The 2011 Richard C. Heyser Memorial Lecture: "Where Did the Negative Frequencies Go?" “What Happened to the Negative Frequencies?” & Science, a Digression

"What Happened to the Negative Frequencies?"
Nothing happened to them, of course, as I will show, but you mustn't forget that they are always there.

Everyone in this room will be familiar with the acronym "FFT." The Fast Fourier Transform is both elegant and ubiquitous. It allows us to move with ease between time-based and frequency-based views of audio events. You will all be familiar with the following example. Here is the waveform of a short section of a piece of music:

And here is the spectrum of that music:

This usefulness of the FFT algorithm—or, more properly, the Discrete Fourier Transform—is everywhere you look in modern audio. If all we had to use were the tubed wave analyzers of my university lab, a life in audio would be very different and very difficult. But I don't like to use tools without understanding how they work—my physics lecturer at university used to yell that we must always try to examine matters from first principles—so in 1981, when got my first PC, a 6502-based BBC Model B, I wrote a BASIC program to perform FFTs, based on an algorithm I found in a textbook. (The computer took around five minutes to perform a 512-point FFT—debugging the program took forever!) This was the process I followed in that program:

1) Capture the discrete-time impulse response of the system

My FFT satori was to realize that you needed to restrict, to window, the impulse response, then stitch its end to its beginning.

Now you have a continuous wave with a fundamental frequency equal to the reciprocal of the length of the windowed impulse, and the FFT algorithm gives the frequency-domain equivalent of that continuous waveform. However, this was the spectrum I obtained from that program:

You get two spectra: one with positive frequencies, corresponding to ei (angular frequency), the other with the same amplitudes of the same frequencies but with a negative sign, corresponding to e–i. You can visualize this as the spectrum being symmetrically "mirrored" on the other side of DC. The negative spectrum is discarded—or, more strictly, you extract the real part of what is a complex solution, and subsequently work with the modulus of the spectrum; ie, the sign is ignored.

But note the assumptions you have made: 1) the fabricated continuous wave extends to ±Infinity, which is untrue—even a Wagner opera has to eventually end—and 2) what happens at the point where the end of the impulse response is stitched to the beginning? What if there is a discontinuity at that point that mandates you having to apply some sort of mathematical "windowing" function to remove the discontinuity from the impulse response data? (Programs using FFTs should have a "Here Lie Monsters" pop-up when you apply the transform before you've checked that you're using the right window for your intended purpose.) And you have made a value judgment only to use the positive frequencies. While having done so will not matter if, for example, you are concerned only with the baseband behavior of a digital system, it will matter under different circumstances—as I will show when I get on to digital systems.

Note also that the frequency resolution of the spectrum is directly related to the length of the time window you used. If that window is 5 milliseconds in length, the datapoints in the transformed spectrum are spaced at 200Hz intervals, which is not a problem in the treble but a real problem in the midrange and bass.

The title of my lecture is therefore a metaphor: You cannot assume that the assumptions you make as an engineer will be appropriate under all circumstances. You almost need to know the result of a calculation before you perform it.

Science: A Digression
I referred earlier to using tubes in my engineering education; pocket calculators were not introduced until after I graduated from university, so my constant companion back then was my slide rule.

My attitude to science was conditioned by my trusty slide rule. Slide rules are elegance personified: not only do you need to have an idea of the order of the answer before you perform the calculation, a slide rule prevents you from getting hung up on irrelevant decimal places in that answer. And you can't forget that, even when you obtain an answer, it is never absolute, but merely a useful approximation.

In physics, you learned to believe one impossible thing before breakfast every day. The strangeness starts when you learn that a coffee cup with a handle and the donut next to it are topologically identical. When you learn that a stream of electrons fired one at a time at a pair of slits in a barrier create the same interference pattern as if they had all arrived simultaneously. And by the time you get to string theory, it's all strange: if you read Leonard Susskind's books, you will find that string theory predicts that every fundamental particle in the universe is represented by a Planck-length tile on the surface of the universe, and that the surface area of the universe just happens to be exactly equal to the sum of the areas of those tiles. Nothing in audio is that strange!

However, when Susskind writes things like "quantum gravity should be exactly described by an appropriate superconformal Lorentz invariant quantum field theory associated with the AdS boundary," my eyes glaze over, and I reach for that donut and coffee I mentioned earlier. But when you study physics, deep down you grasp that "Science" never provides definitive answers, or even proof. Naomi Oreskes and Erik M. Conway wrote in their 2010 book, Merchants of Doubt: "History shows us clearly that science does not provide certainty. It does not provide proof. It only provides the consensus of experts, based on the organized accumulation and scrutiny of evidence."

And that evidence is open-ended. Even with that scrutiny, there are always the outliers, the things that don't fit, that are brushed aside. I am reminded of the old story, which I believe I first heard from Dick Heyser, of the drunk looking for his keys under a street lamp. A passerby joins in the search, and after a fruitless few minutes, asks where the drunk has dropped them. "Over in the bushes," answers the drunk, "but it's too dark to look there."

The philosopher Karl Popper said, "Science may be described as the art of systematic oversimplification." This as true in audio as it is in science: As Richard Heyser explained in 1986, when it comes to correlating what is heard with what is measured, "there are a lot of loose ends!" It is dangerous to be dismissive, therefore, of observations that offend what we would regard as common sense. In Heyser's words, "I no longer regard as fruitcakes people who say they can hear something and I can't measure it—there may be something there!"

Which brings me to the subject of testing and listening

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Comments
bernardperu's picture
Why would you read Stereophile anyway?

JhonnyR

 

If Stereophile has the principle of not doing DBT and for you doing so is a matter of principle why oh earth would you even care to read Stereophile? Your statement above suggests you have very extensive knowledge of the content of Stereophile's articles. So you keep reading stereophile's reviews knowing they are irremediably flawed?? How perverse is this?

 

The real question lies on your double blind tested hatred. 

 

Whatever your answer is I will have to assume that your hatred is a broken record and it will just go on.

GeorgeHolland's picture
He probably reads them as I

He probably reads them as I do for the laughs.smiley

JohnnyR's picture
You Didn't Comment..........

......on your error filled post above so why should I comment on your assumptions? If you wish to worship Fearless Leader then by all means bow down and do so. I myself prefer to ask questions and be skeptical about his "methods" and the reason "why" he doesn't test ALL items under review.

John Atkinson's picture
You're Welcome

Andreasmaaan wrote:
It's a pity that some proponents of DBT as the only valid methodology have used the comments thread here to launch personal attacks against JA.

Indeed. I am not sure why I have become a lightning rod for these audio skeptics. Perhaps it is because it is a matter of religious belief on their part - having had extensive experience of blind testing, I rejected it and thus became a heretic in their eyes. Or perhaps it is just our courtesy in allowing them the space on this website to express themselves.

Andreasmaaan wrote:
JA does measure every piece of gear his reviewers review - a practice which ensures that the opinions of the reviewers are grounded in objective data, or otherwise as the case may be.

As my reviewers do not see the measurements until after they have written the review, I take my hat off to them.

Andreasmaaan wrote:
To cut what is risking becoming a lengthy expression of indignation short: thank-you JA for a wise and thought-provoking read.

You're welcome and my thanks to everyone else who appreciated the preprint. It was an honor to have been invited by the Audio Engineering Society to give this lecture - it's not often that you get the opportunity to look back over a 4-decade career!

A Happy New Year to everyone who surfs this site.

John Atkinson

Editor, Stereophile

JohnnyR's picture
Ahhhhhhhhhh Typical Atkinson Answers

You ignore the errors and just plow along blissfully gobbling up the unwarranted praise.

"JA does measure every piece of gear his reviewers review"

As I pointed out above, along with other errors on the part of Andreasmaaan,you do NOT test every piece of gear your reviewers review, but then you pretended that you do in your response. Only certain items are tested by you. Please get your own facts straight if that's even possible for you to do so.

Regadude's picture
Johnny brand speakers

Johnny, I wish you a great new year 2013, in which you sell 10 times more of your home made speakers than in 2012. 

Since you sold zero of your speakers in 2012, you need to sell 10 X 0=0 speakers in 2013 to meet this objective. Maybe you can hire George Holland as your director of marketing and sales... 

Get cracking, you have only got 364 days left to sell your quota...

JohnnyR's picture
Happy 2013 Raghead

[flames deleted by John Atkinson]

Unlike yourself Raghead, I will be DESIGNING and BUILDING my own speakers and learning. You do know that word "learning"? Hmmmm maybe you stopped after age 10. You on the other hand will be on here *yawning* and acting the fool and spending your money on Stereophile reviewed crapola.yes

Regadude's picture
So only your speakers are good?

So according to you Johnny, everything Stereophile reviews is crapola?! There is not one good piece of equipment that has been reviewed by Stereophile? Hundreds of reviews, not one good product?

But you, in your basement, build quality speakers... 

I have challenged you before (as well as George) to name your gear. What kind of gear merits the Johnny seal of approval? 

As always, you always ignore my challenge (George chickened out to). It seems you can only complain, but cannot even provide a list of your gear. 

I guess ever seeing pictures of your "famous" speakers is also out of the question. 

All talk and no action. 

Regadude's picture
Johnny the pussy

So Johnny, where are your speakers pics and specs?

In your imagination.... [flame deleted by JA]

GeorgeHolland's picture
Johnny never offered to show

Johnny never offered to show you anything other than his backside which I think is approriate considering the childish actions of yourself.What does it matter what our audio components are? Anything we mention would be ridiculed by yourself no doubt. Sorry but I'm not playing your silly game. Good God grow up and act like a man for once will you? angry

John Atkinson's picture
Getting Facts Straight

JohnnyR wrote:
As I pointed out above . . . you do NOT test every piece of gear your reviewers review, but then you pretended that you do in your response. Only certain items are tested by you. Please get your own facts straight if that's even possible for you to do so.

Out of morbid curiosity, I looked at the past 15 issues of Stereophile - we published 98 full reviews and follow-up reviews. Of those 98 reviews, 83 were accompanied with a full set of measurements, or 84.5%. So even if not every review includes measurements, the vast majority do, which supports Andreasmaaan's point.

I have explained to you before, in other comment threads, why we do not publish measurements of analog playback components, which is due to lack of resources. I have also explained to you why we do not publish measurements of tweaks, which is that it it is difficult to determine exactly what to measure. I don't see any reason to reopen those discussions, but that you continue to be fixated on these two issues and continue to raise them is sad.

And, as I have warned you, JohnnyR, I am continuing to delete comments from you, and others, that are nothing more than flames aimed at other readers.

John Atkinson

Editor, Stereophile

JohnnyR's picture
Get Out Your Dictionary

"Reviews" means just that, any item reviewed by your staff. If you think "tweaks" don't qualify then they should be labled as "Opinions" not a review.angle All of those full reviews you speak of were what speakers, amps, preamps, cd transposrts and the like?

Yeah I know why you don't test tweaks, that would involve you using those horrible SBT or DBT. That's a really nice system you have set up. "We don't believe in DBTs or do them or can't afford to do them"......"We don't know what to test on the tweaks"..........add the two up and you get "We don't have to show that tweaks actually do anything other than what our "reviewer" subjectivly alludes to in their sighted biased listening using their golden ears"......EXCUSES Mr Atkinson plain and simple. no

Once more I ask you to read and comment about this link:

http://forums.audioholics.com/forums/loudspeakers/83412-diy-loudspeakers...

You seem to think DIY can't do anything right in your previous post in answer to Raghead. Insulting intelligent people that actually build better speakers than the status quo can produce is beyond being pompus in your case.THAT is what's SAD.

Don't you have some real work to be doing instead of being on the forums? I thought that was Ariel's job. Let him earn his pay.

ChrisS's picture
Circle Game

JRusskie,

Look up "fixation" and "perseveration" in your dictionary. "Absolute truths" regarding audio products seems to exist only in your basement and in your head.

John Atkinson's picture
What?

JohnnyR wrote:
John Atkinson wrote:
I looked at the past 15 issues of Stereophile - we published 98 full reviews and follow-up reviews. Of those 98 reviews, 83 were accompanied with a full set of measurements, or 84.5%. So even if not every review includes measurements, the vast majority do, which supports Andreasmaaan's point.

"Reviews" means just that, any item reviewed by your staff. If you think "tweaks" don't qualify then they should be labled as "Opinions" not a review.angle

I am not sure what you mean. You were objecting to Andreasmaan's statement that Stereophile accompanies its reviews with measurements as not being true. I offered the analysis above to show that he was correct.

JohnnyR wrote:
Once more I ask you to read and comment about this link:

http://forums.audioholics.com/forums/loudspeakers/83412-diy-loudspeakers...

You need to consult with your fellow traveler GeorgeHolland, who criticized me about linking to other sites :-)

JohnnyR wrote:
You seem to think DIY can't do anything right in your previous post in answer to [Regadude.] Insulting intelligent people that actually build better speakers than the status quo can produce is beyond being pompus in your case.THAT is what's SAD.

With respect, you are arguing with the voices in your head here. I haven't written anything about DIY speaker designers not being able to do "anything right." What I have said is that a DIY designer, like yourself, doesn't subject his loudpeakers to the scrutiny of a disinterested marketplace, something professional designers do as a matter of course. So the question whether or not DIY designs are superior to commercial designs isn't put under independent scrutiny. It remains unsupported conjecture, as far as I am concerned.

John Atkinson

Editor, Stereophile

GeorgeHolland's picture
Mr Atkinson, you could 

Mr Atkinson, you could  compare DIY against manufactured speakers but of course you won't. Might upset those that can't make good speakers and I'm not talking about the DIY cheeky

Johnny plainly stated that "tweaks" are reviewed in your magazine and online yet they aren't tested, hence you don't test every component under review. You have a reading comprehension problem? How many components reviewed by Jason Serinus have been tested? uh huh, right.

So now you wuss out about commenting on links? That proves you had nothing to say in the first place about it so you shouldn't have opened your mouth.Cop out time for Mr Atkinson.

The only unsupported conjecture on here are your abilities to print honest reviews. Your ability to avoid testing  alot of audio components is well known by now. Very impressive spinning you do on here.wink

John Atkinson's picture
Reading Comprehension

GeorgeHolland wrote:
Mr Atkinson, you could  compare DIY against manufactured speakers but of course you won't. Might upset those that can't make good speakers and I'm not talking about the DIY cheeky

Back in the early 1990s, Stereophile's Corey Greenberg was one of the judges for a DIY loudpeaker competition in San Francisco. There were 2 winning designs and as part of the prize, those 2 speakers were subject to a full set of measurements in Stereophile, published in the March 1992 issue. The speakers' measured performance was good but not great and certainly didn't embarass professional designers.

GeorgeHolland wrote:
Johnny plainly stated that "tweaks" are reviewed in your magazine and online yet they aren't tested, hence you don't test every component under review. You have a reading comprehension problem?

I don't believe so. JohnnyR was wrong, in that we very rarely publish full reviews of tweak products. Of the 98 reviews published in the magazine January 2012 through March 2013, there was just one review of a "tweak," Robert Deutsch on the HiFi Tuning Supreme Fuses in May 2012. ( I don't count the AudioQuest power strip reviewed by Kal Rubinson's review in December 2012 as Kal didn't make any comment on its sound quality, only on its utllity.)

GeorgeHolland wrote:
How many components reviewed by Jason Serinus have been tested? uh huh, right.

The correct answer is none. But that doesn't support JohnnyR's case either, as Jason's reviews are published on the website Secrets of Home Theater and High Fidelity, which has no connection with Stereophile. I hardly believe I am obliged to support reviews of products in competing publications with measurements in Stereophile.

John Atkinson

Editor, Stereophile

JohnnyR's picture
Wow Nostalgia Time Kids

"Back in the early 1990s, Stereophile's Corey Greenberg was one of the judges for a DIY loudpeaker competition in San Francisco. There were 2 winning designs and as part of the prize, those 2 speakers were subject to a full set of measurements in Stereophile, published in the March 1992 issue. The speakers' measured performance was good but not great and certainly didn't embarass professional designers."

Golly "only" 20 years ago and as we all know, nothing new has happened since then to make speaker design any better (Insert BIG sarcasm face here) cheeky Talk about a lame example Atkinson.

Yeah yeah, "full reviews" whatever, Plenty of "little revews" though all the freakin time and all they are , are OPINIONS but you know what? You let your reviewers get away with murdering the intergrity of the audio world with their stupid banter about  "blacker backgrounds,lifted veils and HUGE improvements in the sound" without them having to justify their claims, Pretty slick there pal.

Oh and Jason Serinus "reviews" as posted about the recent RMAF isn't connected to Stereophile? Give us all a break, Your EXCUSES are even lamer this time than usual.

Yep typical Fearless Leader goobley gook. Did you major in Universisty in the double speak of Orwell's 1984? If not then you are gulity of plagiarizing the concept.

John Atkinson's picture
Not making sense

JohnnyR wrote:
Golly "only" 20 years ago and as we all know, nothing new has happened since then to make speaker design any better...

Again you are arguing with the voices in your head, JohnnyR. Of course a lot has happened in the past 20 years, especially the advent of low-cost measuring equipment. But you're missing my point, which is, as I wrote earlier today, that a DIY designer doesn't subject his loudpeakers to the scrutiny of a disinterested marketplace, something professional designers do as a matter of course. You have repeatedy claimed that your speakers are as good as if not better than commercial designs. However, unlike engineers like Kevin Voecks, Paul Barton, Richard Vandersteen, Jeff Joseph, etc, your ability to pay your mortgage and feed your family doesn't depend on your skill as a speaker designer. Their's does, and it makes a difference.

Look, if you seriously believe your speaker designs are fully competitve with commercial designs, email me your measurements. I'll get back to you with my interpretation of what they mean, assuming they are as comprehensive as what I publish with every Stereophile speaker review. Put up or shut up.

JohnnyR wrote:
Yeah yeah, "full reviews" whatever, Plenty of "little revews" though all the freakin time and all they are , are OPINIONS but you know what? You let your reviewers get away with murdering the intergrity of the audio world with their stupid banter about  "blacker backgrounds,lifted veils and HUGE improvements in the sound" without them having to justify their claim...

You're not making sense. "Andreasmaaan" was discussing reviews in the magazine being accompanied by measurements. I was discussing reviews in the magazine being accompanied by measurements. You seem to be discussing something that exists only in your mind. And unlike Professor Dumbledore's quote, that doesn't mean it's real :-)

JohnnyR wrote:
Oh and Jason Serinus "reviews" as posted about the recent RMAF isn't connected to Stereophile?

That was a show report. No-one other than yourself equates a magazine's coverage of a show with its formal equipment reviews. Are you seriously suggesting that magazines shouldn't publish show reports without including a full set of measurement data for each room they report on? Really? There isn't an audio magazine or website that does that.

John Atkinson

Editor, Stereophile

JohnnyR's picture
BLAH BLAH BLAH

Go have another pink drink Fearless Leader.

Me let YOU interpet my frequency response? Don't make me LAUGH. You can't even interpept Wilson't ragged ass frequency response laugh

[usual flames deleted by John Atkinson]

John Atkinson's picture
Making my point

JohnnyR wrote:
John Atkinson wrote:

Look, if you seriously believe your speaker designs are fully competitve
with commercial designs, email me your measurements. I'll get back to
you with my interpretation of what they mean, assuming they are as
comprehensive as what I publish with every Stereophile speaker review.

Me let YOU interpet my frequency response? Don't make me LAUGH.

You make my point for me, JohnnyR. It was a serious offer on my part. Professional loudspeaker designers send out their products for review and for the public scrutiny of possible customers. Amateur speaker designers, such as yourself, are often afraid to let anyone else judge their work.

John Atkinson

Editor, Stereophilee

Regadude's picture
There you go

I have been challenging Johnny tp provide pictures and or the specs of his speakers. He has never met my challenge.

Now the editor of Stereophile offers to look over his data. Johnny again makes up excuses to avoid providing any information on his heavenly speakers.

Big talker...

bernardperu's picture
Contamination

Dear Jon,

What should have been an enlightening discussion of your great lecture has been contaminated by ill-conceived comments. Some individuals have posted comments that have created an environment of nastiness. 

I would seriously urge you to start deleting ill-conceived comments from now on, whether they use flamable language or not. Some people are just turned on by the flames, but others aren't. And those others would much rather focus on the joy of engaging in the music and exchanges related to it.

You do not run a state-owned site but a private one that is likely to become a lot less lucrative (in all senses) if such nasty comments are to become visible.

I am not even pointing out names and we all know who I am talking about.

I am just giving you feedback that comes from an avid reader of Stereophile online. 

Finally, the correction of my use of English was mentioned by one of these individuals. Please allow me to say that English is my second language and it is oxidized. Sorry about that. 

Thanks!

 

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