d. Thus, it would appear at first glance that, for specified values of detection probability and false alarm probability, one could arbitrarily select the dwell time to achieve any desired mean acquisition time. Upon closer examination, one realizes that indeed this is not possible with the fallacy lying in the fact that, for a given PEA and pre-detection signal-to-noise ratio, PD is implicitly a function of Td To place this statement in evidence, we begin by evaluating PD and PEA in terms of the PN acquisition system parameters for the simple case of no code Doppler or Doppler derivatives. When signal is present (i.e., the cell being searched corresponds to a sam- ple value on the PN correlationc curve), then the input to the square-law envelope detector can be expressed in the form15 x(t) = s(t) n(t) = V2A cos(wot + (s) + - V2n,(t)sin(wit + 4) (1.58) where (1.59) 15 To keep the presentation simple, we shall, at this point, ignore partial correlation effects pro- duced by the filtering of the product of incoming and local PN waveforms over less than a full code period. Later in the chapter, we shall present both exact and approximate approaches for accounting for these effects and their significance. Pseudonoise Code Acquisition in Direct-Sequence Receivers In (1.58), A is the rms signal amplitude,16 w0 the radian carrier frequency, and are band-limited, independent, low-pass, zero-mean Gaussian noise processes with variance o2 = NoB/2, where No is the single-sided noise spectral density and B is the noise bandwidth of the pre-detection band- pass filter. The output of the square-law envelope detector in Figure 1.6, in response to the input x(t) of (1.58), is (ignoring second harmonics of the carrier): (1.60) and has a non-central chi-squared pdf which is given by (1.61) otherwise where (1.62) is the pre-detection signal-to-noise ratio. In the absence of signal, i.e., A = 0, (1.61) reduces to the central chi-squared pdf: (1.63) otherwise which characterizes the square-law output in all search cells that contain 