  The $\beta$-encoder is an analog circuit that converts an input signal $x \in
[0,1]$ into a finite bit stream $\{b_i\}$. The bits $\{b_i\}$ are correlated
and therefore are not immediately suitable for random number generation, but
they can be used to generate bits $\{a_i\}$ that are (nearly) uniformly
distributed. In this article we study two such methods. In the first part the
bits $\{a_i\}$ are defined as the digits of the base-2 representation of the
original input $x$. Under the assumption that there is no noise in the
amplifier we then study a question posed by Jitsumatsu and Matsumura on how
many bits $b_1, \ldots, b_m$ are needed to correctly determine the first $n$
bits $a_1,\ldots,a_n$. In the second part we show this method fails for random
amplification factors. Nevertheless, even in this case, nearly uniformly
distributed bits can still be generated from $b_1,\ldots,b_m$ using modern
cryptographic techniques.
