the analog signal captured from the antenna of a Lowell Digisonde 4D (digisonde.com/pdfs/Digisonde4DManual_LDI-web1-2-6.pdf) and then digitized as IQ pairs into an 8GB bin file for the IARPA PINS Challenge (iarpa.gov/challenges/pins.html). I want to look at the spectrogram of the whole signal, as in Figure 2 of this writeup: topcoder.com/challenges/30088355. I thought it would be easier to make Figure 2 if I had the real signal in the original upsampled rate. When I make a spectrogram of the IQ signal it doesn't look like Figure 2. $\endgroup$ – Lars Ericson Jun 18 '19 at 13:27
  • $\begingroup$ But the analog signal that gets digitized is the I and the Q signal – the RF bandpass signal doesn't exist anywhere, digitally, and contains no additional info compared to the baseband signal – only a relabeled frequency axis. $\endgroup$ – Marcus Müller Jun 19 '19 at 6:19
  • $\begingroup$ I was going through this exercise because when I do a spectrogram of the recording I just get a very bandy picture, where Figure 2 has clear tracks for the chirps. I found an article on Chirp Reception and Interpretation by a Dutch ham radio enthusiast (websdr.ewi.utwente.nl:8901/chirps/article) which, under the heading "Supresssing non-chirping signals", shows a bandy picture and then gives a good trick: Null out all the constant frequency high-power bands, and the chirps will remain. This is under the heading of "SDR chirpfilters", which I have to look up. $\endgroup$ – Lars Ericson Jun 19 '19 at 11:09
  • $\begingroup$ Also by the way the IARPA PINS challenge is still open for another week or so,. This question is actually worth $25,000. I won't be sad if you jump in and grab that easy money ahead of me. $\endgroup$ – Lars Ericson Jun 19 '19 at 11:11
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I gather you are trying to reconstruction a real (non-imaginary) version of your I & Q signal. The following scheme should work:

  1. Up sample your IQ from 10 MHz to 24 MHz. This involves zero padding and applying a low-pass filter to remove the extra images that appear due to the zero padding. See below for more details.
  2. Mix the signal with $\exp(j2\pi7\times 10^6t)$. This shifts the signal to the 7 MHz.
  3. Take the Real component of the result i.e. throw away the imaginary component.

For the upsampling process you have a couple of options. You can upsample by 12 and then decimate by 5. You would up sample by 12, apply the low-pass filter, and then down sample by 5. The filter would need a pass band from 0 - 4MHz (exact upper frequency depends on the nature of your IQ signal). The stop band would need to be at 5 MHz. You might be able to get away with a higher stopband frequency - e.g. if you have no frequency content between 4 MHz and 5 MHz, you could use a stopband starting at 6 MHz. The filter design is slightly dependent on the frequency analysis of your IQ signal and you would have to figure out your requirements for the filter specifications e.g. pass-band ripple and stopband attenuation.

For practical purposes you probably want to break the up sample by 12 into two separate steps e.g. Up sample by 4 and then up sample by 3 - each step would have its own filter applied.

An alternative method is to up sample by a factor of 3 or 4, apply the low-pass filter and then use a spline filter to interpolate the signal to required 24MHz sampling rate.

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