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Part of the book series: Springer Topics in Signal Processing ((STSP,volume 9))

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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.

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Notes

  1. 1.

    The dependency on time is omitted for brevity. In practice, the signals acquired using a spherical microphone array are usually processed in the short-time Fourier transform domain, as explained in Sect. 3.1, where the discrete frequency index is denoted by \(\nu \).

  2. 2.

    If the real SHT is applied instead of the complex SHT, the complex spherical harmonics \(Y_{lm}\) used throughout this chapter should be replaced with the real spherical harmonics \(R_{lm}\), as defined in Sect. 3.3.

  3. 3.

    In practice, since the processing is performed in the short-time Fourier transform domain, this integral is approximated with a sum over discrete frequency indices \(\nu \).

  4. 4.

    As in Sect. 5.1, the dependency on time is omitted for brevity.

  5. 5.

    As noted earlier in the chapter, if the real SHT is applied instead of the complex SHT, the complex spherical harmonics \(Y_{lm}\) used throughout this chapter should be replaced with the real spherical harmonics \(R_{lm}\), as defined in Sect. 3.3.

  6. 6.

    It should be noted that this relationship is dependent upon the chosen mode strength definition (see Sect. 3.4.2). If a \(4 \pi \) factor is included in \(b_l(k)\), as in [31], the relationship becomes \(P_{\mathcal {M}_{\text {ref}}}(k) = \sqrt{4 \pi } \frac{P_{00}(k)}{b_0(k)}\).

  7. 7.

    The operations involved in the proof are linear, and the proof therefore holds for any number of spherical waves.

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Correspondence to Daniel P. Jarrett .

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

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 relationshipFootnote 6

$$\begin{aligned} P_{\mathcal {M}_{\text {ref}}}(k) = \frac{P_{00}(k)}{\sqrt{4 \pi } \, b_0(k)}. \end{aligned}$$
(5.74)

Proof

We assume, without loss of generality,Footnote 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.

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Jarrett, D.P., Habets, E.A.P., Naylor, P.A. (2017). Acoustic Parameter Estimation. In: Theory and Applications of Spherical Microphone Array Processing. Springer Topics in Signal Processing, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-42211-4_5

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