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12th January 2018, 11:34 PM  #181  
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12th January 2018, 11:39 PM  #182  
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13th January 2018, 01:32 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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13th January 2018, 01:38 AM  #184 
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POST #18
D. Infinite Line Source: timedomain impulse response (continued) The timedomain 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 timedomain 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 ... 
13th January 2018, 07:30 PM  #185 
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POST #19
D. Infinite Line Source: timedomain impulse response (continued) timedomain 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 vintegral '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) rearranging 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 reexpress : 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 timedomain impulse response for an Infinite Line Source one or two more posts, and the "tutorial" section of the thread is done 
13th January 2018, 08:17 PM  #186 
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POST #19a
D. Infinite Line Source: timedomain impulse response (continued) A slightly different form for the timedomain 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. 
13th January 2018, 10:51 PM  #187 
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Graphs soon?

14th January 2018, 06:45 PM  #188 
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Perhaps someone can post some plots of the timedomain 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 timedomain impulse response is simply the Inverse Fourier Transform of the frequencydomain 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. 
14th January 2018, 07:31 PM  #189  
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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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Earl Geddes Gedlee Website Last edited by gedlee; 14th January 2018 at 07:35 PM. 

14th January 2018, 08:02 PM  #190 
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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 1storder RC low pass filter. When you examine the mathematical form of the stepresponse, 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 stepfunction has units of "volts"). What does the mathematical formulation of the impulseresponse 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 timedomain impulse response of the simple RC lowpass 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 discretetime domain but too damn abstract in the continuoustime 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 timedomain 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 08:11 PM. 
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