  In this paper, we present an alternative proof showing that the maximal
aperiodic autocorrelation of the $m$-th Rudin-Shapiro sequence is of the same
order as $\lambda^{m}$, where $\lambda$ is the real root of $x^{3} + x^{2} - 2x
- 4$. This result was originally proven by Allouche, Choi, Denise, Erd\'elyi,
and Saffari (2019) and Choi (2020) using a translation of the problem into
linear algebra. Our approach simplifies this linear algebraic translation and
provides another method of dealing with the computations given by Choi.
Additionally, we prove an analogous result for the maximal periodic
autocorrelation of the $m$-th Rudin-Shapiro sequence. We conclude with a
discussion on the connection between the proofs given and joint spectral radius
theory, as well as a couple of conjectures on which autocorrelations are
maximal.
