Hadamard Feedback Matrix

A second-order Hadamard matrix:

$$\mathbf{H}_2 \mathrel{\stackrel{\mathrm{\Delta}}{=}} \frac{1}{\sqrt{2}}\left[\begin{array}{rr} 1 & 1\\ -1 & 1 \end{array}\right],$$

Higher order Hadamard matrices defined by recursive embedding:

$$\mathbf{H}_4 \mathrel{\stackrel{\mathrm{\Delta}}{=}} \frac{1}{\sqrt{2}}\left[\begin{array}{rr} \mathbf{H}_2 & \mathbf{H}_2\\ -\mathbf{H}_2 & \mathbf{H}_2 \end{array}\right].$$

• Since $H_3$ does not exist, the FDN example figure above can be redrawn for $N=4$ , say, (instead of $N=3$ ), so that we can set $Q=H_4$

• The Hadamard conjecture posits the existence of Hadamard matrices $H_N$ of order $N=4k$ for all positive integers $k$ .

• ``As of 2008, there are 13 multiples of 4 less than or equal to 2000 for which no Hadamard matrix of that order is known. They are: 668, 716, 892, 1004, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964.'' [http://en.wikipedia.org/wiki/Hadamard_matrix]