correct typo and math format

Author: narahahn

examples/modal_beamforming_rigid_circular_array.py

@@ -1,6 +1,6 @@
 """
     Compute the plane wave decomposition for an incident broadband plane wave
-    on an open circular array using a modal beamformer of finite order.
+    on an rigid circular array using a modal beamformer of finite order.
 """
 
 import numpy as np
@@ -18,22 +18,20 @@
 # get uniform grid (microphone positions) of number M
 pol, weights = micarray.modal.angular.grid_equal_polar_angle(M)
 
-# get circular harmonics matrix for sensors
+# pressure on the surface of a rigid cylinder for an incident plane wave
 bn = micarray.modal.radial.circular_pw(Nsf, k, r, setup='rigid')
 D = micarray.modal.radial.circ_diagonal_mode_mat(bn)
 Psi_p = micarray.modal.angular.cht_matrix(Nsf, pol, weights)
 Psi_pw = micarray.modal.angular.cht_matrix(Nsf, pw_angle)
 p = np.matmul(np.matmul(np.conj(Psi_pw.T), D), Psi_p)
 p = np.squeeze(p)
 
-# get circular harmonics matrix for a source ensemble of azimuthal plane wave
+# plane wave decomposition using modal beamforming
 Psi_p = micarray.modal.angular.cht_matrix(N, pol)
 Psi_q = micarray.modal.angular.cht_matrix(N, pol_pwd)
-# get radial filters
 Bn = micarray.modal.radial.circular_pw(N, k, r, setup='rigid')
 Dn, _ = micarray.modal.radial.regularize(1/Bn, 100, 'softclip')
 D = micarray.modal.radial.circ_diagonal_mode_mat(Dn)
-# compute plane wave decomposition
 A_pwd = np.matmul(np.matmul(np.conj(Psi_q.T), D), Psi_p)
 q_pwd = np.squeeze(np.matmul(A_pwd, np.expand_dims(p, 2)))
 q_pwd_t = np.fft.fftshift(np.fft.irfft(q_pwd, axis=0), axes=0)

micarray/modal/angular.py

@@ -102,7 +102,7 @@ def cht_matrix(N, pol, weights=None):
         1 & \cdots & e^{i\varphi[0]} & e^{iN\varphi[0]} & e^{-iN\varphi[0]} & \cdots & e^{-i\varphi[0]} \\
         1 & \cdots & e^{i\varphi[1]} & e^{iN\varphi[1]} & e^{-iN\varphi[1]} & \cdots & e^{-i\varphi[1]} \\
         \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
-        1, \cdots, & e^{i\varphi[Q-1]} & e^{iN\varphi[Q-1]} & e^{-iN\varphi[Q-1]} & \cdots & e^{-i\varphi[Q-1]} \\
+        1 & \cdots & e^{i\varphi[Q-1]} & e^{iN\varphi[Q-1]} & e^{-iN\varphi[Q-1]} & \cdots & e^{-i\varphi[Q-1]}
         \end{array} \right]
 
     Parameters
@@ -117,7 +117,7 @@ def cht_matrix(N, pol, weights=None):
     Returns
     -------
     Psi : (2N+1, Q) numpy.ndarray
-        Matrix of spherical harmonics.
+        Matrix of circular harmonics.
     """
     pol = util.asarray_1d(pol)
     if pol.ndim == 0: