# Frequency-ratio at baseband IQ-mixing

I was wondering by what logic or mathematics the frequency-ration $$\frac{f_{lo}}{f_s}$$ in the sine-functions comes into existence?

Source

• 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). Mar 19, 2017 at 15:35
• 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. Mar 19, 2017 at 15:38
• 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". Mar 19, 2017 at 15:40
• 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. Mar 19, 2017 at 15:52
• I corrected the image to an optically readable version; the first attempt of doing so was a mistake. Mar 19, 2017 at 15:53

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).

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).