Performance bounds in Synchronization for low Signal-to-Noise Ratios

In this contribution we consider the Cramer-Rao bound (CRB) for the estimation of the synchronization parameters of a noisy linearly modulated signal with random data symbols. We explore three scenarios, i.e., (i) joint estimation of carrier phase, carrier frequency and time delay, irrespective of the data; (ii) joint estimation of carrier frequency and time delay, irrespective of the data and the carrier phase; and (iii) estimation of carrier frequency, irrespective of the data, the carrier phase and the timing. Because of the presence of the random data (and, in scenarios (ii) and (iii), also of random synchronization parameters), the exact computation of the corresponding CRBs is extremely difficult. Instead, here we derive a simple closed-form expression for the limit of these CRBs at low signal-to-noise ratio (SNR), which holds for arbitrary PAM, PSK and QAM constellations. INTRODUCTION The Cramer-Rao bound (CRB) is a lower bound on the error variance of any unbiased estimate, and as such serves as a useful benchmark for practical estimators [1]. In many cases, the statistics of the observation depend not only on the vector parameter to be estimated, but also on a nuisance vector parameter we do not want to estimate. The presence of this nuisance parameter makes the computation of the CRB very hard, if not impossible. A typical example where a nuisance vector parameter occurs is the observation of a noisy linearly modulated waveform, that is a function of a time delay, a carrier frequency offset, a carrier phase and a data symbol sequence. In [2], the CRBs for estimating the frequency offset and the carrier phase from matched filter output samples have been computed for BPSK and QPSK, considering the data symbols as nuisance parameters and assuming the timing to be known; different constellations yield different expressions for these CRBs. In order to avoid the computational complexity caused by the nuisance parameters, a modified CRB (MCRB) has been derived in [3, 4]. The MCRB is much simpler to evaluate than the CRB, but is in general looser than the CRB. In [5] the high-SNR limit of the CRB has been evaluated analytically, and has been shown to coincide with the MCRB when estimating the delay, the frequency offset or the carrier phase of the linearly modulated waveform. In the presence of coding, synchronization algorithms must operate at low SNR, so that the high-SNR limit of the CRB might no longer be a relevant benchmark. In [6], the low-SNR limit of the CRB related to timing recovery has been presented, assuming a slowly varying carrier phase. The low-SNR limit of the CRB for estimating the carrier phase and frequency, from matched filter output samples taken at the correct decision instants, has been investigated in [7]. In this contribution we derive a simple expression for the low-SNR limit of the CRBs for (i) joint phase, frequency and timing estimation, (ii) joint frequency and timing estimation, and (iii) frequency estimation. The resulting expressions are valid for arbitrary PAM, PSK, and QAM constellations, and for an arbitrary square-root Nyquist transmit pulse. The adopted signal model and the considered scenarios are different from those investigated in [6, 7]. Finally, from this low-SNR limit of the CRB and the known high-SNR limit of the CRB, we derive an approximate expression of the true CRB.