What the sampling rate should be for passband and baseband signals?

I feel dumb for not grasping this is first couple of times this was explained to me, but I feel like I will grasp this with a simple question.

If I have a passband signal at a center frequency of 3 GHz and a bandwidth of 200 MHz, would the sampling rate need to be 2 * (3 GHz + (200 MHz / 2)) = 6.3 GHz. Since the highest frequency is 3 GHz + (200 MHz / 2).

Then at baseband, the sampling will need to be 400 MHz since the highest frequency is 200 MHz.

The confusion arises when I heard the sampling rate needs to be twice the bandwidth, but that is after mixing correct?

First let me be clear on the two options for sampling in the First Nyquist zone, where our signal of interest exists somewhere in the bandwidth of DC to $$f_s/2$$ where $$f_s$$ is the sampling rate. Later I will go into the details on undersampling, where the conclusion of the minimum sampling rate needed (for purposes of meeting the Nyquist criterion) are not changed.

Below is a plot showing two cases; a complex baseband signal and a real IF (intermediate frequency) or passband signal.

For the first case when we are doing complex sampling (I and Q or real and imaginary, with two samplers in actual implementation when sampling directly), with the bandwidth $$B$$ referring here to the two-sided bandwidth extending into the positive and negative frequencies, the minimum sampling rate required must be greater than $$B$$, adding margin for realizable filtering.

For the second case with passband sampling, and here as shown at the lowest possible IF frequency (which also needs margin for realizable filtering), the bandwidth $$B$$ is the same bandwidth as the complex baseband signal prior to be frequency translated to the real IF. In this case the sampling rate must be greater than $$2B$$, adding margin for realizable filtering.

As the Nyquist Criteria states, the sampling rate must be twice the bandwidth of the signal plus margin for achievable filtering to ensure no aliasing (since we cannot implement brickwall filters). What we see as a common feature from the plots above, the sampling rate in both cases is twice the highest frequency for this case when our signal occupies the entire frequency of interest.

A reasonable margin is 20%, more or less depending on filter complexity and allowable delay. So in this case with a bandwidth of 200 MHz, if the signal was at complex baseband occupying $$\pm 100$$ MHz, the minimum sampling rate would be 200*1.2 = 240 MSps, more or less depending on what we actually want to take on for filter complexity and delay.

When sampling a received signal at a 3 GHz passband, the consideration is the analog input bandwidth of the ADC; if the analog input bandwidth exceeds 3GHz, then with a proper band-pass anti-alias filter and sufficient spectral purity on the sampling clock (under-sampling requires better phase noise!), we can select a sampling rate that meets the Nyquist criteria AND reasonably centers the signal within our sampling rate for either a real digital IF signal or complex digital baseband signal. This gets into deeper details of undersampling that I have detailed in this post here.

As a realistic example, I will work through my thought process for the receiver case of direct sampling a 200 MHz passband signal that is at a 3 GHz carrier. My assumption is I have already located an ADC that has an analog bandwidth in excess of 3 GHz, and will meet the dynamic range requirements as given by an effective number of bits (ENOB). I will have reviewed what realizable filter choices I have at the 3 GHz carrier frequency consistent with these considerations for anti-alias filter design. To avoid using two ADC for complex sampling (which is another possibility), I will use a digital IF frequency that is supported by the anti-alias filter used. Assuming the analog bandpass filter allows for a minimum IF frequency of $$f_{IF}=120$$ MHz, with $$f_{RF}=3$$ GHz, then the sampling rate chosen would meet the following criteria:

$$|f_{RF} \pm N f_s| = f_{IF}$$

$$f_{IF}> 120 \text { MHz}$$ $$f_{RF} = 3 \text { GHz}$$ $$f_s = 4f_{IF}$$

The last requirement is actually limited to be $$f_s>2f_{IF}+B$$ given the details provided above, but restricting the digital IF to be $$fs/4$$ centers the spectrum within the available digital bandwidth, from which we can consider small changes for other sampling rates with the final restriction as $$f_s>2f_{IF}+B$$. Combining the above requirements provides for possible solutions as:

$$f_s = \frac{f_{RF}}{N \pm 1/4}$$

$$f_s > 480 \text { Msps}$$

Therefore $$N$$ must be greater than or equal to 6. With $$N=6$$ we get the choices for sampling rate to be either $$3e9/5.75 \approx 521.7 \text { Msps}$$ or $$3e9/6.25 = 480 \text { Msps}$$ (Happy coincidence it came out exact, and as stated early we can shift the actual sampling rate with reason so that the signal is still within the first Nyquist zone and not affected by the achievable rejection of our actual bandpass filter used.).

Confirming this, if the sampling rate was 521.7 Msps, the 6th harmonic of the sampling clock would be at 3.130 GHz, resulting in a 130 MHz digital IF frequency which is well supported by the 521.7 Msps. In the second case with a sampling rate of 480 MHz, the 6th harmonic of the sampling clock would be at 2.88 GHz resulting in a -120 MHz digital IF (and we would have spectral reversal in this case which is no issue to correct for digitally).

As far as achieving a 3 GHz passband as an analog output (meaning we are using a DAC), there are additional considerations beyond meeting the Nyquist criteria outlined above and the analog output bandwidth of the DAC, but the 3 GHz output can be similarly achieved with lower sampling rates through the use of undersampling, or mixing after the DAC to frequency translate the signal. Traditional DAC's will have a Sin(x)/x roll-off due to the zero-order hold in the reconstruction, but modern RFSoc devices include DACs with features to support higher direct outputs at lower sampling rates by extending the first null of the Sin(x)/x roll-off. This is done with techniques that include return-to-zero and mix-mode sampling. In return-to-zero the DAC output is returned to zero half way through the sample duration, and in "mix-mode" the polarity of the signal is inverted half way through the sampling clock.

The confusion arises when I heard the sampling rate needs to be twice the bandwidth, but that is after mixing correct?

Nyquist just said you need to sample at a minimum of twice the bandwidth, but not that it had to be after mixing. So -- in theory -- there are schemes out there that would allow you to obtain $$400\cdot10^6$$ samples per second and be able to fully reconstruct your signal from them.

Note that the Nyquist rate isn't a design goal (usually). For most signal processing you should think of it more as a fiery barrier that's best to be treated with extreme respect. You should design your system -- complete with anti-aliasing filters -- and then answer the question "how strong of an unwanted signal will corrupt my desired signal, if it's this far off in frequency"?

So once you chose your actual sampling rate (call it $$F_s$$), you could:

• Run your signal through a perfect 90$$^\circ$$ phase shifter, and sample the original and the phase-shifted version at $$\frac 1 2 F_s$$ each -- that gives you unique samples at a rate of $$F_s$$.
• But it ignores the fact that no phase shifter is perfect, nor are any two ADC channels perfectly matched.
• Buy an off-the-shelf quadrature downconverter, and run its output into two ADCs, each sampling at $$\frac 1 2 F_s$$. Same downsides as above, except you're using an off-the-shelf part with -- hopefully -- known characteristics.
• Heterodyne your signal down by $$3 \mathrm{GHz} \pm \frac 1 2 F_s$$, then sample by $$F_s$$. This has the cost of filters and radio circuitry, but the advantage that you're sampling slower.
• Choose a sampling frequency such that $$3 \mathrm{GHz} \pm \frac 1 2 F_s = n F_s$$, for some integer $$n$$, and use an ADC or external sampler with a super-narrow sampling window and super-low clock jitter. This has the same effect as the one above.
• Sample initially at some $$F_f \ge 6 \mathrm{GHz} + F_s$$, then use some really simple filter and decimate function (such as CIC filters). This needs insanely fast ADCs (you may not be able to fine them), but you'll benefit from coding gain so they don't need as high a bit count.