# Golay complementary sequences in 802.11ad

The Golay complementary sequences consist of two sequences of the same length $N$ such that there acyclic autocorrelation sequences have sidelobes equal in magnitude but opposite in sign. So, when the two autocorrelations are summed up, the result has a peak of $2N$ and zero sidelobe level. As an example, $Ga_{128}$ and $Gb_{128}$ are two complementary Golay sequences of length 128. If I sum up the autocorrelations of $Ga_{128}$ and $Gb_{128}$, the sidelobes are exactly zero.

The IEEE 802.11ad protocol uses Golay sequences for channel estimation. However, it doesn't simply use the basic $Ga_{128}$ and $Gb_{128}$ sequences. Instead it concatenates these sequences to generate following longer sequences:

$Gu_{512} = [-Gb_{128}\phantom{1} -Ga_{128}\phantom{1} Gb_{128}\phantom{1} -Ga_{128}]$ $Gv_{512} = [-Gb_{128}\phantom{1} Ga_{128}\phantom{1} -Gb_{128}\phantom{1} -Ga_{128}]$

(a) However, when I add the autocorrelations of $Gu_{512}$ and $Gv_{512}$, I don't get zero sidelobes. So, what is the value of such longer sequences?

(b) Regardless of the sidelobe performance, I would assume that only $Gu_{512}$ and $Gv_{512}$ sequences should be transmitted. However, the IEEE 802.11ad single-carrier (SC) modulated signal transmits a third sequence consisting of $-Gb_{128}$. Any explanation why this different length signal should be transmitted for channel estimation apart from $Gu_{512}$ and $Gv_{512}$?

Any explanation would be greatly appreciated. ~ryan

## 1 Answer

For part (a), I discovered that the IEEE 802.11 protocol doesn't recommend adding auto-correlations of $Gu_{512}$ and $Gv_{512}$. Instead the correct method is to autocorrelate the first 256 bits of each of these sequences with the next 256 bits. So, $Gu_{512}$ and $Gv_{512}$ are not complementary Golay sequences themselves. But, essentially, both $Gu_{512}$ and $Gv_{512}$ consist of two 256-bit Golay complementary pairs.

~ryan