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Infinite Line Source: analysis
Infinite Line Source: analysis
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Old 16th November 2017, 11:01 PM   #1
werewolf is offline werewolf  United States
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Default Infinite Line Source: analysis

Well it's been a long time since i offered something like a tutorial on this message board, so it's high time to maybe give a little something back to this wonderful community, once again and so, with your kind permission and indulgence, let's begin!

I'll highlight the posts where i'm advancing the tutorial (so to speak), to allow for plenty of discussion in the time and space "in between" ...


A. Point Source: frequency-domain pressure response
B. Point Source: time-domain impulse response
C. Infinite Line Source: frequency-domain pressure response
D. Infinite Line Source: time-domain impulse response

I thought we'd start with the idealized point source, to establish both a nomenclature and methodology. I'm an engineer (or ... was one, once upon a time), rather than a physicist, and that training tends to drive my perspective and analysis. The analysis will get a bit "heavy" in Fourier math ... it's rather unavoidable ... but i'll do my best to minimize detours and stay focused on the goal, which is to hopefully highlight some pros & cons of the idealized infinite line source, compared to the idealized point source.

Regarding practical implications, for right now i will only offer that which many already understand: the "image" theory of reflections quickly reveals that a floor-to-ceiling line array (with drivers of appropriate size & spacing) does a reasonable job of approximating the idealized infinite line source

Much of what i'll be presenting is already known. There may be some new info, particularly regarding the time-domain impulse response of an infinite line source (which i haven't seen developed), but at the very least i may offer a somewhat concise "repository" of idealized infinite line source info. I hope it's valuable on it's own merits, but i also hope this thread may be a good starting point, reference, or framework for future discussions regarding the analysis of line sources of finite length ... a much more difficult and fascinating topic!

Without further ado ...
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Old 17th November 2017, 12:01 AM   #2
werewolf is offline werewolf  United States
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A. Point Source: Frequency Domain Pressure Response

A simple radiating point source, an "elemental monopole", radiating into "4pi" space, with a constant volume acceleration = Ao. It's been analyzed to death but cherished forever for it's beauty, simplicity, building-block utility and sonic virtue

Our point source radiates uniformly in ALL directions in space (4pi radiation) ... so, as a function of "space", symmetry dictates that the pressure can only be dependent on distance "r" from the source, with no regard for "direction". As a function of both time and space, the pressure is often written as :

p(r,t) = [Ao*(rho)/(4pi*r)]*exp[j(w*t-k*r)]
(eqn 1)


rho = density of the medium (air)
exp[x] = classic complex exponential
w = radian frequency
t = time
r = distance from source
k = wavenumber = w/c, where c= speed of sound

Personally, i hate this form of the expression. HATE IT.

This is MUCH better :

p(r,t) = {Ao*exp[jwt]} * {[rho/(4pi*r)]*exp[-jwTd]}
(eqn 2)


Td = time delay = r/c

Let's take a moment to understand (eqn 2), and explain why i LOVE it:

The first term in brackets, {Ao*exp[jwt]}, is the excitation. In fact, it's nothing more than a complex sinusoidal excitation feel free to think of it as a complex form of sin[wt] (it's really equal to cos[wt] + j*sin[wt]). It's our little elemental monopole vibrating away at a certain frequency, "w", with an Amplitude of Ao

(Aside : we use complex exponentials a lot in signal processing, rather than more simple, readily recognizable, "real" sinusoids, for one reason : they are so-called eigenfunctions of linear, time invariant systems ... and wonderfully simple things happen when linear, time invariant systems are excited by eingenfunctions. i can elaborate, if anyone wishes ...)

The second term in brackets, {[rho/(4pi*r)]*exp[-jwTd]}, has no time-dependency, and represents what we'll call the transfer function of our little "system". It simply represents an attenuation factor of "1/r", along with the Fourier representation of "time delay", Td. More on this, in short order ...

Being an engineer, I really like separating the time-dependent excitation from the time-independent transfer function ... even though (eqn 1) and (eqn 2) are identical. The reason is that the concise identification of the transfer function allows us to inverse-Fourier-Transform it, to reveal the valuable time-domain impulse response

But alas, this post is already long enough i'll stop for now ....
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Old 17th November 2017, 09:33 AM   #3
wesayso is offline wesayso  Netherlands
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I'm getting out the popcorn and drinks for this one...
Use Science to design your speakers and they will sound like a piece of Art...
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Old 17th November 2017, 06:33 PM   #4
The G.O.A.T.145 is offline The G.O.A.T.145  Canada
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Me too
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Old 17th November 2017, 06:48 PM   #5
BYRTT is offline BYRTT  Denmark
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Infinite Line Source: analysis
One more seat warmed thanks
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Old 17th November 2017, 10:06 PM   #6
Overkill Audio is offline Overkill Audio  United Kingdom
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Default Great intro to fascinating subject

Looking forward to learning more on both floor to ceiling line arrays and mini arrays.
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Old 17th November 2017, 10:38 PM   #7
mrk7 is offline mrk7  Australia
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Originally Posted by werewolf View Post
... and wonderfully simple things happen when linear, time invariant systems are excited by eingenfunctions. i can elaborate, if anyone wishes ...)
Elaborate away!
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Old 17th November 2017, 11:38 PM   #8
werewolf is offline werewolf  United States
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great audience! Thanks guys
We'll keep rolling ....
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Old 18th November 2017, 12:23 AM   #9
werewolf is offline werewolf  United States
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Originally Posted by mrk7 View Post
Elaborate away!
(this post will be a short detour, so it won't get a post number)

The processing of signals by linear, time invariant systems ALWAYS begins with this thing called "convolution". Let me state right now ... the concept of convolution is much, MUCH easier to understand with discrete-time signals In fact, it's not too crazy to suggest that we should all BEGIN the study of signal processing with DSP, so that we can all understand this crazy thing called "convolution" much better.

Anyway ... "convolution" is, at its heart, simply the time-domain formulation of how a linear, time-invariant system (analog or digital) responds to a signal But a few definitions first:

Linear System: a linear system is simply a system whose output will 'double' in size (or amplitude), if you 'double' the input size. Of course, you may pick other scaling factors rather than 'two' (one of our favorites is zero ... if you put NOTHING into a system, and still get an output, then that system is NOT linear). You may also ADD a few inputs, and a linear system will respond with the SUM of the individual outputs. Over a wide range of signals, circuits composed of passive elements like resistors, capacitors and inductors are "linear", whereas most of our friendly active elements like transistors tend to be "non-linear" (but, even these elements tend to approach acceptably linear behavior with small signals, which allows clever circuit designers to use feedback to create surprisingly linear-like circuits with non-linear elements).

Time Invariant System: a system is said to be time-invariant if delaying the input simply results in a delayed ... but otherwise unchanged ... output. Most systems behave this way, but a great example of a NON time-invariant system is the 'sampling' process ... yes, that process that turns analog signals into digital ones

So what is convolution?

First, convolution simply recognizes that we can "consider" ANY arbitrary input signal to be a summation of SCALED and DELAYED signals we call "impulses" (very short duration blips or bursts). This is where DSP really shines impulses in the analog world are a bit more abstract, and we'll return to them "anon".

Second, convolution recognizes that a linear, time-invariant system must respond to these scaled and delayed impulses with ... you guessed it ... scaled and delayed "impulse responses" an LTI system simply has no choice and we immediately begin to appreciate why the so-called time-domain "impulse response" is so important in system analysis and design.

Here's the (analog, rather than digital) mathematical formulation for what i just wrote :

y(t) = INT[h(b)*x(t - b)]db


x(t) = time domain input signal
h(t) = time domain 'impulse response' of our LTI (linear, time-invariant) system
y(t) = time domain output signal
INT[f(b)]db = integral of function f(b), where "b" may be a dummy integration variable

The whole study of linear, time-invariant systems could end, right there. After all, we have a mathematical way to figure the OUTPUT of any LTI system, given the INPUT. All we need, is the time-domain "impulse response" of the system ... h(t).

But alas, engineers are a LAZY lot. We don't like integration, if we can avoid it. And that "convolution integral" (above) is kinda messy ... doesn't offer much "insight" into system behavior. But that clever little integral does lend itself to an interesting observation ... up next (yes this short detour will take one more post)

Last edited by werewolf; 18th November 2017 at 12:26 AM.
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Old 18th November 2017, 12:50 AM   #10
werewolf is offline werewolf  United States
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Let's see what happens when we apply a time-domain "exponential" signal to our LTI system :

x(t) = exp[s*t]

where "s" can be any factor, real or complex (in the most general case).

We find the output signal, y(t), from our convolution integral :

y(t) = INT[h(b)*x(t - b)]db
y(t) = INT[h(b)*exp[s*(t-b)]]db
y(t) = INT[h(b)*exp[s*t]*exp[-s*b]]db

notice : that exp[s*t] factor inside the integral is independent of the integration variable "b", so it can be pulled out of the integral :

y(t) = {exp[s*t]} * {INT[h(b)*exp[-s*b]]db}

Note that the first term above is exactly the input signal, and the second term is independent of the variable "t" (time). Let's re-write the above:

y(t) = {exp[s*t]} * H(s)

viola !!!!!

We've just discovered something VERY interesting, and EXTRAORDINARILY powerful :

When an LTI system is excited by an exponential signal, the output is the exact same signal, multiplied by a (complex-looking) scaling factor The ugly "integration" of convolution has become relatively simple "multiplication".

For this reason, we give exponential signals a special name : "eigenfunctions" of LTI systems

OK, ok ... you should be asking ... "So what? How often do we excite LTI systems with so-called exponentials?"

Great question! Turns out we don't often excite LTI systems with single exponentials ... but remember, LTI systems behave quite well when we "sum" input signals

OK, ok ... so now you should be asking ... "So what? How many input signals can be represented as a sum of (possibly complex) exponentials?"

Great question! The answer is ...


/end detour

time to return to Point Sources and Infinte Line Sources

Last edited by werewolf; 18th November 2017 at 01:15 AM.
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