[<<][math][>>][..]Wed Jul 4 15:42:53 EDT 2012

How does this[1] translate to the SNR of s? We have the shape of the noise, but not the maximal amplitude. I'm interested in SNR to compute the channel capacity, i.e. how much information can be encoded in the Fs signal as opposed to the raw bitstream r. Is there any capacity lost by this encoding? It seems that this is not an easy answer, as it involves assumptions to make the nonlinearity go away. What about this: - in r = s + e', the signal component s is negligible. - e' is highpass, which means it has no DC component, so the instantaneous energy is known, and constant: half of the bits are 1, half are 0. Setting 1/2 in the middle, the RMS is 0.5, power is 0.5^2. - given the shape of e' and the the total energy, the frequency-dependent energy can be computed. So it seems that as long as s doesn't have a DC component, it is straightforward to compute the absolute noise envelope, which is constant as long as s remains small wrt. e' If s does have a DC component, the linearization doesn't work. The more DC there is, the less "room" there is for the AC component. A better assumption would be to say that s is highpass (we still need that assumption to compute the total energy) but that it's cutoff is much lower than that of e'. Here the total energy (still 50% duty cycle) is distributed over s and e. As long as the cutoff of e' is much lower than nyquist, it seems that the first assumption (setting r ~ e') is sound. The 2nd assumption can still be used to make sure that s doesn't exceed the highest point of e'. What is surprising here is that the presence of a DC component shifts the dynamic range. Actually, the same happens in other amplitude- limited channels. So.. Given the assumptions above the instantaneous of the high pass signal (0.5^2) would approximate a white signal, meaning that the maximum of the PDF is 0.5^2/fs. This seems to be enough to approximate the signal capacity, using the part of e' that's below the maximum, keeping in mind that the DC part is not usable because the linearized channel properties depend on the DC component, but negligible for computing channel capacity. ( It would be interesting to express all those approximations exactly.. The exact formula is probably fairly complex. ) So, to top off all the approximations, let's say that the bandwidth is reduced by x, the oversampling factor, which leads to an increase of x in amplitude dynamic range. Plugging this into Shannon's formula[2] directly gives the asymptotic behaviour in terms of x C = B log (P / N) ~ B_0/x log ( P_0 * x^2 / N) which clearly shows that this is a fairly expensive technique when looking just at the information content of the channel. The reduction in capacity is log x ----- x (where we ignore the constant power x^2) which corresponds to the intuition that we're using bits to represent individual events E_i, and not "sums of events" which only needs bits in the order of log (sum E_i). This high redundancy makes it plausible to believe that the effect of bit errors is minor. It seems to indicate that the "shape" of the signal is largely irrelevant, so we can probably use that to our advantage (decorrelation to allow computation with such signals). So... say x = 100000 which is 5 decades, which is 100dB dynamic range. The information cost of this is about a factor 6000, meaning that only 1/6000 of the information is actually useful. That's a lot of room to put some extra stuff!. Think of this: to change a bit from 0<->1 adds/subtracts energy that can be seen in the base band (impulse response of the reconstruction filter) but to switch the position of 2 adjacent complementary bits has almost no effect in the base band as this blip has no DC component. As a result it is probably possible to add random permutations to the output bits without this being noticed. Anyways, this also makes it clear why it's probably best to use some steeper filters as they bring the noise floor down. With o the order of the filter this becomes: o log x ------- x NEXT: Revisit the logic operations on (non-correlated) S/D signals + check HF noise modulated into base-band. [1] entry://20120704-134439 [2] http://en.wikipedia.org/wiki/Channel_capacity

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