By Lisa Jacobsen
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Extra resources for Analytic theory of continued fractions III
If we think of the different frequency components as the original coordinate system, the new frequency components are merely the old ones multiplied by factors. The coordinate transformation matrix is thus essentially diagonalized in terms of these coordinates. The Jacobian of the transformation is (for n sine and n cosine components) n J = ∏ jY fi j2 i=1 where the fi are equally spaced through the band W . This becomes in the limit exp 1 W Z W log jY f j2 d f : Since J is constant its average value is the same quantity and applying the theorem on the change of entropy with a change of coordinates, the result follows.
The region ofp high probability is a sphere of radius nN. As n ! ∞ the probability of being outside a sphere of radius nN + approaches zero and 1n times the logarithm of the p volume of the sphere approaches log 2 eN. In the continuous case it is convenient to work not with the entropy H of an ensemble but with a derived quantity which we will call the entropy power. This is defined as the power in a white noise limited to the same band as the original ensemble and having the same entropy. In other words if H 0 is the entropy of an ensemble its entropy power is 1 N1 = exp2H 0 2 e In the geometrical picture this amounts to measuring the high probability volume by the squared radius of a sphere having the same volume.
To TW log N P+N Formulas similar to C = W log for the white noise case have been developed independently N by several other writers, although with somewhat different interpretations. We may mention the work of N. Wiener,7 W. G. Tuller,8 and H. Sullivan in this connection. In the case of an arbitrary perturbing noise (not necessarily white thermal noise) it does not appear that the maximizing problem involved in determining the channel capacity C can be solved explicitly. However, upper and lower bounds can be set for C in terms of the average noise power N the noise entropy power N1 .
Analytic theory of continued fractions III by Lisa Jacobsen