# Abrazolica

RC values for a Pink Noise Filter

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

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

This post as a pdf

© 2010-2023 Stefan Hollos and Richard Hollos