Apply Near-Field Acoustic Holography to Identify the Noise Source of Pass-by Vehicles

Abstract

When using Near-field Acoustic Holography (NAH) to identify the noise source of a pass-by vehicle in a test—room, the hologram aperture must be at least as large as the source aperture, requiring a large element array. The reconstruction of NAH is an ill-posed inversion problem that requires a regularization procedure. The commonly used Tikhonov regularization procedures require a significant amount of computing time for a large hologram array. In this work, a fast and robust regularization procedure is developed for NAH on the basis of a statistical energy constraint equation (SECE) that links the hologram and the reconstruction sound pressures. This procedure is able to identify the optimal cutoff wave number for an existing exponential filter in a single measurement event without a prior knowledge of the noise. It is tested via numerical simulation for an exponential filter function in an NAH at various sound frequencies, hologram distances and signal-to-noise ratios (SNR). The SECE procedure is applied to identify the noise source on the right side of a vehicle in a semi-anechoic chamber. The results are compared with those obtained with the Far-field filter, generalized cross validation (GCV), L-curve and the Morozov discrepancy principle (MDP) methods.

F2012-J05-021

Appendix

Using Eqs. (10), (11) and (12), the correlation between the hologram and measured pressure $E\left[ {\hat{p}_{h} (x,y)\hat{p}_{h}^{*} (x^{\prime},y^{\prime})} \right]$ can be derived as

$$ \begin{gathered} E\left[ {\hat{p}_{h} (x,y)\hat{p}_{h}^{*} (x^{\prime},y^{\prime})} \right] = E[\{ (1 + \varepsilon_{a} (x,y))e^{{i\varepsilon_{\phi } (x,y)}} p_{h} (x,y)\} \{ (1 + \varepsilon_{a} (x^{\prime},y^{\prime}))e^{{ - i\varepsilon_{\phi } (x^{\prime},y^{\prime})}} p_{h}^{*} (x^{\prime},y^{\prime})\} ] \hfill \\ = E[e^{{i\varepsilon_{\phi } (x,y)}} e^{{ - i\varepsilon_{\phi } (x^{\prime},y^{\prime})}} p_{h} (x,y)p_{h}^{*} (x^{\prime},y^{\prime})] + E[\varepsilon_{a} (x,y)\varepsilon_{a} (x^{\prime},y^{\prime})e^{{i\varepsilon_{\phi } (x,y)}} e^{{ - i\varepsilon_{\phi } (x^{\prime},y^{\prime})}} p_{h} (x,y)p_{h}^{*} (x^{\prime},y^{\prime})] \hfill \\ = E\left[ {e^{{i\varepsilon_{\phi } }} } \right]E\left[ {e^{{ - i\varepsilon_{\phi } }} } \right]p_{h} (x,y)p_{h}^{*} (x^{\prime},y^{\prime}) + (1 - E\left[ {e^{{i\varepsilon_{\phi } }} } \right]E\left[ {e^{{ - i\varepsilon_{\phi } }} } \right])p_{h} (x,y)p_{h}^{*} (x^{\prime},y^{\prime})\delta (x - x^{\prime})\delta (y - y^{\prime}) \hfill \\ \;\;+ E[\varepsilon_{a}^{2} ]p_{h} (x,y)p_{h}^{*} (x^{\prime},y^{\prime})\delta (x - x^{\prime})\delta (y - y^{\prime}) \hfill \\ = e^{{ - \sigma_{\phi }^{2} }} p_{h} (x,y)p_{h}^{*} (x^{\prime},y^{\prime}) + (1 + \sigma_{a}^{2} - e^{{ - \sigma_{\phi }^{2} }} )p_{h} (x,y)p_{h}^{*} (x^{\prime},y^{\prime})\delta (x - x^{\prime})\delta (y - y^{\prime}). \hfill \\ \end{gathered} $$

(A1)

Equations (2) and (3) show that the angular spectrum $P(k_{x} ,k_{y} )$ and spatial sound pressure $p(x,y)$ form a Fourier Transform pair. Using Eq. (13), the mean value of the measured hologram angular spectrum $\hat{P}_{h} (k_{x} ,k_{y} )$.

$$ \begin{gathered} E[(\hat{P}_{h} (k_{x} ,k_{y} )] = E[\int_{\infty }^{\infty } {dx\int_{ - \infty }^{\infty } {dy\hat{p}_{h} (x,y)e^{{ - i(k_{x} x + k_{y} y)}} } } ] \hfill \\ = \int_{\infty }^{\infty } {dx\int_{ - \infty }^{\infty } {dyE[\hat{p}_{h} (x,y)]e^{{ - i(k_{x} x + k_{y} y)}} } } = e^{{ - \sigma_{\phi }^{2} /2}} P_{h} (k_{x} ,k_{y} ). \hfill \\ \end{gathered} $$

(A2)

The bias error of the measured hologram angular spectrum $\hat{P}_{h} (k_{x} ,k_{y} )$ is

$$ b[\hat{P}_{h} (k_{x} ,k_{y} )] = E\left[ {\hat{P}_{h} (k_{x} ,k_{y} )} \right] - P_{h} (k_{x} ,k_{y} ) = (e^{{ - \sigma_{\phi }^{2} /2}} - 1)P_{h} (k_{x} ,k_{y} ). $$

(A3)

Using Eq. (A1), the mean value of $\left| {\hat{P}_{h} (k_{x} ,k_{y} )} \right|^{2}$ can be derived as

$$ \begin{gathered} E\left[ {\left| {\hat{P}_{h} (k_{x} ,k_{y} )} \right|^{2} } \right] = E[\left| {\int_{\infty }^{\infty } {dx\int_{ - \infty }^{\infty } {dy\hat{p}_{h} (x,y)e^{{ - i(k_{x} x + k_{y} y)}} } } } \right|^{2} ] \hfill \\ = E[(\int_{ - \infty }^{\infty } {dx\int_{ - \infty }^{\infty } {dy\hat{p}_{h} (x,y)e^{{ - i(k_{x} x + k_{y} y)}} } } )(\int_{ - \infty }^{\infty } {dx\int_{ - \infty }^{\infty } {dy\hat{p}_{h}^{*} (x,y)e^{{i(k_{x} x + k_{y} y)}} } } )] \hfill \\ = \int_{ - \infty }^{\infty } {dx\int_{ - \infty }^{\infty } {dy\int_{ - \infty }^{\infty } {dx^{\prime}\int_{ - \infty }^{\infty } {dy^{\prime}} } E[\hat{p}_{h} (x,y)\hat{p}_{h}^{*} (x^{\prime},y^{\prime})]e^{{ - i(k_{x} x + k_{y} y - k_{x} x^{\prime} - k_{y} y^{\prime})}} } } \hfill \\ = \int_{ - \infty }^{\infty } {dx\int_{ - \infty }^{\infty } {dy\int_{ - \infty }^{\infty } {dx^{\prime}\int_{ - \infty }^{\infty } {dy^{\prime}} } \{ E\left[ {e^{{i\varepsilon_{\phi } }} } \right]E\left[ {e^{{ - i\varepsilon_{\phi } }} } \right]p_{h} (x,y)p_{h}^{*} (x^{\prime},y^{\prime}) \,+\, } } \hfill \\ (1 + E[\varepsilon_{a}^{2} ] - E\left[ {e^{{i\varepsilon_{\phi } }} } \right]E\left[ {e^{{ - i\varepsilon_{\phi } }} } \right])p_{h} (x,y)p_{h}^{*} (x^{\prime},y^{\prime})\delta (x - x^{\prime})\delta (y - y^{\prime})\} e^{{ - i(k_{x} x + k_{y} y - k_{x} x^{\prime} - k_{y} y^{\prime})}} . \hfill \\ \end{gathered} $$

(A4)

Using Parseval’s theorem, $E\left[ {\left| {\hat{P}_{h} (k_{x} ,k_{y} )} \right|^{2} } \right]$ is finally derived as

$$ \begin{gathered} E\left[ {\left| {\hat{P}_{h} (k_{x} ,k_{y} )} \right|^{2} } \right] = E\left[ {e^{{i\varepsilon_{\phi } }} } \right]E\left[ {e^{{ - i\varepsilon_{\phi } }} } \right]\left| {P_{h} (k_{x} ,k_{y} )} \right|^{2} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + (1 + E[\varepsilon_{a}^{2} ] - E\left[ {e^{{i\varepsilon_{\phi } }} } \right]E\left[ {e^{{ - i\varepsilon_{\phi } }} } \right])\int_{ - \infty }^{\infty } {dk_{x} \int_{ - \infty }^{\infty } {dk_{y} } } \left| {P_{h} (k_{x} ,k_{y} )} \right|^{2} \hfill \\ = e^{{ - \sigma_{\phi }^{2} }} \left| {P_{h} (k_{x} ,k_{y} )} \right|^{2} + (1 + \sigma_{a}^{2} - e^{{ - \sigma_{\phi }^{2} }} )\int_{ - \infty }^{\infty } {dk_{x} \int_{ - \infty }^{\infty } {dk_{y} } } \left| {P_{h} (k_{x} ,k_{y} )} \right|^{2} . \hfill \\ \end{gathered} $$

(A5)

Using Eqs. (A2) and (A5), the variance of $\hat{P}_{h} (k_{x} ,k_{y} )$ is

$$ \begin{gathered} Var[\hat{P}_{h} (k_{x} ,k_{y} )] = E\left[ {\left| {\hat{P}_{h} (k_{x} ,k_{y} )} \right|^{2} } \right] - \left| {E\left[ {\hat{P}_{h} (k_{x} ,k_{y} )} \right]} \right|^{2} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern 1pt} {\kern 1pt} = (1 + \sigma_{a}^{2} - e^{{ - \sigma_{\phi }^{2} }} )\int_{ - \infty }^{\infty } {dk_{x} \int_{ - \infty }^{\infty } {dk_{y} } } \left| {P_{h} (k_{x} ,k_{y} )} \right|^{2} . \hfill \\ \end{gathered} $$

(A6)

Next, the mean value and variance of the reconstruction angular spectrum $\hat{P}_{z} (k_{x} ,k_{y} )$ are derived. Using Eq. (1) and (A2), the mean value of $\hat{P}_{z} (k_{x} ,k_{y} )$ is:

$$ \begin{gathered} E[\hat{P}_{z} (k_{x} ,k_{y} )] = E[\hat{P}_{h} (k_{x} ,k_{y} )G(k_{x} ,k_{y} ,z - z_{h} )] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = e^{{ - \sigma_{\phi }^{2} /2}} G(k_{x} ,k_{y} ,z - z_{h} )P_{h} (k_{x} ,k_{y} ). \hfill \\ \end{gathered} $$

(A7)

The bias error of $\hat{P}_{z} (k_{x} ,k_{y} )$ is:

$$ b[\hat{P}_{z} (k_{x} ,k_{y} )] = E\left[ {\hat{P}_{z} (k_{x} ,k_{y} )} \right] - P_{z} (k_{x} ,k_{y} ) = (e^{{ - \sigma_{\phi }^{2} /2}} - 1)G(k_{x} ,k_{y} ,z - z_{h} )P_{h} (k_{x} ,k_{y} ). $$

(A8)

Using Eqs. (A5) and (A6), the mean value of $\left| {\hat{P}_{z} (k_{x} ,k_{y} )} \right|^{2}$ is

$$\begin{gathered} E\left[ {\left| {\hat{P}_{z} (k_{x} ,k_{y} )} \right|^{2} } \right] = E[\left| {\hat{P}_{h} (k_{x} ,k_{y} )G(k_{x} ,k_{y} ,z - z_{h} )} \right|^{2} ]\; = E[\left| {\hat{P}_{h} (k_{x} ,k_{y} )} \right|^{2} \left| {G(k_{x} ,k_{y} ,z - z_{h} )} \right|^{2} ] \hfill \\ \; = \left| {G(k_{x} ,k_{y} ,z - z_{h} )} \right|^{2} \{ e^{{ - \sigma_{\phi }^{2} }} \left| {P_{h} (k_{x} ,k_{y} )} \right|^{2} \,+\, (1 + \sigma_{a}^{2} - e^{{ - \sigma_{\phi }^{2} }} )\int_{ - \infty }^{\infty } {dk_{x} \int_{ - \infty }^{\infty } {dk_{y} } } \left| {P_{h} (k_{x} ,k_{y} )} \right|^{2} \} \hfill \\ \; = e^{{ - \sigma_{\phi }^{2} }} \left| {P_{z} (k_{x} ,k_{y} )} \right|^{2} \,+ \,\left| {G(k_{x} ,k_{y} ,z - z_{h} )} \right|^{2} Var[\hat{P}_{h} (k_{x} ,k_{y} )]. \hfill \\ \end{gathered} $$

(A9)

Using Eq (A7) and (A9), the variance of $\hat{P}_{z} (k_{x} ,k_{y} )$ is

$$ \begin{gathered} Var[\hat{P}_{z} (k_{x} ,k_{y} )] = E\left[ {\left| {\hat{P}_{z} (k_{x} ,k_{y} )} \right|^{2} } \right] - \left| {E\left[ {\hat{P}_{z} (k_{x} ,k_{y} )} \right]} \right|^{2} \hfill \\ {\kern 1pt} = \left| {G(k_{x} ,k_{y} ,z - z_{h} )} \right|^{2} Var[\hat{P}_{h} (k_{x} ,k_{y} )] \hfill \\ = \left| {G(k_{x} ,k_{y} ,z - z_{h} )} \right|^{2} (1 + \sigma_{a}^{2} - e^{{ - \sigma_{\phi }^{2} }} )\int_{ - \infty }^{\infty } {dk_{x} \int_{ - \infty }^{\infty } {dk_{y} } } \left| {P_{h} (k_{x} ,k_{y} )} \right|^{2} . \hfill \\ \end{gathered} $$

(A10)