CN101251412A - Method for rebuilding circulation calm sound source by overlapping spherical wave - Google Patents

Abstract

The invention relates to a method of reconstructing a circular stable sound source to be measured through spherical wave superposition in the signal reconstruction technical field. Through setting the sound source to be measured, a microphone array and a reference source microphone, the invention synchronously records the acquired reference signal and holographic measuring point sound pressure, obtains a reference phase from the reference signal, obtains the relative phase relation of the holographic measuring point source pressure acquired by the microphone array through the phase relation between the reference signal and the holographic measuring point source pressure, and finally obtains the self-spectrum correlated density matrix of the sound source signal to be measured through the spherical wave basis function reconstruction computing. Through the reconstruction of the sound source signal to be measured on the holographic measurement surface acquired by the microphone array, the method can analyze the sound source to be measure with random shape and obtain the three-dimensional spectrum correlated density distribution of the sound source to be measured. Compared with the traditional method of reconstructing the stable sound source to be measured through spherical wave superposition, the method can be applicable to the circular stable sound source to be measured with breakthrough on the requirements of the shape of the sound source to be measured.

Description

Adopt the spherical wave stack to rebuild the method for cyclo-stationary sound source

Technical field

What the present invention relates to is a kind of method of signal reconstruction technical field, particularly a kind of method that adopts the spherical wave method of superposition to rebuild the cyclo-stationary sound source.

Background technology

In order to control noise effectively, before noise reduction measure is implemented, must at first carry out the noise source diagnosis, determine each position, overriding noise seedbed and characteristic thereof.Along with the development of modern signal processing technology ground, spectral analysis technology, relevant and partial coherence analysis technology, sound intensity analytical technology and sound near-field holography technology etc. have obtained developing by leaps and bounds.

The cyclo-stationary signal is the special non-stationary signal of a class because the cyclic stationary of self uniqueness, make single acquisition to record have the cycle ergodic property, increased this class non-stationary signal method of analysis.The cyclo-stationary signal has crucial realistic meaning in engineering is used, for example rotating machinery is because the physical arrangement of symmetry or near symmetrical and periodic working motion pattern, its sound field has the obvious periodic time varying characteristic, and sound-source signal has cyclostationarity.

Find by prior art documents, Wang Z. and Wu S.F. are at " The Journal ofthe Acoustical Society of America " (1997,102 (4): write articles " Helmholtz equation least-squares method for reconstructing theacoustic pressure field " (" Acoustical Society of America's magazine ": the research that Helmholtz equation least square method is used to rebuild sonic pressure field) 2020-2032), this article proposes to utilize the particular solution of Helmholtz equation under spherical co-ordinate---the sonic pressure field of gang's spherical wave basis function match reality, and determines that with least square method optimum spherical wave basis function launches item number and the coefficient of representing spherical wave basis function shared weight in sound field.This method can be analyzed the radiation characteristic of noise source effectively, the prediction exterior acoustic radiation, and have counting yield height, measuring point few, be easy to the advantage that engineering survey and experiment are carried out.Afterwards, round this technology, launched many researchs, and comprised applied environment, algorithm improves, and the spherical wave basis function is chosen, and precision improves, measuring system etc.People such as Wu S.F. have also delivered some similar documents, have studied with the influence factor of Helmholtz equation least square method holographic reconstruction sound source surficial acoustic field and improve one's methods.Above-mentioned all working is based on all that steady sound field carries out, and therefore, is necessary to propose new technology, is used for the analysis of non-stationary sound source.Yet, for general non-stationary sound source, become when the statistical property parameter of sound-source signal is, thereby also replace ensemble average with regard to unrenewable time average, make data acquisition very difficult, be difficult to analyze the characteristic of sound source.

Chinese patent publication number in the prior art document: 1487500, name is called: " adopting the method for near field acoustic holography technology identification non-stationary sound source ", this technology was carried out analysis for the cyclo-stationary sound source on plane, but can not be applied to random appearance.In further retrieving, find as yet and the identical or similar bibliographical information of theme of the present invention.

Summary of the invention

The objective of the invention is to overcome the deficiencies in the prior art, a kind of method that adopts the spherical wave method of superposition to rebuild the cyclo-stationary sound source is provided, the present invention rebuilds by the near-field holography of cyclo-stationary sound source, the characteristic and the three-dimensional sound field that are picked out the cyclo-stationary sound source by the sound-source signal to be measured that measures on the holographic measurement face distribute, and are applicable to the cyclo-stationary sound source of random appearance.

The present invention is achieved through the following technical solutions, occasion in the steady sound source of complex loops, adopt the cyclo-stationary statistic to substitute the spectrum component that obtains behind traditional Fourier transform, select the physical quantity of spectral density function, proposed to be used for the near-field holography technology of the steady sound source of analysis cycle as sound field rebuilding.At first arrange with reference to microphone, extract and analyze reference signal; Arrange that again microphone array carries out scanning survey to the holographic measurement face, gather holographic measurement face data; Then, utilize cyclo-stationary near-field holography technology, rebuild the distributed in three dimensions that obtains sound field based on the spherical wave stack.

The present invention specifically may further comprise the steps:

The first step, the sound source to be measured of a cyclo-stationary sounding is set, sound source volume to be measured is V, and sound source to be measured surface is S _s, the holographic measurement face is S _h, during sound source volume V to be measured takes up space arbitrarily the length ratio of both direction smaller or equal to 1: 4, holographic measurement face S _hBe sound source to be measured surface S _sA near arbitrary face.Holographic measurement face S _hWith sound source to be measured surface S _sConformal, but for ease of measuring S _hAlso can be S _sOutside one non-conformal.

Second step, layout microphone array are at S _sOutside holographic measurement face S _hThe equal mode for cloth of last employing is provided with microphone array and covers whole sound source to be measured, so that gather the high-frequency information of sound source to be measured; Arrange near sound source to be measured that in addition the fixing reference source microphone of a relative position is in order to gather reference signal; The contained microphone of microphone array adds up to L.

Reference signal and holophonic field signal that the 3rd step, synchronous recording reference source microphone and microphone array collect, and determine the locus of each microphone in the microphone array to be stored in the memory device in the lump with the space orientation instrument.

The 4th step, obtain fixed phase from reference signal, utilize the phase relation between reference signal and the holographic measuring point acoustic pressure, the relative phase that can obtain the holographic measuring point acoustic pressure that microphone array collects concerns.Specific as follows:

At first analyze holographic measuring point acoustic pressure and reference signal, by the spectral density function of reference signal:

S _rr ^α(f)，

Wherein f is the frequency of sound source characteristic to be measured, and α is a cycle frequency, and subscript r represents reference signal, and rr represents the spectral density function of reference signal, promptly from spectral density function;

Calculate the relevant density matrix of cross-spectrum of reference signal and holographic measuring point acoustic pressure then:

(S _Rp ^α(r _h, f)) _{L * 1}And (S _Pr ^α(r _h, f)) _{L * 1},

Wherein: r _hThe locus of each microphone on the expression microphone array, L is the contained microphone sum of microphone array, subscript r represents reference signal, subscript p represents holographic measuring point acoustic pressure, rp represents the relevant density function of cross-spectrum of reference signal spectral component (f+ α/2) and holographic measuring point spectra of sound pressure component (f-α/2), and pr represents the relevant density function of cross-spectrum of holographic measuring point spectra of sound pressure component (f+ α/2) and reference signal spectral component (f-α/2);

Spectral density function S in conjunction with reference signal _Rr ^α(f), obtain the relevant density matrix of spectrum certainly of holographic measuring point acoustic pressure:

(S _pp ^α(r _h，f)) _L×1，

Wherein: subscript pp represents the spectral density function of holographic measuring point acoustic pressure;

Obtain formula (1) at last:

${\left(S_{\mathrm{pp}}^{\mathrm{\α}}\left(f\right)\right)}_{L\×1}={\left(S_{\mathrm{pr}}^{\mathrm{\α}}\left(f\right)\right)}_{L\×1}\·{\left(S_{\mathrm{rp}}^{\mathrm{\α}}\left(f\right)\right)}_{L\×1}/S_{\mathrm{rr}}^{\mathrm{\α}}\left(f\right)---\left(1\right)$

Wherein " " represents the dot product of vector, and this formula has reflected the time domain phase relation of reference signal and holographic measuring point acoustic pressure, is used to carry out phase-locking.

The 5th goes on foot, is rebuild by the spherical wave method of superposition and calculate the relevant density matrix of the spectrum certainly that obtains sound-source signal to be measured.

Set sound source to be measured surface S _sCertainly the relevant density matrix of the spectrum of the sound-source signal to be measured that a last N node sends is:

(S _pp ^α(r _s，f)) _N×1，

Wherein: r _sRepresent sound source to be measured surface S _sLast N node locus separately, N is a natural constant;

The relevant density matrix of cross-spectrum of setting reference signal and sound-source signal to be measured is:

(S _Rp ^α(r _s, f)) _{N * 1}And (S _Pr ^α(r _s, f)) _{N * 1},

Since the loss of the space phase of each holographic measuring point acoustic pressure relation on the holographic measurement face, S _Pp ^α(r _s, f) can't be directly by S _Pp ^α(r _h, f) rebuild.But, S _Rp ^α(r _h, f) and S _Pr ^α(r _h, f) still keeping the space phase relation, can be directly used in reconstruction.Because sound source to be measured is the cyclo-stationary sounding, so the weight coefficient of spherical wave basis function also is that cyclo-stationary changes.

Setting r (f) is the spectrum component of reference signal, and p (f) is the spectrum component of holographic measuring point acoustic pressure, and q (f) is the spectrum component of spherical wave basis function weights coefficient, then

S _Rp ^α(r _h, f) be r (f+ α/2) and p ^*The function of (f-α/2);

S _Pr ^α(r _h, f) be p (f+ α/2) and r ^*The function of (f-α/2);

Wherein: S _Rp ^α(r _h, f) and S _Pr ^α(r _h, f) be the relevant density matrix of cross-spectrum of reference signal and holographic measuring point acoustic pressure;

Q _Rp ^α(f) be r (f+ α/2) and q ^*The function of (f-α/2);

Q _Pr ^α(f) be q (f+ α/2) and r ^*The function of (f-α/2);

Wherein: subscript * is a conjugate transpose, Q _Rp ^α(f) and Q _Pr ^α(f) be the relevant density of cross-spectrum between spherical wave basis function weights coefficient and the reference signal, S _Rp ^α(r _s, f) should be by S _Rp ^α(r _h, f) go up reconstruction and obtain in frequency (f-α/2), and S _Pr ^α(r _s, f) should be by S _Pr ^α(r _h, f) go up reconstruction and obtain in frequency (f+ α/2).

Therefore, can obtain following with spherical wave Superposition Formula (2)-(3) of the relevant density of cross-spectrum as variable:

$S_{\mathrm{pr}}^{\mathrm{\α}}(r,f)=\mathrm{\ρc}\underset{j=1}{\overset{J}{\mathrm{\Σ}}}{Q_{\mathrm{pr}}^{\mathrm{\α}}}_j\left(f\right){\mathrm{\Ψ}}_j(r,f+\mathrm{\α}/2)---\left(2\right)$

${S_{\mathrm{rp}}^{\mathrm{\α}}}^{*}(r,f)=\mathrm{\ρc}\underset{j=1}{\overset{J}{\mathrm{\Σ}}}{Q_{\mathrm{rp}}^{\mathrm{\α}}}_j^{*}\left(f\right){\mathrm{\Ψ}}_j(r,f-\mathrm{\α}/2)---\left(3\right)$

Wherein: ρ, c are respectively the medium average density and the velocity of sound; Ψ _jBe the spherical wave basis function, J is the expansion item number of spherical wave basis function, Q _{Pr j} ^αAnd Q _{Pr j} ^αBe the relevant density of cross-spectrum between corresponding spherical wave basis function weights coefficient and the reference signal.

The form of spherical wave basis function is expressed as follows with formula (4):

${\mathrm{\ψ}}_j(r,\mathrm{\θ},\mathrm{\φ},k)=\sqrt{\frac{(2n+1)(n-m)!}{4\mathrm{\π}(n+m)!}}{\×h}_n\left(\mathrm{kr}\right)\×P_n^m\left(\cos \mathrm{\θ}\right)\×\left\{\begin{array}{cc}\cos \mathrm{m\φ}& l=\mathrm{0,2,4},\·\·\·\\ \sin \mathrm{m\φ}& l=\mathrm{1,3,5},\·\·\·\end{array}\right.---\left(4\right)$

Wherein: j, m, n and l are respectively nonnegative integer, and the pass between them is j=n ²+ n+m+1, l=j-n ²h _n(kr) be ball Hankel function, k is a wave number; P _n ^m(cos θ) follows the Legendre function; J launches item number for the spherical wave basis function.Form with matrix can be expressed as formula (2) and (3):

$\mathrm{\ρc}{\left[\mathrm{\Ψ}(f+\mathrm{\α}/2)\right]}_{L\×J}{\left\{Q_{\mathrm{pr}}^{\mathrm{\α}}\left(f\right)\right\}}_{J\×1}={\left\{S_{\mathrm{pr}}^{\mathrm{\α}}\left(f\right)\right\}}_{L\×1}---\left(5\right)$

$\mathrm{\ρc}{\left[\mathrm{\Ψ}(f-\mathrm{\α}/2)\right]}_{L\×J}{\left\{{Q_{\mathrm{pr}}^{\mathrm{\α}}}^{*}\left(f\right)\right\}}_{J\×1}={\left\{{S_{\mathrm{rp}}^{\mathrm{\α}}}^{*}\left(f\right)\right\}}_{L\×1}---\left(6\right)$

Utilize the least square solution method, make to be shown below the sound source error minimum to be measured of rebuilding in the hope of coefficient with formula (5) and (6):

${\left\{Q_{\mathrm{pr}}^{\mathrm{\α}}\left(f\right)\right\}}_{J\×1}={\left(\mathrm{\ρc}\right)}^{-1}{\left({\left[{\mathrm{\Ψ}}_{+}\right]}_{L\×J}^H{\left[{\mathrm{\Ψ}}_{+}\right]}_{L\×J}\right)}^{-1}{\left[{\mathrm{\Ψ}}_{+}\right]}_{L\×J}^H{\left\{S_{\mathrm{pr}}^{\mathrm{\α}}\left(f\right)\right\}}_{L\×1}---\left(7\right)$

${\left\{{Q_{\mathrm{rp}}^{\mathrm{\α}}}^{*}\left(f\right)\right\}}_{J\×1}={\left(\mathrm{\ρc}\right)}^{-1}{\left({\left[{\mathrm{\Ψ}}_{-}\right]}_{L\×J}^H{\left[{\mathrm{\Ψ}}_{-}\right]}_{L\×J}\right)}^{-1}{\left[{\mathrm{\Ψ}}_{-}\right]}_{L\×J}^H{\left\{{S_{\mathrm{rp}}^{\mathrm{\α}}}^{*}\left(f\right)\right\}}_{L\×1}---\left(8\right)$

In the formula subscript "+" and "-" to represent analysis frequency respectively be (f+ α/2) and (f-α/2).Obtaining Q _Rp ^α(f) and Q _Pr ^α(f) afterwards, just can use formula (2) and (3) to rebuild sound field, obtain the S at sound source to be measured surface and r place, outside arbitrfary point _Pr ^α(r, f) and S _Rp ^α(r f), utilizes formula (1) at last, just can obtain the relevant Density Distribution of spectrum of sound source to be measured.

The present invention utilizes the periodicity of the uniqueness of cyclo-stationary sound source to be measured, on the basis of the near-field holography technology of and steady sound source to be measured theoretical at cyclo-stationary, cyclo-stationary near field acoustic holography technology has been proposed, pass through to rebuild to analyze the sound source to be measured of random appearance by the holographic measuring point acoustic pressure that collects, obtain the relevant Density Distribution of three-dimensional spectrum of sound source to be measured.Rebuild steady sound source method to be measured with traditional spherical wave stack and compare, the present invention can be applicable to cyclo-stationary sound source to be measured, has broken through the requirement to sound source profile to be measured simultaneously.

Embodiment

Below embodiments of the invention are elaborated: present embodiment has provided detailed embodiment and process being to implement under the prerequisite with the technical solution of the present invention, but protection scope of the present invention is not limited to following embodiment:

1, adopt a spherical loudspeaker sounding to form a cyclo-stationary sound source to be measured, arrange to be used to extract reference signal by a microphone near loudspeaker, the driving source of described loudspeaker is:

V＝Acos(2πf ₁t)*noise(t)，

Wherein: A=1, f1=300, noise is connected with coloured noise for band.

2, at first 20 microphones are arranged to linear microphone array, gather holographic measuring point acoustic pressure in the enterprising line scanning of holographic measurement face successively, finally form 20 * 20 holographic measurement face array; Utilize the reference source microphone to gather reference signal simultaneously; The sound-source signal to be measured of each whole 20 microphone passages of synchronous acquisition and reference signal and the locus of writing down microphone are stored in the computing machine simultaneously.

3, holographic measuring point acoustic pressure and the reference signal that collects by playback chosen the frequency f=120Hz and the cycle frequency α=600Hz that can reflect sound source characteristic to be measured.

4, calculate selected frequency f and compose relevant density matrix S certainly with the reference signal on the cycle frequency α _Rr ^α(f), the relevant density matrix S of cross-spectrum of reference signal and holographic measuring point acoustic pressure _Rp ^α(r _h, f) and S _Pr ^α(r _h, f), and utilize formula (1) to obtain the relevant density matrix S of spectrum certainly of holographic measuring point acoustic pressure _Pp ^α(r _h, f).

5, utilize formula (2) to (8) that sound field is rebuild, and then obtain sound source to be measured surface S _sGo up the relevant Density Distribution S of spectrum of sound-source signal to be measured _Pp ^α(r _s, f).

Can find by analysis, utilize the steady sound source to be measured of the described step analysis cycle of present embodiment, can undergo reconstruction by sound-source signal to be measured on the holographic measurement face and obtain the cyclo-stationary component information of whole three-dimensional sound source to be measured, thereby analyze the feature of the steady sound source to be measured of whole circulation all sidedly.Point out that in passing when the cycle frequency of circulation sound source was 0, above-mentioned steps was steady near field acoustic holography technology, also can rebuild the stationary components information that obtains whole three-dimensional sound field.

Claims (4)

1, a kind of method that adopts the spherical wave stack to rebuild steady sound source to be measured is characterized in that, may further comprise the steps:

The first step, the sound source to be measured of a cyclo-stationary sounding is set, sound source to be measured surface is S _s, the holographic measurement face is S _h, during sound source volume V to be measured takes up space arbitrarily the length ratio of both direction smaller or equal to 1: 4, holographic measurement face S _hBe sound source to be measured surface S _sA near arbitrary face;

Second step, arrange microphone array, near sound source to be measured, arrange one with respect to the changeless reference source microphone of sound source position to be measured in order to gather reference signal;

Reference signal and holographic measuring point acoustic pressure that the 3rd step, synchronous recording reference source microphone and microphone array collect, and determine the locus of each microphone in the microphone array to be stored in the memory device in the lump with the space orientation instrument;

The 4th step, obtain fixed phase from reference signal, utilize the phase relation between reference signal and the holographic measuring point acoustic pressure, the relative phase that can obtain the holographic measuring point acoustic pressure that microphone array collects concerns;

The 5th goes on foot, is rebuild by the spherical wave method of superposition and calculate the relevant density matrix of the spectrum certainly that obtains sound-source signal to be measured.

2, the method for steady sound source to be measured is rebuild in employing spherical wave stack according to claim 1, it is characterized in that the layout microphone array described in second step is at holographic measurement face S _hThe equal mode for cloth of last employing is provided with microphone array and covers sound source to be measured.

3, the method for steady sound source to be measured is rebuild in employing spherical wave stack according to claim 1, it is characterized in that, the relative phase relation of the holographic measuring point acoustic pressure described in the 4th step specifically comprises:

At first analyze holographic measuring point acoustic pressure and reference signal, by the spectral density function of reference signal:

S _rr ^α(f)，

Wherein f is the frequency of sound source characteristic to be measured, and α is a cycle frequency, and subscript r represents reference signal, and rr represents the spectral density function of reference signal, promptly from spectral density function;

Calculate the relevant density matrix of cross-spectrum of reference signal and holographic measuring point acoustic pressure then:

(S _Rp ^α(r _h, f)) _{L * 1}And (S _Pr ^α(r _h, f)) _{L * 1},

Wherein: r _hThe locus of each microphone on the expression microphone array, L is the contained microphone sum of microphone array, subscript r represents reference signal, subscript p represents holographic measuring point acoustic pressure, rp represents the relevant density function of cross-spectrum of reference signal spectral component (f+ α/2) and holographic measuring point spectra of sound pressure component (f-α/2), and pr represents the relevant density function of cross-spectrum of holographic measuring point spectra of sound pressure component (f+ α/2) and reference signal spectral component (f-α/2);

Spectral density function S in conjunction with reference signal _Rr ^α(f), obtain the relevant density matrix of spectrum certainly of holographic measuring point acoustic pressure:

(S _pp ^α(r _h，f)) _L×1，

Wherein: subscript pp represents the spectral density function of holographic measuring point acoustic pressure;

Obtain at last: ${\left(S_{\mathrm{pp}}^{\mathrm{\α}}\left(f\right)\right)}_{L\×1}={\left(S_{\mathrm{pr}}^{\mathrm{\α}}\left(f\right)\right)}_{L\×1}\·{\left(S_{\mathrm{rp}}^{\mathrm{\α}}\left(f\right)\right)}_{L\×1}/S_{\mathrm{rr}}^{\mathrm{\α}}\left(f\right)---\left(1\right)$

Wherein " " represents the dot product of vector.

4, the method for steady sound source to be measured is rebuild in employing spherical wave stack according to claim 1, it is characterized in that the reconstruction described in the 5th step is calculated, and specifically comprises:

1. set sound source to be measured surface S _sCertainly the relevant density matrix of the spectrum of the sound-source signal to be measured of a last N node is:

(S _pp ^α(r _s，f)) _N×1，

Wherein: f is the frequency of sound source characteristic to be measured, and α is a cycle frequency, r _sRepresent sound source to be measured surface S _sGo up the locus of each node, N is a natural constant;

2. the relevant density matrix of cross-spectrum of setting reference signal and sound-source signal to be measured is:

(S _Rp ^α(r _s, f)) _{N * 1}And (S _Pr ^α(r _s, f)) _{N * 1},

3. setting r (f) is the spectrum component of reference signal, and p (f) is the spectrum component of holographic measuring point acoustic pressure, and q (f) is the spectrum component of spherical wave basis function weights coefficient, then

S _Rp ^α(r _h, f) be r (f+ α/2) and p ^*The function of (f-α/2);

S _Pr ^α(r _h, f) be p (f+ α/2) and r ^*The function of (f-α/2);

Wherein: S _Rp ^α(r _h, f) and S _Pr ^α(r _h, f) be the relevant density matrix of cross-spectrum of reference signal and holographic measuring point acoustic pressure;

Q _Rp ^α(f) be r (f+ α/2) and q ^*The function of (f-α/2);

Q _Pr ^α(f) be q (f+ α/2) and r ^*The function of (f-α/2);

Wherein: subscript * is a conjugate transpose, Q _Rp ^α(f) and Q _Pr ^α(f) be the relevant density of cross-spectrum between spherical wave basis function weights coefficient and the reference signal, S _Rp ^α(r _s, f) should be by S _Rp ^α(r _h, f) go up reconstruction and obtain in frequency (f-α/2), and S _Pr ^α(r _s, f) should be by S _Pr ^α(r _h, f) go up reconstruction and obtain in frequency (f+ α/2);

4. obtain following with spherical wave Superposition Formula (2) and (3) of the relevant density of cross-spectrum as variable:

$S_{\mathrm{pr}}^{\mathrm{\α}}(r,f)=\mathrm{\ρc}\underset{j=1}{\overset{J}{\mathrm{\Σ}}}{Q_{\mathrm{pr}}^{\mathrm{\α}}}_j\left(f\right){\mathrm{\Ψ}}_j(r,f+\mathrm{\α}/2)---\left(2\right)$

${S_{\mathrm{rp}}^{\mathrm{\α}}}^{*}(r,f)=\mathrm{\ρc}\underset{j=1}{\overset{J}{\mathrm{\Σ}}}{Q_{\mathrm{rp}}^{\mathrm{\α}}}_j^{*}\left(f\right){\mathrm{\Ψ}}_j(r,f-\mathrm{\α}/2)---\left(3\right)$

Wherein: ρ, c are respectively the medium average density and the velocity of sound; Ψ _jBe the spherical wave basis function, J is the expansion item number of spherical wave basis function, Q _{Pr j} ^αAnd Q _{Rp j} ^αBe the relevant density of cross-spectrum between corresponding spherical wave basis function weights coefficient and the reference signal;

5. the form of spherical wave basis function is expressed as follows with formula (4):

${\mathrm{\ψ}}_j(r,\mathrm{\θ},\mathrm{\φ},k)=\sqrt{\frac{(2n+1)(n-m)!}{4\mathrm{\π}(n+m)!}}{\×h}_n\left(\mathrm{kr}\right)\×P_n^m\left(\cos \mathrm{\θ}\right)\×\left\{\begin{array}{cc}\cos \mathrm{m\φ}& l=\mathrm{0,2,4},\·\·\·\\ \sin \mathrm{m\φ}& l=\mathrm{1,3,5},\·\·\·\end{array}\right.---\left(4\right)$

Wherein: j, m, n and l are respectively nonnegative integer, and the pass between them is j=n ²+ n+m+1, l=j-n ²h _n(kr) be ball Hankel function, k is a wave number; P _n ^m(cos θ) follows the Legendre function;

Form with matrix can be expressed as formula (2) and (3):

$\mathrm{\ρc}{\left[\mathrm{\Ψ}(f+\mathrm{\α}/2)\right]}_{L\×J}{\left\{Q_{\mathrm{pr}}^{\mathrm{\α}}\left(f\right)\right\}}_{J\×1}={\left\{S_{\mathrm{pr}}^{\mathrm{\α}}\left(f\right)\right\}}_{L\×1}---\left(5\right)$

$\mathrm{\ρc}{\left[\mathrm{\Ψ}(f-\mathrm{\α}/2)\right]}_{L\×J}{\left\{{Q_{\mathrm{pr}}^{\mathrm{\α}}}^{*}\left(f\right)\right\}}_{J\×1}={\left\{{S_{\mathrm{rp}}^{\mathrm{\α}}}^{*}\left(f\right)\right\}}_{L\×1}---\left(6\right)$

6. utilize the least square solution method, make to be shown below the sound source error minimum to be measured of rebuilding in the hope of coefficient with formula (5) and (6):

${\left\{Q_{\mathrm{pr}}^{\mathrm{\α}}\left(f\right)\right\}}_{J\×1}={\left(\mathrm{\ρc}\right)}^{-1}{\left({\left[{\mathrm{\Ψ}}_{+}\right]}_{L\×J}^H{\left[{\mathrm{\Ψ}}_{+}\right]}_{L\×J}\right)}^{-1}{\left[{\mathrm{\Ψ}}_{+}\right]}_{L\×J}^H{\left\{S_{\mathrm{pr}}^{\mathrm{\α}}\left(f\right)\right\}}_{L\×1}---\left(7\right)$

${\left\{{Q_{\mathrm{rp}}^{\mathrm{\α}}}^{*}\left(f\right)\right\}}_{J\×1}={\left(\mathrm{\ρc}\right)}^{-1}{\left({\left[{\mathrm{\Ψ}}_{-}\right]}_{L\×J}^H{\left[{\mathrm{\Ψ}}_{-}\right]}_{L\×J}\right)}^{-1}{\left[{\mathrm{\Ψ}}_{-}\right]}_{L\×J}^H{\left\{{S_{\mathrm{rp}}^{\mathrm{\α}}}^{*}\left(f\right)\right\}}_{L\×1}---\left(8\right)$

In the formula subscript "+" and "-" to represent analysis frequency respectively be (f+ α/2) and (f-α/2), obtaining Q _Rp ^α(f) and Q _Pr ^α(f) afterwards, just can use formula (2) and (3) to rebuild sound field, obtain the S at sound source to be measured surface and r place, outside arbitrfary point _Pr ^α(r, f) and S _Rp ^α(r f), utilizes the relative phase relation of holographic measuring point acoustic pressure at last, just can obtain the relevant Density Distribution of spectrum of sound source to be measured.