Acoustic Parameter Estimation

Abstract

This chapter introduces methods for the estimation of two important acoustic parameters using spherical microphone arrays: the direction of arrival of sound from a localized sound source, and the signal-to-diffuse energy ratio at a particular position in a sound field. Later in the book, it will be seen that these quantities can be used for signal enhancement purposes.

Portions of Sect. 5.1.5 and the Appendix were first published in [13], and are reproduced here with the author’s permission.

Appendix: Relationship Between the Zero-Order Eigenbeam and the Omnidirectional Reference Microphone Signal

Property 5.1

Let $P_{lm}(k)$ denote the SHT, as defined in (3.6), of the spatial domain sound pressure $P(k,\mathbf {r})$, where $\mathbf {r}$ denotes the position (in spherical coordinates) with respect to the centre of a spherical microphone array with mode strength $b_l(k)$. Let $P_{\mathcal {M}_{\text {ref}}}(k)$ denote the sound pressure which would be measured using an omnidirectional microphone $\mathcal {M}_{\text {ref}}$ at a position corresponding to the centre of the sphere, i.e., at the origin of the spherical coordinate system; $P_{\mathcal {M}_{\text {ref}}}(k)$ is then related to the zero-order eigenbeam $P_{00}(k)$ via the relationship⁶

$$\begin{aligned} P_{\mathcal {M}_{\text {ref}}}(k) = \frac{P_{00}(k)}{\sqrt{4 \pi } \, b_0(k)}. \end{aligned}$$

(5.74)

Proof

We assume, without loss of generality,⁷ that the sound field is composed of a single unit amplitude spherical wave incident from a point source at a position $\mathbf {r}_{\text {s}} = (r_{\text {s}}, \varOmega _{\text {s}})$.

In the absence of the sphere, the sound pressure measured at the origin of the spherical coordinate system due to a single spherical wave incident from a point source at a position $\mathbf {r}_{\text {s}} = (r_{\text {s}}, \varOmega _{\text {s}})$ is given by (4.7), i.e.,

$$\begin{aligned} P_{\mathcal {M}_{\text {ref}}}(k)&= \frac{e^{-ik\left| \left| \mathbf {r}_{\text {s}}\right| \right| }}{4 \pi \left| \left| r_{\text {s}}\right| \right| }\end{aligned}$$

(5.75a)

$$\begin{aligned}&= \frac{e^{-ikr_{\text {s}}}}{4 \pi r_{\text {s}}}. \end{aligned}$$

(5.75b)

In the spatial domain, the sound pressure $P(k,\mathbf {r})$ at a position $\mathbf {r}$ due to the spherical wave is given by (4.10), and can be written using (2.23) as

$$\begin{aligned} P(k,\mathbf {r}) = -i k \sum _{l=0}^\infty i^{-l} b_l(k) h_l^{(2)}(kr_{\text {s}}) \sum _{m=-l}^l Y_{lm}^*(\varOmega _\text {s}) Y_{lm}(\varOmega ), \end{aligned}$$

(5.76)

where $h_l^{(2)}$ is the spherical Hankel function of the second kind and of order l. From the definition of the SHT (3.7), $P_{00}(k)$ is given by

$$\begin{aligned} P_{00}(k) = \int _{\varOmega \in \mathcal {S}^2} P(k,\mathbf {r}) Y_{00}^*(\varOmega ) \text {d}\varOmega . \end{aligned}$$

(5.77)

By substituting (5.76) into (5.77), we find

$$\begin{aligned} P_{00}(k) = \int _{\varOmega \in \mathcal {S}^2} -i k \sum _{l=0}^\infty i^{-l} b_l(k) h_l^{(2)}(kr_{\text {s}}) \sum _{m=-l}^l Y_{lm}^*(\varOmega _\text {s}) Y_{lm}(\varOmega ) Y_{00}^*(\varOmega ) \text {d}\varOmega . \end{aligned}$$

(5.78)

Using the orthonormality of the spherical harmonics (2.18) and the fact that $Y_{00}(\cdot ) = 1/\sqrt{4 \pi }$, we can simplify (5.78) to

$$\begin{aligned} P_{00}(k)&= -i k b_0(k) h_0^{(2)}(kr_{\text {s}}) Y_{00}^*(\varOmega _\text {s})\end{aligned}$$

(5.79a)

$$\begin{aligned}&= -\frac{i k}{\sqrt{4 \pi }} b_0(k) h_0^{(2)}(kr_{\text {s}}). \end{aligned}$$

(5.79b)

Finally, using the fact that $h_0^{(2)}(x) = \frac{- e^{-i x}}{i x}$ [46, Eq. 6.62] and (5.75b), we can simplify (5.79b) to

$$\begin{aligned} P_{00}(k)&= \frac{i k}{\sqrt{4 \pi }} \, b_0(k) \, \frac{e^{-i kr_{\text {s}}}}{i kr_{\text {s}}}\end{aligned}$$

(5.80a)

$$\begin{aligned}&= \sqrt{4 \pi } \, b_0(k) \, \frac{e^{-i kr_{\text {s}}}}{4 \pi r_{\text {s}}}\end{aligned}$$

(5.80b)

$$\begin{aligned}&= \sqrt{4 \pi } \, b_0(k) \, P_{\mathcal {M}_{\text {ref}}}(k), \end{aligned}$$

(5.80c)

and therefore Property 5.1 holds.