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Tue Sep 4 14:07:56 CEST 2012

Where is the low frequency noise in S/D approximation?

( Context: understanding properties of the binary sequence produced by
           a Sigma/Delta modulator. )

Supposedly (following from linear approximation and white quantisation
noise assumptions) the frequency envelope of the S/D approximation
noise is F^{-1}, where F is the loop filter, e.g. lowpass one pole or
integrator.

Then I seem to have a paradox.  Suppose we're approximating a constant
signal s[n] = 2/3 with the periodic sequence b = 1,1,0,...  This gives
a periodic approximation error e = 1/3, 1/3, -2/3, ...  (following b =
s + e)

This b is the output of a S/D modulator with F = 1 / z-1, a pure
(summing) integrator with pole at DC: w_0 = 0, e^{j w_0} = z_0 = 1.

Where is the low frequency component of e?  The signal is periodic
with period 3, meaning all frequency components are multiples of fs/3
which is high, meaning there are no low frequency components in e.

Is this because linearizated analysis is too crude an approximation?

This is a proper S/D signal?  It is the
output of a S/D modulator with F = 1 / z-1, a pure (summing)
integrator with pole at DC: w_0 = 0, e^{j w_0} = z_0 = 1.




Is it possible to construct an error signal that does have a
low-frequency component?  I'm guessing that this would be the case for
irrational constant signals.

Just as in the analysis above, if s[n] is constant, the frequency
content of b and e are the same, so it is enough to limit the analysis
to b.

Picking F = (z-1)^-1 allows a similar simple accumulation model to
generate the binary sequence.  As irrational number, pick 1/sqrt(2).

What is the sequence?

This can be represented by a graph of y = D int(x/sqrt(2)).  Each
impulse is a 1.

This sequence will be a-periodic, which means that it will have
non-vanishing arbitrarily low frequency content: for any period p, no
matter how large, we can never have

   1/sqrt(2) - \sum_{k=1}^p b[k] = 0



To construct an approximation to a constant signal that has some some
low frequency content in the error signal, we simply need to pick the
period large enough.  If s[n] = q/p with q<p, then the output will be
periodic with period p.

So can it be seen that at least the amplitude of low frequency content
is proportional to 1/p, following the conclusion of the linear
analysis?

An example:
 3/11  0  3  6  9  1  4  7 10  2  5  8  0
-8        0  0  0  1  0  0  0  1  0  0  1

The DC component obviously corresponds to 3/11.  The frequency
component corresponding to 3/11 fs is:

          e^3x + e^7x + e^10x   with x = j 2pi/11




Conclusion: for constant approximation there is going to be
low-frequency content apart from some exceptions because:

- (real)  most real numbers are irrational
- (n-bit) most integers are "fairly" coprime with 2^n

( fairly meaning they have a factor 2^m with m << nb bits in word )



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