I was wondering by what logic or mathematics the frequency-ration $$ \frac{f_{lo}}{f_s} $$ in the sine-functions comes into existence?
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1$\begingroup$ by the way, I inverted the colors in your graphic to make it readable at all – it's still not very great to decipher. If these are your graphics: I'd try with less different colors, and higher contrast between text and background; so, black text on light colors is usually best to read. If these are not your graphics, it might be worth pointing out where they're from (you called it "Figure 1.7c", but not from where). $\endgroup$– Marcus MüllerMar 19, 2017 at 15:35
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1$\begingroup$ Notice that this is not a very usual receiver design (aside from expensive measurement equipment) – mixing with $f_{LO}$ in analog domain allows you to use a much lower $f_s$, which makes the system much easier to construct, lower in bandwidth, thus lower in noise power, and cheaper. $\endgroup$– Marcus MüllerMar 19, 2017 at 15:38
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1$\begingroup$ What you do find in the wild is the anti-alias filter being a band-pass instead of a low-pass (as in the figure), and the ADC running at an $f_s$ that is significantly below $f_{LO}$, building an undersampling system. But then you'd not need the oscillators to run at $\frac{f_{RF}}{f_s}$, but at $\frac{f_{RF}-n\cdot f_s}{f_s},\,n \in\mathbb N$, as the undersampling process shifts the bandpass region by a multiple of the sampling rate "for free". $\endgroup$– Marcus MüllerMar 19, 2017 at 15:40
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$\begingroup$ By the way, the figure is inconsistent with the text of the PDF: "The architecture of the RTL-SDR corresponds to Fig. 1.7b" in the text, "100s of MHz" in the figure – the RTL-SDR's ADC runs at a couple MHz, not multiple hundreds. That would be expensive. $\endgroup$– Marcus MüllerMar 19, 2017 at 15:52
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$\begingroup$ I corrected the image to an optically readable version; the first attempt of doing so was a mistake. $\endgroup$– StarhowlMar 19, 2017 at 15:53
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that's pretty simple: in digital domain, real-world frequencies have no inherent "meaning".
That is, a period of a real-world sine of frequency 10 Hz might be 2, 200, 1233 or whatever samples long, depending on the sampling rate.
Thus, periods in DSP can only be measured in samples; for example, a sine of period 1 second has a period of 5.4 samples if the sampling rate was 5.4 S/s.
Now, logically, that also means that frequencies can only be related to the sampling rate. Hence, your $\frac{f_{LO}}{f_\text{sample}}$ (which is a unitless thing!) oscillator corresponds to an analog oscillator of frequency $f_{LO}$ (which has Hertz as a unit).
With your question in mind, it strikes me as odd that you're looking at a figure describing a receiver that mixes in the digital domain – in software defined radio, most receivers try to minimize the rate at which the ADC needs to run, simply for cost, efficiency, complexity, ADC dynamic range, and noise bandwidth reasons, and thus have the IQ mixer on the analog side – I'd call your design a bit of a "speciality". So, if this is you learning about direct conversion receivers, you might have picked an unlucky reference. If this is you learning about different types of digital receivers: That is extremely close to the theoretical perfect Software Radio, that does all the signal processing in digital domain.
Also note, in the figure, $f_s \overset !\ge 2\cdot f_{max}$ of the low-pass filter, not only one time that frequency (I assume "GHz" just means "order of magnitude is Gigahertzes"). That's why direct sampling is really an expensive technology for high-frequency signals (and why undersampling with filters or an IF makes a lot more sense, most of the time, since you don't care about everything happening between 0 Hz and the maximum frequency of your signal of interest, usually, but only about some limited bandwidth around a center frequency).
answered Mar 19, 2017 at 15:25
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$\begingroup$ Are there any good websites, books, scripts or papers to read on the sampling ratio for digital mixing? $\endgroup$– StarhowlMar 19, 2017 at 16:04
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$\begingroup$ All the theory there is – frequencies only being represented as ratio to the sampling rate – is in my answer. As to basic theory of signals in analog and digital domain, which will lead to an understanding of the three different receiver architectures in your source's fig 1.7: any textbook on digital signals and systems. Try Oppenheim's "Digital Signal Processing", chapter 4 (sampling signals), and Sklar's "Digital communications", chapter 2.4 (formatting analog information). $\endgroup$ Mar 19, 2017 at 16:12
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$\begingroup$ Oppenheim's work is already partly part of me; it's not focused on modulation. I will have a look at Sklar's work in time! $\endgroup$– StarhowlMar 19, 2017 at 16:16