Golay Code Theory

The length $L$ bilevel sequences $a(n)$ and $b(n)$ are Golay if and only if the following condition holds, where $\star$ is the autocorrelation operator [1]:

$$a(n) \star a(n) + b(n) \star b(n) = 2L \delta (n)$$ (2)

$\delta (n)$ is the Kronecker delta function. Recall that (2) can also be written using $\ast$, the convolution operator.

$$a(-n) \ast a(n) + b(-n) \ast b(n) = 2L \delta (n)$$ (3)

Given that $a_L(n)$ and $b_L(n)$ are Golay, it turns out that $a_{2L}(n) = [ \begin{array}{cc}a_L(n) &b_L(n) \end{array} ]$ and $b_{2L}(n) = [ \begin{array}{cc} a_L(n) & -b_L(n) \end{array} ]$ are also Golay. This means that Golay sequences can be constructed recursively given Golay seed sequences such as $a_2(n) = [ \begin{array}{cc} 1&1 \end{array} ]$ and $b_2(n) = [ \begin{array}{cc} 1&-1 \end{array} ]$. See the MATLAB/Octave source code generate_golay.m for details. Notice also that the resulting bilevel sequences consist of only $1$'s and $-1$'s. This means that the signal contains the maximum possible power level given that $\vert s(n)\vert \leq 1 \forall n$. This property is helpful in combatting measurement noise.

Let $r_a(n) = a(n) \ast h(n)$ be the response due to the Golay code input $a(n)$, and let $r_b(n) = b(n) \ast h(n)$ be the response due to the Golay code input $b(n)$. Due to (2), the impulse response $h(n)$ may be determined as follows:

$$h(n) = \frac{1}{2L}( a(n) \star r_a(n) + b(n) \star r_b(n) )$$ (4)

See golay_response.m for more details.