[<<][math][>>][..]Tue Sep 4 14:07:56 CEST 2012

( 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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