add 2.5D WFS reference contour examples #184
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| { | ||||||
| "cells": [ | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "# 2.5D WFS Referencing Schemes\n", | ||||||
| "\n", | ||||||
| "This notebook illustrates the usage of the SFS toolbox for the simulation of different 2.5D WFS referencing schemes.\n", | ||||||
| "A dedicated referencing scheme allows correct amplitude alongside a reference contour within the listening area.\n", | ||||||
| "For the theory please check\n", | ||||||
| "Ch 3.1-3.3 in <cite data-cite=\"Start1997\">(Start1997)</cite>,\n", | ||||||
| "Ch. 4.1.3 in <cite data-cite=\"Firtha2019\">(Firtha2019)</cite> and\n", | ||||||
| "<cite data-cite=\"Firtha2017\">(Firtha2017)</cite>." | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": null, | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "import matplotlib.pyplot as plt\n", | ||||||
| "import numpy as np\n", | ||||||
| "import sfs" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "## Circular loudspeaker arrays" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": null, | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "R = 1.5 # radius [m] of circular loudspeaker array\n", | ||||||
| "array = sfs.array.circular(N=64, R=R) # with N loudspeakers\n", | ||||||
| "grid = sfs.util.xyz_grid([-2, 2], [-2, 2], 0, spacing=0.02)\n", | ||||||
| "\n", | ||||||
| "xs = -4, 0, 0 # virtual point source on negative x-axis\n", | ||||||
| "wavelength = 1 / 4 # m" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": null, | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "def sound_field(d, selection, array, secondary_source, grid, xref):\n", | ||||||
| " p = sfs.fd.synthesize(d, selection, array, secondary_source, grid=grid)\n", | ||||||
| " fig, [ax_amp, ax_lvl] = plt.subplots(2, 1, sharex=True)\n", | ||||||
| " fig.set_figheight(fig.get_figwidth() * 3/2)\n", | ||||||
| " sfs.plot2d.amplitude(p, grid, vmax=2, vmin=-2, ax=ax_amp)\n", | ||||||
| " sfs.plot2d.level(p, grid, vmax=6, vmin=-6, ax=ax_lvl)\n", | ||||||
| " sfs.plot2d.level_contour(p, grid, levels=[0], colors='w', ax=ax_lvl)\n", | ||||||
| " for ax in ax_amp, ax_lvl:\n", | ||||||
| " sfs.plot2d.loudspeakers(array.x, array.n, selection, size=0.125, ax=ax)\n", | ||||||
| " ax_lvl.scatter(*xref[selection, :2].T, marker='o', s=20, c='lightsalmon',\n", | ||||||
| " zorder=3)\n", | ||||||
| " plt.tight_layout()\n", | ||||||
| " return p" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": null, | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "xs = sfs.util.asarray_of_rows(xs)\n", | ||||||
| "frequency = sfs.default.c / wavelength # Hz\n", | ||||||
| "omega = 2 * np.pi * frequency # rad/s\n", | ||||||
| "normalize_gain = 4 * np.pi * np.linalg.norm(xs)" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "### Line as reference contour\n", | ||||||
| "\n", | ||||||
| "The reference contour is calculated according to eqs. (24), (31), (52) in <cite data-cite=\"Firtha2017\">(Firtha2017)</cite>. \n", | ||||||
| "The code assumes a virtual point source on x-axis.\n", | ||||||
| "The reference contour is a straight line on y-axis." | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": null, | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "xref_line = 0\n", | ||||||
| "cosbeta = (array.n @ [1, 0, 0]).reshape(-1, 1)\n", | ||||||
| "xref = array.x + \\\n", | ||||||
| " (xs - array.x) * (xref_line + R * cosbeta) / (xs[0, 0] + R * cosbeta)\n", | ||||||
| "\n", | ||||||
| "d, selection, secondary_source = sfs.fd.wfs.point_25d(\n", | ||||||
| " omega, array.x, array.n, xs, xref=xref)\n", | ||||||
| "p_line = sound_field(\n", | ||||||
| " d * normalize_gain, selection, array, secondary_source, grid, xref)" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "The level plot includes a white 0 dB isobar curve.\n", | ||||||
| "The orange-like dots represent the stationary phase points at which amplitude correct synthesis is to be expected.\n", | ||||||
| "These dots shape the line reference contour.\n", | ||||||
| "Note that the isobar curve is not perfectly aligned along line reference contour due to diffraction artifacts." | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "### Circle as reference contour\n", | ||||||
| "\n", | ||||||
| "This reference contour is a circle with its origin at xs and a radius |xs|. This contour is obtained with more straightforward vector calculus than the previous example." | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": null, | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "# reference contour is a circle with origin xs and radius |xs|\n", | ||||||
| "xref_dist = np.linalg.norm(xs)\n", | ||||||
| "# calc reference contour xref(x0), cf. [Firtha19, eq. (24), (31)]\n", | ||||||
| "xref = xs + xref_dist * sfs.util.normalize_rows(array.x - xs)\n", | ||||||
| "d, selection, secondary_source = sfs.fd.wfs.point_25d(\n", | ||||||
| " omega, array.x, array.n, xs, xref=xref)\n", | ||||||
| "p_circ = sound_field(\n", | ||||||
| " d * normalize_gain, selection, array, secondary_source, grid, xref)" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "### Reference point" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "The default handling in\n", | ||||||
| "`point_25d(omega, x0, n0, xs, xref=[0, 0, 0], c=None, omalias=None)`\n", | ||||||
| "uses just a reference point xref, and more specifically this default point is the origin of the coordinate system.\n", | ||||||
| "This single point xref, the virtual source position xs and the loudspeaker array geometry together determine the reference contour without further user access to it.\n", | ||||||
| "This handling is chosen due to convenience and practical relevance when working with circular loudspeaker arrays.\n", | ||||||
| "\n", | ||||||
| "The example below shows the resulting reference contour for the default case.\n", | ||||||
| "In the example it looks similar to the line reference contour, but is in general not exactly the same.\n", | ||||||
| "For example, please try a virtual point source that is far away from the array." | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": null, | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "d, selection, secondary_source = sfs.fd.wfs.point_25d(\n", | ||||||
| " omega, array.x, array.n, xs)\n", | ||||||
| "p_point = sound_field(\n", | ||||||
| " d * normalize_gain, selection, array, secondary_source,\n", | ||||||
| " grid, np.array([[0, 0, 0]]*array.x.shape[0]))" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "Points with amplitude correct synthesis need to be stationary phase points, theoretically.\n", | ||||||
| "Within the listening area, these points are found on rays that start at the virtual point source and intersect with active loudspeakers.\n", | ||||||
| "The chosen points together shall shape a smooth contour, i.e. the reference contour.\n", | ||||||
| "\n", | ||||||
| "The example below shows a reference point xref that does not meet any ray (the gray lines in the level plot) alongside the stationary phase holds with its corresponding loudspeaker.\n", | ||||||
| "\n", | ||||||
| "The single point referencing scheme results in 0 dB isobar curve that closely passes the chosen xref point.\n", | ||||||
| "In practice this typically works with sufficient precision once the position of xref is appropriately chosen (i.e. not too close, not too far, not to off-center from the active loudspeakers etc.)." | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": null, | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "xref = [0, 0.1175, 0] # intentionally no stationary phase point\n", | ||||||
|
||||||
| "xref = [0, 0.1175, 0] # intentionally no stationary phase point\n", | |
| "xref = 0, 0.1175, 0 # intentionally no stationary phase point\n", |
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| "normalize_gain = 4 * np.pi * np.linalg.norm(xs - np.array(xref))\n", | |
| "normalize_gain = 4 * np.pi * np.linalg.norm(xs - xref)\n", |
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