You need to read about WHT. It is just a fourier transform obtained for $\pm 1$ valued sequences. A fourier transform is usually normalized in different ways. Some authors like to divide by $2^n$ (since the sum is that of a function taking $2^n$ inputs) in the forward direction, others split the normalization into $2^{n/2}$ in both forward and reverse directions, to make the map an isometry.
References:
- Claude Carlet's Chapter on Boolean functions is detailed, you can start there and read Section 2.
- There is a nice book by Cusick and Stanica see here.
Consider a function $f:\{0,1\}^n \rightarrow \{0,1\}$ a binary boolean function.
Sometimes it is convenient to work with $(-1)^{f(x_1,\ldots,x_n)}$ instead. There are $2^n$ points in its domain, hence the normalization factor.
The Walsh transform is usually written as
$$\hat{f}(a)=\sum_{x\in{F^n_2}}(-1)^{f(x)+\langle a,x\rangle}.$$
The function $L_a(x)=\langle a,x\rangle=a_1 x_1+\cdots+a_n x_n$ is a linear multivariate function of $(x_1,\ldots,x_n)$.
The function $$f(x)+\langle a,x\rangle=f(x_1,\ldots,x_n)+a_1 x_1+\cdots+a_n x_n$$
equals $0$ mod 2 if $f(x)=\langle a,x\rangle$ and $1$ mod 2 otherwise.
The sum
$$
\hat{f}(a)=\sum_{(x_1,\ldots,x_n) \in \mathbb{F}_2^n} (-1)^{f(x)+\langle a,x\rangle},
$$
is equal to $2^n-2 d_H(f,L_a)$ and computes the correlation between the function $f$ and the linear function $L_a$. Here $d_H(f,L_a)$ is the Hamming distance between those two functions as $x$ varies over $\mathbb{F}_2^n.$ Each one of these $2^n$ correlations (for different $a$) can be obtained by a matrix multiplication.
Since the mapping is invertible, a $2^n\times 2^n$ Hadamard matrix appears as below: $$\left[(-1)^{\langle a,x \rangle}\right]$$
where $a$ indexes rows and $x$ indexes the columns in the standard order.
Due to the kronecker structure of the Hadamard matrix, a fast transform and inverse transform exist. The functions that have the maximum hadamard coefficient $\hat{f}(a)$ minimized are called bent functions and are important in coding theory and cryptography. By Parseval's relation, $$\sum_{a \in \mathbb{F}_2^n} |\hat{f}(a)|^2 =2^{2n},$$
and bent functions have $|\hat{f}(a)|=2^{n/2}.$ Note $f(0)=0$ means a function is balanced, bent functions are not.
Under the uniform distribution on the domain $\{0,1\}^n$ the Walsh coefficients can be shown to be binomially distributed, and with proper normalization (like in the other question you asked) will behave like standard normal random variables.
