Orthogonal filters have applications in many areas of signal processing. In this paper, the author presents a simple and explicit expression for real orthogonal filters.

If the dimension of the filter is L, then the new parametrization is optimal; namely, it depends on L/2+1 real numbers, which is exactly the degree of freedom of the problem.

The steps of the clever construction are as follows:

Denote h₁, a₁, a₂, ... , a_L/2 the parameters. With a₁, a₂, ... , a_L/2 a lower triangular matrix is defined. Put a = (a₂, ... , a_L/2)^t, b =(I+AA^t)^-1a and c = -A^tb. Then, by Theorem 2, h = (h₁, ... , h_L) is an orthogonal filter if and only if h₂=h₁a₁, h_even = h₁ b and h_odd = h₁c, where h_even = (h₄, h₆, ... , h_L-2, h_L)^t and h_odd = (h₃, h₅, ... , h_L-1)^t. By normalization of the general formula, one gets an expression for paraunitary filters as well.

The results are made completely explicit for four-taps low-pass paraunitary filters.