rom ²4 (1.111) Substituting (1.107) in (1.111) and carrying out the required differentations of the N-dimensional polynomials H(z) and C(z) gives after much simplifi- cation [7] X (Tdj - Td,j-1) (1.112) Superscript(2)For convenience we set Tdo =0. 24The notation zArd represents a vector whose j-th component, j =1,2,...,Nis = The Multiple Dwell Serial PN Acquisition System which for q 1 simplifies to KNPFASIN(TdN TACQ = (1.113) In (1.112) and (1.113), the Kronecker delta function has the usual definition (1.114) Also, for N = 1 (a single dwell system) and K = K, (1.113) reduces to (1.4), as it should. Comparing the forms of (1.113) and (1.4), it is apparent that for the same false alarm penalty time, i.e., Ktd = KN(TdN - Td,N-1), the N-dwell system can yield a smaller mean acquisition time than the single dwell system if (1.115) The ability to design the N-dwell system to satisfy (1.115) depends upon the functional relationship between the conditional false alarm probabilities and the dwell times. More will be said about this relationship shortly. The generating function of (1.107) can also be used to obtain an approx- imate expression for the acquisition time variance o'AcQ of the N-dwell sys- tem. In particular, (1.116) Taking the required second partial derivatives using U(z) defined in (1.107) and (1.108) and making the assumption of large q, then together with T ACQ and (1.113) one obtains after much simplification the relation [7] (12/1/1) (1.117) Again for the special case of N = 1 and K = K, (1.117) reduces to (1.7). Once again comparing (1.117) and (1.7) we observe that if (1.115) is satisfied, the N-dwell system yields a smaller acquisition time variance than the single dwell sytem. In fact, for large q, both TACQ and the standard Pseudonoise Code Acquisition in Direct-Sequence Receivers deviation ACO are directly proportional to the same function F(N) of false alarm probabilities and dwell times, namely, (1.118) To proceed further with the evaluation of the first two moments of acqui- sition time one mu