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)  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)  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 $$0<|k_x|<|\wc |\,$$.

  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.
  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).