4. Special Geometries: NFC-HOA and SDM

The integral equation (3.1) states a Fredholm equation of first kind with a Green’s function as kernel. This type of equation can be solved in a straightforward manner for geometries that have a complete set of orthogonal basis functions. Then the involved functions are expanded into the basis functions \(\psi_n\) after [MF81], p. (940) as

(4.1)\[G(\x-\x_0, \w) = \sum_{n} \tilde{G}_n(\w) \psi_n^*(\x_0) \psi_n(\x)\]

(4.2)\[D(\x_0, \w) = \sum_n \tilde{D}_n(\w) \psi_n(\x_0)\]

(4.3)\[S(\x, \w) = \sum_n \tilde{S}_n(\w) \psi_n(\x),\]

where \(\tilde{G}_n, \tilde{D}_n, \tilde{S}_n\) denote the series expansion coefficients, \(n \in \mathbb{Z}\), and \(\langle\psi_n, \psi_{n'}\rangle = 0\,\) for \(n \ne n'\). If the underlying space is not compact the equations will involve an integration instead of a summation

(4.4)\[G(\x-\x_0, \w) = \int \tilde{G}(\mu, \w) \psi^*(\mu, \x_0) \psi(\mu, \x) \d\mu\]

(4.5)\[D(\x_0, \w) = \int \tilde{D}(\mu, \w) \psi(\mu, \x_0) \d\mu\]

(4.6)\[S(\x, \w) = \int \tilde{S}(\mu, \w) \psi(\mu, \x) \d\mu,\]

where \(\d\mu\) is the measure in the underlying space. Introducing these equations into (3.1) one gets

(4.7)\[\tilde{D}_n(\w) = \frac{\tilde{S}_n(\w)}{\tilde{G}_n(\w)}.\]

This means that the Fredholm equation (3.1) states a convolution. For geometries where the required orthogonal basis functions exist, (4.7) follows directly via the convolution theorem [AW05], eq. (1013). Due to the division of the desired sound field by the spectrum of the Green’s function this kind of approach has been named SDM [AS10]. For circular and spherical geometries the term NFC-HOA is more common due to the corresponding basis functions. “Near-field compensated” highlights the usage of point sources as secondary sources in contrast to Ambisonics and HOA that assume plane waves as secondary sources.

The challenge is to find a set of basis functions for a given geometry. In the following paragraphs three simple geometries and their widely known sets of basis functions will be discussed.

4.1. Spherical Geometries

The spherical harmonic functions constitute a basis for a spherical secondary source distribution in \({\mathbb{R}}^3\) and can be defined after [GD04], eq. (12.153) [1] as

(4.8)\[\begin{gathered} Y_n^m(\theta,\phi) = (-1)^m \sqrt{\frac{(2n+1)(n-|m|)!}{4\pi(n+|m|)!}} P_n^{|m|}(\sin\theta) \e{\i m\phi} \; \\ n = 0,1,2,... \;\;\;\;\;\; m = -n,...,n \end{gathered}\]

where \(P_n^{|m|}\) are the associated Legendre functions. Note that this function may also be defined in a slightly different way, omitting the \((-1)^m\) factor, see for example [Wil99], eq. (6.20).

The complex conjugate of \(Y_n^m\) is given by negating the degree \(m\) as

(4.9)\[Y_n^m(\theta,\phi)^* = Y_n^{-m}(\theta,\phi).\]

For a spherical secondary source distribution with a radius of \(R_0\) the sound field can be calculated by a convolution along the surface. The driving function is then given by a simple division after [Ahr12], eq. (3.21) [2] as

(4.10)\[\begin{gathered} D_\text{spherical}(\theta_0,\phi_0,\w) = \\ \frac{1}{R_0^{\,2}} \sum_{n=0}^\infty \sum_{m=-n}^n \sqrt{\frac{2n+1}{4\pi}} \frac{\breve{S}_n^m(\theta_\text{s},\phi_\text{s},r_\text{s},\w)} {\breve{G}_n^0(\frac{\pi}{2},0,\w)} Y_n^m(\theta_0,\phi_0), \end{gathered}\]

where \(\breve{S}_n^m\) denote the spherical expansion coefficients of the source model, \(\theta_\text{s}\), \(\phi_\text{s}\), and \(r_\text{s}\) its directional dependency, and \(\breve{G}_n^0\) the spherical expansion coefficients of a secondary monopole source located at the north pole of the sphere \(\x_0 = (\frac{\pi}{2},0,R_0)\). For a point source this is given after [SS14], eq. (25) as

(4.11)\[\breve{G}_n^0(\tfrac{\pi}{2},0,\w) = -\i\wc \sqrt{\frac{2n+1}{4\pi}} \hankel{2}{n}{\wc R_0},\]

where \(\hankel{2}{n}{}\) describes the spherical Hankel function of \(n\)-th order and second kind.

4.2. Circular Geometries

The following functions build a basis in \(\mathbb{R}^2\) for a circular secondary source distribution, compare [Wil99]

(4.12)\[\Phi_m(\phi) = \e{\i m\phi}.\]

The complex conjugate of \(\Phi_m\) is given by negating the degree \(m\) as

(4.13)\[\Phi_m(\phi)^* = \Phi_{-m}(\phi).\]

For a circular secondary source distribution with a radius of \(R_0\) the driving function can be calculated by a convolution along the surface of the circle as explicitly shown by [AS09a] and is then given as

(4.14)\[D_\text{circular}(\phi_0,\w) = \frac{1}{2\pi R_0} \sum_{m=-\infty}^\infty \frac{\breve{S}_m(\phi_\text{s},r_\text{s},\w)} {\breve{G}_m(0,\w)} \, \Phi_m(\phi_0),\]

where \(\breve{S}_m\) denotes the circular expansion coefficients for the source model, \(\phi_\text{s}\), and \(r_\text{s}\) its directional dependency, and \(\breve{G}_m\) the circular expansion coefficients for a secondary monopole source. For a line source located at \(\x_0 = (0,R_0)\) this is given as

(4.15)\[\breve{G}_m(0,\w) = -\frac{\i}{4} \Hankel{2}{m}{\wc R_0},\]

where \(\Hankel{2}{m}{}\) describes the Hankel function of \(m\)-th order and second kind.

4.3. Planar Geometries

The basis functions for a planar secondary source distribution located on the \(xz\)-plane in \(\mathbb{R}^3\) are given as

(4.16)\[\Lambda(k_x,k_z,x,z) = \e{-\i(k_x x + k_z z)},\]

where \(k_x\), \(k_z\) are entries in the wave vector \(\k\) with \(k^2 = (\wc )^2\). The complex conjugate is given by negating \(k_x\) and \(k_z\) as

(4.17)\[\Lambda(k_x,k_z,x,z)^* = \Lambda(-k_x,-k_z,x,z).\]

For an infinitely long secondary source distribution located on the \(xz\)-plane the driving function can be calculated by a two-dimensional convolution along the plane after [Ahr12], eq. (3.65) as

(4.18)\[D_\text{planar}(x_0,y_0,\w) = \frac{1}{4{\pi}^2} \iint_{-\infty}^\infty \frac{\breve{S}(k_x,y_\text{s},k_z,\w)}{\breve{G}(k_x,0,k_z,\w)} \Lambda(k_x,x_0,k_z,z_0) \d k_x \d k_z,\]

where \(\breve{S}\) denotes the planar expansion coefficients for the source model, \(y_\text{s}\) its positional dependency, and \(\breve{G}\) the planar expansion coefficients of a secondary point source after [SS14], eq. (49) with

(4.19)\[\breve{G}(k_x,0,k_z,\w) = -\frac{\i}{2} \frac{1}{\sqrt{(\wc )^2-k_x^2-k_z^2}},\]

for \((\wc )^2 > (k_x^2+k_z^2)\).

For the planar and the following linear geometries the Fredholm equation is solved for a non compact space \(V\), which leads to an infinite and non-denumerable number of basis functions as opposed to the denumerable case for compact spaces [SS14].

4.4. Linear Geometries

The basis functions for a linear secondary source distribution located on the \(x\)-axis are given as

(4.20)\[\chi(k_x,x) = \e{-\i k_x x}.\]

The complex conjugate is given by negating \(k_x\) as

(4.21)\[\chi(k_x,x)^* = \chi(-k_x,x).\]

For an infinitely long secondary source distribution located on the \(x\)-axis the driving function for \({\mathbb{R}}^2\) can be calculated by a convolution along this axis after [Ahr12], eq. (3.73) as

(4.22)\[D_\text{linear}(x_0,\w) = \frac{1}{2\pi} \int_{-\infty}^\infty \frac{\breve{S}(k_x,y_\text{s},\w)}{\breve{G}(k_x,0,\w)} \chi(k_x,x_0) \d k_x,\]

where \(\breve{S}\) denotes the linear expansion coefficients for the source model, \(y_\text{s}\), \(z_\text{s}\) its positional dependency, and \(\breve{G}\) the linear expansion coefficients of a secondary line source with

(4.23)\[\breve{G}(k_x,0,\w) = -\frac{\i}{2} \frac{1}{\sqrt{(\wc )^2-k_x^2}},\]

for \(0<|k_x|<|\wc |\,\).

[1]	Note that \(\sin\theta\) is used here instead of \(\cos\theta\) due to the use of another coordinate system, compare Figure 2.1 from [GD04] and Fig. 2.1.

[2]	Note the \(\frac{1}{2\pi}\) term is wrong in [Ahr12], eq. (3.21) and eq. (5.7) and omitted here, compare the errata and [SS14], eq. (24).