[hpsdr] "Taylor Corrected DDS"

[Bob McGwier]

This is really straightforward and was what we used for the sinusoidal 
oscillator back in the visual basic code for the SDR-1000 in our 
numerically controlled oscillator before we figured out how fast modern 
desktop/laptop CPU's are when doing floating point arithmetic.   In 
fact, we did not need it at all for our PC based NCO's.  But in FPGA and 
DDS's,   it is a boost and this set of tricks should be filed away in 
your repertoire even if ultimately they do not prove useful here.

It is based upon nothing more sophisticated than  sin(a+b) and cos(a+b) 
and trigonometric identities.

The "Taylor corrected"  is all marketing hype.  You are making a first 
order correction to the sinusoid from a table lookup and that means you 
have done a first order Taylor series expansion that amount to nothing 
more than linear interpolation where the interpolating steps are 
quantized.  It sounds very sexy.  It is not.

That said,  this will be a really great experiment for us to conduct 
with our ALTERA parts (to shove it to Xilinx for doing this kind of hype).


sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

and

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

And the magic is over  ;-).


There has never been a simpler problem that is more illustrative of WHY 
one should think of these things as complex numbers.  These same two 
identities are one trivial identity in complex numbers.

exp(j (a+b))  = exp(ja)exp(jb)  (because the arguments are exponents)  
is the entire thing (if you follow Euler's identity in its   exp(j C)  
=   cos(C) + j sin(C) you will see it immediately).

So let us suppose that we wish to have 4096 points in our main look up 
table.  This is 2^12 points so we are looking at 12 bits of our 
accumulator for A.  There is NO advantage in our procedure for doing 
more than just truncating the phase Phi to 12 bits and calling the 
truncation a, an integer.

The residual phase error not taken into account by our truncated phase 
is   Phi - a.   Notice since we truncated Phi to twelve bits,  Phi - a 
is ALWAYS nonnegative.  This is a definite plus.  Let us add a new table 
of complex numbers 256 long and it will be

exp(j b)  where b takes on the values

b =    2 * i  * pi /(256*4096)  where i can take on the values  0,1, 
..., 255.    Pick the i  from 0 - 256 so that  Phi - a -  
(2*i*pi/(256*4096))is MINIMIZED.   If i is 256,  add one  phase step to 
a  and make b = 0.   Otherwise,   b is in our little 256 long table of 
complex numbers.

The final complex sinusoid output is   exp(ja)exp(jb)  and we have 
tremendously reduced the total harmonic distortion and phase truncation 
spurs.   But all we have is two table lookups.    In simple terms,   you 
have broken up the phase circle in 256*4096 or 2^20 pieces BUT, using 
the trig identities above  we can do the entire 2^20 different points 
with a single 2^12  lookup and a 2^8 lookup.  So in two table lookups,  
you get a very nice complex number with much lower phase truncation crud 
than the 2^12 lookup.


So,  what is the catch?   They have NOT told to you in the dds  paper 
from Xilinx how to make the analog signals, the voltages for the RF 
combine for the final output.  Notice that the entire block diagram set 
ends in the table lookups.  The hard part is what must then follow.  For 
the real channel  we need

cos(a)cos(b) - sin(a)sin(b) for example.  This is a summer and two 
mixers running at RF frequencies and we expect them to be perfectly 
balanced, etc.  We need to figure out how to do the two multiplies and a 
subtract to take advantage of the trivial trig identities and we need to 
do a pair of these, again perfectly balanced and identical group delay 
to produce signals in quadrature.   Running at tens to hundreds of MHz,  
this is a nontrivial issue.   Analog devices has spent tons of money 
learning how to make these nonlinear processes produce linear results.  
We do not have this at our disposal.  We are better off spending a few 
tens of dollars to get a DDS and feeding it to a PLL if you want spur 
free (and phase noise increased) operation.


Bob
N4HY






-- 
Robert W. McGwier, Ph.D.
805 Bunn Drive
Princeton, NJ 08540
(609)-924-4600
(sig required by employer)