  We explore a plausible mechanism that the hemispherical power asymmetry in
the CMB is produced by the spatial variation of the primordial sound speed
parameter. We suggest that in a generalized approach of the $\delta N$
formalism the local e-folding number may depend on some other primordial
parameters besides the initial values of inflaton. Here the $\delta N$
formalism is extended by considering the effects of a spatially varying sound
speed parameter caused by a super-Hubble perturbation of a light field. Using
this generalized $\delta N$ formalism, we systematically calculate the
asymmetric primordial spectrum in the model of multi-speed inflation by taking
into account the constraints of primordial non-Gaussianities. We further
discuss specific model constraints, and the corresponding asymmetry amplitudes
are found to be scale-dependent, which can accommodate current observations of
the power asymmetry at different length scales.
