#' Compute the time-shift parameter tau.
#'
#' @param t Sampling times.
#' @param omega Angular frequency.
#'
#' @return Time-shift parameter tau.
#'
#' @examples
#' tau(1:10, 2*pi)
#'
#' @export
tau <- function(t, omega) {
# two.omega[j] = 2 * omega[j]
two.omega <- 2 * omega
# two.omega.t[j,i] = 2 * omega[j] * t[i]
two.omega.t <- two.omega %o% t
# sin.sum[j] = Sum from i=1..N of sin(two.omega.t[j,i])
sin.sum = rowSums(sin(two.omega.t))
# cos.sum[j] = Sum from i=1..N of cos(two.omega.t[j,i])
cos.sum = rowSums(cos(two.omega.t))
# Consistent with the Fortran code
# two.omega.tau[j] = atan(sin.sum[j] / cos.sum[j])
two.omega.tau <- atan(sin.sum / cos.sum)
# # Consistent with the paper.
# # two.omega.tau[j] = atan2(sin.sum[j], cos.sum[j])
# two.omega.tau <- atan2(sin.sum, cos.sum)
# Consistent with the Fortran code
# two.omega.tau[j] / (2*pi)
two.omega.tau / (2*pi)
# # Consistent with the paper.
# # two.omega.tau[j] / (2*omega[j])
# two.omega.tau / two.omega
}
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