From: "Klaus von der Heide" <v.d.heide@on-line.de>
To: rsgb_lf_group@blacksheep.org
Date: Thu, 24 Jan 2002 11:16:44 +0100
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Subject: Re: LF: Re: Digital Sine Generators
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Hello all,

Zim wrote:

> Thank you for your good humour and patience in answering my dumb questions.
> It's embarrassing for me to show my ignorance in public, but it's worth it
> if
> I learn something useful.

The mathematician would say:  
The only dumb question is the empty question.
 
> For a long time now I've been trying to understand the practical trade-offs
> in DDS design.
> 
> So let's see if I have this DDS Distortion thing right ...........
> 
> Memory Quantisation:
> ----------------------
> 
> If all of the bits in a DDS latch are used to drive the Sine ROM, then there
> is NO problem with memory quantisation.
> 
> If the Latch has many bits (e.g. to get fine frequency steps) then the ROM
> will have to be excessively large. If only the higher order bits are used to
> drive the ROM, then the wave generated will have a degree of roughness.

Yes, this roughness is additional to the inevitable value 
quantization of the memory words which you mention further below.  
The effect of address quantization often is much greater.
 
> In the special case where the low order (unused) bits do not change (e.g.
> where the matching low order Increment bits are at zero), then there is also
> no quantisation error (is this correct?).

Yes and no. See above. The quantization of the memory words remains. 
But the effort to get this smaller by longer words is linear in 
complexity.  That means, each further bit gives a gain of 6 dB. 
In contrast, the effort to get the effect of address quantization 
smaller is exponential.  To gain 6 dB, you have to double the memory.
  
> Question: If we use a modern DDS chip, but limit the steps used to those
> which only exercise the ROM address lines, does this give us a more pure
> signal ?

Yes. But that is a severe limitation. And modern DDS chips often have 
implemented extra hardware for dithering which smoothes out the 
effect over ALL frequencies.  You cannot switch off that.
 
> In practical DDS design, there are various methods to increase the
> resolution without increasing the ROM size. One method is to interpolate the
> "missing" readings. The simplest would be to draw a straight line between
> the two readings and then subdivide it into as many points as are required.

Yes, interpolation is the best way to reduce the error.  But linear 
interpolation is not sufficient.  Implementation of a Taylor 
approximation is easy since the derivations of sin are cos -sin -cos 
sin and so on.  So you have all the coefficients from your sine 
table.  But it requires an arithmetic unit.

> D/A Quantisation:
> -------------------
> 
> This is caused by the limit to the number of bits in each ROM location.

Yes, we mentioned that above.  When the phase step is chosen such 
that only a few memory words are addressed periodically, the spectrum 
of the periodical quantization error (that's not a sine!) 
concentrates the power into equidistant spikes, that are mirrored at 
half the sampling frequency and at 0 again and so on.  This also can 
be smoothed out by dithering. But a dither generally raises the noise 
level.


Now, let us review the method with the Chines Remainder Theorem:
We have no address quantization.  But we have value quantization 
of the memory words.  And additionally we have some rounding error 
of the arithmetic that computes the final result. That's the same 
if we use a Taylor-approximation.  I found that the DDS with the 
Chinese Remainders is easier to implement and much more efficient 
on a DSP than a classic DDS with interpolation.


73 de Klaus, DJ5HG