  Discovering the mechanism underlying the ubiquity of $"1/f^{\alpha}"$ noise
has been a long--standing problem. The wide range of systems in which the
fluctuations show the implied long--time correlations suggests the existence of
some simple and general mechanism that is independent of the details of any
specific system. We argue here that a {\it memoryless nonlinear response}
suffices to explain the observed non--trivial values of $\alpha$: a random
input noisy signal $S(t)$ with a power spectrum varying as $1/f^{\alpha'}$,
when fed to an element with such a response function $R$ gives an output
$R(S(t))$ that can have a power spectrum $1/f^{\alpha}$ with $\alpha <
\alpha'$. As an illustrative example, we show that an input Brownian noise
($\alpha'=2$) acting on a device with a sigmoidal response function $R(S)=
\sgn(S)|S|^x$, with $x<1$, produces an output with $\alpha = 3/2 +x$, for $0
\leq x \leq 1/2$. Our discussion is easily extended to more general types of
input noise as well as more general response functions.
