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I'm currently dealing on the so called “Walsh Function” (WF) that is written like: $$f^W(a)=\sum_{x\in{F^n_2}}(-1)^{f(x)+(a,x)}$$ What I know is that this function is used to approximate an arbitrary function like $f: F^n_2 \to F_2$ and that the $a$ in the upper function is the so called Walsh Coefficient.

But whats in detail about the WF? What can I do with it exactly? How do the elements in the set of the WF look like? As far as I know is this set called the Walsh Spectrum!? How can an arbitrary function be approximated by the WF?

And what exactly is the the Walsh Transformation (WT). I know there is a method to conduct the WT, the so called Fast-Fourier-Transformation. Why do I do a WT?

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There is some stuff on this over on dsp.SE (search for Hadamard also, not just for Walsh) that might get you some of the information that you seek. – Dilip Sarwate May 5 '15 at 19:57
up vote 2 down vote accepted

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.

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Thank you very much, that gives me some insight. One more question: As far as I understand for every $\langle a,x\rangle$ exists the Walsh Coefficient $\hat{f}(a)$ where are some of the WC are maximum. I know that those WC can be found for each $a$ by the Fast-Fourier-Transformation. Can you tell how this method basically works? – Ovomaltine May 6 '15 at 5:40
See the example in block diagram in Wikipedia at – kodlu May 6 '15 at 6:30

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