I am a computer scientist and not a mathematician. I do not know how to edit questions on this site, so I am rephrasing a problem I came across while reading on the explicit construction of expander graphs by Margulis using discrete Fourier transform:
Let $H$ be a finite Abelian group. It has $\vert H \vert$ distinct characters, where character of $ H $ is a homomorphism $\chi : H \to \mathbb C$, that is $\chi (g+h) = \chi (g) \chi (h)$ for all $g, h \in H$. The characters of $H= {\mathbb Z_n \times \mathbb Z_n} $ are exactly $ \chi_a (x) = \omega ^ {\langle a, x \rangle} $ for $a, x \in \mathbb Z_n ^2 $, where $\omega = e^\frac {2 \pi i}{n}$. These characters form an orthonormal basis for the space of functions $f: H \to \mathbb C$.
Given $f : \mathbb Z_n ^2 \to \mathbb C $ we can write the Fourier coefficients of $f$ as
$$\hat{f} (x) = \langle f, \chi _ x \rangle = \frac {1}{\vert H \vert } \sum_ {y \in H} \overline {f(y)} \chi _ x {(y)}$$
Furthermore:
Let $A$ be a $2 \times 2$ non-singular matrix over $\mathbb Z_n$ and $b\in \mathbb Z_n ^2$.
Define $g(x) = f(Ax+b)$.
Then the Fourier coefficients of $g$ are given by
$$\hat{g} (y) = \omega ^ {- \langle A^{-1} b, y \rangle} \hat{f} ({(A ^ {-1})^ T } y) $$
Can anyone kindly show me how the last equation was obtained but in simple steps?