Real signal from I,Q, sample rate and center frequency

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 $$R=R[0],...,R[T\times F_{up}-1]$$.

• 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? – Marcus Müller Jun 18 '19 at 7:12
• 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. – Lars Ericson Jun 18 '19 at 13:27
• 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. – Marcus Müller Jun 19 '19 at 6:19
• 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. – Lars Ericson Jun 19 '19 at 11:09
• 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. – Lars Ericson Jun 19 '19 at 11:11

2. Mix the signal with $$\exp(j2\pi7\times 10^6t)$$. This shifts the signal to the 7 MHz.