# REPORT ON MULTI MICROPHONE METHOD AND LINER ACOUSTICS
## **MULTI MICROPHONE METHOD**
* classical wave equation
$$\frac{\partial^2 p'}{\partial t^2}= c^2\frac{\partial^2 p'}{\partial x^2}$$
here
$$p'=p(x)e^{i\omega t}$$
where
$e^{i\omega t}$ denotes time harmonicity
substituting $p'$ in above wave equation
$$\frac{i\omega^2 p(x) e^{i\omega t} }{c^2}= \frac{d^2 (p e^{i\omega t})}{dx^2 }$$
$$\frac{d^2 p}{dx^2}+\frac{\omega^2 p}{c^2}=0$$
since
$$\frac{\omega}{c}=k$$
we have
$$\frac{d^2 p}{dx^2}+k^2 p = 0$$
above equation is helmhotz equation
* the solution for above equation is given as
$$p= Ae^{ikx}+ Be^{-ikx}$$
* in above equation the first term is left running wave i.e, reflected wave and second term is right running wave i.e, incident wave
* The sound field for a plane wave propagating in the duct can be expressed as
$$ p(f) = p_+(f) exp(i\Gamma_+x) + p_-(f)exp(i\Gamma_-x) $$
* where $\Gamma_+$ and $\Gamma_-$ denote the incident and reflected plane wave propagation constants, respectively
here
$$\Gamma_+=\frac{-(k-i\delta)}{1+M}$$
$$\Gamma_-=\frac{k-i\delta}{1-M}$$
* where ***k*** is the wave number, ***M*** is the mean flow mach number, and ***$\delta$*** is the attenuation constant.
* attenuation constant originates from the viscothermal and turbulent effects as
$$\delta=\delta_v+\delta_t$$
where,
$$\delta_v = \frac{\omega}{2\sqrt2a_0c_0}\left[\sqrt\frac{\mu}{\rho_0\omega} + (\gamma-1)\sqrt\frac{K}{\rho_0\omega c_p}\right]$$
and
$$\delta_t = \frac{\Psi*M}{2a_0}\left[1+\frac{\Psi'Re}{2\Psi}\right]$$
here,
$\omega$ : angular frequency
$a_0$ : ratio of duct area to perimeter (if a circular duct of diameter D is used, $a_0 = D/4$)
$c_0$ : speed of sound
$\rho_0$ : density of the fluid medium
$\mu$ : shear viscosity coefficient
$\gamma$ : specific heat ratio
$K$ : heat conduction coefficient
$c_p$ : specific heat at constant pressure
$Re$ : Reynolds number of the flow
$\psi$ : coefficient of friction for turbulent flow
$\psi'$ : $\frac{\partial\psi}{\partial Re}$
if a smooth duct is involved, $\psi$ can be calculated from the prandtl universal resistance law as given by
$$\frac{1}{\psi}= 2log_{10}\left(Re\sqrt\psi\right)-0.8$$
* based on the above equation and assembling all $P_j(f)$ measured at $x_j$ which can be seen in below figure
[^1]
* the over determined linear system of equations can be written as
$$Ax=b$$
* where
$$ A=\begin{bmatrix} exp(iΓ_+x_1)&exp(iΓ_-x_1) \\ exp(iΓ_+x_2)&exp(iΓ_-x_2)\\\vdots&\vdots\\exp(iΓ_+x_n)&exp(iΓ_-x_n) \end{bmatrix} $$
$$ x=\begin{bmatrix} p_+(f) \\ p_-(f) \end{bmatrix} $$
$$ b=\begin{bmatrix} p_1(f)\\p_2(f)\\\vdots&\\p_n(f) \end{bmatrix} $$
* the least-squares solution of above system of equations can be obtained by using the moor–penrose generalized inverse
$$x=A^+b$$
## **Condition Number $\kappa(A)$**
* condition number is used to tell whether the system is stable or not
* It is defined as
$\kappa(A)=||A||||A^{-1}||$
* here $||A||$ = norm $[A]$ means absolute value of the largest row sum
* if $A$ is singular i.e, $det(A) = 0$ then $\kappa(A)= ∞$
* let us consider system of equations $Ax=b$
- for small change in $b$ if there is small change in $x$ or $A$ then the system is well conditioned i.e, condition number is less
- for large change in $b$ if there is large change in $x$ or $A$ then the system is ill conditioned i.e, condition number is large or tending to $∞$
- if $\kappa(A)=1$ it is well conditioned (stable)
- if $\kappa(A)>1$ it is ill conditioned (unstable)
* condition number of $A$ depends on how close $A$ is to being singular
* ill-conditioning occurs if and only if the two equations give two nearly parallel lines, so that their intersection point (the solution of the system) moves substantially if we raise or lower a line just a little
[^2]
[^2]: Kreyszig Erwin, 'Advanced Engineering Mathematics', $10^{th}$ ed., pp.864-72
* consider a system of equations
$$0.9999x - 1.0001y = 1$$$$x - y = 1$$
solution of this system of equations is $x=0.5$ & $y=-0.5$.
* whereas if we consider a system of equations as
$$0.9999x - 1.0001y = 1$$$$x - y = 1 + \epsilon$$
solution of this system of equations is $x=0.5 + 5000.5\epsilon$ & $y=-0.5+4999.5\epsilon$
* this shows that the system is ill-conditioned because a change on the right of magnitude $\epsilon$ produces a change in the solution of magnitude $5000\epsilon$ approximately. We see that the lines given by the equations have nearly the same slope.
* **Well-conditioning** can be asserted if the main diagonal entries of $A$ have large absolute values compared to those of the other entries. Similarly if $A^{-1}$ and $A$ have maximum entries of about the same absolute value.
* **Ill-conditioning** is indicated if $A^{-1}$ has entries of large absolute value compared to those of the solution (about 5000 in above example) and if poor approximate solutions may still produce small residuals.
* in three-dimensional space, two points with position vectors $x$ and $x'$ have distance $|x-x'|$ from each other. For a linear system $Ax=b$, this suggests that we take $||x - x'||$ as a measure of inaccuracy and call it the **distance** between an exact and an approximate solution, or the error of $x'$.
* for $Ax=b$ consider
1. inaccuracy in matrix entries
one can prove $$\frac{||\delta x||}{||x||} = \kappa(A)\frac{||\delta A||}{||A||}$$
this shows, if the system is well-conditioned, small inaccuracies $\frac{||\delta A||}{||A||}$ can have only a small effect on the solution. However, in the case of ill-conditioning, if $\frac{||\delta A||}{||A||}$ is small, $\frac{||\delta x||}{||x||}$ may be large.
2. inaccuracies in right side
one an show, when A is accurate, an inaccuracy $\delta b$ of $b$ causes an inaccuracy $\delta x$ satisfying
$$\frac{||\delta x||}{||x||} \leq \kappa(A)\frac{||\delta b||}{||b||}$$
Hence $\frac{||\delta x||}{||x||}$ must remain relatively small whenever $\kappa(A)$ is small
* example of condition number by considering a matrix
$$ A=\begin{bmatrix}2&-1&1\\1&0&1\\3&-1&4\end{bmatrix} $$
$$ A^{-1}=\begin{bmatrix}0.5&1.5&-0.5\\-0.5&2.5&-0.5\\-0.5&-0.5&0.5\end{bmatrix}$$
* here norm of **$A$** maximum sum of absolute values of element in a row of $A$
$$||A||=|3|+|1|+|4|= 8$$
* norm of **$A^{-1}$** maximum sum of absolute values of element in a row of $A^{-1}$
$$||A^{-1}||=|-0.5|+|2.5|+|-0.5|= 3.5$$
* hence $$|cond(A)|=||A||||A^{-1}||=8*3.5=27$$
* since $cond(A)>1$ the above system is ill conditioned and not stable but there is no sharp dividing line between “well-conditioned” and “ill-conditioned,” generally the situation will get worse as we go from systems with small $\kappa(A)$ to system with larger $\kappa(A)$ now always $|\kappa(A)|\geq1$ so that values of 10 or 20 or so give no reason for concern, whereas $\kappa(A)=100$ say, calls for caution
## Moore Penrose Pseudoinverse ($A^+$)
* matrix must be a square matrix to find its inverse
* Moore Penrose pseudoinverse is a procedure to calculate inverse for rectangular matrices
* common use of pseudoinverse is to compute a 'best fit' (least sqaure) solution to a system of linear equations that lacks a unique solution
* $A^+$ based on svd method
- if colums of A are linearly independent
- $A^+$ = ($A^T$$A$)$^{-1}$$A^T$
- if rows of A are linearly independent
- $A^+$ = $A^T$($A$$A^T$)$^{-1}$
- if $A$ has rank deficiency
- using singular value decomposition of $A$
- $A$ = $U$$S$$V^T$
- $A^+$ = $V$$S^{-1}$$U^T$
* In the overdetermined linear system of equations mentioned above
- $A_{n*2}$
- $S_{n*2}$ , $S^{-1}$$_{2*n}$ : real diagonal matrices
- $U_{n*n}$ , $V_{2*2}$ : unitary matrices
- matlab code to get moore penrose pseudoinverse:
* example of matrix taken in the code -
$$ A=\begin{bmatrix}1&2&4&5\\2&4&8&10\\4&8&16&20\end{bmatrix} $$
---
```
clc;
A = [1 2 4 5 ; 2 4 8 10 ; 4 8 16 20];
[m,n] = size(A);
r = rank (A);
a = A*A';
b = A'*A;
e = eig(a);
l=zeros(n,m);
if r==m
p = A'/a;
elseif r==n
p = b\A';
else
[U,S,V] = svd(A);
S1 = round(S,5);
for i=1:1:m
for j=1:1:n
if S1(i,j)== 0
l(j,i)= S1(i,j);
else
l(j,i) = 1/S1(i,j);
end
end
end
p = V*l*U'
end
p1 = pinv(A)
```
---
## Multi Microphone Method
$$x = A^+b$$
where $A^+$ = $V$$S^{-1}$$U^T$
$$ b' = b + m = Ax + m = Ax' $$
where
- **$m$** : measurement error in $b$
- ' : represents estimated value
hence,
estimation error : $x'$-$x$ = $A^+$$m$
expected value squared norm of the estimation error
$$E\left[||x'-x||^2\right] = E\left[(A^+m)^H(A^+m)\right]$$
$$E\left[||x'-x||^2\right] = σ^2Tr\left((S^{-1})^T (S^{-1})\right)$$
$$E\left[||x'-x||^2\right] = σ^2SF^2$$
where,
SF : singularity factor
$$SF = \sqrt{\sum_j S_j^{-2}}$$
* in order to get a small estimation error, **SF** should be held to the minimum possible.
* the sensitivity of the method to the measurement input errors can be obtained by using the **SF**
* singular condition derived
$$\left(\sum_jexp\left[i\left(\frac{2k}{1-M^2}\right)x_j\right]\right)\left(\sum_jexp\left[-i\left(\frac{2k}{1-M^2}\right)x_j\right]\right)-n^2=0$$
* if this condition is met, a large error can occur in solution of $x$ and smallest and non zero wave number in above equation is *critical wave number $k_{cr}$*
* when $n=2$ i.e, two microphone method corresponding singular condition can be derived from above equation
$$cos\left(\frac{2kx_{21}}{1-M^2}\right)= 1$$
here $x_{21}$= distance between microphones 1 and 2
* in order to avoid the sensor placement at singular points, choose $x_{21}$ satisfying the criterion given by
$$kx_{21}<\pi(1-M^2)$$
* SF for $n=2$ case is smallest when $k/k_{cr}=0.5$ goes to $\infty$ for $k/k_{cr}=0$ & $1$.
[^1]
[^1]: [Seung-Ho Jang](https://asa.scitation.org/author/Jang%2C+Seung-Ho) _and_ [Jeong-Guon Ih](https://asa.scitation.org/author/Ih%2C+Jeong-Guon) , "On the multiple microphone method for measuring in-duct acoustic properties in the presence of mean flow", The Journal of the Acoustical Society of America 103, 1520-1526 (1998) [https://doi.org/10.1121/1.421289](https://doi.org/10.1121/1.421289)
* when $n=3$ corresponding singular condition which can be derived from above equation
$$cos\left(\frac{2kx_{21}}{1-M^2}\right)+cos\left(\frac{2kx_{32}}{1-M^2}\right)+cos\left(\frac{2kx_{31}}{1-M^2}\right)= 3$$
* all three expressions $$\left(\frac{kx_{21}}{1-M^2}\right)$$ $$\left(\frac{kx_{32}}{1-M^2}\right)$$ $$\left(\frac{kx_{31}}{1-M^2}\right)$$ must be integer multiples of $\pi$.
* here $\frac{x_{32}}{x_{21}}=\frac{q}{r}$ , where $\frac{q}{r}$ is an irreducible fraction
* $SF_{max}$ **(singularity factor)** increases as r and q increases and $SF_{max}$ **(singularity factor)** is smallest when **r=1 and q=1**
* when $n=5$ corresponding singular condition which can be derived from above equation
$$cos\left(\frac{2kx_{21}}{1-M^2}\right)+cos\left(\frac{2kx_{32}}{1-M^2}\right)+cos\left(\frac{2kx_{43}}{1-M^2}\right)+cos\left(\frac{2kx_{54}}{1-M^2}\right)+cos\left(\frac{2kx_{31}}{1-M^2}\right)+cos\left(\frac{2kx_{42}}{1-M^2}\right)+cos\left(\frac{2kx_{41}}{1-M^2}\right)+cos\left(\frac{2kx_{53}}{1-M^2}\right)+cos\left(\frac{2kx_{52}}{1-M^2}\right)+cos\left(\frac{2kx_{51}}{1-M^2}\right)= 10$$
[^1]
* every set of measurement points having uniform spacings provides smaller singularity factors than those having nonuniform spacings at all frequencies except the ranges near to $\frac{k}{kcr} = 0$ and $\frac{k}{kcr}= 1$.
* as the number of measuring points increases, the SF within effective frequency span becomes small, especially for $\frac{k}{kcr}= 0.5$
* the frequency range can be widened and the singularity can be reduced by increasing the number of measuring points, the efficiency of these favorable effects becomes marginal when $n$ is greater than about $7$
## Problem Statement
* **consider a duct of length $1m$ , fix $5$ microphone locations in the duct and $frequency-range = 1000Hz-4000Hz$**
* condition to be satisfied
$$kx_{21}<\pi(1-M^2)$$
 [^1]
* for $n=5$ $\delta$$\phi=0.16$ and
$$\delta\phi<k/k_{cr}<1-\delta\phi$$
$$0.16<k/k_{cr}<0.84$$
* we can take optimum $\frac{k}{kcr}$ in between $0.16-0.84$
* in the below reference mentioned the critical frequency is taken as $3034Hz$ (*after that frequency there are irregularities in the graph of reflection coefficient vs frequency*)
fig. Comparison of the magnitudes of the pressure reflection coefficient of an unflanged open pipe with mean flow ($M = 0.1$) & $n = 5$. ———, measured by using the equidistant measuring points; – – –, theory
* we take optimum $\frac{k}{k_{cr}}$ as $0.35$ since frequency range $1000Hz-4000Hz$
* we consider mean flow with **mach number $M= 0.1$**
$$kx_{21}<\pi(1-M^2)$$
$$kx_{21}<\pi(0.99)$$
* for optimum case we consider $\frac{k}{k_{cr}}=0.35$
$$\frac{k}{kcr}=\frac{f}{f_{cr}}=0.35$$
$$\frac{f}{3034Hz}=0.35$$
$$f=1061.9Hz$$
$$k=\frac{2\pi*f}{c}=\frac{2\pi*1061.9Hz}{346.3m/s}$$
$$k=19.2571m^{-1}$$
* hence,
$$x_{21}<\frac{\pi*0.99}{19.2571}$$
$$x_{21}<0.161426mm$$
$$x_{21}<161.426mm$$
* since
$$x_{21}<161.426mm$$
we consider
$$x_{21}=x_2-x_1=160mm$$
* If we consider $x_{21}>161.426mm$, singularity factor is more and the system is unstable
* since $x_{21}=160mm$ and these distances are placed logarithimally i.e, decreased logarithimcally
* hence
$$x_{32}= 160-ln(160)$$
$$x_{32}=154.925mm$$
we consider,
$$x_{32}=155mm$$
* $$x_{43}=155-ln(155)$$
$$x_{43}=149.95mm$$
we consider,
$$x_{43}=150mm$$
* $$x_{54}=150-ln(150)$$;
$$x_{54}=144.94mm$$
we consider,
$$x_{54}=145mm$$
* hence,
distance between microphones 1 and 2 = $x_{21}=160mm$
distance between microphones 2 and 3 = $x_{32}=155mm$
distance between microphones 3 and 4 = $x_{43}=150mm$
distance between microphones 1 and 2 = $x_{54}=145mm$
distance between last microphone to acoustic liner = $200mm$
distance from loudspeaker to first microphone = $190mm$
* plot of **frequency vs condition number of matrix $A$**
* matlab code used for the plot:
```
clear
clc
f=zeros(1,3001);
cond_num=zeros(1,3001);
k=zeros(1,3001);
a=zeros(1,3001);
b=zeros(1,3001);
c=zeros(1,3001);
d=zeros(1,3001);
e=zeros(1,3001);
sf=zeros(1,3001);
x1 = 0.81;
x2 = 0.65;
x3 = 0.495;
x4 = 0.345;
x5 = 0.2;
c1 = 346.02753;
for j= 1:1:3001
f(1,j) = 999 + 1*j; %frequency 1000 Hz to 4000 Hz
k(1,j) = 2*pi*f(1,j)/c1; %calculating wave number
a(1,j) = 1i*k(1,j)*x1;
b(1,j) = 1i*k(1,j)*x2;
c(1,j) = 1i*k(1,j)*x3;
d(1,j) = 1i*k(1,j)*x4;
e(1,j) = 1i*k(1,j)*x5;
A = [exp(-a(1,j)) exp(a(1,j)); exp(-b(1,j)) exp(b(1,j)); exp(-c(1,j)) exp(c(1,j)); exp(-d(1,j)) exp(d(1,j)); exp(-e(1,j)) exp(e(1,j))];
[U,S,V] = svd(A); %singular value decomposition of matrix A
S(S~=0);
m = diag(S);
n = m.^2;
q = 1./n;
sf(1,j)= sqrt(sum(q)); %calculating singularity factor
cond_num(1,j) = cond(A); %calculating condition number of matrix A
end
plot(f,sf,'r-','LineWidth' ,2) %plot of singularity factor vs frequency
hold on
plot(f,cond_num,'b-','LineWidth', 2) %plot of condition number vs frequency
ylim([0 12])
xlabel ('frequency (f)')
ylabel ('condition number [\kappa(A)] and singularity factor')
title ('condition number and singularity factor vs frequency')
legend('singularity factor','condition number')
grid on
```
* taking condition number and frequency range $1000Hz-4000Hz$ we get the following plot
* plot of singularity factor$(S.F)$ and frequency range $1000Hz-4000Hz$
* plot of $k/k_{cr}$ and singularity factor frequency range $1000Hz-4000Hz$
* in the matrix A
$$ A=\begin{bmatrix} exp(iΓ_+x_1)&exp(iΓ_-x_1) \\ exp(iΓ_+x_2)&exp(iΓ_-x_2)\\\vdots&\vdots\\exp(iΓ_+x_n)&exp(iΓ_-x_n) \end{bmatrix} $$
* the values $x_1$,$x_2$,$x_3$,$x_4$,$x_5$ are shown in below figure
## Anechioc Termination Design
* the design of the anechoic chamber is shown below
* anechoic chamber design is taken reference from the following image shown below
* reference of above image is from the following paper
[^3]
* with above reference we can take mineral wool as material for anechoic termination
* in the above design of anechoic termination we consider a linear expansion chamber
* from the above reference we consider the length of chamber $L\geq5d$
hence, $L\geq700$
we consider length of anechoic chamber $L=900mm$
* diameter of anechoic chamber at starting is same as impedance tube diameter, since the chamber has linear expansion the diameter of chamber increases
* diameter of chamber at half of its length($\frac{L}{2}=450mm$)is given by = $\sqrt{d^2+dH}=548mm$
here $H=2m$ which is shown in above reference
* diameter of chamber at the ending($L=900mm$) is given by = $d+H=2140mm$
* graph of pressure reflection coefficient ($|R|$) vs frequency ($f$) for the above configuration
[^3]
[^3]: W. NEISE AND F. ARNOLD, "on sound power determination for flow in ducts"‚DLR-Institute for Antriebstechnik, Abteilung turbulenzforschung Berlin,
Muller-Breslau-Strasse 8,D-10623 Berlin, Germany.
* from the above graph we can say that for intial frequencies below 200Hz the reflection coefficient is 0.6 and for frequencies above 200Hz we get reflection coefficient below 0.2 which is the basic criteria for anechoic chamber to have reflection coefficient ($|R|=0.2$)
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