---
# System prepended metadata

title: REPORT ON THE RESULTS OBTAINED FROM THE EXPERIMENT (MULTI MICROPHONE METHOD) USING pwelch METHOD (POWER SPECTRAL DENSITY)

---

# REPORT ON THE RESULTS OBTAINED FROM THE EXPERIMENT (MULTI MICROPHONE METHOD) USING pwelch METHOD (POWER SPECTRAL DENSITY)

## REPORT OF THE ABSOLUTE CALIBRATION

* For the absolute calibration we require acoustic calibrator (MODEL-CAL200) and the microphone is gently inserted into the acoustic calibrator for the calibration 
* Output level of the calibrator = $94$ (or) $114  dB$ - tolerance(+- $0.2dB$) at $1KHz$ (+-1% tolerance) frequency
* we have performed two cases for the absolute calibration
    - Nominal level 114 dB with signal generated from the calibration (near distance)
    - Nominal level 114 dB with signal generated from the calibration (far distance)
    - Near distance means the calibrator is kept nearer to the apparatus i.e, DAQ, Signal conditioner, test rig and far distance the calibrator is kept at far distance from the apparatus

### Nominal level 114 dB with signal generated from the calibration (far distance)

* Output level
  
| Nominal level| rms pressure | Test result |  Lower limit      | upper limit    | lying in the range    | result    |
| -------- | -------- | -------- | ---    | --- | --- | --- |
| 114 dB   | 9.0182 Pa    | 113.818dB    | 113.80dB   |114.20dB     | yes    | pass    |

* Frequency

| Nominal level| rms pressure | Test result |  Lower limit      | upper limit    | lying in the range    | result    |
| -------- | -------- | -------- | ---    | --- | --- | --- |
| 114 dB   | 9.0182 Pa    | 1000.58Hz    | 990 Hz   |1010 Hz     | yes    | pass    |

* plot of pressure-time domain
   - For the time interval 50sec to 90sec the p-t graph is constant, it is shown below
![](https://i.imgur.com/uFm5Fyi.jpg)

* The graph of pressure frequency domain after doing fft
![](https://i.imgur.com/7NO8g4A.jpg)

* phase plot for the time interval is shown as below (radians)
![](https://i.imgur.com/04EPcDC.jpg)


### Nominal level 114 dB with signal generated from the calibration (near distance)

* Output

| Nominal level| rms pressure | Test result |  Lower limit      | upper limit    | lying in the range    | result    |
| -------- | -------- | -------- | ---    | --- | --- | --- |
| 114 dB   | 10.6195Pa    | 114.05014dB    | 113.80dB   |114.20dB     | yes    | pass    |

* Frequency

| Nominal level| rms pressure | Test result |  Lower limit      | upper limit    | lying in the range    | result    |
| -------- | -------- | -------- | ---    | --- | --- | --- |
| 114 dB   | 10.6195Pa| 1000.59Hz    | 990Hz   |1010Hz     | yes    | pass    |

* plot of pressure-time domain
   - For the time interval 10sec to 40sec the p-t graph is constant, it is shown below
![](https://i.imgur.com/QVlvhC5.jpg)

* The graph of pressure frequency domain after doing fft
![](https://i.imgur.com/mbzo2j4.jpg)

* phase plot for the time interval is shown as below (radians)
![](https://i.imgur.com/ZhP8ubo.jpg)


## RELATIVE CALIBRATION OF MICROPHONES 

### RESULTS OBTAINED

* We have done three cases for the relative calibration i.e, three types of signals were given to the speaker 
    - sine wave with amplitude 2mV, frequency 200Hz and time 100sec
    - chirp signal with amplitude 1mV, frequency from 250Hz-2500Hz and time 100sec
    - chirp signal with amplitude 2mV, frequency from 250Hz-2500Hz and time 100sec
    
### sine wave with amplitude 2mV, frequency 200Hz and time of 100sec

* the pressure time series data of the 5 microphones which includes absolutely calibrated microphone is shown below

 ![](https://i.imgur.com/2oOqnip.jpg)


* the pressure time series data of the five microphones with time shown upto 1sec in the box.
 
![](https://i.imgur.com/LkZjZCb.jpg)

 
* the pressure time series data of the five microphones with time shown upto 5sec in the box.

 ![](https://i.imgur.com/BjjiEnX.jpg)
 
* the pressure frequency domain of the five microphones is shown below 

 ![](https://i.imgur.com/vKKHp8Z.jpg)

* the detailed view of pressure frequency domain for the five microphones is shown below 

![](https://i.imgur.com/j9QNeuo.jpg)

  
* rms value of pressures of the microphones are shown below

     | rms value of the pressure   | value |
     | -----------                 | ----------- |
     | rms voltage of input signal | $\sqrt2$ mV
     | absolutely calibrated       | $5.0945$ Pa   |
     | microphone 1                | $4.3759$ Pa|
     | microphone 2                | $3.8967$ Pa            |
     | microphone 3                | $6.2414$ Pa           |
     | microphone 4                | $4.2038$ Pa           |

### chirp signal with amplitude 1mV, frequency from 250Hz-2500Hz and time 100sec

* the pressure time series data of the 5 microphones which includes absolutely calibrated microphone is shown below

![](https://i.imgur.com/QwSROm5.jpg)

* the pressure time series data of the five microphones with time shown upto 1sec in the box

![](https://i.imgur.com/uKKZkiw.jpg)

* the pressure time series data of the five microphones with time shown upto 5sec in the box

![](https://i.imgur.com/92mVwKB.jpg)

* the pressure frequency domain of the five microphones is shown below 

![](https://i.imgur.com/IM8wX9E.jpg)

* the detailed view of pressure frequency domain for the five microphones is shown below 

![](https://i.imgur.com/l6ilFaB.jpg)



* rms value of pressures of the microphones are shown below

     | rms value of the pressure   | value |
     | -----------                 | ----------- |
     | rms voltage of input signal | $0.7053$ mV
     | absolutely calibrated       | $2.4462$ Pa   |
     | microphone 1                | $2.7339$ Pa|
     | microphone 2                | $2.7433$ Pa            |
     | microphone 3                | $3.0999$ Pa           |
     | microphone 4                | $3.2027$ Pa           |

### chirp signal with amplitude 2mV, frequency from 250Hz-2500Hz and time 100sec

* the pressure time series data of the 5 microphones which includes absolutely calibrated microphone is shown below

![](https://i.imgur.com/nEEa8Dg.jpg)

* the pressure time series data of the five microphones with time shown upto 1sec in the box

![](https://i.imgur.com/PE6JD94.jpg)

* the pressure time series data of the five microphones with time shown upto 5sec in the box

![](https://i.imgur.com/aUUJgn5.jpg)


* the pressure frequency domain of the five microphones is shown below 

![](https://i.imgur.com/pls7b86.jpg)

* the detailed view of pressure frequency domain for the five microphones is shown below 

![](https://i.imgur.com/0vCaFGI.jpg)


* rms value of pressures of the microphones are shown below

     | rms value of the pressure   | value |
     | -----------                 | ----------- |
     | rms voltage of input signal | $\sqrt2$ mV
     | absolutely calibrated       | $3.1395$ Pa   |
     | microphone 1                | $3.5455$ Pa|
     | microphone 2                | $3.5019$ Pa            |
     | microphone 3                | $3.6339$ Pa           |
     | microphone 4                | $3.9622$ Pa           |

### Calibration coefficients of microphones with respect to absolutely calibrated microphone

#### Sine wave with amplitude 1mV, frequency from 250Hz-2500Hz and time 100sec

* magnitude of pressure with respect to absolutely calibrated microphone ($\frac{p_{ref}}{p_j}$)
![](https://i.imgur.com/tdUJS7f.jpg)

* magnitude of phase with respect to absolutely calibrated microphone ($\phi_{ref}$-$\phi_{j}$)
![](https://i.imgur.com/YBdAGei.jpg)


#### Chirp signal with amplitude 1mV, frequency from 250Hz-2500Hz and time 100sec

* magnitude of pressure with respect to absolutely calibrated microphone ($\frac{p_{ref}}{p_j}$)

![](https://i.imgur.com/6E765x7.jpg)

* magnitude of phase with respect to absolutely calibrated microphone ($\phi_{ref}$-$\phi_{j}$)

![](https://i.imgur.com/Y9t4UcC.jpg)


#### Chirp signal with amplitude 2mV, frequency from 250Hz-2500Hz and time 100sec

* magnitude of pressure with respect to absolutely calibrated microphone ($\frac{p_{ref}}{p_j}$)

![](https://i.imgur.com/BRg0CmO.jpg)

* magnitude of phase with respect to absolutely calibrated microphone ($\phi_{ref}$-$\phi_{j}$)

![](https://i.imgur.com/HhEFVPL.jpg)


## RESULTS OF THE TEST RIG USING MMM (MULTI MICROPHONE METHOD)

* The microphones are kept at axial locations as follows
  $x_1$ = 220mm
  $x_2$ = 380mm
  $x_3$ = 510mm
  $x_4$ = 620mm
  $x_5$ = 700mm (all the distances are measured from the source i.e, speaker)
* The tube (test rig) is of length $1mm$

### WITH FOAM AT SPEAKER END

#### Chirp signal with amplitude 1mV, frequency from 250Hz-2500Hz and time 50sec

* the pressure time series data of the 5 microphones is shown below
![](https://i.imgur.com/FoSokD2.jpg)

* the frequency domain of the five microphones is shown below
![](https://i.imgur.com/8Ox03aU.jpg)

* the power spectral density of the five microphones is shown below
![](https://i.imgur.com/x9DXQMJ.jpg)

* the plot of the reflection coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (calculated using pwelch method)
![](https://i.imgur.com/VIOnENN.jpg)

* the plot of the absorption coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (calculated using pwelch method)
![](https://i.imgur.com/n3i7baM.jpg)

* the plot of the real and imaginary part of impedance vs frequency range at the open end i.e, at $x=0$
![](https://i.imgur.com/x34d3DZ.jpg)

* the plot of pressure along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/IKYJaZu.jpg)

* the plot of velocity along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/vOSBh1i.jpg)


#### Chirp signal with amplitude 2mV, frequency from 250Hz-2500Hz and time 50sec

* the pressure time series data of the 5 microphones is shown below
![](https://i.imgur.com/4JeJXKj.jpg)

* the frequency domain of the five microphones is shown below
![](https://i.imgur.com/biFFgU1.jpg)

* the power spectral density of the five microphones is shown below
![](https://i.imgur.com/uS1Z639.jpg)

* the plot of the reflection coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (calculated using pwelch method)
![](https://i.imgur.com/Mc000aA.jpg)

* the plot of the absorption coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (calculated using pwelch method)
![](https://i.imgur.com/jAHEKQG.jpg)

* the plot of the real and imaginary part of impedance vs frequency range at the open end i.e, at $x=0$
![](https://i.imgur.com/FjKkEKK.jpg)

* the plot of pressure along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/J8hTvWm.jpg)

* the plot of velocity along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/XuRhnVz.jpg)


#### Chirp signal with amplitude 3mV, frequency from 250Hz-2500Hz and time 50sec

* the pressure time series data of the 5 microphones is shown below
![](https://i.imgur.com/oWgct9R.jpg)

* the frequency domain of the five microphones is shown below
![](https://i.imgur.com/Uiq93Em.jpg)

* the power spectral density of the five microphones is shown below
![](https://i.imgur.com/kuOAZ2t.jpg)

* the plot of the reflection coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (calculated using pwelch method)
![](https://i.imgur.com/TOhTsWM.jpg)

* the plot of the absorption coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (calculated using pwelch method)
![](https://i.imgur.com/NoRYYjF.jpg)

* the plot of the real and imaginary part of impedance vs frequency range at the open end i.e, at $x=0$
![](https://i.imgur.com/y2tuN5r.jpg)

* the plot of pressure along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/my5LOer.jpg)

* the plot of velocity along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/FfClU1v.jpg)


### WITH OUT FOAM AT SPEAKER END

#### Chirp signal with amplitude 1mV, frequency from 250Hz-2500Hz and time 50sec

* the pressure time series data of the 5 microphones is shown below
![](https://i.imgur.com/2aQ7Wts.jpg)

* the frequency domain of the five microphones is shown below
![](https://i.imgur.com/oWjNdNZ.jpg)

* the power spectral density of the five microphones is shown below
![](https://i.imgur.com/ZXW7XHC.jpg)

* the plot of the reflection coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (using pwelch method)
![](https://i.imgur.com/vIAGnl9.jpg)

* the plot of the absorption coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (using pwelch method)
![](https://i.imgur.com/PtN4UdY.jpg)

* the plot of the real and imaginary part of impedance vs frequency range at the open end i.e, at $x=0$
![](https://i.imgur.com/NPj1F7O.jpg)

* the plot of pressure along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/cyfArKp.jpg)

* the plot of velocity along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/nhasVxM.jpg)


#### Chirp signal with amplitude 2mV, frequency from 250Hz-2500Hz and time 50sec

* the pressure time series data of the 5 microphones is shown below
![](https://i.imgur.com/XyRfckV.jpg)

* the frequency domain of the five microphones is shown below
![](https://i.imgur.com/P6GKUyf.jpg)

* the power spectral density of the five microphones is shown below
![](https://i.imgur.com/WBvYvZh.jpg)

* the plot of the reflection coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (using pwelch method)
![](https://i.imgur.com/uKuz33E.jpg)

* the plot of the absorption coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (using pwelch method)
![](https://i.imgur.com/qPCrsig.jpg)

* the plot of the real and imaginary part of impedance vs frequency range at the open end i.e, at $x=0$
![](https://i.imgur.com/Bz4OiBG.jpg)

* the plot of pressure along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/QRd2IJ5.jpg)

* the plot of velocity along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/ap2nh4k.jpg)


#### Chirp signal with amplitude 3mV, frequency from 250Hz-2500Hz and time 50sec

* the pressure time series data of the 5 microphones is shown below
![](https://i.imgur.com/l89IMp9.jpg)

* the frequency domain of the five microphones is shown below
![](https://i.imgur.com/Mg8RxLF.jpg)

* the power spectral density of the five microphones is shown below
![](https://i.imgur.com/FaXNDYM.jpg)

* the plot of the reflection coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (using pwelch method)
![](https://i.imgur.com/V30a61L.jpg)

* the plot of the absorption coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (using pwelch method)
![](https://i.imgur.com/aqAl9OL.jpg)

* the plot of the real and imaginary part of impedance vs frequency range at the open end i.e, at $x=0$
![](https://i.imgur.com/Wak8fSk.jpg)

* the plot of pressure along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/8pc2i8n.jpg)

* the plot of velocity along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/3qZEfZ1.jpg)


#### Chirp signal with amplitude 4mV, frequency from 250Hz-2500Hz and time 50sec

* the pressure time series data of the 5 microphones is shown below
![](https://i.imgur.com/wbJSVf8.jpg)

* the frequency domain of the five microphones is shown below
![](https://i.imgur.com/xBV5MC1.jpg)

* the power spectral density of the five microphones is shown below
![](https://i.imgur.com/Vw7ApTX.jpg)

* the plot of the reflection coefficient vs frequency range i.e, 250Hz-2500Hz is shown below (using pwelch method)
![](https://i.imgur.com/pVhdMmh.jpg)

* the plot of the absorption coefficient vs frequency range i.e, 250Hz-2500Hz is shown below
![](https://i.imgur.com/ZEJjpBI.jpg)

* the plot of the real and imaginary part of impedance vs frequency range at the open end i.e, at $x=0$
![](https://i.imgur.com/B0God3T.jpg)

* the plot of pressure along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/GnUuOSY.jpg)

* the plot of velocity along the length of tube for time interval of $t=0-50sec$ with a step of $t=10sec$
![](https://i.imgur.com/kXSbu7d.jpg)