How Many Frequencies Are Enough To Characterize Nonlineariti

[Gary Brown]

Hi,

About thirty years ago I was part of a research project to
analyze nonlinearities in receivers. The approach was to
combine frequencies, calculate harmonics from emperical
coefficients, then calculate higher order harmonics. But I
don't remember how many frequencies are enough to
adequately characterize the nonlinear behavior. Does
anyone out there know?

FWIW: The analysis was for IF sections of receivers but I am
interested in its application to audio.

Thanks,
Gary

 

[Kevin Aylward]

Gary Brown wrote:

Hi,

About thirty years ago I was part of a research project to
analyze nonlinearities in receivers. The approach was to
combine frequencies, calculate harmonics from emperical
coefficients, then calculate higher order harmonics. But I
don't remember how many frequencies are enough to
adequately characterize the nonlinear behavior.

How long is a piece of string?

Does
anyone out there know?

If the system has an error spec, i.e., it must be say, 0.01% accurate,
then that will mean that the thd needs to be less than 0.01%, or
thereabouts. There is no general method, it all depends on the specific
application.

FWIW: The analysis was for IF sections of receivers but I am
interested in its application to audio.

Many audio applications are not designed with respect to any objective
distortion critear ata all. Its a smallest numbers marketing game.

Kevin Aylward
salesEXTRACT@anasoft.co.uk
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

"quotes with no meaning, are meaningless" - Kevin Aylward.

 

[John Woodgate]

I read in sci.electronics.design that Gary Brown
<garyjbrown@charter.net> wrote (in <1062bo4jujruaa8@corp.supernews.com&gt
about 'How Many Frequencies Are Enough To Characterize Nonlinearities?',
on Wed, 24 Mar 2004:

About thirty years ago I was part of a research project to analyze
nonlinearities in receivers. The approach was to combine frequencies,
calculate harmonics from emperical coefficients, then calculate higher
order harmonics. But I don't remember how many frequencies are enough
to adequately characterize the nonlinear behavior. Does anyone out
there know?

FWIW: The analysis was for IF sections of receivers but I am
interested in its application to audio.

There are several Audio Engineering Society papers about multitone
testing. www.aes.org
--
Regards, John Woodgate, OOO - Own Opinions Only.
The good news is that nothing is compulsory.
The bad news is that everything is prohibited.
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk

 

[Peter O. Brackett]

Gary:

Dr. Norris Hekimian, [RIP] an old Bell Labs scientist and the founder of
Hekimian Laboratories said, "it takes 4 tones"!

Heh, heh...

Four is the number of tones used by the Hekimian test method which he
patented many years ago. Check the USPTO www site for inventors named
"Hekimian" to find the patent number. The well known Hekimian Four Tone
Test Set has been widely used by telephone companies for decades to measure
the non-linear distortion of telephone plant, specifically to characterize
telephone channels for high speed digital data transmission which place much
higher demands on linearity than voice transmission does.

But truthfully... the answer to your question is that "it all depends" [on
the non-linearities and the degree of linearity that the application and end
user can tolerate].

"Linearity" means "straight". What you are trying to do when you
characterize non-linear distortion is to measure the deviation of, say, a
gain function from a mathematical straight line.

As the saying goes, "beauty is in the eye of the beholder" and so is
non-linearity!

There are many ways in which such a measurement of linearity error can be
performed, and there are many ways in which the deviation from linearity can
be expressed.

For instance if the device is a voltage amplifier then the ideal linear gain
would be say:

Vo = G Vin

Where G is the gain, and clearly a plot of Vo versus Vin is a straight line
of slope G.

Ultimately the deviation from linearity could be measured directly and a
mathematical difference or error function taken and plotted graphically. If
one then made careful measurements of the actual values of Vo for actual
applied values of Vin, and plotted this experimentally determined function,
let's call it "f(Vin)" then the non-linear error e(Vin) is the simple
difference:

e(Vin) = [GVin - f(Vin)] and you are done... no tones involved!

The difference could be expressed absolutely as abover, or perhaps in
relative or percentage terms e.g. e(Vi)/Vin, etc...

However because the error e is often extremely small in relative terms such
direct measurements are tedious at best and fraught with great
instrumentation difficulties or impossibilities at worst. With many
applications and equipment such direct measurements are impossible!

When it comes to measuring non-linearities, "the devil is in the details"!

Unfortunately many [most] approaches to measuring non-linearity are often
proprietary are not standardized and often based upon "quick and dirty"
methods intended only to get an "indication" or an approximation to the
error. Seldom is the end measurement or result of these "schemes" related
back to the actual error e(Vin)! I know of no examples where this has been
done. Anyone?

Following mathematical power series methods [e.g. Taylor's Series from
elementary differential calculus] one can "represent" the acutal
measurements of the gain function of a practical device Vo = f(Vin) by a
power series say:

f(Vin) ~ F0 + F1*Vin + F2*Vin^2 + F3*Vin^3 + ... to as many terms as the
user sees necessary. More is good!

Since Joseph Fourier discovered his famous series a couple of centuries ago
many have noted that if a "pure" sinusoidal waveform [sine wave] is applied
to a device with non-linearity, say expressed mathematically as a Taylor's
Series then simple trigonometric function manipulations and identities from
high school trigonometry will show that the output must contain "harmonics"
of the original pure sinusoid. The output waveform will no longer be a pure
sinusoid rather it will contain the original pure sinusoid plus several pure
sinusiodal harmonics of it [Harmonic Distortion].

And so many modern methods of measuring non-linearity are based based upon
frequency or spectral analysis.

Apply a pure tone [to measure Harmonic Distortion] or several pure tones [to
measure Harmonic plus Intermodulation Distortion] to the input and then
measure all the tones you find in the output and subtract out the expected
original tone. The remainder are the results of the distortion. The power
in the remainder of the tones, the ones that shoyuld not be there, are often
totaled up and called THD or Total Harmonic Distortion. If only one tone
was applied the result is called simple Harmonic Distortion. If more than
one tone was applied the distortion results are due to both simple harmonic
generation and the generation of, or intermodualtion of, the various several
tones.

How you relate THD to the error e(Vin) is a very good question, but
undoubtedly for a device to be linear i.e. "good" the THD should be as small
as possible.

Norris Hekiminan used four tones, for many radio receiver and audio
applications a simple two-tone test is usual, and so two-tone tests are all
that are used in many applications. Quite often manufacturers or commercial
labs will publish the tone frequencies that they use and present the results
of laboratory tests often in terms of estimated or "projected" intercept
points [IPs] where the distortion produced tone(s) power would equal the
linear tone(s) power if one applied a sufficiently high enough input power
level, one that cannot even be safely applied! IP's are usually given in
dB.

Sometimes the output signal is simply digitized by a "sufficiently accurate"
A/D [Analog to Digital] converter and then the result is transformed over to
the frequency domain by an FFT [Fast Fourier Transform] algorithm so that
all the tones can be seen on a spectral plot and read out from the digital
output. This FFT technique is in fact the one most often used by A/D
manufacturers.

These THD, IP and FFT measurements can then be used by consumers or buyers
to compare equipment. The lower the THD or the higher the IP, or the fewer
and the lower the tones in the FFT the better! There are however no
definitive ways to relate THD and/or IP back to the actual error e(Vin)!
Beauty is in the eye of the beholder.

There is some information on such radio receiver testing in some ARRL
publications available on the ARRL Web site at http://www.arrl.org. Many
A/D manufacturers like Analog Devices Inc, Linear Technologies Corp, Cirrus
Logic/Crystal Semiconductor, etc... publish their methodologies for testing
A/D's. Google to their sites and download their application notes and data
sheets.

For high end audio applications there are a few companies that manufacture
distortion test equipment for high end audio who
have published information on such tests. Audio Precision is one such
company, more can be found by googling
around the web.

It is easy to show mathematically, and one can even write down, a
particular e(Vin) for which one tone could completely characterize the
error, but it is also easy to show that for an arbitrary error e(Vin) an
infinity of tones may be required to exactly reproduce the error!

So in answer to your first question, one tone does some good [Harmonic
Distortion], two tones are better [Intermodulation Distortion], many tones
[Harmonic plus Intermodulation] are much better!

--
Peter
Professional Consultant
Signal Processing and Analog Electronics
Indialantic By-the-Sea, FL


"Gary Brown" <garyjbrown@charyter.net> wrote in message
news:1062bo4jujruaa8@corp.supernews.com...

Hi,

About thirty years ago I was part of a research project to
analyze nonlinearities in receivers. The approach was to
combine frequencies, calculate harmonics from emperical
coefficients, then calculate higher order harmonics. But I
don't remember how many frequencies are enough to
adequately characterize the nonlinear behavior. Does
anyone out there know?

FWIW: The analysis was for IF sections of receivers but I am
interested in its application to audio.

Thanks,
Gary

 

[John Woodgate]

I read in sci.electronics.design that Peter O. Brackett <no_such_address
@ix.netcom.com> wrote (in <NWc8c.5177$HP.2368@newsread2.news.atl.earthli
nk.net&gt about 'How Many Frequencies Are Enough To Characterize
Nonlinearities?', on Wed, 24 Mar 2004:

So in answer to your first question, one tone does some good [Harmonic
Distortion], two tones are better [Intermodulation Distortion], many
tones [Harmonic plus Intermodulation] are much better!

There's more. The linearity is very often indeed frequency-dependent.
Measurement at one frequency does not take this into account. Two
frequency tests can do so, but only crudely. There are two standard
forms of test (IEC/EN 60268-3). Modulation distortion (aka SMPTE
distortion) measures most usefully linearity at low frequencies, using a
higher frequency as a 'search tone'. Difference-frequency distortion
(aka CCIF distortion) measures most usefully linearity at high-
frequencies.

For the features of multi-tone testing, see the AES papers I mentioned
previously. The results may present as a spectrum so complex that
analysis is difficult.
--
Regards, John Woodgate, OOO - Own Opinions Only.
The good news is that nothing is compulsory.
The bad news is that everything is prohibited.
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk