Hi Cinaed,
On 08/16/2017 04:14 AM, Cinaed Simson wrote:
I would replace the 2 serial rational resamplers with one, namely,
(8/1)*(400000/614400)=(8/1)*400/614.4) = 5.208333333333333/1
or
15.625/3
That is, interpolate by 15.625 decimate by 3.
By which you mean "interpolate by 15625 and decimate by 3000", because a
rational resampler can only sample up and down by integer amounts" :)
If I'm not doing this wrong in my head, that is equivalent to
interpolating by 125, decimating by 24; that is, in fact, what the
rational resampler will do internally if you parameterize it with
interp=15625, decim=3000, but it's usually a good idea to find the
minimum prime factors necessary to express the quotient, just so one
sees how bad the computational effort will be.
The larger of the two numbers (125,24) will dictate how sharp the
anti-aliasing or anti-imaging filter in the resampler must be, and that
sharpness is what makes the filter long, and potentially hard to do in
real time.
1/125 isn't really the nicest of all (nyquist rate-relative) transition
widths, but it should still be quite manageable.
In extreme cases, and I don't think this is one just yet, but I haven't
done the math or tested it, you'd actually switch from using a rational
resampler to using an arbitrary resampler:
While rational resamplers are based on the idea that there's a rational
ratio between in- and output rate, so that you can just interpolate and
decimate by integer factors, other resamplers do exist.
Essentially, they are based on the idea that since the input signal, as
Nyquist tells us, must be band-limited, there's at least a hypothetical
"theoretical" continuous signal that is equivalent to the digital
signal. Find a formula for that continuous signal, calculate arbitrary
signal values between the original sample points, and you can do *any*
resampling ratio, including ∛2, e, π⁄2, whatever, any real number.
If you look at the description closely, "calculate arbitrary … between
original sample points", those are interpolators. There's different
approaches to interpolation functions; one would be to have a bank of
filters that delay the input signal by a fraction of the sample period,
and then pick (and if necessary, somehow combine) the outputs closest to
the new sampling point we actually need. That's what the PFB arbitrary
resamplers do – take the two delayed versions "closest" to that
fractional delay we need, linearly interpolate¹.
Of course, this comes at the cost of continuously having to feed a whole
bank of filters with the input signal. And that's a large cost. But
before someone implements a 1023/1000-rational resampler with thousands
of taps, having maybe 64 filters to have enough fractionally delayed
versions of the input signal to keep the jitter tolerably low isn't all
that bad.
Best regards,
Marcus
¹ one could be smarter than linear, but I didn't do the math whether
that is a good idea. Didn't read any papers, but IIRC, harris' book only
does the math for linear interpolation; intuitively, sinc interpolation
would be "correcter", but there was beauty in the zeros of the sinc²(t)
transform of the triangle.
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