Sampling and Nyquist/Antialiasing filters

[Magnus Danielson]

>>>>> "HDT" == Harvey Devoe Thornburg <harv23 at leland.Stanford.EDU> writes:

Hi Harvey and others!

 >> 
 >> >>>>> "DRB" == Duane R Balvage <dbalvage at ptdcs2.ra.intel.com> writes:
 >> 
 >> Is this the reason people have considered oversampling? Certainly.
 >> 
 DRB> They are using 4 pole input filters (I think - have to check the schems)
 DRB> so the slopes are a bit gradual, taking off some of the "sparkle" of
 DRB> the samples because they have to have their corner frequency set low 
 DRB> enough to avoid aliasing. Any suggestions???
 >> 
 >> The benefit of 48 db extra per octave allows us to move the cutoff
 >> upwards. I made some quick calculations and assuming that we require
 >> the same damping at the nyquist point, we can move it to
 >> 
 >> 41 kHz	18.85 kHz
 >> 32 kHz	14.4 kHz
 >> 12 kHz	10.6 kHz
 >> 
 >> This using a very crude method (basically using a Bode-plot fashion of
 >> view and let the nyquist frequency be the common intersection point) .
 >> 

 HDT> This assumes that all of the poles are at the same frequency, right?

Nope. It just assumes that their common responce resulted in the -3.01
dB point at the given frequencies and that their common slope where
-6.02 dB per octave and pole.

 HDT> Maybe it is possible to distribute them somehow such that the slope
 HDT> is steeper at the Nyquist frequency, or such that the response is
 HDT> an even better approximation to a "brickwall" lowpass.  Elliptic
 HDT> filters give you the best approximation, in the sense of minimizing
 HDT> the maximum of the difference between your filter's response and the
 HDT> desired "brickwall" response.

Well, actually, the -6 dB slop per octave _AND POLE_ is kind of
inherent if the poles sits in a brick wall optimisation like
Butterworth or Chebyshev (both are likely candidates here).
It is however true that Elliptic filters have even steper slopes, but
this is due to the fact that they also places zeros near the slope
region in the stop band instead of placing them in infinity (as with
Butterworth, Bessel or Chebyshev does).

Also, note that he was using integrated switch capacitor filters these
are very likely to be either Butterworth (maximum passband flatness)
or Chebyshev (maximum steepness). Elliptic/Cauer filters is rarely
seen in these filters, but they are fully realizable aswell.

 HDT> Unfortunately, elliptic filters as well as cascaded one poles have 
 HDT> a highly nonlinear phase response.  This means certain frequencies
 HDT> will be delayed relative to others. I would guess this becomes a 
 HDT> bad problem near the Nyquist frequency. There is another type of filter
 HDT> (called Bessel filter) which has approximately linear phase, but a more
 HDT> gradual cutoff.

Yes, Bessel/Thomson filters is really the maximum flat groupdelay
approximation to the brickwall filter.

In these respects the Chebyshev filters are just terrible and a
Butterworth is much smoother and Bessel further better, especially
when you get up to higher degrees (where Butterworth filters gets a
stonger bump in the groupdelay at the cutoff frequency).

 HDT> IMHO, nonlinear phase is not so important for a sampler, but for
 HDT> "high-end" or mixing applications, recording engineers swear by it.

On the other hand, there is not such a big difference after all in how
you design, and my experience tells me that smother, quieter filters
are usually prefered in the long run. The less strange things that
happends with the sound, the better. Yeat again, we are talking about
what happends in the over range, and dropping in a few zeros to kick
out distorsion effect is well worth it.

 HDT> Also in communications it is important not to distort the shape of 
 HDT> the waveform; most of the "shape" information is contained in the pahse. for 
 HDT> these reasons all of the DAC's I've seen employ Bessel filters, something 
 HDT> like 6-8th order(*).  if this were my project I would try elliptic filters.

When you talk about telephone cursuits you must also recall that for
longer runs they do echo-cancelling, and the screwier they get the
frequency and phase/group-delay responce, the hard work it
becomes. They have pure economics and reality of life pointing fingers
for them.

As for real communication curcuits (like SDH/SONET/PDH lines) long
runs of lines require many transponders which must resample the data
with a recovererd clock. Due to jitter accumulation and requirements
for low bit error rates is peaking in the phase responce of the clock
recovery an critical issue (an peaking of 0.05 dB is an issue for
these people).

Cheers,
Magnus