Erratum: Cyclic statistic estimators with uncertain cycle frequencies (IEEE Transactions on Information Theory (2017) 63:1 (649-675) DOI: 10.1109/TIT.2016.2614321)

IN the above article [1], expression (A128), in the proof of Theorem 17 on page 672, has a missed term. Here, the missed term is considered and its convergence to zero is proved. The (conjugate) cyclic correlogram R(T)x (α, τ) is nonzero only for τ ∈ [-T,T]. Thus, from (42b) and (43), we have the following corrected version of (A128) (normalized bias of the frequency-smoothed (conjugate) cyclic periodogram with estimated (conjugate) cycle frequency) {equation presented} where {equation presented} The term B1 is considered in (A128), but for the integration interval which is R in (A128) while it is (-T,T) in (C6). The convergence to zero of B1 can be proved observing that {equation presented} with V (T)(α(T), τ) defined in [1, Eq. (39)], and using the arguments of [1] (see (A129)-(A133)). The term B2 is missed in (A128). Its convergence to zero is proved here. The first-order Taylor series expansion of q(Δf τ) for τ ∈ [-T,T] is {equation presented} where q (t) is the first-order derivative of q(t) and τ depends on τ, Δf, and T, and is such that τ Δ (0, τ) if τ > 0 and τ Δ (τ, 0) ifτ < 0. Since q(0) = 1 [1, Assumption 2], we have {equation presented}If first T →∞ with finite Δf and then Δf → 0, the term B2 does not approach zero. Let be Δf = ΔfT = T-s with s > 0. For T →∞, we have ΔfT → 0. Moreover, T ΔfT = T 1-s. Thus, the condition T ΔfT →∞ is satisfied provided that s < 1. Therefore, the condition on s is 0 < s < 1. Assumptions 1)q (t) bounded {equation presented} A sufficient condition assuring the validity of both 2) and 3) is {equation presented} Under Assumptions 1-3, from (C6), we have {equation presented} where k1 and k2 are constants. In the right-hand side of (C7), as T →∞, the first term approaches zero provided that s > 1/3 and the second one provided that s/2 + r > 1/2. This last condition is automatically satisfied since r > 1/2 (Assumption 3). Finally, the following minor corrections are needed in [1]. In (A2), the integrand function must be multiplied by a(t/Z). In (A3), the integrand function must be multiplied by a(t1/Z) a(∗)(t2/Z). In (A4), the term in the second line must be multiplied by ||a||2∞. In (A10), terms of inequalities from 2nd to 4th must be multiplied by ||a||2∞. In (A11), 2nd and 3rd terms of inequalities must be multiplied by ||a||2∞.. (Formula Presented).