Does a language stay regular if you re-encode the input? $\tau$ be a transformation that, applied to a word of $\Sigma_a^\star$, replaces each character with a word from a possibly different $\Sigma_b^\star$ and concatenates the resulting words. For instance, given a $\tau$ with the following translation table:

A -&gt; AA
B -&gt; AB
C -&gt; BA
D -&gt; BB


$\tau(ABC)=AAABBA$ and $\tau(DACB)=BBAABAAB$. Note that the words may not all have the same length. A translation table such as

A -&gt; FOO
B -&gt; BAR
C -&gt; BAZ
D -&gt; QUUX


is also valid.

Now, is $\{w'|w\in L\land w'=\tau(w)\}$ regular iff $L$ is a regular language as well?
The answer is yes. This is generally called homomorphism, and regular languages are closed under homomorphism. Moreover, regular languages are closed under inverse homomorphism also. More precisely,
if $\tau$ is a homomorphism from alphabet $\Sigma$ to alphabet $\Delta$ and $L$ is a regular language over $\Delta$, then $\tau^{-1}(L)$ is also a regular language.

You can take a look here for more details, or the best is to take a look at this book, which contains proofs about basic properties of regular languages.