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Infinite Line Source: analysis
Infinite Line Source: analysis
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Old 12th January 2018, 10:34 PM   #181
Markw4 is online now Markw4  United States
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Originally Posted by werewolf View Post
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.
Of course, there are arguments to the contrary. People are more sensitive to phase and absolute phase at lower frequencies, particularly for percussive sounds. Some people probably more so than others. Maybe most of that is below frequencies at which a line array would be operated?
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Old 12th January 2018, 10:39 PM   #182
werewolf is offline werewolf  United States
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Originally Posted by Markw4 View Post
Of course, there are arguments to the contrary. People are more sensitive to phase and absolute phase at lower frequencies, particularly for percussive sounds. Some people probably more so than others. Maybe most of that is below frequencies at which a line array would be operated?
Very reasonable points ... on all counts.
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Old 13th January 2018, 12:32 AM   #183
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I think that there is evidence that a smaller phase shift across the acoustic band above say 500 Hz has some effect, albeit the effect is not huge, kind of a "tweek". It's not a major effect like linear frequency response is.
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Old 13th January 2018, 12:38 AM   #184
werewolf is offline werewolf  United States
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POST #18

D. Infinite Line Source: time-domain impulse response (continued)

The time-domain impulse response for our Infinite Line Source is :

h(t) = (rho/2pi)*INT[(1/R)*delta[t - R/c]]dz, from z = 0 to +infinity

where "R" is a function of the integration variable, "z" :

R = sqrt[r^2 + z^2]

And we recognize that the impulse response is simply the continuous sum of the weighted and delayed impulses from all of the little "dz" elements that form our infinite line All we need to do, is solve this integral ...


Given the presence of the delta function inside the integral, we'd like to get the integral in the form of :

INT[ f(v)*delta[t - v] ]dv

so that we can exploit the "sampling property" of the delta function (discussed earlier in this thread, POST #4).


So, let's perform a straightforward substitution of variables :

v = R/c = {sqrt[r^2 + z^2]}/c

from which it immediately follows that :

z = sqrt[(cv)^2 - r^2]

dv/dz = z/{c*sqrt[r^2 + z^2]}

dz/R = (c/z)*dv

and the integration limits in the new variable "v" are :

z = 0 ======> v = r/c
z = infinite ===> v = infinite


Our time-domain impulse response for the Infinite Line Source is now given by this new integral in "v":

h(t) = (rho/2pi)*INT[ c*delta[t - v]/sqrt[(cv)^2 - r^2] ]dv, from v = r/c to v = +infinity


we're almost done ...
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Old 13th January 2018, 06:30 PM   #185
werewolf is offline werewolf  United States
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POST #19

D. Infinite Line Source: time-domain impulse response (continued)

time-domain impulse response :

h(t) = (rho/2pi)*INT[ c*delta[t - v]/sqrt[(cv)^2 - r^2] ]dv, from v = r/c to v = +infinity

thanks to the "sampling property" of the delta function :

h(t) = (rho/2pi)*INT[ c*delta[t - v]/sqrt[(ct)^2 - r^2] ]dv, from v = r/c to v = +infinity

pulling all terms with no "v" dependency outside the integral :

h(t) = (rho/2pi)*{c/sqrt[(ct)^2 - r^2]} * INT[delta[t - v]]dv, from v = r/c to v = +infinity


The remaining integral in the above expression is now quite trivial
Note the lower limit of the integration variable, "r/c". The delta function inside that integral means that :

For t<(r/c): the argument of the delta function is less than zero for all included "v" ... and so the integral = 0.
For t>(r/c): the v-integral 'sweep' of the delta function will 'trigger' the impulse ... and so the integral = 1.

Which means that the remaining integral in the above expression = u[t - r/c]
(where u[t] is the unit step function)

re-arranging a few terms, leaves us with the wonderful result we've been seeking :

h(t) = (rho/2pi) * {u[t - r/c]/sqrt[t^2 - (r/c)^2]}

where:

r = distance from the Infinite Line Source to our measurement point




we can simplify even further, and re-express :

Let Td = (r/c), so that Td is the 'perpendicular' (or shortest) time delay from the Infinite Line Source to our measurement point:

h(t) = 0, for t<Td
h(t) = rho/{2pi*sqrt[t^2 - (Td)^2]}, for t>Td

THIS ^^ is the time-domain impulse response for an Infinite Line Source


one or two more posts, and the "tutorial" section of the thread is done
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Old 13th January 2018, 07:17 PM   #186
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POST #19a

D. Infinite Line Source: time-domain impulse response (continued)

A slightly different form for the time-domain impulse response ... i like this one even better :

h(t) = 0, for T<Td
h(t) = rho/{2pi*Td*sqrt[(t/Td)^2 - 1]}, for t>Td

where Td = 'perpendicular' (or shortest) time delay from the Infinite Line Source to our measurement point.

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Old 13th January 2018, 09:51 PM   #187
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Graphs soon?
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Old 14th January 2018, 05:45 PM   #188
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Originally Posted by Dr1v3n View Post
Graphs soon?
Perhaps someone can post some plots of the time-domain impulse response from POST #19a, for a few values of Td ... maybe Td = 1, 2, 5, 10 ... where (rho/2pi) can be 'normalized' to unity (or any other convenient value).

The impulse response "tail" will be immediately apparent. Please note that:

1/{Td*sqrt[(t/Td)^2 - 1]} ~ 1/t (for t >> Td)


And please remember ... the time-domain impulse response is simply the Inverse Fourier Transform of the frequency-domain transfer function (aka frequency response). One is completely determined by the other ... the same information is presented in either domain, time or frequency. For the Infinite Line Source ... we see ~ 3dB drop for every doubling of frequency and/or distance in the frequency domain, which dictates (or, is dictated by) ~ 1/t behavior in the time domain.
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Old 14th January 2018, 06:31 PM   #189
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Quote:
Originally Posted by werewolf View Post
Perhaps someone can post some plots of the time-domain impulse response
Done

I think that I'll show the finite cylindrical source in Mathcad also as it makes for a nice clean presentation.
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Last edited by gedlee; 14th January 2018 at 06:35 PM.
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Old 14th January 2018, 07:02 PM   #190
werewolf is offline werewolf  United States
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Thank you Earl !!!

Guys (and girls), i really need to explore something in yet another 'sidebar'. The issue i want to discuss, briefly, is triggered by an observation of UNITS.

If you look at the impulse response (presented in its most compact form in POST #19a), you'll notice (ignore the "rho" term, for now) that the units are "1/(time)". What's up with that? Is that even right?

The answer is ... YES, it's right

The Dirac impulse ... delta(t) ... is really a mathematical abstraction, defined more by its integral than anything. When you integrate the Dirac impulse, over time, you get something with real, physical meaning ... namely, the unit step function ... u(t). It's not really "wrong", then, to consider that the "units" of the Dirac impulse itself are "1/(time)".

A simple, familiar example : consider the simple 1st-order RC low pass filter. When you examine the mathematical form of the step-response, you'll find the classic exponential rise, asymptotically approaching the value of 1 ... unitless "one" (or, the step response will have units of "volts" if the input step-function has units of "volts"). What does the mathematical formulation of the impulse-response look like? Recognizing that differentiation and integration are linear operators, whose operations are "preserved" across linear systems, we can answer that by taking the first derivative (wrt time) of the step response We'll find another exponential ... asymptotically approaching zero ... but now, with units of (1/time). YES, the time-domain impulse response of the simple RC low-pass filter ALSO has units of (1/time).

... just another reason why studying digital signal processing BEFORE analog signal processing makes sense The first was the concept of convolution, which is really easy to understand in the discrete-time domain but too damn abstract in the continuous-time domain And now, we find ourselves dealing with the abstract nature of the Dirac impulse yet again, in something as seemingly straightforward as "impulse response".

If anyone is interested, we can develop the time-domain STEP response of an Infinite Line Source (by simply integrating, in time, the impulse response) ... to reveal something a bit more practical or "tangible", with units that make physical sense ...

Last edited by werewolf; 14th January 2018 at 07:11 PM.
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