High frequency, narrow banded beamforming

nedlrichards

I am very impressed by the Bela ecosystem, and I am hoping that it will simplify a project for me. I just wanted to run my thinking past the experts, to see if it makes sense before moving forward.

I am interested in beamforming at audio frequencies up to 40 kHz, and would like to use python to write all code. I have a current audio capture setup (Eidrol UA-101) that uses the python package sounddevice, but I am happy to be flexible on which package I use with Bela equipment.

The essential challenge is to take a FFT on 2 or 3 channels every little bit, perhaps with 50% overlap between FFTs. This result is then reduced to the few bins around the frequency of interest. This seems to me possible with the ALSA option (96 kHz) of the CTAG FACE, or the 88.2 kHz analog in option of the Bela expander capelet. I don't anticipate that cpu load will be an issue, so I am tempted to use the FACE for simplicity.

Has anyone worked on a project using python and the FACE? Is it as straightforward as using a rapberry pi and a USB audio capture system, or are there additional complications I should be aware of? Thank you for your time 🙂

giuliomoro

When using the FACE with ALSA drivers, you are back in a regular Linux ecosystem, so sounddevice and other python packages should work just fine.

Incidentally, if you are discarding most of the results of the FFT, and only end up looking at a few bins, would it not be cheaper to simply apply the DFT for a number of selected frequency bins? Pre-computing the twiddle factors should give reasonably cheap results.

nedlrichards

Thanks for getting back to me. I had one other thought that came to me last night which may be too much. If the user takes care of bandpass filtering and all that, will sampling a 40 kHz signal at 48 kHz create the theoretically predicted aliasing? I would be happy to use the lower sampling rate and get savings that way, but I don't know how the sigma-delta ADCs on the FACE will handle intentional aliasing.

As a follow up, I looked into twiddle factors but they are a little new to me. I can certainly save effort precomputing and using a few rows in the matrix representation of a DFT. Hope I got that right, and I appreciate the idea.

giuliomoro

nedlrichards , will sampling a 40 kHz signal at 48 kHz create the theoretically predicted aliasing? I

If you sample with the audio converters, which use sigma-delta technology, then the built-in digital filter will filter it out. From the CTAG codec datasheet, it seems that when sampling at 96kHz the bandwidth of the filter will go up above 50kHz(see page 11), so that would work to sample a 40kHz signal.

If you use Bela's own analog in (which use the Successive Approximation Register technology), then those are not filtered, and so you should be able to sample a 40kHz signal, even when sampling at 22.05kHz: let's not forget that the sampling theorem talks about a bandwidth of Fs/2, and not a maximum frequency of Fs/2 ! Because of the lack of flitering, these ADCs are noisier. For more info on SAR vs Sigma-delta converters on Bela (and not only), see here.

nedlrichards As a follow up, I looked into twiddle factors but they are a little new to me. I can certainly save effort precomputing and using a few rows in the matrix representation of a DFT. Hope I got that right, and I appreciate the idea.

Remember the DFT definition:

alt text

Now, because of Euler's identity, you see that you can compute the value of the DFT at a given frequency bin k as the sum of the product of your signal x and a pair of sinewaves, that is, e.g., in MATLAB:

n = 1:length(x));
X(k) = sum(x .* cos(2*pi*k*n/N) - 1j * sin(2*pi*k*n/N))