Commun. Tech., COM-18, no. 6, pp. 740-750, December 1970. F.S. Nakamoto, R. W. Middlestead, and C.1 R. Wolfson, "Impact of time and fre- quency errors on processing satellites with MFSK modulation," ICC '81 Conference Record, June 14-18, 1981, pp. 37.3.1-37.3.5, Denver, CO. Time and Frequency Synchronization of Frequency-Hopped Receivers [14] J.I. Marcum, Table of Q Functions, ASTIA Document No. AD116551, January 1, 1950, RAND Corporation, Santa Monica, CA. APPENDIX 3A: TO PROVE THAT A FREQUENCY ESTIMATOR BASED UPON ADJACENT SPECTRAL ESTIMATES TAKEN AT INTEGER MULTIPLES OF 1/T CANNOT BE UNBIASED Let k/T denote the observed frequency and j/T the nominal signal frequency. Then i/T Ill (k - j)/T represents the frequency spacing of the observed fre- quency relative to the nominal signal frequency. In the presence of a nor- malized frequency error B=XT, the pdf of the spectral estiamtes ri is given 0; otherwise (3A.1a) where (3A.1b) Suppose we consider constructing an estimator B of B which is only a func- tion of the spectral estimates at the nominal signal frequency and two adja- cent ones, i.e., ro, r+1, and r-_1. Thus, we propose that (3A.2) where g is an arbitrary function and we have shown the dependence of the spectral estimates on the normalized frequency error B. We wish to prove that aE{B|B}/aB|B=0 # B,i.e., B as given by (3A.2) is biased. We do this by proving something stronger, namely that (3A.3) where (3A.4) To prove (3A.3) we must first find the conditional pdf p(B|B). If B is given by (3A.3), then there exists some other function f such that (3A.5) Appendix 3A Define r++(B)== r__((B) =v ro(B) = f(@,u,v). (3A.6) Then, the Jacobian of the transformation is given by (3A.7) dv. (3A.8) Since r+1,r-1 and ro are independent random variables, then from (3A.8) we du dv. Substituting (3A.1) into (3A.9) and making use of (3A.4) and (3A.6) gives E{B|B} [ " "w" IOV2yf(B,u',v') sincB|)dß du' dv' (3A.10) Time and Frequency Synchronization of Frequency-Hopped Receivers where u'=u/o V = v/o = (3A.11) and y has previous