Abrazolica

In our Creating Noise book we show how to use RC ladder networks to create pink noise. An \(n^{th}\) order network is shown below.

The R and C values are given by the following (equations 37 and 38 in the book).

\[C(n,i) = \frac{4i-1}{(2(n-i)+1)16^i}\frac{\binom{2(n+i)}{n+i}}{\binom{2(n-i)}{n-i}}\] \[R(n,i) = \frac{(4i-3)(2(n-i)+1)16^i}{4(n-i+1)(n+i)}\frac{\binom{2(n-i)}{n-i}}{\binom{2(n+i)}{n+i}}\]

Using these equations is a bit cumbersome so here are some asymptotic versions of the equations which will go into the next edition of the book.

\[C(n,i) = \frac{4i-1}{(2(n-i)+1)}\sqrt{\frac{n-i}{n+i}}\] \[R(n,i) = \frac{(4i-3)(2(n-i)+1)}{4(n-i+1)\sqrt{(n+i)(n-i)}}\] \[C(n,i)R(n,i) = \frac{(4i-1)(4i-3)}{4(n-i+1)(n+i)}\]

These equations obviously only work for \(i<n\). For \(i=n\) the asymptotic equations are

\[C(n,n) = \frac{4n-1}{\sqrt{2\pi n}}\] \[R(n,n) = \frac{(4n-3)\sqrt{2\pi n}}{8n}\]

The last resistor in the chain has the asymptotic formula \[R_L(n)=\sqrt{2\pi n}\]

In the limit \(n\rightarrow\infty\) with \(i/n\rightarrow x\) both the R and C equations have the following form

\[\frac{2x}{\sqrt{1-x^2}}\]

where \(x\) has the range \(0<x<1\). In this limit the ladder becomes a continuous transmission line and the total capacitance or resistance (measured from the beginning of the line) is found by integrating the above equation

\[C(x)=R(x)=2\left(1-\sqrt{1-x^2}\right)\]

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