Extrapolatoin

[narahahn]

Example for modal analysis and sound field extrapolation

• capture the sound field of a monochromatic plane wave by an open circular array

• compute the circular harmonics expansion coefficients up to order N. As shown below, The CHT coefficients pm is obtained by multiplying the regularized radial weights dn with the IDFT of the captured signal p. No matrix computation is needed. The reason for using IDFT (not DFT) is due to our choice of the circular harmonics as np.exp(-1j*n*phi).
  

  sfa-numpy/examples/sound_field_extrapolation_open_circular_array_mono.py

  Lines 36 to 38 in 3da5e01

  	bn = micarray.modal.radial.circ_radial_weights(N, k*R, setup='open')
  	dn, _ = micarray.modal.radial.regularize(1/bn, threshold, 'softclip')
  	pm = dn * np.fft.ifft(p)

• extrapolate the sound field on 2-dimensional grid points
  

  sfa-numpy/examples/sound_field_extrapolation_open_circular_array_mono.py

  Lines 40 to 43 in 3da5e01

  	# Sound field extrapolation
  	basis = special.jn(order[:, np.newaxis, np.newaxis], k * r[np.newaxis, :, :]) \
  	* np.exp(-1j*order[:, np.newaxis, np.newaxis] * phi[np.newaxis, :, :])
  	s = np.tensordot(pm, basis, axes=[0, 0])

• results
  

CAUTION
One line in regularize was commented out so that it works for mono-frequency cases. This issue is being handled in #4.

sfa-numpy/micarray/modal/radial.py

Lines 179 to 183 in 3da5e01

	# dn[0, 1:] = dn[1, 1:]
	dn = dn * hn
	if not np.isfinite(dn).all():
	raise UserWarning("Filter not finite")
	return dn, hn

[narahahn]

Additional note:
Using IDFT for circular harmonics expansion works only if the microphone array is uniformly distributed on a circle and one of them is placed exactly at phi=0. If there is an angular offset, then a phase shift should be applied to the coefficients.