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Traditionally, high-order statistics are used for blind estimation of nonminimum phase finite impulse response (FIR) channels. In this paper we present a new, alternative approach, that uses first- or second-order derivatives of the observations' second generalized characteristic function, evaluated at arbitrary (off-origin) locations. The estimation of these derivatives reduces plainly into specially-weighted empirical mean and covariance. We show that despite the addition of some nuisance parameters, this approach generates more equations than unknowns, and thus enables a well-averaged least-squares solution. A simulation example demonstrates the potential improvement in estimation accuracy over cumulants-based estimation.