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8th July 2012, 06:55 PM  #31  
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8th July 2012, 07:15 PM  #32  
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And what about the corresponding high pass? I could subtract the output of the lowpass filter from the signal, to give me the high pass response, presumably. Would the equivalent filter still have some sort of Gaussian classification, or would it be classified as something else? (Many thanks for your patience) 

8th July 2012, 07:25 PM  #33  
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You can vary the width of your Gaussian lowpass filter, and thus change its cutoff frequency, but you cannot change the steepness of the cutoff. If you attempt to do so, what you end up with will no longer be a Gaussian. Quote:


8th July 2012, 09:45 PM  #34 
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Consider the following:
S: Sinc (Sinus cardinal) delivers the most selective lowpass when used as FIR coefficients, unfortunately with a lot of preshoot and ringing in time domain. Dirac minus Sinc delivers the most selective highpass when used as FIR coefficient, unfortunately with a lot of preshoot and ringing in time domain. G: Gaussian delivers a quite selective lowpass when used as FIR coefficient, with the advantage of no preshoot and no ringing in time domain. Dirac minus Gaussian FIR only delivers a 2ndorder (24db/octave slope) highpass, which is insufficient. F: We have the socalled "FlatTop" window. What if we use the "FlatTop" shape, not as window, but as FIR lowpass coefficients? What are the properties of a Dirac minus "FlatTop", as complementary highpass? A new lowpass FIR could be computed, combining the three above approaches. What kind of combination? I would suggest: (S at the power of s) * (G at the power of g) * (F at the power of f) With s, g and f as real numbers. Start with s=g=f=1 maybe. Let us define "F6db" the desired 6 dB crossover frequency. S would feature a 3dB cutoff frequency equal to "F6db" * sf G would feature a 3dB cutoff frequency equal to "G3db" * gf F would feature a 3dB cutoff frequency equal to "F3dB" * ff Start with sf=gf=ff=1 maybe. How to iterate the s, g, f, sf, gf, ff values? You would check for a 0.2 dB corridor in the lowpass passband, a 0.2 dB corridor in the highpass passband, a 36dB/octave slope for the lowpass, and a 36dB/octave slope for the highpass. Other constraints?  the FIR would feature 128 taps maximum  quite welcome would be the FIR to feature lots of zero coefficients  the desired FIR "F6dB" could not be arbitrary, but an integer fraction of the sampling frequency, say Fs/12 (4 kHz for Fs=48 kHz) Some relaxes? A "don't care" scheme at frequencies where both the targeted amplitude and the realized amplitude are less than 40 dB. 
8th July 2012, 10:29 PM  #35 
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You appear to be reinventing the wheel. Look up "windowbased filter design". It can be applied in two ways: prototype frequency response is convolved with a window for improved impulse response; or, by duality, prototype impulse response is convolved with a window for improved frequency response.

9th July 2012, 12:10 AM  #36 
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Say Fs = 48 kHz.
Say you want the 6dB cutoff to be 4 kHz. Say you want a 36dB/octave slope. You want the FIR lowpass to feature a monotonic amplitude response. You want the FIR lowpass to feature a linear phase. Those design parameters are especially appropriated when designing the FIR in frequency domain. It is like operating a graphic equalizer with equally spaced frequencies from DC to Fs/2. You draw the amplitude response at will, following the targeted amplitude response, and for each frequency point, you set the phase in order to remain phase linear. However, we get an infinity of FIRs implementing the above constraints, as we have said nothing about the transition region: is it Bessel, Butterworth, or anything else? We will now select one and only one FIR in the infinity of available FIRs. We will now focus on the transition band. We will start from a maximallyflat design emulating a phaselinear 36dB/octave Butterworth. We will then assess the complementary highpass, equal to Dirac less the lowpass. If the complementary highpass doesn't exhibit the targeted 36dB/octave slope and a symmetrical transition band (compared to the lowpass), we will soften the FIR lowpass transition band. Then reassess the complementary highpass. And so on, iteratively. There should be rules enabling great success if asking for particular feature sets. Looking at the Philips DSS930 design, I have the impression that the Fs/12 6dB point combined to the 36dB slope is one particular, high efficiency feature set. I would like to know more about this. By designing the FIR in frequency domain, you have the temptation of relying on a power of two FIR lenght. This may not be optimal. There is a great probability that the high efficiency feature sets get mathematically coupled to particular FIR lenghts, not fitting into the power of two numbers. The whole thing gets thus complicated, because the exact FIR lenght may be of great importance. Physically, there is major problem with phaselinear FIRs having a power of two lenghts. Physically, you would always design a phaselinear FIR as having an odd amount of coefficients. In a 3 tap FIR, the central coefficient would describe the main energy locus, while the two side coefficients would describe the energy leak over time, before and after the main energy locus. You would add precision to the energy leak description using a 5 tap FIR, then 7 tap FIR, and so on. This makes me say that the Fs/12 6dB point combined to the 36dB slope in the Philips DSS930, is a particular feature set leading to a great computing efficiency, but at this stage, we don't know the exact FIR lenght. Philips says a 79 tap FIR lenght for the lowpass, and a 30 tap FIR lenght for the highpass. That's quite bizarre. With different FIR lengths, how could they exactly derive the complementary highpass, from the lowpass? Anyway, we are dealing with medium length FIRs. We don't want to execute them in the frequency domain. We thus need to translate them back in time domain, in an exact way, using the appropriate math. This was about the crossover. Now, if we want to add the driver compensation feature (a Bode Plot inversion), nothing prevents embedding it into the final FIR. If we need a better frequency resolution for compensating the driver, nothing prevents from basing the whole design on a longer FIR, featuring a length that's compatible with the efficiency set we are starting from. I guess that if the high efficiency FIR was N taps, that a (2N+1) FIR lenght or (3N+1) FIR lenght won't harm the efficiency pattern. Possibly, in the Philips DSS930 design, the 30 tap tweeter FIR managed to do both the highpass crossover, and the tweeter Bode plot inversion. Possibly, what Philips calls a 30 tap FIR, is in reality a 31 tap FIR, as a 31 tap FIR is only requiring 30 memory cells. Possibly, in the Philips DSS930 design, a 30 tap woofer FIR was too short for embedding the woofer Bode plot inversion. When looking at the woofer unfiltered frequency response featuring a steep extinction above 5 kHz, for sure such woofer was exhibiting a quite long ringing at 4.5 KHz or so, even if such natural ringing was shortened using a dedicated, IIRbased notch. Possibly Philips needed to extend the woofer FIR length to 79 tap (1.65 ms), for encompassing the 4.5 kHz residual ringing untill a decent extinction. The 1.65 ms FIR duration equal 7.4 periods at 4.5 kHz. 
9th July 2012, 12:12 AM  #37  
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9th July 2012, 12:36 AM  #38 
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@steph and berchin
(Still trying to catch up to a fraction of what you know) Something I missed totally earlier was the possibility of the high pass being an exact complement of the low pass in the crossover region, so that the ringing of the woofer is 'neutralised' by the opposite ringing of the tweeter (give or take the discrepancies due to offaxis listening). If this is the case (or can be made the case), is the issue of overshoot one of a waste of power and/or needless displacement of the driver? 
9th July 2012, 12:49 AM  #39 
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9th July 2012, 12:56 AM  #40  
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The real issue is the fact that, even though the lowpass and highpass timedomain responses may cancel mathematically, they do not necessarily cancel acoustically. So your first thought was correct. [1] Parseval's Relation 

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