# Filtering after a phase domain ADC

While researching receiver architectures for the demodulating constant envelope signals like a GFSK, I stumbled over the concept of phase domain ADCs, where after

$$\boxed{\text{LNA}}{\longrightarrow}\boxed{\text{Mixer}}{\longrightarrow}\boxed{\text{Filter}}{\longrightarrow}\boxed{\text{ADC}}$$

instead of the classical IQ signals $$i[n]$$ and $$q[n]$$ you get directly the phase $$\phi[n]$$ from the ADC, but you loose the amplitude information about your $$A*sin(2\pi f t + \delta)$$ signal. How can I mathematically model filtering of my phase domain signal $$\phi[n] = 2\pi f t + \delta$$?

Assume I have an interferer at a higher frequency than my desired GFSK signal. I can derive the phase to get the instantaneous frequency: $$\frac{d}{dn}\phi[n]=f[n]$$. Then intuitively, I would low pass filter with a cutoff frequency at the symbol rate of my GFSK, which should remove the higher order frequency component coming from the interferer. Is this intuition correct? Derivation is a linear operation and so is the filtering, but transforming a signal from IQ representation to polar representation (with a CORDIC for example) is not. I struggle with the notion and reasoning as I am used to perform such operations on $$i[n]$$ and $$q[n]$$ and not in phase domain. Going back from $$\phi[n]$$ to IQ does not seem useful as I lost the amplitude information.

Do you know any references or books which treat this subject?

First as a disclaimer to my answer; I am not aware of references or books on use of such a device, and I have little actual experience with "Phase Domain ADC's". However this caught my interest and I was hoping someone with more experience would post. Since no-one else has yet posted, I am offering my perspective below. I do understand GFSK demodulation and data conversion well enough to see the shortcomings and now possible advantages of a phase-domain ADC, which I hope may on its own be helpful and/or elicit other answers. It is interesting to note that my opinion was initially quite negative, but in the effort of writing this out I came to understand more about the possible utility of such a device.

I agree with the OP regarding the challenge in demodulating from phase alone, when the "phase domain ADC" is under the presence of any significant interference anywhere within the observable bandwidth at the ADC input. This is its primary limitation so the critical consideration is filtering prior to the phase domain ADC rather than after: this isn’t the also necessary anti-alias filtering but similar in that if it is insufficient ahead of the ADC, there is nothing that can be done to recover with later filtering afterward. It's utility is the opportunity for significant cost, power, size (on an IC, translating then back to both the first two utilities) for cases when the interference can be sufficiently mitigated prior to the ADC input. It is feasible that whatever I would typically want to do in the digital domain (I have experience in both the analog and digital domains, so see it both ways, although I still want to do it digitally :) ), can be done in the analog especially when we want to give up some flexibility- and great progress is being made in tunable analog, MEMs, etc that can open up other ways of thinking to get the same result.

Specifically, it is necessary that interference be removed prior to the phase-domain ADC input, which is the technical point I will continue to make below.

Significant interference simply means an interference with a peak amplitude level that is as large or larger than the amplitude of the modulated signal of interest, anywhere within the observable bandwidth of the phase-domain ADC. If such interference is possible, then front-end filtering will be necessary prior to the phase domain ADC input for demodulation to be possible. In comparison, when using a traditional ADC both the amplitude and phase are sampled, in which case the interference if not directly in band and if within the ADC's dynamic range, can still be filtered out in the digital domain. This is not possible if phase demodulation is done before the interference is removed (A "phase-domain ADC" demodulates the phase).

Assuming the phase demodulation process is well understood, then what I describe would be intuitive by visualizing the process on an IQ diagram (complex plane) showing the modulated signal versus time as I attempt to depict in the simpler graphics below.

To describe the graphics: the first plot is a GFSK modulated signal as it would appear on the IQ plane under condition of no interference. Over time the signal would map out the entire circle as the phase rotates either left or right depending on "positive" or "negative" frequency relative to the carrier (phase modulation and no amplitude modulation). Naively, we might think at this point that a device that demodulates the phase only, such as a "phase domain ADC", would be perfect for this!

But then in the center, I show what would start to happen if there was a single tone interference off further away from the carrier (just out of the primary bandwidth of the signal). What we see in my sketch is the interferor (as a single tone) rotating at a faster rate as the primary modulated signal moves along its trajectory. Bottom line- it induces amplitude modulation (AM) as well as phase modulation (PM). For this middle picture, the result after the phase-domain ADC extracts phase only would be a small induced ripple in the phase modulation due to the interference, but we would still be able to demodulate the primary phase modulation that existed without the interference. So, no problem yet. It's still predominantly linear and the interference that results will still spectrally be at its original location (can be filtered out), and further we have the advantage that we removed half of the interference power! (The phase demodulator, as a hard-limiter, has removed all AM and the interference was half AM, half PM).

With that understood, it is hopefully clearer why the third plot would be so destructive in comparison. Here the much larger interference also induces AM and PM, and the AM is also removed by the phase-domain ADC. However the AM has now crossed the origin leading to highly non-linear phase results (the phase for example can immediately jump 180 degrees as we cross the origin). If we did maintain both AM and PM with an alternate traditional data-converter (assuming the signal + interference is all within the linear range of the converter), then this signal with all of its distortion as shown in the third plot, as it exists at the ADC input will still be similar digitally- we can indeed filter out the interference component that is not directly in our modulated signal bandwidth digitally (much more effectively than in the analog), and then after that we can then, with no interference present, demodulate the phase.

That said, as long as the implementation ahead of the phase-domain ADC provides the functionality I describe in the last paragraph in the analog domain (proper filtering), then demodulation will be possible and a much simpler (lower power, lower cost) receiver may result.

• Thanks for you answer :-) In a previous job we employed a phase ADC in a older technology node product, but in newer nodes we moved to a more traditional IQ demodulating scheme. As it puts more burden on the analog side, this approach is more suitable for older nodes than for newer ones where you try to move more functionality to the digital as it scales better with the techno. I just tried to pick up this old idea and wanted to see if we could combine it with some DSP to alleviate the stringent filter design in analog, but I also concluded that we cannot.
– njg
Commented Aug 6, 2023 at 13:57
• @njg thanks for your comments. Do you concur with my answer? Commented Aug 6, 2023 at 15:52