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Re: DDS question: why sine lookup?

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On 2025-05-14 11:35, Waldek Hebisch wrote:
> In sci.electronics.design Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> wrote:
>> On 2025-05-13 19:25, Waldek Hebisch wrote:
>>> In sci.electronics.design john larkin <jl@glen--canyon.com> wrote:
>>>> A DDS clock generator uses an NCO (a phase accumulator) and takes some
>>>> number of MSBs, maps through a sine lookup table, drives a DAC and a
>>>> lowpass filter and finally a comparator. The DAC output gets pretty
>>>> ratty near Nyquist, and the filter smooths out and interpolates the
>>>> steps and reduces jitter.
>>>>
>>>> But why do the sine lookup? Why not use the phase accumulator MSBs
>>>> directly and get a sawtooth, and filter that?
>>>>
>>>> The lowpass filter looks backwards in time for a bunch of ugly samples
>>>> to average into a straight line. The older sine samples are the wrong
>>>> polarity! If the filter impulse response is basically zero over the
>>>> period of the sawtooth, and we compare near the peak, we'll average a
>>>> lot of steps and forget the big sawtooth reset.
>>>
>>> Sine is close to optimal for high quality DDS.  The math is
>>> as follows.  First, your DAC has some response in time
>>> domain, but for purpose of computation one can assume that
>>> at clock tick number n it generates delta peak centered
>>> at nT_0 with amplitude f(nT_0/T_1) where f is function stored in
>>> lookup table, T_0 is period of digital clock and T_1 is desired
>>> period.  Mathematically
>>>
>>> S(t) = \sum_n f(nT_0/T_1)\delta(t - nT_0) = \sum_n f(t/T_1)\delta(t - nT_0)
>>>
>>> where summation is over all integer n.
>>>
>>> Fourier transform of this is
>>>
>>> C\sum_l \sum_m c_l \delta(\omega - 2m\pi/T_0 - 2l\pi/T_1)
>>>
>>> where we have double summation over integer l and m, c_l is
>>> l-th Fourier coefficient of f and C is a constant.
>>> Sine has only 2 Fourier components, so formula simplifies to
>>>
>>> (1/2)C\sum_m (\delta(\omega - 2m\pi/T_0 - 2\pi/T_1) +
>>>                 \delta(\omega - 2m\pi/T_0 + 2\pi/T_1))
>>>
>>> With aggressive filtering high freqency components can be
>>> made arbitrarily small, so after filter Fourier transform
>>> is
>>>
>>> (M/2)C(\delta(\omega - 2\pi/T_1) + \delta(\omega+ 2\pi/T_1)) +
>>>      small distortion
>>>
>>> where M represents transmitance of the filter at frequency
>>> 1/T_1.  Back in time domain signal is
>>>
>>> M\sin(t) + small distortion
>>>
>>> The point is that distortion, hence phase noise can be made
>>> arbitrarily small.
>>>
>>> What happens with different f?  When T_0/T_1 is irrational,
>>> the sum 2m\pi/T_0 + 2l\pi/T_1 can take values arbitrarily
>>> close to 0.  In particular, there will be combinations of
>>> l and m such that this sum is in the interval [-\pi/T_1, \pi/T_1],
>>> so we will get low frequency terms with wrong frequency.
>>> Assuming fixed low pass filter such terms can not be filered
>>> out.  How bad this is?  For sawtooth the second Fourier
>>> coefficient has maginitude equal to half of the magnitude
>>> of the first coefficient, so one can expect distortion
>>> of order 50%, which looks quite bad.  Using symmetric
>>> troangular weave, second Fourier coefficient is 0 and
>>> third has magnitude 1/9 of magnitude of the first
>>> coefficient, which is much better, but still limits
>>> possible quality.
>>>
>>>
>> Distortion and phase noise are only obliquely related.
> 
> Well, signal to the comparator is f(t) + r(t) where r(t)
> is distortion.  Assuming that distortion is reasonably
> small and regular we have
> 
> phase error \approx -r(t_0)/f'(t_0)
> 
> where t_0 is zero of f.  To minimize phase error you can
> try to make f'(t_0) big, but John is working over an
> octave, so not much possibility here.
> 
> So we need small r(t_0).  In general desired frequency and
> and frequency of digital clock are uncorelated, so zeros
> of f will be randomly distributed over quasi periods of
> r, which means that to have small avarage error you
> need small average of absolute value of r.  Similarly
> smal maximal error need small maximum of r.
> 
> Of course, there are constant factors because simple Fourier
> computation works exactly only for energy.  Those constants
> are hard to compute but in practice do not tend to be really
> large (say of order of 2 or 3).  So at low accuracy there
> may be some room to use different function than sine.
> For higher accuracy the above calculation gives too big
> term to ignore.

"Distortion" to me means harmonics and IMD.  The usual small-signal 
analysis isn't too useful when the interference is at frequencies 
comparable to or greater than the fundamental.

And it isn't really clock interference that's in view with a DDS, 
because there's a zero-order hold, which in principle nulls out the 
clock and all of its harmonics.  (You have to have a reconstruction 
filter of some sort anyway.)

What kills you with DDS is the nasty, very high-order subharmonics due 
to truncation of the phase word.  Power supply junk is often of the same 
order, but its easier to get rid of--the truncation sidebands  extend 
down to the very low baseband.

Cheers

Phil Hobbs

-- 
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510

http://electrooptical.net
http://hobbs-eo.com

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DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-06 08:48 -0700
  Re: DDS question: why sine lookup? Jeroen Belleman <jeroen@nospam.please> - 2025-05-06 21:00 +0200
    Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-06 12:33 -0700
    Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-06 16:46 -0400
      Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-06 15:50 -0700
        Re: DDS question: why sine lookup? Arie de Muijnck <noreply@ademu.nl> - 2025-05-07 11:44 +0200
          Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-07 06:32 -0700
            Re: DDS question: why sine lookup? Bill Sloman <bill.sloman@ieee.org> - 2025-05-08 01:58 +1000
      Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-07 06:55 -0700
        Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-07 11:31 -0400
          Re: DDS question: why sine lookup? Bill Sloman <bill.sloman@ieee.org> - 2025-05-08 02:11 +1000
  Re: DDS question: why sine lookup? Gerhard Hoffmann <dk4xp@arcor.de> - 2025-05-06 21:27 +0200
    Re: DDS question: why sine lookup? "Don" <g@crcomp.net> - 2025-05-06 20:10 +0000
    Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-06 13:35 -0700
  Re: DDS question: why sine lookup? Bill Sloman <bill.sloman@ieee.org> - 2025-05-07 16:27 +1000
  Re: DDS question: why sine lookup? Martin Brown <'''newspam'''@nonad.co.uk> - 2025-05-07 20:32 +0100
    Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-07 13:01 -0700
      Re: DDS question: why sine lookup? bitrex <user@example.net> - 2025-05-07 20:27 -0400
        Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-07 19:21 -0700
          Re: DDS question: why sine lookup? Bill Sloman <bill.sloman@ieee.org> - 2025-05-08 22:49 +1000
          Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-08 14:20 -0400
            Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-08 11:58 -0700
              Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-08 16:24 -0400
                Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-08 14:18 -0700
                Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-08 22:28 +0000
                Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-08 16:24 -0700
                Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-08 23:57 +0000
                Re: DDS question: why sine lookup? piglet <erichpwagner@hotmail.com> - 2025-05-09 04:48 +0000
                Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-09 12:53 -0700
                Re: DDS question: why sine lookup? Bill Sloman <bill.sloman@ieee.org> - 2025-05-10 18:53 +1000
                Re: DDS question: why sine lookup? Lasse Langwadt <llc@fonz.dk> - 2025-05-10 19:52 +0200
                Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-10 11:53 -0700
                Re: DDS question: why sine lookup? dplatt@coop.radagast.org (Dave Platt) - 2025-05-10 16:50 -0700
                Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-10 17:39 -0700
                Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-11 16:27 -0400
                Re: DDS question: why sine lookup? Bill Sloman <bill.sloman@ieee.org> - 2025-05-11 16:55 +1000
                Re: DDS question: why sine lookup? Lasse Langwadt <llc@fonz.dk> - 2025-05-11 22:03 +0200
                Re: DDS question: why sine lookup? Bill Sloman <bill.sloman@ieee.org> - 2025-05-09 15:52 +1000
        Re: DDS question: why sine lookup? Lawrence D'Oliveiro <ldo@nz.invalid> - 2025-05-10 00:44 +0000
      Re: DDS question: why sine lookup? Martin Brown <'''newspam'''@nonad.co.uk> - 2025-05-09 09:32 +0100
  Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-10 17:00 -0700
  Re: DDS question: why sine lookup? antispam@fricas.org (Waldek Hebisch) - 2025-05-13 23:25 +0000
    Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-13 20:22 -0400
      Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-13 18:40 -0700
      Re: DDS question: why sine lookup? antispam@fricas.org (Waldek Hebisch) - 2025-05-14 15:35 +0000
        Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-14 15:05 -0400
          Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-14 14:53 -0700
            Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-14 19:16 -0400
              Re: DDS question: why sine lookup? john larkin <jl@glen--canyon.com> - 2025-05-14 17:10 -0700
                Re: DDS question: why sine lookup? Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-05-15 12:40 -0400
          Re: DDS question: why sine lookup? Bill Sloman <bill.sloman@ieee.org> - 2025-05-15 17:23 +1000

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