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 gather you are trying to reconstruction a real (non-imaginary) version of your I &amp; Q signal. The following scheme should work:


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