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

GeorgeHolland's picture

No you may address me as George or Mr Holland. Georie is an attempt at making fun but then again that's about all you know to do anyway.

Please ask a relevant question or just shut it.

The lamb would first have to prove they can hear a difference with a dbt

The shepard can buy whatever he wants.

The big bad wolf tried to convince both that the expensive boutique amp was the one to buy

They told him to shove it and bought a less expensive but well built amp and they all lived happily ever after except for Mr Big Bad who soon folded his shop due to no sales.

ChrisS's picture

So Georgie,

Let's make this question relevant....

Let's say we're testing two amplifiers with two listeners. The first listener is an 18 year old young lady who is trained in classical piano at a Grade 10 music conservatory level. She can hear that Amp A has an excellent range 16hz-40khz through the test system, but even though Amp B doesn't have the same range, she likes the "sound" of it better.  The second listener is the shepherd boy who's grown up now. He's 54 years old, likes big band jazz, but his hearing has been damaged by working with heavy machinery without hearing protection. He can't hear a difference between the two amps.


Which amplifier should the ex-shepherd buy?

GeorgeHolland's picture

The 18 year old can hear to 40KHz?  Were her parents bats?

"Like" doesn't have anything to do with blind testing. You don't pick which one you "like" you see if you can tell WHICH amp is playing. You don't even know how a dbt test is run I can see already.

 Who cares which amp they buy?  Maybe the people selling them do but that is completly irrevelant to dbt. You have no clue as to what blind testing is all about.

ChrisS's picture


Are you a real person or a computer generated figment from JRusskie's russian clone of an old IBM PC? Do you know anyone with normal hearing? Do you know how real people shop or do you isolate yourself in the sanctuary of your closed mind and order everything on-line after reading extensive reviews in Consumer Reports?

ChrisS's picture

You call those proper Double Blind studies?

GeorgeHolland's picture

Oh so you are an expert on double or single blind studies. The ABX system is a proven dbt way to do things. Just because the results have you so upset, you claim the people doing the testing are doing it wrong? Laughable. Tell me some more jokes.

ChrisS's picture

...Swinging down the street so fancy-free..."


In fact, Georgie, my major(s) for my undergraduate degree were in Developmental Psychology (including Perception) and Statistics (including Research Methodology). So yes, the set up and methodology shown in those links are crap and the results are laughable...

ChrisS's picture

Get thee to a local college and enroll in a first year research methodology course. Have fun learning!

GeorgeHolland's picture

Either address me as George or STFU you stupid little boy. I think you majored in being a twit and smart ass. Who can take anything you say as serious? Grow the fuck up already. You act out like a lil boy with the IQ of a rock.

Go tell the people who make the ABX test system what you just said and see how they laugh you out of the room. You bring nothing to this discussion other than what you don't agree with , with zero facts to back up your claims. Come on show us all how the ABX test methods aren't any good or why the blind testing done in the othe link was faulty. Better yet tell Harman Kardon that their blind testing techniques are faulty and not worth doing.

ChrisS's picture


You must be running out of neurons if you don't trust your own eyes and ears. Yes, the facts are out there.

JohnnyR's picture

.is what SBT and DBT are all about. Using your eyes to test audio products? Well yes I can SEE that YOU would have to look so you would know which one is "better".

ChrisS's picture

Whose ears? Why?

John Atkinson's picture

GeorgeHolland wrote:
I find it sad that Stereophile keeps saying that DBT or even SBT are not a valid way to test those claims.

Please do not put words in my mouth. That is not what I have said. What I _have_ written is that to design a blind test that limits the variables to just that which you are interrsted in and that produces valid results when there is a small but real audible difference is complicated and time-consuming. The literature is full of poorly designed and performed blind tests that have been proclaimed by audio skeptics as "proving" that there are no audible differences. Such people demonstrate both their ignorance of the Scientfiic Method and their unquesitoning faith in "Scientism."

John Atkinson

Editor, Stereophile

GeorgeHolland's picture

JohnnyR was right more EXCUSES.

It's pretty simple Mr Atkinson but then having the will or gumption to put a dbt into the line of testing is the first thing you have to have,

You obviously don't have that or just don't care so it's a moot point anyways.

JohnnyR's picture

.nor ever will when money is involved.

dalethorn's picture

I would hate to see magazines and websites like Stereophile become intimidated by naysayers who demand "proof" of everything they say, in advance or after the fact. There's a lot of that in mainstream media, due no doubt to controversial topics and false information being fed to reporters. But come on, people - this isn't a mainstream news outfit reporting life or death stories. We have here an incredibly rich article full of facts that can be researched and questioned with references that are well established over time. Instead we have people questioning the author's motives or his pursuit of truth? I think people who are looking for "The truth" should be looking in a religious forum, not a hi-fi forum. There's very little you can "prove" on these topics - the value here is the very informed opinion that costs you nothing.

Ariel Bitran's picture

nice comment Dale. as an attendee of this lecture, i can tell you it was surely enlightening.

GeorgeHolland's picture

I think you have it all backwards there friend.....

"I think people who are looking for "The truth" should be looking in a religious forum, not a hi-fi forum. There's very little you can "prove" on these topics - the value here is the very informed opinion that costs you nothing."

Religion is based upon belief and subjectivists cling to their belief, they don't go looking for proof or the scientific method. No you can't prove very much when the reviewers use subjective say so instead of actually measuring the units. Opinons are a dime a dozen or even less than that and worth next to nothing. Just look at all the opinions here.

dalethorn's picture

No, George, you don't get it. You're still stuck in religion, looking for proof of something. Here you get 'information' only, and if you want 'proof' of something, you have to do the work in proving it to yourself. What *you* believe, outside of yourself, is purely opinion. Perhaps all these people looking for truth or proof are just lazy, and trying to intimidate others into doing the work for them. Like bullies.

JohnnyR's picture

......the facts there pal. Your "arguement" is FLAT.  The only "info" Stereophile shows us is what the WANT to show us. Cables, power cords, magic bowls are off the list of even testing them in anyway what-so-ever. The reason? Ohhhhh we really don't know how to test those duhhhhh. How lame an EXCUSE is that?  Stereophile is SUPPOSED to be a magazine for information NOT excuses.

"Perhaps all these people looking for truth or proof are just lazy, and trying to intimidate others into doing the work for them. Like bullies."

 BWAHAHAHAHAH!!! that has to be on my "Top 10 WTF Things of 2012"

 You DIDN'T just say that did you???? So let me get this straight, Stereophiles job as you see it is to just fling out "say so" and it's up to the readers to wade through the muck and mire of those reviews to try and grasp one little bit of truth? Yeah right, you must really love it then because they rarely show any truth at all.

Ariel Bitran's picture

why is it so hard to accept that double-blind listening tests are difficult to achieve as JA has explained in his lecture?

the fact that our existences are commandeered by individual perception based on thousands of variables makes it very easy for me to understand, just as how one person may enjoy spicy foods but not grapefruit or where some may hear too much bass and others not enough. so many VARIABLES!!! culture, upbringing, what sounds you are surrounded by, traffic signals, your genetic structure, your actual physical position when listening. perception is a learned skill that we do not choose to accept, it just happens and it is different for every single person.

i think THESE are the sort of differences between individuals that make DBT difficult: everyone hears differently. there is no absolute sound.

the best example of how an ear and sonic preference can change is in the study of language and sounds. the chinese language has a completely different set of sounds to that of the english language, thus their speaking intonation, laughter, and music reflect their cultural and sonic inclinations. eastern and western and andean and greek and celtic and ... and ... all use completely different scales based on their preferences of sound learned over time through language and their environments.

Thus, i often wonder do hi-fi listeners across the globe prefer different sounding systems based on their installed sonic memory? or is there a constant in terms of preference across the globe? probably not. or even more interestingly, can one find similarities in preferences in sound based on linguistic sounds of an individual region? are the frequencies accented in the german language more easily noticed by a german in his hi-fi? DBTs are a waste of time. instead of focusing why not, it is much more fun to focus on the why.

the heart of all of this lies within JA's question: where do the negative frequencies go? there are aspects to our perception of sound that simply cannot be measured because they are based on individual perception which is different for every single one of us.

GeorgeHolland's picture

If blind testing is so difficult then how did the people that I linked to manage to do so?  Harman Kardon does blind testing at the drop of a hat. Go ask them how they do it so easily. Mr Atkinson's refusal to do so is simply an excuse as to not have to bring up why cables, power cords and other snake oil is indeed snake oil. He can merrily go along his way as he has for years now ignoring such products and letting his reviewers say whatever BS they want about the sham products and not have to worry one bit. He just doesn't care is the bottom line.

ChrisS's picture

Georgie Porgie,

That you cite these links as authoritative sources indicates the level of your understanding of testing methodology.

JohnnyR's picture

......nothing to back up whatever it is you are trying to say but it is amusing.

ChrisS's picture

Research methodology courses are taught in colleges and universities all over the world, even Russia...

Let us know when you and Georgie take one.

JohnnyR's picture

Nothing to cite other than your own wandering silly posts? Thought so.

Regadude's picture

Well little Johnny, at least Chris is A REAL psychologist. He's not a, you know, a hobbyist like yourself...

JohnnyR's picture

........credentials from ChrisSy just say so. Oh lets see I think I'll be a nuclear scientist now just because I say I am on the forums. There now it's a done deal. Besides how he acts out is more like a 3 year old than an adult. Some professional he is and tell us all again just what your expertise is? Trolling perhaps?

Regadude's picture

...credentials from Johnny the hobbyist speaker designer. Let's see some pictures of your Johnny brand speakers! Post some pics, or provide a link to a site where we can see these speakers of yours.

I demand to see these speakers of yours! 

ChrisS's picture

If JRusskie has a misguided and limited understanding of DBT, and....

Georgie has a misguided and limited understanding of DBT, then....

Are JRusskie and Georgie one and the same person?

Has anyone seen them in the same room together? Hmmmm.

Please, one of you (I guess it doesn't matter which...) ask Harman Kardon how they do their DBT's and how they use the results.

Thank you.


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