No worries!
I think it works for my presentation to start with a point-source 'monopole' of constant volume acceleration, because such a source "offers" a flat (constant-with-frequency) frequency response at any point in space. Shortly, we will be building an infinite line source with a continual line of such 'monopoles', and the frequency response of the infinite line can then be readily compared to the "ideal" (flat) response of the single monopole.
It's hard to 'cross' disciplines, sometimes. My background is engineering, and that gives me a certain "perspective" on the world. I try to "fit" things into my background ... in this case, signal processing ... but if i try too hard, and force-fit something that doesn't belong (or, is flat-out 'wrong') ... please don't hesitate to correct me!
It really is my intention to learn, as well as teach ...
"truncating the infinite" ....
It always poses some difficult challenges, and clever solutions, in any field that involves Fourier math:
- in signal processing, we deal with the consequences of "truncating the infinite" in the sub-field of "spectral estimation", the field of estimating the power spectrum (and statistics) of an infinitely long signal from a finite sample of that signal. Hann Windows, Blackman Windows ... all designed to provide some degree of "tapering" or "shading" to modify the frequency-domain "sinc" function of the rectangular window.
- in optics, various forms of lens/mirror shading have been studied to modify the Airy disc ... which is a sinc-like function that results from the lens/mirror only observing a finite section of an (essentially) infinitely-wide star beam (yes, lenses take Fourier Transforms too!).
- in audio, various forms of "tapering" or "shading" a finite line array have been studied, to address the effects of truncating an infinite line source. The infinite line source is not "ideal", all by itself (although it does have some wonderful merits, as we will see), but truncating the infinite line introduces even further challenges. Keele's work is very compelling in this regard, but probably best left for another thread ... perhaps, the Finite Line Source thread
as Earl said, the math gets a bit complex ... but the goals and results are well-worth discussing!
By the way guys, Earl Geddes, Don Keele ... these guys are GIANTS in acoustics. My background is more on the electronics and signal processing side of audio ... so i'm honored, and humbled, to be in their company as well
It always poses some difficult challenges, and clever solutions, in any field that involves Fourier math:
- in signal processing, we deal with the consequences of "truncating the infinite" in the sub-field of "spectral estimation", the field of estimating the power spectrum (and statistics) of an infinitely long signal from a finite sample of that signal. Hann Windows, Blackman Windows ... all designed to provide some degree of "tapering" or "shading" to modify the frequency-domain "sinc" function of the rectangular window.
- in optics, various forms of lens/mirror shading have been studied to modify the Airy disc ... which is a sinc-like function that results from the lens/mirror only observing a finite section of an (essentially) infinitely-wide star beam (yes, lenses take Fourier Transforms too!).
- in audio, various forms of "tapering" or "shading" a finite line array have been studied, to address the effects of truncating an infinite line source. The infinite line source is not "ideal", all by itself (although it does have some wonderful merits, as we will see), but truncating the infinite line introduces even further challenges. Keele's work is very compelling in this regard, but probably best left for another thread ... perhaps, the Finite Line Source thread
as Earl said, the math gets a bit complex ... but the goals and results are well-worth discussing!By the way guys, Earl Geddes, Don Keele ... these guys are GIANTS in acoustics. My background is more on the electronics and signal processing side of audio ... so i'm honored, and humbled, to be in their company as well
So far, it looks like the discussion might be framed as having been about point sources and line sources, and about how areas of wavefronts emanating therefrom expand over distance, thereby spreading out acoustic energy.
A possibly interesting way of thinking about other types of curved arrays, etc., from a very basic perspective might be to think about the wavefronts they produce, and how similar wavefronts might be produced by an array of point sources, perhaps in a planar array or other basic configuration. Also related, how physical dimensions of a finite sized phased array would be expected to affect wavefront evolution over distance as a function of frequency.
A possibly interesting way of thinking about other types of curved arrays, etc., from a very basic perspective might be to think about the wavefronts they produce, and how similar wavefronts might be produced by an array of point sources, perhaps in a planar array or other basic configuration. Also related, how physical dimensions of a finite sized phased array would be expected to affect wavefront evolution over distance as a function of frequency.
Free from commercial bias.
The fact that you are not "in acoustics" makes your opinion so valuable and is why I am so interested in your findings and research.
You clearly have a powerful mind well focused on audio electronics and a willingness to share valuable information without any commercial bias.
All of the "giants" you mention had / have major commercial interests and advocated products / solutions to benefit their employers or their own product range / type.... Thats way of the world, nothing wrong with that but well worth noting!
The one exception I have found to the commercial interest bias on this site is Tom Danley.
He has posted a huge amount of valuable speaker design information and is a genius....Proven by building a multi million $ audio business based on his own ideas and hard work... Not writing a book or becoming a consultant to a multinational... Not saying there bad either, just noting the difference!
Anyway, back over to you Werewolf!
Cheers
Derek.
By the way guys, Earl Geddes, Don Keele ... these guys are GIANTS in acoustics. My background is more on the electronics and signal processing side of audio ... so i'm honored, and humbled, to be in their company as well
The fact that you are not "in acoustics" makes your opinion so valuable and is why I am so interested in your findings and research.
You clearly have a powerful mind well focused on audio electronics and a willingness to share valuable information without any commercial bias.
All of the "giants" you mention had / have major commercial interests and advocated products / solutions to benefit their employers or their own product range / type.... Thats way of the world, nothing wrong with that but well worth noting!
The one exception I have found to the commercial interest bias on this site is Tom Danley.
He has posted a huge amount of valuable speaker design information and is a genius....Proven by building a multi million $ audio business based on his own ideas and hard work... Not writing a book or becoming a consultant to a multinational... Not saying there bad either, just noting the difference!
Anyway, back over to you Werewolf!
Cheers
Derek.
Having taken a class in Fourier Optics (its very close to acoustics!) let me just correct you a bit here. If the lens is circular then the transforms are Bessel Transforms which result in a circ function that is like a rotated sinc function (but Bessel instead of cosine). If the lens is rectangular then a 2d Fourier transform would apply (but those are rare). It's called Fourier optics even though for real lenses its really Bessel optics!
PS: I don't sell speakers anymore so does that mean that I can be trusted now? What does someone have to do to get their credibility back?
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You don't think that Tom is biased for his own products? How could that be? He doesn't believe that the way he does things is the best way?
Tom and I go way-way back to living in Chicago at the same time. I went to parties at his house. Tom is a great guy and very sharp. But to say that he is less biased than Don or I is simply unfounded. We all believe in what we are doing and we say so.
POST #4
B. Point Source: time-domain impulse response
There's no avoiding it. We can run, but we can't hide. We have to introduce the Dirac Delta function
Strictly speaking, it's not even a "function". Oh well ... tomayto, tomahto. Mathematicians hate it, engineers love it.
The Dirac Delta, or unit impulse function, is defined as :
delta(t) = 0, for all values of t except for t=0. At t=0, the function is "infinite" (or, perhaps better, "undefined").
It has the following properties :
INT[delta(t)]dt, from t=(-infinity) to t=(+infinity), = 1. The integral of the impulse function up to t=0 is zero, but as soon as the integration variable, t, sweeps past t=0, the integral becomes one.
So, in non-mathematical terms, delta(t) as zero "width", but it has infinite "height" ... and it has an "area" of exactly 1
We can also define it as the first derivative of the unit step function:
d/dt {u(t)} = delta(t)
It drives pure mathematicians crazy (that's reason enough to love it), but its use in the calculus of signal processing (and apparently, quantum mechanics) is entirely consistent ... so there's no real cause for concern.
In analog signal processing, when we speak of the "impulse" ... we're talking about the Dirac Delta. A "pulse" function of vanishing width, but growing height ... always maintaining the same area of 1 ... even as the width reduces to zero, and the height becomes unbounded.
One more thing to mention ... the Dirac Delta has a "sampling property" that comes in very handy. When you multiply some function, call it f(t), by an impulse that occurs at some point in time T, defined as delta(t - T), the integral equals (or reveals) the value of the function at t=T :
INT[f(t)*delta(t - T)]dt
= INT[f(T)*delta(t - T)]dt
(since the value of the impulse is zero everywhere, except at t=T)
= f(T)*INT[delta(t - T)]dt
(f(T) can be pulled out of integral, since it no longer depends on the integration variable, t)
= f(T)
(since the integration of the delta function is unity)
We will use this "sampling property" when we develop the time-domain impulse response of the Infinite Line Source. Also worth noting ... the "sampling property" of the impulse function forms the entire foundation for the mathematics of digital audio
next up : the time-domain impulse response of our simple little point source
B. Point Source: time-domain impulse response
There's no avoiding it. We can run, but we can't hide. We have to introduce the Dirac Delta function
Strictly speaking, it's not even a "function". Oh well ... tomayto, tomahto. Mathematicians hate it, engineers love it.
The Dirac Delta, or unit impulse function, is defined as :
delta(t) = 0, for all values of t except for t=0. At t=0, the function is "infinite" (or, perhaps better, "undefined").
It has the following properties :
INT[delta(t)]dt, from t=(-infinity) to t=(+infinity), = 1. The integral of the impulse function up to t=0 is zero, but as soon as the integration variable, t, sweeps past t=0, the integral becomes one.
So, in non-mathematical terms, delta(t) as zero "width", but it has infinite "height" ... and it has an "area" of exactly 1
We can also define it as the first derivative of the unit step function:
d/dt {u(t)} = delta(t)
It drives pure mathematicians crazy (that's reason enough to love it), but its use in the calculus of signal processing (and apparently, quantum mechanics) is entirely consistent ... so there's no real cause for concern.
In analog signal processing, when we speak of the "impulse" ... we're talking about the Dirac Delta. A "pulse" function of vanishing width, but growing height ... always maintaining the same area of 1 ... even as the width reduces to zero, and the height becomes unbounded.
One more thing to mention ... the Dirac Delta has a "sampling property" that comes in very handy. When you multiply some function, call it f(t), by an impulse that occurs at some point in time T, defined as delta(t - T), the integral equals (or reveals) the value of the function at t=T :
INT[f(t)*delta(t - T)]dt
= INT[f(T)*delta(t - T)]dt
(since the value of the impulse is zero everywhere, except at t=T)
= f(T)*INT[delta(t - T)]dt
(f(T) can be pulled out of integral, since it no longer depends on the integration variable, t)
= f(T)
(since the integration of the delta function is unity)
We will use this "sampling property" when we develop the time-domain impulse response of the Infinite Line Source. Also worth noting ... the "sampling property" of the impulse function forms the entire foundation for the mathematics of digital audio
next up : the time-domain impulse response of our simple little point source
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Ultimately, the ability to integrate the acoustic impact of an infinite line array on a point in space is interesting but may be insufficient (or even misleading) for our interests as human listeners. Our eardrums are not points in space. They are surrounded by our ears, heads, and bodies. The 7th page of the linked pdf below shows the HRTF of a KEMAR mannequin (designed to be similar to human HRTFs) for varying elevations, and an azimuth of 0 degrees. Viewing the right ear, a difference of just 10 degrees in elevation can be quite substantial. I suspect that the variations with elevation are likely even more pronounced at other azimuths, based on the geometry of most human ears.
http://www.acoustics.asn.au/conference_proceedings/AAS2009/papers/p8.pdf
In order to calculate the vibrations of a human eardrum caused by a line array, appropriate transfer functions need to be applied to sounds arriving from different locations. Even if we do that, perception is still different from excitation of the eardrum.
Recently I attended the capital audio festival and auditioned the Bob Carver ALS loudspeaker. I was surprised how little precision there was to the image. I hadn't expected to perceive sounds as if they were being generated by the full height of the speakers, but that's the perception I had. Perhaps this was caused because the varying HRTFs across a large span of elevations were all excited by that speaker. Maybe there was another cause. It's at least something to consider.
http://www.acoustics.asn.au/conference_proceedings/AAS2009/papers/p8.pdf
In order to calculate the vibrations of a human eardrum caused by a line array, appropriate transfer functions need to be applied to sounds arriving from different locations. Even if we do that, perception is still different from excitation of the eardrum.
Recently I attended the capital audio festival and auditioned the Bob Carver ALS loudspeaker. I was surprised how little precision there was to the image. I hadn't expected to perceive sounds as if they were being generated by the full height of the speakers, but that's the perception I had. Perhaps this was caused because the varying HRTFs across a large span of elevations were all excited by that speaker. Maybe there was another cause. It's at least something to consider.
Interesting post. Thanks for sharing!
Regarding the perceived image size, I've found that our eyes have a huge influence on what he hear, especially related to image size. Large speakers are usually perceived to produce large images and small speakers are perceived to produce small images. Yet, this is just our eyes playing tricks on our minds. I say that because I've compared a small speaker, my Tannoy dual concentric 5" speakers, to a large speaker, the Altec VOTT system, in my home. You couldn't tell which speakers were playing when sited next to each other (they both had their responses measured and equalized and sounded similar to a large degree).
At first, the line array image also seems tall, but as you get used to the sound, and especially if you close your eyes and forget which speakers are playing, the image tends to be very precise and not at all tall. This is my impression after living my arrays for a couple of years. I'll add that my arrays are floor-to-ceiling and have full FIR correction to produce a balanced, neutral sound at the listening position. In my cursory look at the ALS specs, I did not see any equalization or correction for room effects.
It depends on how the arriving wavefront of sound hits you. That's all there is, a disturbance in the air somewhere caused by something, but your ears only know about the wavefront as it impinges upon them.
Similar to how your eyes see according to the light arriving at them. Doesn't matter how the light is produced, what arrives at your eyes is all there is to sense. The source can only be inferred from that.
If a speaker system produces a succession of wavefronts identical to a live performance, there can be no way to tell a difference blindfolded, HRTF or not. So, looking at basic speaker induced wave geometry can start to provide some insights into increasingly sophisticated speaker systems.
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