CLOSED-FORM EXPRESSIONS OF THE TRUE CRAMER-RAO BOUND FOR PARAMETER ESTIMATION OF BPSK, MSK OR QPSK WAVEFORMS Jean-Pierre Delmas Institut National des T´ el´ ecommunications 9 rue Charles Fourier, 91011 Evry Cedex, France phone: + (33)-1-60 76 46 32, fax: + (33)-1-60 76 44 33, email: jean-pierre.delmas@int-evry.fr web: www-citi.int-evry.fr/∼delmas/ ABSTRACT This paper addresses the stochastic Cramer-Rao bound (CRB) pertaining to the joint estimation of the carrier fre- quency offset, the carrier phase and the noise and data pow- ers of binary phase-shift keying (BPSK), minimum shift key- ing (MSK) and quaternary phase-shift keying (QPSK) mod- ulated signals corrupted by additive white circular Gaussian noise. Because the associated models are governed by simple Gaussian mixture distributions, an explicit expression of the Fisher Information matrix is given and an explicit expression for the stochastic CRB of these four parameters are deduced. Refined expressions for low and high-SNR are presented as well. Finally, our proposed analytical expressions are numer- ically compared with the approximate expressions previously given in the literature. 1. INTRODUCTION The stochastic Cramer-Rao bound (CRB) is a well known lower bound on the variance of any unbiased estimate, and as such, serves as useful benchmark for practical estimators. Unfortunately, the evaluation of this CRB is mathematically quite difficult when the observed signal contains, in addition to the parameters to be estimated, random discrete data and random noise. A typical example of such a situation that has been studied by many authors (see e.g., [1] and the references therein) is the observation of noisy linearly modulated wave- forms that are a function of deterministic parameters such that the time delay, the carrier frequency offset, the carrier phase, noise and data powers, as well as the data symbol se- quence. Because the analytical computation of this CRB has been considered to be unfeasible, a modified CRB (MCRB) which is much simpler to evaluate than the true CRB has been introduced in [2]. But this MCRB may not be as tight as the true 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 inves- tigated. But unfortunately, these asymptotic expressions do not apply at moderate SNR, for which only combined ana- lytical/numerical (see e.g., [5, 6, 1]) approaches are available until now. In this paper, we investigate an analytical expression of the stochastic CRB associated with the joint estimation of the carrier frequency offset, the carrier phase and the noise and data powers of BPSK, QPSK or MSK modulated signals corrupted by additive white circular Gaussian noise, which is valid for arbitrary SNR. This paper is organized as follows. After formulating the problem in Section 2, an explicit ex- pression of the Fisher information matrix (FIM) associated with all the deterministic parameters is given in Section 3. Because the carrier frequency offset and the carrier phase pa- rameters are decoupled from the signal noise and data pow- ers parameters, simple explicit expressions for the stochastic CRB of these four parameters are deduced. Refined expres- sions for low and high-SNR are presented as well. Finally, in Section 4, our proposed analytical expressions are numer- ically compared with the previously given approximate ex- pressions. 2. PROBLEM FORMULATION Consider BPSK, QPSK or MSK modulated signals. The re- ceived signals are bandpass filtered and after down-shifting the signal to baseband, the in-phase and quadrature com- ponents are paired to obtain complex signals. We assume Nyquist shaping and ideal sample timing so that the inter- symbol interference at each symbol spaced sampling in- stance can be ignored. In the presence of frequency offset and carrier phase, the signals at the output of the matched fil- ters yield the observation vector y =(y k 0 , ..., y k 0 +K-1 ), with y k = as k e i2π kν e iφ + n k , for k = k 0 , ..., k 0 + K - 1. {s k } is a sequence of independent identically distributed (IID) data symbols taking values ±1 [resp. ± √ 2/2 ± i √ 2/2] with equal probabilities for BPSK [resp., QPSK] modulations and for MSK modulations are de- fined by s k+1 = is k c k where c k is a sequence of independent BPSK symbols with equal probabilities where the original value s k 0 remains unspecified in the set {+1, +i, -1, -i}. The deterministic unknown parameters a, ν and φ repre- sent the amplitude, the carrier frequency offset normalized to the symbol rate and the carrier phase at k = 0. Finally, the sequence {n k } consists of IID zero-mean complex circular Gaussian noise variables 1 of variance σ 2 . The symbols s k are assumed to be independent from n k . If no a priori information is available concerning the transmitted symbols, the distribution of y is parameterized by θ def =(ν , φ , a, σ ). We note that the MSK modulation is modelled equivalently (see e.g., [7]) by s k = i k-k 0 b k s k 0 where b k is another sequence of independent BPSK symbols {-1, +1} with equal probabilities. Consequently, similarly to the BPSK and QPSK modulations, (y k ) k=k 0 ,...,k 0 +K-1 are 1 Note that many papers consider the parameters a 2 and σ 2 denoted usu- ally as the symbol energy E s and the noise power spectral density N 0 as known. They usually suppose a unit variance for the noise and use the ra- tio ε def =(E s /N 0 ) 1/2 as the modulation amplitude, but in practice these two parameters are unknown. 14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIP