What is the right way to extract particular frequency phase information from FFT?

Question

Signals acquisition:

Two proximity sensors are positioned Orthogonal to each other. These would be measuring the vibrations of a rotating shaft.

Problem:

Each of these 2 signals has two dominant frequency components (2441 Hz & 762.9 Hz ) in it. I don't know the phase and amplitude information. I'm interested in finding the phase of the signal at 2441 Hz and 762.9 Hz. I did the following process to achieve amplitudes and phases in MATLAB

1. Applying FFT to the raw signal and calculating Amplitude and Phase.

   %fft for X signal
   fftx=fft(X,NFFT);
   % absolute value of the signal 
   Xn = abs(fftx);
   A = mean(Xn);  % I took the mean to get single value.
   phase = unwrap(angle(fftx));
   phase_mean = mean(alpha);
   

2. Filtering raw signal to specific frequency prior to applying FFT using Butterworth filter. ( In this example I filtered to 2441 Hz)

   [b,a] = butter(2, [ 2440/(fs/2),2441/(fs/2)],'bandpass');
   X_filtered = filter(b,a,X);
   % Applying FFT to filtered signal
   fft_filtered = fft( X_filtered,NFFT);
   Xn_filtered = abs(fft_filtered);
   A_filtered = mean(Xn_filtered);
   phase_filtered = unwrap(angle(fft_filtered));
   phase_filtered_mean = mean(phase_filtered); 
   

3. I did the same process as point 2 to filter to another frequency say 762.9 Hz.

4. My sampling frequency and other things are defined as follows.

   L = length(X); % 50050 is length of signal
   NFFT = 2^nextpow2(length(X)); % Zero padding to nearest N power 2
   %Define frequency axis
   fs = 1e7; % Sampling frequency
   df = fs/NFFT; % frequency resolution
   dt = 1/df; % time resolution
   X_detrend = detrend(X,0); % Removing DC Offset
   

   I've been reading a lot of other posts on the FFT phase and Amplitude.But I'm confused with most of them. In my case, I need to extract frequency particular information (amplitude and phase) to do further calculation. I'm still unsure if using the filters is right way? Will FFT and filters alter the signal information in any way? Could someone enlighten in understanding the efficient way to extract amplitude and phase information from a signal of particular frequency component.

Answer 1

You don't specify the sampling rate, but I'll assume it's high enough. With only two frequencies, your best bet is to define your DFT frame on a whole number multiple, of one, possibly both. In this case the ratio of frequencies is about 3.2, which is a ratio of 16 to 5. So if you select a DFT length with 10 of the lower frequency's wavelengths, and 32 of the higher frequencies, or some greater multiplesv when you take the DFT, the frequencies should be adequately centered on their respective bins. Then you can get a good read on the phase and amplitude directly from their respective bin values. They should be far enough apart that leakage or harmonics will not significantly interfere with your results.

In case with a real signal, where the frequency is known and not bin centered, the phase and amplitude can be calculated by a method I describe in my blog article "Phase and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1".

You can also reduce the impact of noise by using the average method of my blog article "Exponential Smoothing with a Wrinkle" which will smooth your signal without shifting the phase values. The amplitude will be attenuated but the article gives a formula for recovering the original amplitude from the smoothed one.

Hope this helps.

Ced

Answer 2

To estimate spectral component phase angle relative to shaft angle using an FFT of length N, first gather a sequence of N samples such that the shaft angle is known with accuracy at sample number N/2, then window with a Nuttal window and do an FFT shift before the FFT. The atan2() of the FFT bin result closest to the desired spectral component will produce an estimate of the phase relative to the center of the original sample window. Adjust that phase by the known shaft angle at that N/2 sample point to get a spectral phase relative to shaft angle.

If you know the spectral frequency or use a frequency peak estimator, then the phase can also be interpolated between bins for increases phase estimation accuracy. Interpolate phase by using a Sinc interpolation seperately on the real (even) and imaginary (odd) FFT result components before taking the arctangent.