Hello,
i ask you to tell me, provided a constant sampling interval, how hi a sampling rate f is necessary in order to capture any signal of F dynamic range, retaining p power bandwidth and introducing neither ringing nor aliasing. Clear is, that only lopasses of first order, possibly cascaded, may be used. Also clear is, that the filter must attentuate signal components above half sampling rate to belo dynamic range.
If only one reactance is allowed, it gets f=2*p*F, well two Gigasamples/second for p and F of the stereo Compact Disc. This also is the formula for F equal 2, in case of which a cascade of lopasses is of no use. Optimizing for greater F, one uses multiple reactances and must take into account passband response of a first-order lopass. For example, for F equal 8 (~18dB) one would apply three equal lopasses, one of which attentuates ruffly -1dB at p, -3dB at 2p and -6dB at 4p. Compared to using only one reactance, one can halve f.
I guess, one uses as many reactances as the logarithm of F to base of 2, which is 16 for the Compact Disc. I also guess, that now f must be 2p times the root of F. For a New stereo Compact Disc one needs f=2*2*16KHz*256, well 15 Megasamples/s.
If one allows moderate ringing at and above p and does not make a fuzz about F, f=MagneticFrequency=2*Pi*p.
i ask you to tell me, provided a constant sampling interval, how hi a sampling rate f is necessary in order to capture any signal of F dynamic range, retaining p power bandwidth and introducing neither ringing nor aliasing. Clear is, that only lopasses of first order, possibly cascaded, may be used. Also clear is, that the filter must attentuate signal components above half sampling rate to belo dynamic range.
If only one reactance is allowed, it gets f=2*p*F, well two Gigasamples/second for p and F of the stereo Compact Disc. This also is the formula for F equal 2, in case of which a cascade of lopasses is of no use. Optimizing for greater F, one uses multiple reactances and must take into account passband response of a first-order lopass. For example, for F equal 8 (~18dB) one would apply three equal lopasses, one of which attentuates ruffly -1dB at p, -3dB at 2p and -6dB at 4p. Compared to using only one reactance, one can halve f.
I guess, one uses as many reactances as the logarithm of F to base of 2, which is 16 for the Compact Disc. I also guess, that now f must be 2p times the root of F. For a New stereo Compact Disc one needs f=2*2*16KHz*256, well 15 Megasamples/s.
If one allows moderate ringing at and above p and does not make a fuzz about F, f=MagneticFrequency=2*Pi*p.
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Grasso filters
A cascade of two first-order filters of equal corner frequency is also called a second-order filter with Q=1/2 or a second-order Linkwitz filter. But a cascade of more first-order filters of equal corner frequency has no name yet, or has it? It is useful, as it has the fastest transition from pass- to stopband of all filters without ringing. The key is equal corner frequency of each first-order filter element.
Calculating it, one chooses n elements, that each one attentuates for 6dB at the frequency, at and above or below which everything shall be discarded. Power bandwidth is then calculated as (root of ((nth root of 2) minus 1) times corner frequency of a single element), see Filter". Should resulting power bandwidth be too small, then one must think over one's goals of dynamic range, power and discard frequencies, because it ain't getting any steeper without ringing anyway.
A cascade of two first-order filters of equal corner frequency is also called a second-order filter with Q=1/2 or a second-order Linkwitz filter. But a cascade of more first-order filters of equal corner frequency has no name yet, or has it? It is useful, as it has the fastest transition from pass- to stopband of all filters without ringing. The key is equal corner frequency of each first-order filter element.
Calculating it, one chooses n elements, that each one attentuates for 6dB at the frequency, at and above or below which everything shall be discarded. Power bandwidth is then calculated as (root of ((nth root of 2) minus 1) times corner frequency of a single element), see Filter". Should resulting power bandwidth be too small, then one must think over one's goals of dynamic range, power and discard frequencies, because it ain't getting any steeper without ringing anyway.
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No, they have maximum flat group delay. Still this does not mean freedom from ringing.grasso789 said:
Thanks for not ignoring me anymore, DF96! Gaussian filters cause pre-ringing, because their impulse response rises as an asymptote to flat. Perfectly this only works with an infinite time delay. In contrary to that, a minimum-phase filter or allpass, say an analogue filter, shows an impulse response, which jumps out of flat, not necessarily vertically but with a lift-off point. This does not need a time delay.
Since you're thinking theoretically, not practically, I suspect you may be making the problem more complex than is necessary. The required sample rate is determined by the signal's bandwidth. If the maximum signal frequency falls to the level of the system quantization floor by frequency F, then the system Nyquist frequency can be set to F. So long as the sample rate is => F*2, no anti-alias filter is required. In other words, the signal would be incapable of provoking aliasing, therefore, no anti-alias filter is required. Anti-alias filters become necessary where the signal's maximum bandwidth is not completely known or guaranteed.
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And is it? Natural acoustic sources are not strictly bandwidth-limited, neither microphones are. A microphone may record a pair of cymbals with -12 dB at 32KHz. In order to provide for this fact, we must either set sampling rate to at least 64 KHz or further lopass-filter the signal, further than what the microphone already does.
In what way are natural acoustic sources not bandwidth limited? Even if they weren't, microphones and their amplifiers are certainly bandwidth limited. The Nyquist frequency only need extend to the frequency where their output has fallen to the level of the channel quantization noise floor, and even that may easily prove a more demanding requirement than necessary for an application intended for human auditory perception.
The only study on the ultrasonic spectrum of live instruments I readily found via Google is by James Boyk. His spectrum analyzer test limit was 102KHz, where the output was roughly 70dB below the peak output. My eyeball estimation of the charted roll-off rate is that it would be down roughly 100dB by 200kHz. If we double that figure to 400kHz for an additional octave of margin, an roughly 1MHz sample rate seems likely adequate to capture any music without recourse to an anti-alias filter. Without suffering audible aliasing.
If you like, double the sample rate again to 2MHz. Whatever is the minimum required rate, it seems orders of magnitude lower than the 2GHz you had calculated for supporting the ultra wide transition band of an non-ringing anti-alias filter.
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