The point(sample, memory location)- skipping nature of DDS

 
 DDS: Direct Digital Synthesis = point-skipping at  constant  fc  update(clock)-rate
[ vs. traditional PPC: point-per-clock synthesis at variable clock-rate ]


Let N data points covering one complete cycle of a (sine) waveform (a block of N samples) be stored in the memory (LUT: look-up table). The address pointer of LUT is a "D step size, modulo N (mod N, overflowing)" phase ACCumulator.
D, a positive integer, is the frequency tuning word (or simple the tuning number, or the frequency word, FCW: frequency control word, FIW: frequency input word, FSW: frequency setting word ...)

f / fc = D / N,       D prime to N,  and   D < N/2.

          As the output frequency is increased, the number of samples per (sinusoid) cycle decreases.
Notes:
  1. If  D is not prime to N and g > 1 is the gcd: greatest common divisor of  D and N,  then the same frequency  f = (D/N).f c  can be generated with less memory by replacing  the tuning word of D by D/g  and  the numerical period of N by N/g.
  2. If  D > N/2  then an alias frequency appears, and there is a phase inversion ("negative frequency") in the even Nyquist zones; i.e. the phase ACC works in a "phase decrement" mode.
Using a "D step size, mod N (overflowing)" phase ACCumulator as address pointer to N size LUT is equivalent with an uniform(U)-permutation (temporal rearrangement) of waveform samples within a N-block waveform.
(1) U-permutation (temporal "scrambling" or time "shuffling") is a simple reversible mapping of sample position i into position s, with

s = D.i (mod N),       i = 0, 1 , ..., N-1 and D is prime to N.

Note: the tuning number D should be prime to the block length N  (which is the numerical period of samples) for the mapping one to one.

The figures illustrate this concept with a simple example (N = 8, and D = 1, D = 3). Marks in the permutation matrix are spread uniformly, hence the name.
If  D > N/2 one can visually explore the aliasing phenomenon.
And, try D = 2 value - not prime to N (you get a shorter numerical period).


(2) U-permutations produce a uniform rearrangement in the DFT spectrum as well. ( U-permutation in the time-domain causes an U-permutation in the frequency-domain.)
Only the position of each component in the spectrum changes!

Note: see the References

This means that changing frequency tuning word ( D prime to a given N ! ) results in "scrambling" of the same spectral lines: the spectrum structure will differ only in the location of the components and not in the magnitude. 
( See:  PTperm.avi, generated from Mathcad worksheet .)

(3) The inverse mapping, i.e. the sample re-ordering or ModT (re)plot (Modulo Time plot, temporal "descrambling") is performed by another (inverse) U-permutation so that  J.D = 1 (mod N):

i = J.s (mod N),       s = 0, 1, ..., N-1.

Notes:
  1. The condition that  J.D = 1 (mod N)   results from the requirement that J.s = J . (D.i) = i, for all i. This condition is equivalent to finding an J ( and k ) such that  J.D = k.N + 1,  where both J and k are integers.
  2.  See:  inverse.mcd  Mathcad worksheet  ( for computing J ).

J is multiplicative inverse of D (and in the matrix Figure the value of  J = 3 one can find graphically, too).
The inverse mapping unrolls the spectrum as well.


References: 

 e-mail: papay@hit.bme.hu