- Thread Starter
- #21
I'll discuss in this post the sources of errors of the NFS measurements. Please be reminded that these are my personal opinions based on my (limited) understanding of the theory and Klippel's implementation. I can be wrong and probably have left out others too.
Fitting and approximation errors
The mathematical model of the speaker is based on spherical wave expansion functions. When the sound sources are compact, the spherical wave functions are capable of approximating them efficiently (i.e. requiring only the lower order functions). However, when the sound sources are of high aspect ratio (i.e. either long and thin and/or flat and thin), the spherical wave functions are less efficient and the approximation errors will be higher.
So when do we have high aspect ratio sound sources? Remember if we measure in a reflective room, the reflections are also sound sources the spherical wave functions need to account for, and these reflections extend the size and alter the shape of the total/combined sound source. Thus, having a large amount of reflections will have a negative effect.
On 'fitting' errors, I don't know exactly how Klippel estimates them. From what I've read, it seems to me those are residuals from the least squares fit. Residuals are usually not considered a very good estimate of the generalization errors due to the possibility of over-fitting. Klippel didn't mention using any kind of regularization in their least squares fits. Without regularization, the computed fitting coefficients are not constrained, and can be susceptible to measurement errors which can cause over-fitting.
Reference on regularization:
A very math heavy paper by Earl Williams on the use of regularization in nearfield acoustical holography.
https://www.researchgate.net/public..._Journal_of_the_Acoustical_Society_of_America
Systematic errors
Setup error of the speaker is one source of systematic error. It is easy to mis-aim the reference axis by a few degrees, especially for a small, non-rectangular-box-shaped, speaker.
A second source of systematic error is positioning error of the microphone. The Klippel robot is a very simple 3-axis robot with precision linear and rotary actuators. Positioning error should therefore be very well controlled. The largest sources of the robot positioning error, in my estimation, are in the alignment/parallelism of the axis of the vertical linear actuator to the rotational axis, and the perpendicularity of the horizontal linear actuator to the vertical linear actuator and how well it intersects the rotational axis.
Other systematic errors may include microphone calibration errors, non-uniform directional response of the microphone, etc.
Noise and environmental conditions
The mathematics of the NFS is entirely in the frequency domain. It basically assumes/requires that all measurements be made in the exact same conditions (time invariant). That means if we have varying environmental noise, errors will be introduced into the model. The NFS mitigates this issue by measuring at nearfield to maximize signal-to-noise ratio, and to use double layer measurements to estimate the actual signal-to-noise ratio. Movements of the NFS robot during the test also cause some changes to the sound diffractions/reflections, and violate the time invariant requirement.
Environmental temperature can also affect the results as the speed of sound is proportional to the square root of the absolute air temperature. Changes in the speed of sound alter the ratio between wavelength to physical dimension, and can change the behavior of diffractions and acoustical interference.
Characterization of the system
Please do not consider these as my recommendations to Amir. I am expressing my opinions only.
If I have an NFS (or if I were to build one), here are the steps I will take to characterize the effects of these sources of errors.
Sound field separation
Erect some large plywood boards to act as reflectors around the NFS setup. Compare test results with a few, with many, and without these reflectors.
Positioning effects (system repeatability)
Test the speaker upright, sideways, and upside down. Compare results.
Environmental conditions
Compare the test results of the same speaker tested at different room temperatures.
Play a constant noise in the room and run the test. Repeat with a varying noise. Compare results to no added noise.
Fitting and approximation errors
The mathematical model of the speaker is based on spherical wave expansion functions. When the sound sources are compact, the spherical wave functions are capable of approximating them efficiently (i.e. requiring only the lower order functions). However, when the sound sources are of high aspect ratio (i.e. either long and thin and/or flat and thin), the spherical wave functions are less efficient and the approximation errors will be higher.
So when do we have high aspect ratio sound sources? Remember if we measure in a reflective room, the reflections are also sound sources the spherical wave functions need to account for, and these reflections extend the size and alter the shape of the total/combined sound source. Thus, having a large amount of reflections will have a negative effect.
On 'fitting' errors, I don't know exactly how Klippel estimates them. From what I've read, it seems to me those are residuals from the least squares fit. Residuals are usually not considered a very good estimate of the generalization errors due to the possibility of over-fitting. Klippel didn't mention using any kind of regularization in their least squares fits. Without regularization, the computed fitting coefficients are not constrained, and can be susceptible to measurement errors which can cause over-fitting.
Reference on regularization:
A very math heavy paper by Earl Williams on the use of regularization in nearfield acoustical holography.
https://www.researchgate.net/public..._Journal_of_the_Acoustical_Society_of_America
Systematic errors
Setup error of the speaker is one source of systematic error. It is easy to mis-aim the reference axis by a few degrees, especially for a small, non-rectangular-box-shaped, speaker.
A second source of systematic error is positioning error of the microphone. The Klippel robot is a very simple 3-axis robot with precision linear and rotary actuators. Positioning error should therefore be very well controlled. The largest sources of the robot positioning error, in my estimation, are in the alignment/parallelism of the axis of the vertical linear actuator to the rotational axis, and the perpendicularity of the horizontal linear actuator to the vertical linear actuator and how well it intersects the rotational axis.
Other systematic errors may include microphone calibration errors, non-uniform directional response of the microphone, etc.
Noise and environmental conditions
The mathematics of the NFS is entirely in the frequency domain. It basically assumes/requires that all measurements be made in the exact same conditions (time invariant). That means if we have varying environmental noise, errors will be introduced into the model. The NFS mitigates this issue by measuring at nearfield to maximize signal-to-noise ratio, and to use double layer measurements to estimate the actual signal-to-noise ratio. Movements of the NFS robot during the test also cause some changes to the sound diffractions/reflections, and violate the time invariant requirement.
Environmental temperature can also affect the results as the speed of sound is proportional to the square root of the absolute air temperature. Changes in the speed of sound alter the ratio between wavelength to physical dimension, and can change the behavior of diffractions and acoustical interference.
Characterization of the system
Please do not consider these as my recommendations to Amir. I am expressing my opinions only.
If I have an NFS (or if I were to build one), here are the steps I will take to characterize the effects of these sources of errors.
Sound field separation
Erect some large plywood boards to act as reflectors around the NFS setup. Compare test results with a few, with many, and without these reflectors.
Positioning effects (system repeatability)
Test the speaker upright, sideways, and upside down. Compare results.
Environmental conditions
Compare the test results of the same speaker tested at different room temperatures.
Play a constant noise in the room and run the test. Repeat with a varying noise. Compare results to no added noise.
Last edited: