Indeed!
It would be nice if we could model the summing at the eardrums with a finite set of weighted summed points in space. Have been looking for this for some time now. Has anyone made such approximation?
Marinus
Reality is such a PITA!
"Finite set", maybe, but that would be a very large set depending on how high in frequency you wanted this approximation to be valid. At LFs a single point works, but at 10 kHz the ear is so complex that it would take thousands of weighted points in space to even be approximately correct. Just look at plots of HRTFs and their complexity and imagine needing to achieve this complexity for all angles in space. It would be massively complex. The closest thing to what you are suggesting is the work being done with the Boundary Element Method to simulate HRTFs, but these are not points in space but a complex surface calculation.
We are absolutely calculating the exact effect that you describe: the time of arrival of the same frequency from multiple source points. This is exactly what has been calculated. How many source points did we calculate? Pick a number ... any large number ... we included and calculated more than that
It just so happens that the symmetry of the situation dictates that the response at z=7 is EXACTLY the same, as the response at z=142. Well, how about the response at z=-324? We calculated it also ... including the different arrival times, at the same frequency, from many, many multiple source points ... and it's the same as the response at z=56
We could limit this set by only taking into account the typical listener at the 'ideal' listening position.(so no full room simulation).
POST #16
Infinite Line Source: time-domain impulse response
The impulse response from our infinite line source is given by:
h(t) = (1/2pi)*INT[ {(rho/2pi)*INT[(1/R)*exp[(-jw/c)*R]]dz} *exp[jwt]]dw
where:
- the term in {brackets} is the 'transfer function' H(jw)
- R = sqrt[r^2 + z^2]
What do you do, when taking the integral of another integral? Why, you change the order of integration, of course! After pulling the exp[jwt] term inside the 'dz' integral (we're allowed ... it does depend on 'w', but it has no dependence on 'z'), we have :
h(t) = INT[(1/2pi)*INT[(rho/2pi)*(1/R)*exp[(-jw/c)*R]*exp[jwt]]dw]dz
After re-arranging some terms, and pulling the '(1/R)' term outside the 'dw' integral (we're allowed ... it does depend on 'z', but it has no dependence on 'w'), we have :
h(t) = (rho/2pi)*INT[(1/R)* {(1/2pi)*INT[exp[-jw(R/c)]*exp[jwt]]dw} ]dz
Now we have something very interesting ... because the term in {brackets} is immediately recognizable a gold star to anyone who can identify what it is ...
Infinite Line Source: time-domain impulse response
The impulse response from our infinite line source is given by:
h(t) = (1/2pi)*INT[ {(rho/2pi)*INT[(1/R)*exp[(-jw/c)*R]]dz} *exp[jwt]]dw
where:
- the term in {brackets} is the 'transfer function' H(jw)
- R = sqrt[r^2 + z^2]
What do you do, when taking the integral of another integral? Why, you change the order of integration, of course!
After pulling the exp[jwt] term inside the 'dz' integral (we're allowed ... it does depend on 'w', but it has no dependence on 'z'), we have :h(t) = INT[(1/2pi)*INT[(rho/2pi)*(1/R)*exp[(-jw/c)*R]*exp[jwt]]dw]dz
After re-arranging some terms, and pulling the '(1/R)' term outside the 'dw' integral (we're allowed ... it does depend on 'z', but it has no dependence on 'w'), we have :
h(t) = (rho/2pi)*INT[(1/R)* {(1/2pi)*INT[exp[-jw(R/c)]*exp[jwt]]dw} ]dz
Now we have something very interesting ... because the term in {brackets} is immediately recognizable
a gold star to anyone who can identify what it is ... 
We have a winner!
The term above, in {brackets}, is the Inverse Fourier Transform of ... the Fourier Transform of a delayed impulse
In other words, the term in brackets is equal to : delta[t - R/c]In still other words, as we saw earlier in this thread (POST #5) ... these two functions are Fourier Transform pairs :
delta[t - Td] <----> exp[-jw*Td]
well done bolserst!
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POST 17
D. Infinite Line Source: time-domain impulse response
The {bracketed} term in the expression for the impulse response h(t) (above) is the Inverse Fourier Transform corresponding to pure time delay :
{(1/2pi)*INT[exp[-jw(R/c)]*exp[jwt]dw} = delta[t - R/c]
where delta[t] is the Dirac Impulse (introduced earlier in this thread)
... which means that our expression for the time-domain impulse response of the Infinite Line Source reduces to :
h(t) = (rho/2pi)*INT[(1/R)*delta[t - R/c]]dz, from z = 0 to +infinity
where we remember that "R" is a function of "z", corresponding to:
R = sqrt[r^2 + z^2]
and we expect the resulting solution to be a function of the distance "r" from the line, and also a function of time "t" as well (of course), but not a function of "z" (the variable of integration will naturally disappear).
Before we solve this integral (and yes, there IS a tidy closed-form solution), let's interpret what it's telling us:
It's saying that the time-domain impulse response, at some distance "r" from the Infinite Line Source, is simply the continuous summation (aka integral) of weighted (by the distance, 1/R) and delayed (by the time delay = R/c) impulses from all of the "dz elements" that form our infinite line
D. Infinite Line Source: time-domain impulse response
The {bracketed} term in the expression for the impulse response h(t) (above) is the Inverse Fourier Transform corresponding to pure time delay :
{(1/2pi)*INT[exp[-jw(R/c)]*exp[jwt]dw} = delta[t - R/c]
where delta[t] is the Dirac Impulse (introduced earlier in this thread)
... which means that our expression for the time-domain impulse response of the Infinite Line Source reduces to :
h(t) = (rho/2pi)*INT[(1/R)*delta[t - R/c]]dz, from z = 0 to +infinity
where we remember that "R" is a function of "z", corresponding to:
R = sqrt[r^2 + z^2]
and we expect the resulting solution to be a function of the distance "r" from the line, and also a function of time "t" as well (of course), but not a function of "z" (the variable of integration will naturally disappear).
Before we solve this integral (and yes, there IS a tidy closed-form solution), let's interpret what it's telling us:
It's saying that the time-domain impulse response, at some distance "r" from the Infinite Line Source, is simply the continuous summation (aka integral) of weighted (by the distance, 1/R) and delayed (by the time delay = R/c) impulses from all of the "dz elements" that form our infinite line
The math so far has been developing the radiation characteristics for a single, Infinite Line Source. And the math developed so far doesn't really care what the orientation is, of course ... you can "rotate" the 3-D axes any way you wish
and the response relative to the orientation of the line won't change. Nor has the math paid any attention to the characteristics of human hearing. BUT, to address your question, let's consider a couple things. Ears are on the sides of our heads
for very good lateral localization (yes, humans are also capable of limited vertical localization ... but height cues are mono and don't begin until the treble, where wavelengths become comparable to the dimensions of our outer ears). We don't need to explore the nuances of human hearing (frequency ranges where ITD matters versus IID, for example) to recognize that the left & right ears probably need to hear something at least "slightly" different, for good stereo reproduction So let's ask a couple very basic questions:- For stereo reproduction, we'll probably need more than one Line Source. If the "left" channel Infinite Line Source is horizontal, where will the "right" channel be ?
- Understanding that the response along the length of an infinite line source (uniformly excited) will not vary along the length ... will a horizontal arrangement ultimately offer good lateral cues, consistent with ears on the sides of our heads (for stereo reproduction designed to exploit that human characteristic) ?
The bottom line is that, for stereo, a vertical orientation of the left & right sources is gonna be preferable. We may "lose" height cues ... but they were never really captured in stereo recordings to begin with. Or, we may experience a somewhat weird phenomenon, whereby whatever height our minds "assign" to the reproduction, will tend to move up and down with us, as we stand or sit. But, again, height is not a strong merit of stereo reproduction anyway. What we gain ... and it's a biggie, in my opinion ... is a 3dB reduction in pressure for every doubling of distance, as opposed to 6dB (for a point source). This will substantially widen the 'sweet spot' for stereo listening
Of course, we'll also have to electronically equalize that 3dB drop for every doubling of frequency ...
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In a word, no
but ... we don't need to To equalize the impulse response back to a Dirac impulse ... or, alternatively, to equalize back to "flat" frequency response ... would require enormous (essentially, infinite) dynamic range. We have to remember that a truly "impulsive" impulse response corresponds to a frequency response that's flat ... not just over 20kHz, but "from DC to Daylight".
HOWEVER ... we don't need a flat frequency beyond beyond 20kHz
And there's already plenty of filters in the audio processing chain (below 20Hz, above 20kHz) that will already deviate the impulse response from the idealized Dirac impulse. All we really need from the loudspeaker, is "reasonably flat" frequency response over 20kHz ... and the corresponding impulse response will be just fine and that is something that IS possible ...For example ... recognizing that we probably won't be using a line source in the sub-bass, or maybe even low-bass, regions ... let's say we want to use a line source over the top seven (7) octaves of the human hearing range. The natural frequency response of the Infinite Line Source ... falling at -3dB per octave ... means that we'll need about 20dB of dynamic range to equalize 'flat' over 7 octaves. That's not too crazy, i think ...
Hmmm...if I've already given myself the degrees of freedom to consider an infinitely long array, it doesn't seem like a reality stretch to consider infinite dynamic range too.....just saying
PS...thank you for this thread !
In a word, no
I think there's an underlying question of equalization methods across our 7 octaves of interest. Suppose we have a 10 band equalizer that allows +/- 10 dB. Would that be sufficient to flatten the response to an acceptable approximation? Also, would utilizing that equalizer improve the impulse response? Does it depend on the implementation of the equalizer?
If we use DSP we have a lot of options. We can design linear phase correction filters, or minimum phase correction filters, or something else. Would both of those improve the impulse response? Would one improve it more?
touche' ...Hmmm...if I've already given myself the degrees of freedom to consider an infinitely long array, it doesn't seem like a reality stretch to consider infinite dynamic range too.....just saying
PS...thank you for this thread !
A couple points, regarding the 'infinite' nature of our line source:
- First, it really makes sense to analyze and understand a "theoretical ideal", even if it only exists in the abstract ... as a target or goal, perhaps, to compare against real-world solutions. In this sense, we do the same for 'point-sources', too
- Second, it really is possible to build something approaching an infinite line, if we utilize reflections ... and understand them according to simple 'image theory'
When Earl starts his thread on Finite Line Sources
I hadn't really considered starting a new thread. I figured that once you had done the approximation for an infinite source I would just review how one deals with a finite length line source. This is already done in my book, but I could review it here for discussion.
Personally, i wouldn't attempt the equalization of any line source ... infinite or finite ... with anything BUT an FIR filter. Then again, i've had a very long love affair with FIR filters
but seriously, 3dB per octave is just not worth attempting, with any "old school" parametric or graphic EQs Regarding the time-domain impulse response, couple of thoughts :
- we haven't even developed it yet! But it's coming soon ...
- don't worry too much about the shape of the time-domain impulse response, in any of our audio endeavors. It gets "ruined" by high-pass filters at 10Hz. It gets "ruined" by low-pass filters at 40kHz. It gets "ruined" by all-pass filters, with perfectly flat magnitude but with poles and zeros right in the middle of the audio band. And yet, the ear is remarkably insensitive to all of this
two takeaways, for now :
1. Yes, the time-domain impulse response of the Infinite Line Source will be far from the perfect Dirac impulse. This is, of course, directly connected to the non-flat frequency response ... they are NOT separate issues. They are VERY inter-dependent issues. No, stronger ... they are the EXACT SAME issue
Either one, is completely determined by the other.2. For those willing to pursue Line Sources, my advice is : focus on getting the magnitude response flat (in the frequency domain). The ear is remarkably insensitive to absolute phase (relative phase is important at crossover, or between stereo channels, because relative phase impacts the magnitude when 2 or more signals are summed), which means the ear is also remarkably insensitive to the "shape" of the time-domain impulse response.
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i know
The way i see it, if you're really not making speakers anymore
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