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
JohnnyR's picture

I haven't seen one iota of explanation from yourself yet as to why both of us and Harman Kardon and the other links George posted to are wrong. Still waiting ChrisSy.

ChrisS's picture

Has HK hired you guys as DBT consultants?

 

Hey JRusskie,

Can you answer this one?

If A=B and C=B, then A=?

If you pass the test, then perhaps someone will hire you... But you and Georgie might have to fight over the job.

JohnnyR's picture

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

The sole purpose of DBT is to see if the person listening can distinguish between A and B. If they can't then for all practical purposes there is no difference in the sound from A and B. You and ChrisSy seem to think it's all about what the person "likes". It's a straight forward test method and "likes" has nothing to do with it.

Please explain to us all how Harman Kardon manages to use DBT all the time and do it well? I will be awaiting your reply Ariel.

ChrisS's picture

So every household has a Harman Kardon product? And you and Georgie have living rooms that look like anechoic chambers? No fireplaces, of course....

ChrisS's picture

Hearing a difference between a Harman Kardon product and another product in a anechoic chamber means what to you, Georgie and JRusskie?

Do you know that Ford makes the best trucks in the world?

ChrisS's picture

Are you sure I didn't say 'licks". You know maybe tasting an audio product will yield just as useful results in a DBT.

ChrisS's picture

JRusskie,

Now run out to your nearest Boris' Convenience store and get yourself a can each of Pepsi (do you even have Pepsi in the Former-USSR?) and Coca-Cola and set up your own Pepsi (or whatever passes for cola in Russia) Challenge.

Wiki has a nice explanation of how to do a DBT...

Once you've done your very own Peps(k)i Challenge, please send us your conclusion. We're curious...

The next step now is to get everyone in your subsidized housing project to participate in your Pepski Challenge.

Gather up that data, compare it your own conclusion and let us know how useful that information is.

I'm sure you'll enjoy the challenge of your doing your very own DBT's! (You won't even have to ask Harman Kardon to use their anechoic chamber!)

John Atkinson's picture

JohnnyR wrote:
The sole purpose of DBT is to see if the person listening can distinguish between A and B. If they can't then for all practical purposes there is no difference in the sound from A and B.

And that's the problem with these tests. If a formal blind test gives results that are indistinguishable from what would be given by chance, formal statistical analysis tells us that this result does _not_ "prove" there was no difference in the stimulus being tested, only that if there _was_ a difference, it was _not_ detectable under the conditions of the test. No more general conclusion can be drawn from the results. And as I have said, it is very difficult to arrange so that those conditions don't themselves become interfering variables. Even the fact that it is a test at all can be an interfering variable, as I explain in this lecture preprint.

JohnnyR wrote:
Please explain to us all how Harman Kardon manages to use DBT all the time and do it well?

I have visited Harman's facility in Northridge and their blind testing set-up is impressive. They have worked hard to eliminate interfering variables and their testing is time- and resource-consuming and painstaking. Even so, they have to make compromises. Blind testing of loudspeakers, for example, is almots always performed in mono. And despite the rigor of their testing, you still have anomalous results, like the Mark Levinson No.53 amplifier, which was designed with such testing but fared poorly in the Stereophile review.

While formal blind tests are prone to false negative results - not detecting a difference when one exists - sighted listening is prone to false positives, ie, it detects a difference when none exists or perhaps exaggerates the degree of difference. As neither methodology is perfect, we go with the one that is manageable with our resources. We therefore offer our opinions for readers to reject or accept in the context of their own experience and I believe Stereophile does  a better job of that than any other review magazine or webzine.

If you are uncomfortable with that policy, then you should not read the magazine. And if I remember correctly, JohnnyR, you admitted in earlier discussions on this sute that you neither subscribe to Stereophile, nor do you buy the magazine on the newsstand. So why should anyone pay attention to your opinions on how the magazine conducts itself?

John Atkinson

Editor, Stereophile

JohnnyR's picture

So you are saying that if in a DBT the listeners could NOT tell a difference between two amps using music of their own choice then that doesn't prove the amps sound alike?  Funny stuff there Atkinson. For one who thinks you should trust your ears to evaluate components you just bashed the ONE single TRUE way to test by USING YOUR OWN EARS in a DBT.

"And that's the problem with these tests. If a formal blind test gives results that are indistinguishable from what would be given by chance, formal statistical analysis tells us that this result does _not_ "prove" there was no difference in the stimulus being tested, only that if there _was_ a difference, it was _not_ detectable under the conditions of the test."

Your own words are saying that "if there was a difference it was not detectable under the conditions of the test"........oh you mean like letting the listener use the music of their own choice and switch back and forth endless times between two amps and then guess wrongly enough times so that they can't tell which one was which? LMAO if that's not proof that both amps sound alike then what sort of test WOULD prove that they do?  Come on Atkinson you just don't like DBTs because they would show up so many components that people think sound "oh so better than the rest"

Opinions from you and your reviewers are the Gospel now folks. No need to test anything really just trust good ol'JA and his flunkies. Yay.

"If you are uncomfortable with that policy, then you should not read the magazine. And if I remember correctly, JohnnyR, you admitted in earlier discussions on this sute that you neither subscribe to Stereophile, nor do you buy the magazine on the newsstand. So why should anyone pay attention to your opinions on how the magazine conducts itself?"

Oh just maybe because  a lot of people care for this little thing called the TRUTH? When magazines like your's take liberties with the truth by having shoddy reviews instead of in depth testing, then it's everyone's and anyone's responsibility to speak up when crappy falsehoods are published and the readers are supposed to take it all on faith. That's why. I for one do not take your opinions on anything audio related as worthwhile at all for the simple reasons that you show so much promise when you measure speakers but fail to even bother with the snakeoli products that you let slide under the radar yet let your reviewers give them glowing reviews sans any testing what so ever. Maybe that's the sighted listening bias you just spoke about yet you fail to even try with those type of products to get to the real TRUTH.

ChrisS's picture

JRusskie,

If you like Harman Kardon marketing, but you're not sure if Ford makes the best trucks in the world, then get yourself an F-150 and whatever truck you used to rumble across Afghanistan with, do your DBT (just like  the Pepski Challenge) and let us know what you come up with...

You are marketing TRUTH now? How pure is it?

I know some construction workers who might be interested...

ChrisS's picture

JRusskie, Just looking at your response to John's post and comparing word-for-word what John wrote and what you think he says, there's such a huge world of difference!! There's a war in your head!

[Flame deleted by John Atkinson].

GeorgeHolland's picture

"Even so, they have to make compromises. Blind testing of loudspeakers, for example, is almots always performed in mono."

Well Mr Atkinson the reasoning behind testing speakers in mono is to eliminate the dreaded comb filter affect that would otherwise show up if a stereo pair were auditioned and the listener moved their head even a couple of inces. I'm surprised you didn't mention that fact but then again you think DBTs are hard to do, so if you don't know how to do them then indeed they are hard to do. *Chuckle*  Any DBT done should be auditoned is such a manner. The rest of your "excuses" for not doing them is the same old same old from you, nothing surprising there.

ChrisS's picture

So Georgie Porgie,

Let's say JRusskie is DBT'ing a $1500 speaker and a $500 speaker and can't hear a difference, and you are DBT'ing a $4500 speaker and a $4000 speaker and you happen to have enough working neurons to hear a difference... Which set of speakers should the ex-shepherd construction worker buy?

John Atkinson's picture

GeorgeHolland wrote:
John Atkinson wrote:
Even so, they have to make compromises. Blind testing of loudspeakers, for example, is almost always performed in mono.

Well Mr Atkinson the reasoning behind testing speakers in mono is to eliminate the dreaded comb filter affect that would otherwise show up if a stereo pair were auditioned and the listener moved their head even a couple of [inches].

That is a consideration, of course, but in my opinion a minor one. As I had understood from Floyd Toole back in the day, the additional complexity required  of  Harman's physical speaker shuffling apparatus to do blind speaker testing in stereo was not justified by the results, ie, they felt that the stereo performance could be predicted from the mono results.

I don't agree with that, but more importantly, this illustrates the thesis offered in my lecture, that when you move the testing situation a step away from how the product is going to be used, you can't be sure that the assumptions you make haven't invalidated the test. As I write in the abstract to the lecture, "perhaps some of things we discard as audio engineers bear further examination when it comes to the perception of music."

BTW, I am still waiting for you to acknowledge that the criticism you made of my lecture, that it was not about Richard Heyser, was incorrect.

John Atkinson

Editor, Stereophile

GeorgeHolland's picture

More excuses?  I see what Johnny meant. You are never wrong. I am pretty sure Haram Kardon knows what they are doing. Please address any criticisms to them not me.

I am afraid that comb filtering IS a big deal. That would explain why cables "sound" different. It's not the cable but the listener changing where their head is between "testing"

You will be waiting a long time for any ackowledgement about your "lecture". Stop being the primadonna already.

Regadude's picture

Georgie wrote:

"Stop being the primadonna already."

Look in the mirror and repeat those words!!!! laugh

ChrisS's picture

So how does one differentiate speakers that sound differently, amplifiers that sound differently, pre-amps, turntables, tonearms, cartridges, DAC's, etc., if a turn of one's head makes that much difference?

Where's your reliability, Georgie? Doesn't science depend on reliability?

JohnnyR's picture

I can see what George is up against in here with Tweedle Dum and Tweedle Dee tag teaming and showing their ignorance.

http://www.ethanwiner.com/believe.html

Golly look what he meant.I think Ethan was banned from here ages ago for showing up Fearless Leader and his cronies and out right showing how REAL science works. BWAHAHAHAHAHAH loser boys.

Regadude's picture
John Atkinson's picture

Quote:
http://www.audioholics.com/news/editorials/diy-loudspeakers

Just bookmarking the link for future reference.

John Atkinson

Editor, Stereophile

JohnnyR's picture

......when you start doing a single DBT or even a SBT then you can talk about the "truth". Have you EVER designed and built your own speakers? Nahhhhhhhh you are too lazy or too "busy". Still finding plenty of time though to post online all the time though strangely enough.cheekyTill then you aren't an engineer so take your own advice and don't comment on speaker design anymore.

If you are going to save the link then please also save this link where discussion about it unfolded.

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

As you can see the original post was just one of many OPINIONS about the topic that is if you bother to read it at all. There are various OPINIONS about the topic and notice just how many of the so called "hobbyists" ended up being professional speaker builders. If you just pick and choose certain OPINIONS from the thread then you are guilty of leaving out facts.

For starters read the sixth post down by Jinjuku regarding Jeff's post that pretty much sums up where DIY has progressed.

ChrisS's picture

Not real science either...

GeorgeHolland's picture

Frick and Frack strike again. Regadude and ChrisS always come up with strawman replies and ignore the links posted."Not real science "? How pompus can you get? Mr Winer measured the effects of comb filtering, what did you measure ChrisS the length of your nose when you typed that reply? You dismiss anything people link to yet show us nothing in return. Regadude, posting opinions isn't real science just so you both understand. Now run along lil boys and study real hard, maybe in another 20 years you might be able to hold your own in a discussion.

ChrisS's picture

Has Winer's results been verified?

Did you know that Harman Kardon makes the best audio products in the world? And Ford makes the best trucks, right?

JohnnyR's picture

Or is that above your abilities like thinking?

Go ahead, put on a pink or white noise source and move your head about and tell me the sound doesn't change. You won't bother so forget it ChrisSy.laugh

ChrisS's picture

When moving a microphone while recording a person's voice, the sound changes. Did the voice change?

GeorgeHolland's picture

You never answer a question , you just put forth silly questions of your own. That's what people do when they don't know or are scared to try.

Moving a microphone while recording a person's voice? If that's how you do things then no wonder you don't know what Johnny was talking about. Yes the sound changes as recorded by the microphone so what?  Genius.angle

ChrisS's picture

I'll answer you this one... You and JRusskie always answer your own questions that you pose to everyone in these discussions. There's no need to provide any answer to you. As well, your attitudes and limited knowledge of the application of research methodology make civil and thoughtful discourse impossible.

So my questions to you and JRusskie are formed to reveal how each of you think whenever you provide a response.

You provide enough information for me to say that I find the "best" use of my time in these discussions is to make fun of you and JRusskie.

Ariel Bitran's picture

makes some sense of the whole darn thing.

Regadude's picture

The only duo that strikes here George, is you and Johnny. You both STRIKE OUT!

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