---
# System prepended metadata

title: REPORT ON MULTI MICROPHONE METHOD AND LINER ACOUSTICS

---

# REPORT ON MULTI MICROPHONE METHOD AND LINER ACOUSTICS
## **MULTI MICROPHONE METHOD**
* The sound field for a plane wave propagating in the duct can be expressed as

  *$p(f)$=$p_+(f)$$exp($i$Γ_+$x$)$ + $p_-(f)$$exp($i$Γ_-$x$)$* 
  
* where $Γ_+$ and $Γ_-$ denote the incident and reflected plane wave propagation constants, respectively 
   here
       $Γ_+$=$(k-iδ)/(1+M)$
       $Γ_-$=$(k-iδ)/(1+M)$
* where **k** is the wave number, **M** is the mean flow mach number, and d is the attenuation constant.
* based on the above equation and assembling all $P_j(f)$ measured at $x_j$ which can be seen in below figure

![](https://i.imgur.com/kzLwDH1.png)

* the over determined linear system of equations can be written as

  $Ax=b$
  
* where 

![](https://i.imgur.com/56H65mA.png)

* the least-squares solution of above system of equations can be obtained by using the moor–penrose generalized inverse
    
    $x=A^+b$
    
# CONDITION NUMBER
* condition number is used to tell whether the system is stable or not
* It is defined as
    $cond(A)=||A||||A^{-1}||$
* here $||A||$ = norm $[A]$ means absolute value of the largest row sum
* if $A$ is singular i.e, $det(A) = 0$ then $cond(A)= ∞$
* 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 $cond(A)=1$ it is well conditioned (stable)
    - if $cond(A)>1$ it is ill conditioned (unstable)
* condition number of $A$ depends on how close $A$ is to being singular
* example of condition number by considering a matrix 
![](https://i.imgur.com/ZG3apR8.png)
![](https://i.imgur.com/T862R0k.png)
![](https://i.imgur.com/gPWvBWq.png)
* since $cond(A)>1$ the above system is ill conditioned and not stable

# MOORE PENROSE PSEUDOINVERSE       ($A^+$)
* matrix must be a square matrix to find its inverse
* Moore Penrose psuedoinverse 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:

```
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)
```


