# Only detect the phase of a sinusoid multiplexed with other sinusoids of different frequencies

Is there an easy way to detect the phase of a sinusoid that has been added to other sinudoids of different frequencies (through a frequency-division scheme) without using a superheterodyne receiver accurate enough to reject all other signals? All sinusoids in the system have the same amplitude, and their frequencies and phases (other than phase of sinusoid in question) are precisely known. I was thinking that we can use a wide-band filter, and as long as I know the behaviour of the filter precisely (how much of the neighboring signals it admits, etc.), I can post-process the output of the filter in some way.

The system is currently set up so that the multiplexed signal passes through a filter, and then a phase difference detector (AD8302) compares the filter output's phase with a reference signal appropriate to that frequency.

EDIT: It's good to hear back! To clarify, the goal of the system is to detect the change in phase difference. The target sinusoid will change its phase with respect to the reference phase (very slowly) over time due to some chemical process. We would like to monitor this phase change. The absolute phase between the target sinusoid and the reference is irrelevant to us, and so is the filter phase response (since it equally applies to the signal before and after phase change). All components are analog, and the high frequencies of the sinusoid is currently preventing me from sampling the multiplexed signal and from using DSP techniques directly on the multiplexed signal.

• Quadrature demodulation? Jul 29 '20 at 23:06
• Only FFT can give you the accurate results. Filters are too phase sensitive with high Q. Try here falstad.com/fourier enable Mag/Phase/Log then draw any spectrum Jul 29 '20 at 23:27
• Phase is a relative term and requires a reference, something defined as 0 degrees. In a system with other sines of different frequencies, what would the phase of the desired signal be measured with respect to? Jul 29 '20 at 23:49
• FFT with a square wave fundamental and all sub octaves measured with respect to the master clock might work. Why do you want to do this? Jul 30 '20 at 0:41
• Your assumptions are that your filter is exactly 0 deg phase shift which may be completely wrong if there is an error. What do you expect the summation signal to look like? not white noise but very noisy looking with sharp amplitude peaks at the lowest f/2n Jul 30 '20 at 1:05

This seems like an irrelevant hypothetical problem but I have done the reverse to show what you might have to do.

If you had a 5 stage binary counter with f=320 Hz then subharmonics from f/2 to f/32 with coherent phase (0deg) for each sine and equal amplitude (-20dB) all mixed together. It would look like below in red. The yellow trace shows only the analog component of the lowest frequency by pointing at it.

I did this with Falstad's Fourier Spectrum analyzer/synthesizer and manually adjusted the spectrum per your question for each binary sub-harmonic signal with ~0deg phase difference.

The result was starting to look like a messy recursive sin(x)/x modulated carrier with a peak that was 16dB above the amplitude of each frequency at the frequency of f/32=10Hz. Now to get this spectral density result for amplitude and phase you must do the reverse of what I did.

But what's the point? 