Ultrasonic TDoA / ToA Measurement?

Question

I am trying to make an ultrasonic flowmeter. The device has two ultrasonic tranducers (resonance frequency of about 200kHz) that sample one ultrasonic signal in the upstream directon and then immediately after one signal in the downstream direction. I know a correlation between the upstream and downstream signals might be enough but the problem is that we don't have a lot of processing power available. We have signals that are 1024 samples long each and a full cross correlation would take too long to process. The maximum sampling Rate is about 6MHz if that helps.

Since I don't come from an audio background, I am also finding it hard to figure out at what point the signal really 'arrives' and how to detect it at the resolution that I need. With the SNR that I have, it seems like the exact startng point would be lost in the noise?

No matter what I try, I seem to end up with too many wrong measurements where the time measured is exactly one wave period away from the correct value.

I know this question is far too vague but I would really appreciate any ideas on how I could understand the problem better!

Here's what the signals look like: So the main questions are:

1. What is the exact time of arrival of these two signals at the ultrasonic transducer?
2. What is the differential time of arrival between both the signals?

Answer 1

The FFT-based cross-correlation is done as follows:

TX = np.fft.fft(tx)
RX = np.fft.fft(rx)
CORR = np.multiply(TX, np.conjugate(RX))
corr = np.real(np.fft.ifft(corr))

Here is a NumPy script that tries a solution.

It does the following:

• create a TX and delayed RX signal that similar to your waveform (a sine wave modulated by an envelope cosine with zeros at the front and the back and noise superimposed
• calculate the cross-correlation with SciPy's signal.correlate
• calculate the cross-correlation with FFTs
• find and print out the location of maximum correlation
• if all went well, the resulting number should be the same as the delay between TX and RX

In the graph below, you see the time domain test signal at the top, and the correlation result at the bottom (blue = signal.correlate, orange = FFT-based).

As you can see, there's a shift between the blue and the orange result. Figuring out why is an exercise for the reader (I'm learning this stuff myself). It has to do with cyclical convolutions, zero padding etc. You should be able to get a better understanding than what I'm able to offer by googling "FFT convolution". Here is one such Google result that talks more about this.

In my script, instead of doing it fundamentally right, I went the pragmatic road and changed some fixed constant to make the convolution and the FFT convolution come up with the same result.

The full script finds the maximum value of the 2 correlations, like this:

    max_corr = np.argmax(corr)-len(tx)//2
    max_corr_fft = np.argmax(corr_fft) - len(corr_fft)

And prints the result:

max corr: -129
max corr fft: -129

129 matches the delay that was used to generate the RX signal.