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Old 21st February 2016, 09:37 AM   #1
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Default The Gedlee Metric Demystified

Since the 1930's, measurement of Total Harmonic Distortion (THD) has been recognised as a poor arbiter of sound quality. The Gedlee Metric, Gm, attempts to overcome the shortcomings of traditional THD measurements, and to quantify non-linearity in a way that corresponds much more closely to the psycho-acoustical perception of distortion. It is derived from the Transfer Function T(x) of the DUT. Since, in principal, any signal can be used as an input to determine the Transfer Function, Gm can be said to be a characteristic of the DUT rather than a characteristic of the output signal. Characteristics of the human hearing system such as masking have also been allowed for in the formula for Gm. Experimental results indicate that for low to intermediate levels of non-linear distortion, the Gedlee Metric has a much improved correlation to the subjective perception of distortion, and is a significant predictor of subjective sound quality, when compared to traditional metrics such as THD and IMD.

The original AES papers describing the Gedlee Metric are available on Earl Geddes's website:

Lee and Geddes: Auditory Perception of Nonlinear Distortion, AES October 2003. Part I: http://www.gedlee.com/downloads/Distortion_AES_I.pdf
Part II: http://www.gedlee.com/downloads/Distortion_AES_II.pdf

An undoubted obstacle to the adoption of Gm is that the mathematics involved in calculating Gm are complicated compared to the simplicity of THD calculations. It is also likely that many manufacturers are comfortable with consumers' misguided belief that low THD is a meaningful predictor of sound quality, and design and market their products accordingly.

The calculation of Gm involves two steps: 1) the derivation of a function T(x) which represents the normalised transfer function of an audio black box and which is amenable to differentiation; b) plugging T(x) into the equation for the Gedlee Metric Gm proposed by Geddes and Lee in their AES papers.

Published suggested methods for deriving T(x) have to date utilised either the theory of non-linear systems or trigonometric identities to derive a polynomial expression of T(x). I found both approaches very complicated, so I developed a hassle-free method of calculating Gm, and have created a spreadsheet to facilitate this. The spreadsheet derives the coefficients of a trigonometric expression of T(x). This set of coefficients can then be plugged into a short worksheet for Sage Math Online which then calculates Gm virtually instantly. The spreadsheet contains directions on how to use it, together with a breakdown of the mathematics I have used.

The rapid calculation of Gm is therefore possible, given a sinewave input to the DUT and the amplitude and phase information of the harmonics present in the output.

The spreadsheet file can be downloaded from Google Drive using the link below (the file is too large to post directly on diyaudio).

********************
Gm_Sine_Mixer_and_Coefficient_Calculator_Shared.xs lx

Download from: https://drive.google.com/file/d/0B8A...ew?usp=sharing

This version has been developed in Excel 2016, and is best viewed with Auto-hide Ribbon selected. It should also work without problems in Excel 2010 and Excel 2013. It does not load correctly into LibreOffice Calc, but a version for Calc also exists which I may post in due course if there is any demand for it.
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Questions/comments on the use of this solution are invited, though I am reluctant to get embroiled in discussions such as subjective/objective assessments of audio gear, the "best" objective assessments, etc. I am simply posting here a tool that simplifies a particular calculation. The Gedlee Metric does, however, have the potential to explain and quantify, for example, the difference between alternative circuit architectures, even when THD measurements are the same (see first two examples in the spreadsheet).

Forum contributors who quote THD and other distortions in their postings of their designs are, however, encouraged to use this tool to quote Gm too.

Thanks go to Earl Geddes for his support in this study.

Stephen
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Old 21st February 2016, 10:07 PM   #2
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I definitely would be interested in a Calc version!
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Old 22nd February 2016, 11:17 AM   #3
arnyk is offline arnyk  United States
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Originally Posted by Ludus Tonalis View Post
Fail! Can't download it with Chrome, even after connecting a bunch of seemingly related google apps.
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Old 22nd February 2016, 11:42 AM   #4
SY is offline SY  United States
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Downloaded no problem for me (also Chrome). 4.1 meg.
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Old 22nd February 2016, 11:48 AM   #5
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Downloaded fine with Firefox so tried Chrome and worked fine with that as well, just clicked the download button next to the print button.

So for me resounding Pass!

Edit: SY beat me to it, but I was cross-browser testing.
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Old 22nd February 2016, 05:23 PM   #6
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Originally Posted by BaffleChallenged View Post
I definitely would be interested in a Calc version!
You should be able to download the .ods version for Calc here:

https://drive.google.com/file/d/0B8A...ew?usp=sharing

This version, which does not work properly in Excel, is a bit different from the Excel version. It incorporates two different solutions: Method A is the same as the Excel version; Method B uses line statistics to derive the coefficients for a polynomial expression of the transfer function T(x).

Method B was actually my original idea and predates Method A. It is not as precise as Method A, but in practice the two methods produce the same result for Gm to around 5 or 6 s.f. Method B is, however, more forgiving if the harmonics entered are not in exact quadrature.

The line statistics in Excel are nothing like as accurate as they are in Calc, which is why I removed Method B from the Excel version. But as Method A is an exact solution, this is not a great loss. I've left B in the Calc version principally because my Linux box died in December.
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Old 22nd February 2016, 05:25 PM   #7
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Originally Posted by arnyk View Post
Fail! Can't download it with Chrome, even after connecting a bunch of seemingly related google apps.
Have you managed to download it? I have no issues in Edge. If you PM me with your email address I will email it to you if necessary.
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Old 22nd February 2016, 09:14 PM   #8
gedlee is offline gedlee  United States
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All.

In previous discussions, I cannot tell you how many times I have found that people misunderstood what I/we have done. Even so-called "experts" get it wrong. If the metric is useful, then fine, if it is pointless then that's fine too (personally, I don't do much "distortion" testing as it is not highly correlated with perception), but let's at least make sure that we all understand the principles and not continue to propagate the erroneous ideals that have abounded.

Ludus - thanks for your work. I applaud the effort.
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Old 23rd February 2016, 04:07 AM   #9
arnyk is offline arnyk  United States
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Fail! Can't download it with Chrome, even after connecting a bunch of seemingly related google apps.
Redid the downloads over 12 hours later, and this time I obtained good results.

Thanks!
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Old 23rd February 2016, 04:16 AM   #10
arnyk is offline arnyk  United States
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Originally Posted by gedlee View Post
All.

In previous discussions, I cannot tell you how many times I have found that people misunderstood what I/we have done. Even so-called "experts" get it wrong.
IME, the same thing applies to many situations where people who happen to advocate the importance of perceptually vanishing amounts of common artifacts and spurious responses make big errors trying to apply the effects of perceptual masking to distortion and noise measurements, if they try it at all.

Figures 3 and 4 in your first of the two papers seem to lay out the basics of this situation quite simply and clearly.

It may be a case of perceptual inability to correctly perceive reliable evidence that does not agree with their prejudices and long-held positions.
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