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From: Bill Sloman <bill.sloman@ieee.org>
Newsgroups: sci.electronics.design,comp.dsp
Subject: Re: DDS question: why sine lookup?
Date: Thu, 15 May 2025 17:23:28 +1000
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On 15/05/2025 5:05 am, Phil Hobbs wrote:
> 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.

This is where Phil Hobbs VXCO solution starts looking good. You can only 
shift the VXCO frequency by +/-100pm but within that band you can shift 
it continuously. Admittedly, with a 150MHz VXCO you can only shift 
frequencies up 15kHz entirely continuously, but that should be enough 
for rotating machinery.

And a 150MHx VXCO should offer picosecond jitter, rather better than a 
nanosecond.

-- 
Bill Sloman, Sydney