Low-SNR limit of the Cramer-Rao bound for estimating the carrier phase and frequency of a PAM, PSK, or QAM waveform

Abstract

Abstract-In this letter we consider the Cramer-Rao bound (CRB) for the estimation of the time delay of a noisy linearly modulated signal with random data symbols and random carrier phase. Because of the presence of the nuisance parameters (i.e., data symbols and carrier phase), a closed-form expression of this CRB is very hard to obtain for arbitrary PSK, QAM or PAM constellations and a band-limited transmit pulse. Instead, here we derive a simple expression for the limit of the CRB at low signal-to-noise ratio (SNR), which is a relevant benchmark for timing recovery algorithms operating at small 0 .

Closed-Form Expressions of the Exact Cramer-Rao Bound for Parameter Estimation of BPSK, MSK, or QPSK Waveforms

“…The analytical result 1 is numerically compared with the approximations given in result 2 and to the approximations given in [4] for and of BPSK and QPSK at low SNR.…”

“…(c) Approximate value given in [4]. and 0.005 for and , and consequently, the ratios CRB/MCRB is around one from these values of SNR.…”

“…Then, using the values of the moments of order of zero-mean unit variance Gaussian random variables, we obtain, after tedious but straightforward algebraic manipulations Inserting these expansions in Result 1 allows us to prove the following result. 4 …”

“…However, this MCRB may not be as tight as the exact CRB [3] for joint estimation of all parameters. To circumvent this difficulty, asymptotic expressions at low [4] or high [5] signal-to-noise ratio (SNR) have been investigated. Unfortunately, though, these asymptotic expressions do not apply at moderate SNR, for which only combined analytical/numerical (see, e.g., [1], [5], and [6]) approaches are available until now.…”

Bayesian and hybrid Cramer-Rao bounds for QAM dynamical phase estimation

“…), we obtain matrix : (25) For both the DA and NDA scenarios (see (9) and (10) We now invert the HIM. Starting with and just similarly to Appendix I of [15], we find:…”