I'm top-editing this since it answers the question directly.
The sinc series is fundamentally a $C/x$, so you can extract as many absolutely convergent series out of it as you want, but what is left over is still only conditionally convergent. Also, you can rescale $x$ and it is still a $C/x$ series.
Saying you have a summation to or from infinity is an ...
You've got your answer but let me summarize a bit about your confusion. We can classify signals as being baseband (aka lowpass) or bandpass.
The basic form of Nyquist-Shannon sampling theorem involves bandlimited baseband real signals and says that :
A real, bandlimited to $W$ (Hz), continuous-time signal $x_c(t)$ can be
exactly and uniquely recovered ...
Well, according to Nyquist-Shannon theorem you should sample at a rate which is at least twice the highest frequency you want to capture. This is also referred to as the sampling theorem because it "forms the basis" for sampling.
Now regarding your specific question, I have to say that cos(4*pi) (along with the rest of the components) is just a number which ...
I try to think of it this way: In the time domain, when downsampling occurs, the signal gets compressed; while on upsampling, the signal gets stretched.
Then, from the Fourier transform we know that time stretching means frequency compression, and vice-versa.
It may not be a rigorous answer, but hope this helps!
I gather you are trying to reconstruction a real (non-imaginary) version of your I & 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\...