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
ChrisS's picture
Who's on First?...

Who's doing the listening in your tests, JRusskie? Do you know any 18 year old musicians? Oh, of course not...But you probably keep company with a bunch of construction guys (lucky you!) with damaged hearing. They should all find that there's no difference between any products.

ChrisS's picture
What's on Second? The Man of La Mancha...

Once again, comrade JRusskie, you are on your Quixotic journey down that twisty, winding path for "truth"... Being an upstanding citizen of the former-USSR, you should know about "truth", right?

DBT is SCIENCE, and SCIENCE is ALWAYS RIGHT, especially during the heyday of the Union of Soviet Socialist Republics and you've been hanging on to this idea since the dissolution of the USSR. Hanging on to something you really want to be TRUE.

Comrade JRusskie, just because you want it, doesn't make it so.

Ariel Bitran's picture
Hiya Johnny

just wanted to let you know i haven't forgotten about you.

i've been south of the equator spending time with my father and brother, but now that I'm back in the Stereophile office, i'll answer your question in full a little later.

peace out homeslice.

be nice.

Regadude's picture
Asking Johnny...

Asking Johnny to be nice... Wow, you are very optimistic Ariel! 

GeorgeHolland's picture
Ariel, how about doing your

Ariel, how about doing your job and deleting post and start placing a 30 day BAN on Regadude and ChrisS for spamming this whole article day after day with nothing less than pure drivel and taunts? I've seen kids frums better moderated than this fiasco but it's already turned into a child's forum the way they act. You would think they had some dignity but we haven't seen any yet.

Ariel Bitran's picture
My Policy

I believe the internet is pretty much open range, so I'm happy to just watch folks speak their minds using whatever manner to express themselves.

one thing that bothers me is when a member is unneccessarily rude to someone who is being respectful or even innocently posting unknowing of the snark they may receive. in those instances, I say something so new users are not discouraged to continue posting.

On the other hand, if you're being provocative with your initial comment, I can only expect others to react to your provocation. 

******

The other policy I encourage and have stated many times is as follows 

"don't feed the trolls"

******

Finally, we do delete comments from ANY member we feel is totally off-topic and insulting to an individual member, as we want the conversation to continue and bring light to the subject  at hand rather than turn into a name-calling game. That being said, I'd like to delete Regadude's comment above since it just makes fun of Johnny. Also, I probably shouldn't have typed be nice. i dont know what i was thinking. I think i just wanted a rhyme in there, and also to include a general suggestion of positivity, but deleting RegaDude's comment would delete your reply, and this is a conversation I'd like to keep around.

Regadude's picture
What?

What did I do? In response to your comment of "be nice" to Johnny, I merely joked that you were asking a lot. A lot, because Johnny is often not nice. He is that way, that is a fact. I have no responsibility for his behavior. I just pointed it out.

Don't shoot the messenger!

ChrisS's picture
"...and made them cry"

What is reliability?

Think, Georgie, think.

Ariel Bitran's picture
So I read

the two links provided earlier giving examples of some DBTs.

I found the matrixhifi test to be ignorable: who is the sample? how did they select these people? how are they representative of a population of listeners as a whole? in order to gather significance from these these tests, the first and most important step is determining your sample, sample size, and how you select your sample. this just seems like a bunch of friends having fun. also, since there were multiple components being switched at the same time, system synergies could have been the cause of the weaker sounding more 'hi-fi' system. maybe those components weren't right for each other, but the cheaper system just sounded better. at least in ABX, they only changed one piece at a time

what i found interesting in the ABX test was the user's ability to control the change of system component themselves. this helps eliminate the idea that the listener might feel like they are getting 'duped' or constantly searching/guessing for the difference.

Also regarding the ABX method, the # of times a difference was heard was 33. the # of times no difference was heard: 29. Interestingly, cables were the least discernable. 

DBT is time consuming and for signnificant results you need a large sample size (to represent a large population of listeners). With a small sample size, as in both of these tests, you risk a greatly flawed hypothesis and will lack confidence in your results. 

I don't want to whip out my textbooks, b/c i have other stuff to do, but 17 listeners is not nearly large enough of a sample size to even represent a population of 70,000 Stereophile readers (for example). Then we run into an even bigger problem of "who" is selected-->ie what type of sample you are trying to represent.

I've heard repeatedly that H/K does have a successful DBT model. I'm sure it takes them years to perform each experiment, and it is wildly expensive and time consuming. You need a large sample size for any of this stuff to matter, not a few dudes in a basement.

GeorgeHolland's picture
More excuses from

More excuses from Stereophile?  Why am I not surprised. You scoff and blow off any attempts at doing a DBT by the people I linked to yet when it comes to "reviewing" cables and amps it's suddenly okay to accept a sighted biased review as being true?  Laughable sir but then I suppose you being an emploee have to toe the line. So be it.

I give up. Go on and trust your sighted "tests" while ignoring a DBT or even a SBT. I know your boss won't let you do any. He "knows" it all. *eyeroll*

Regadude's picture
More of the same...

Hey George! Ariel provided you with a more than satisfactory answer. So why are you still complaining? You never heard the expression "agree to disagree"?

Ariel is right; sample size is crucial. Anyone who has a basic knowledge of statistics understands this. 

JohnnyR's picture
Yawn

George was talking about simply doing a DBT or SBT among friends like he linked to. Let's say you and 5 pals think the "Humungo" amp  blows away every other amp you have listened to. So you set up a simple DBT or even a SBT to see if you can RELIABLY pick which amp is which WITHOUT sighted bias. In other words using your EARS???? I have read on the forums here how much people rely upon "What I heard" yet don't trust a simple test to prove that they can.

So lets say the "Humungo" amp is just simply "liquid, lifts 400 veils and makes the back ground blacker", what ever silly terms you wish to use, If so then you should be able to pick it out 100% of the time doing a DBT. If not and lets say you only choose right 50% of the time then that's proof you were only guessing and couldn't tell which was which. So much for saying sample size is critical. If no one can pick the "Humungo" amp from the other one then where are your rationalisations for saying it's better?

I can tell though that none of you have even bothered to try a SBT or DBT so it's all a waste of time speaking to the peanut gallery. I love how much money you both have WASTED on pricey products that only look good and sound the same. Funny as hell, go on spending YOUR money I love ityes

ChrisS's picture
Everyone stand back!

JRusskie just beat himself with his own thinking...

Regadude's picture
Money spent?

 

Johnny wote:

"I can tell though that none of you have even bothered to try a SBT or DBT so it's all a waste of time speaking to the peanut gallery. I love how much money you both have WASTED on pricey products that only look good and sound the same. Funny as hell, go on spending YOUR money I love ityes"

How do you know what I've bought with my money? I will go on spending my money (but not on your plywood speakers you make in your basement), that's what it's made for!

 

JohnnyR's picture
Thnak You For Proving The Old Saying...........

.a fool and his money are soon parted.

Regadude's picture
Well...

Well I am not that much of a fool with my money. I have not yet bought any of your JohnnyR brand speakers! 

ChrisS's picture
"When the boys..."

"...came out to play,

Georgie Porgie ran away!"

Delusions of absolute truths cloaked in "science" aren't much to hide behind, eh Georgie...

I hear our Man of La Mancha, JRusskie, is looking for his Sancho Panza to accompany him on his journey in search for TRUTH.

Happy trails!

ChrisS's picture
Call Me Maybe...

Georgie, Look at all those scientists backing you up!

If you take a college level research methodology course, Georgie, you may not appear so laughable. At this point, there appears to be very little knowledge in what you say about reviewing audio products.

GeorgeHolland's picture
Here is how you review audio

Here is how you review audio products........ looks in Stereophile and buys whatever the product of the month is. Spends too much money but doesn't care. Goes online and talks like a 5 year old and spews insults and THINKS he's smart. Case closed.

Regadude's picture
Name one product...

George, name ONE SINGLE product that I have bought. Just one. You and Johnny have this little fantasy in which you think you know everything and everyone. 

How about you list your gear? If you are so good and knowledgeable about buying audio products, do share with us the products that are George Holland worthy. If you actually have any audio gear...

GeorgeHolland's picture
ZzzzzzzzzZZzzzzzzzzzzzzz

ZzzzzzzzzZZzzzzzzzzzzzzz sorry but conversing with the likes of you is boring and makes my brain hurt considering the amount of BS you spew,

ChrisS's picture
The cure

Take one course of college-level research methodology and call us in the morning...

rl1856's picture
Objectivist- Subjectivist

Do you like what you are hearing ?  If no, move on until you do.  If yes, then shut up and relax.  This is a hobby focused on the enjoyment of the creative output of artists.  It is not about how many proverbial angels can dance on the head of a pin.

Go listen to MUSIC !

hnipen's picture
Thanks John !!!!

Thanks John for a very interesting and exciting presentation, lots of interesting information here and I'm surprised, to say the least, from the lack of positive feedback.

There are many who are skeptical to some of the ways of doing measurements in Stereophile and in some ways I'm one of them too, especially the way speakers are measured so close, large array speakers like the bigger Dunlavy's and some others will not sum up very nicely in this way. We do, however, not live in a perfect world and Stereophile cannot afford an anechoic chamber, so this is probably the best they can do.

I wish John would share more of this kind of information as he has gathered lots of knowledge during a long interesting career at Stereophile and other places.

Go on John :-)

Merry Christmas

Cheers harald

absolutepitch's picture
Heyser lecture

John, thanks for getting this lecture pre-print available for us to read. I have been looking forward to this.

I agree that there is a lot fo information combined into one lecture that anyone would need a lot of time to learn and understand the details. Pardon me for paraphrasing some of your words below.

Regarding the null result of DBTs, your description of the interpretation is in agreement with what I remember from statistics classes. I might add that a statistician would include a probability value or confidence band with the interpretation (something to the effect that 'the null hypothesis of no-difference-detctable is accepted with high probability'), and equally for the case when a difference is detected with high probability. I personally think DBTs should be done for product reviews, but agree that valid DBT's are difficult and time consuming (expensive) to do correctly, as Dr. Toole has shown in his writings.

The example of the 'backwards' impulse being not agreeable to listeners is something I have noticed in reference to digital recording. It's a wave form that does not occur naturally in music production, so reproducing it should sound 'bothersome'.

I also agree with the previous post, that more articles like this would be welcomed, to further highlight how complicated this field really is.

bernardperu's picture
Congratulations, Mr Atkinson!

I have read your essay with great pleasure (all of it!) and I think it is a great example of the Liberal Arts and Science coming together. In the end, it feels like a piece of applied music philosophy, which I find fascinating. It also seems to be free of busines-oriented interests, as your opinion on cables clearly suggests. It is awesome and very unsual to meet an accomplished person who gives priority to his passions and principles over financial interests (as also expressed on your 2012 writing on the CES and Las Vegas). 

I consider myself to be an audiophile that turns off the lights and tries to connect his emotions with the music with a very relaxed mind (this seems to be a category in itself, as the un-relaxed passive listeners who cannot focus on the music on a mid to long term basis tend to be very opinionated). Having said this, I recently purchased a pair of Class D mono amps that can clearly connect me to the music (Hephaestus brand). I have not ever listened to amps which are over 15k. Within similar prices, class D seems to be the better choice (but how relative this can be, Jon!)

I will continue to follow your writings with deep admiration and I thank you for making a difference on my musical experience (which is passed on to my girlfriend and my child). 

 

Bernard

GeorgeHolland's picture
Clueless you are if you think

Clueless you are if you think Mr Atkinson is something special angle

ChrisS's picture
Georgie?

Yoda you are?

Andreasmaaan's picture
Thank-you JA

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. Personally, I found the lecture fascinating and thought-provoking, and I thought that the nuggets of personal history provided a powerful context for the thoughtful opinions expressed. For better or worse, Stereophile doesn't restrict itself to DBTs as their only reviewing tool, but 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. I'm not sure why this approach, coupled with a reliance on an income stream from advertising, seems to place JA in line for so much personal vitriol. If similar attacks were levelled at me in my professional life, I'd be mortified and enraged.

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

JohnnyR's picture
Uhhhhhhhhhhhhh Sorry But You Are..............

WRONG!

"For better or worse, Stereophile doesn't restrict itself to DBTs as their only reviewing tool, but 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"

Do you even READ the reviews? Seriously dude. Stereophile DOESN'T do DBTs at ALL! Atkinson has said repeatedly that they are "too difficult to do". Show me ONE single DBT he has done in a review please. Plus he does NOT "measure every piece of gear his reviewers review". Cables, power cords, record demagnitizers, just to name a few. You obviously have been reading another publication NOT Stereophile.

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