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Wed Jul 4 13:44:39 EDT 2012

Sigma/Delta retake

Let's start from scratch.  Basically, there are two systems I'm
interested in: the first order analog lowpass or integrator, and the
digital integrator (i.e. the PIC carry flag hack).

Some basic assumptions:

- Reconstruction should be a simple (maximally flat) lowpass filter,
  meaning that the digital signal has a direct analog meaning.

- Feedback and reconstruction filter are the same.  I.e. the error
  signal is the effective error.

- First iteration: work with 1st order LP to simplify math, this
  leaves one design parameter: filter cutoff.

- Quantisation noise and signal are not correlated (which is wrong in
  practice!).

- Negative feedback minimizes error with white spectral distribution
  (if signal and noise are not correlated).


Let's get at the basic schematics.  We're interested in the error of
the reconstruction which is present after the negative feedback
summation [i].  First iteration I was thinking to minimise the
reconstruction error, which is o = (s - Fr).
                
    s-->--[-]-e-->-[Q]--->--r
           \--o--<-[F]-/

                    q
                    |
    s-->--[-]-e-->-[+]--->--r
           \--o--<-[F]-/

where F is a lowpass filter, s is the input signal, r is the binary
representation, e is the feedback error signal to be minimized, and o
is the reconstructed output.  The quantizer Q is modeled as a noise
source q.

However, this is not the same as the circuit I find here[1].

    s-->--[-]-e-[F]->-[Q]--->--r
           \--------<----/

which is equivalent to 

    s->-[F]--[-]-e->-[Q]--->--r
              \----<-[F]-/ 

This means that we're minimizing the error between the reconstruction
(filtered representation) Fr and the filtered input Fs.  This makes
perfect sense: we can't reconstruct the high band of s, as that's
where we're going to move the noise.

In practice however, if F is already bandlimited such that s = Fs, the
first schematic will also work.

To see the shape of the noise component e in r, divide out F in the
error equation:

             Fs - Fr = e
              s -  r = e/F = e'

since e is white, e' has the shape of 1/F.


EDIT: The fist schematic is a delta modulator, and [1] uses
integrators instead of lowpass filters.  Explain that.

[1] http://en.wikipedia.org/wiki/Delta-sigma_modulation



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