Phase encoding with Gold codes for wave-equation migration

APPENDIX

We show that phase encoding using Gold codes is equivalent to the linear phase encoding introduced by (Romero et al., 2000). The cross-correlation function of Gold codes is given by equation 13. It corresponds to a spike of amplitude $\pm 2^n$ at lag $\lambda$, which depends on the difference between the number of circular shifts applied to the m-sequences to compute the Gold codes, plus a DC term, $\mp 1$. The phase of the cross-correlation function is given by the phase of the spike, $t_{\lambda} \omega$, and is equal to the phase difference of the input signals. If Gold codes have phases $\gamma_1(\omega)$ and $\gamma_2(\omega)$, the phase of their cross-correlation is

$$\gamma_1(\omega) - \gamma_2(\omega) = t_{\lambda} \omega.$$ (A-1)

According to equation A-1, the phase of the cross-correlation of Gold codes is a linear function of the frequency. Equation A-1 is equal to equation 26 of Romero's paper.

Phase encoding with Gold codes for wave-equation migration