Playing with the FFT a little, maybe some of you might find this interesting…
We can never have enough frequency resolution. Consider analyzing a 50% amplitude modulated 29.6 MHz signal.
The first screenshot shows the regular approach:
Acquisition sample rate = 1 GSa/s;
FFT sample rate = 125 MSa/s;
FFT length = 2 Mpts;
Bin width = 59,6 Hz;
SDS2504X HD_FFT_2Mpts_29.6MHz_0dBm_Excl
At 1 kHz modulation frequency, we can see the sidebands clearly and measure all the amplitude levels accurately, but there’s not much room for an even closer frequency spacing.
We can have a better resolution bandwidth (RBW) if we use undersampling. The analog/digital conversion acts like a mixer, where the input signal gets multiplied by the sample clock.
In the next example, we acquire the waveform with a sufficient sample rate of 100 MSa/s, but feed the FFT with the decimated rate of just 1.25 MSa/s, thus Nyquist for the FFT drops to only 625 kHz.
We are aiming at the mixer product 24 * fc - fs = 24 * 1.25 MSa/s - 29.6 MHz = 400 kHz;
At 400 kHz, it is much easier to get a decent RBW.
SDS2504X HD_FFT_128kpts_DS100MSa_1.25MSa_29.6MHz_0dBm_Excl
We can take this concept even further and can use undersampling already in the acquisition process.
The next example acquires the waveform at only 10 MSa/s (while we would need at least 60) and this gets further decimated to just 125 kSa/s for the FFT. Nyquist of the FFT is now 62.5 kHz.
We are aiming at the mixer product 474 * fc - fs = 474 * 125 kSa/s - 29.6 MHz = 25 kHz;
At only 25 kHz, it is no problem to get just 0.95 Hz bin width and a correspondingly narrow RBW.
SDS2504X HD_FFT_128kpts_DS10MSa_125kSa_29.6MHz_0dBm_Excl
As has been demonstrated, the method of undersampling works quite well and it requires no further utilities. The only drawback is that we cannot choose an arbitrary sample frequency but stick to what the DSO has to offer – and we need to do some calculations to predict at what frequency the signals will ultimately appear.
In the good old times of spectrum analyzer boat anchors, sometimes external mixers would be used to extend their frequency range up to 40 GHz. We could make use of this idea also for our DSO. If we don’t actually want to extend the frequency range but rather get the signal frequencies down to a range where we can have a decent RBW, then we don’t even need an external mixer but can have some fun with the integrated math instead. All we need is an “oscillator signal”. Maybe in future devices we’ll get full-length reference waveforms, so that we can use these instead and don’t even need a physical signal source anymore.
For the time being, one channel of the AWG has been used to supply the 29 MHz oscillator signal, which needs to have an amplitude of +16 dBm in order to have zero loss in the mixing process. Then it’s simply a matter of running the FFT on the product of the two input channels and find the spectrum shifted down by the oscillator frequency, i.e. 29.6 MHz become 600 Hz now.
First look at the next screenshot, that shows both input signals together with the mixer result – modern art created by the SDS2000X HD 😉
SDS2504X HD_MATH_C2xC4
We now acquire the waveform with a sufficient sample rate of 100 MSa/s, but feed the FFT with the mixing result, which should be around 600 kHz, hence an FFT sample rate of just 2.5 MSa/s should be plenty. The calculation is very simple now:
We are aiming at the mixer product fs - fo = 29.6 MHz – 29 MHz = 600 kHz;
SDS2504X HD_FFT_128kpts_DS100MSa_1.25MSa_29.6MHz_0dBm_Excl
Again, it is fairly easy to get a decent RBW.
EDIT: all these undersampling / downconversion approaches also enable us to make do with less FFT points and still have a decent RBW.
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