Re: DDS question: why sine lookup?

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.


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                              Waldek Hebisch