g yields Ne)2 = + (2.21) or, alternately, Ne(ei) = (2.22) where in addition (2.23) The Delay-Locked Loop and denotes the two-sided noise bandwidth of the equivalent low-pass filter H(s), i.e., (2.24) or equivalently the band-pass noise bandwidth of H(s). Also, in (2.22) (2.25) denotes the signal-to-noise ratio in this bandwidth. 2.1.3 Linear Analysis of DLL Tracking Performance When the equivalent loop signal-to-noise ratio is large, the tracking perfor- mance of the DLL can be determined from (2.15) with Dn (E1) replaced by simply Et. Under this assumption and further assuming that T = 0, one can write down, by inspection of (2.15), an expression for the mean-squared tracking jitter, viz., (2.26) Substituting (2.19) and (2.22) in (2.26) yields (2.27) or solving o2 (2.28) where p All SINBL Since for the linear analysis case p is assumed to be large, then to a first approximation (2.28) can be simplified to (2.29) where SL is the "squaring loss" of the DLL i.e., the ratio of signal X signal to signal X noise plus noise X noise distortions. Alternately in terms of the data symbol signal-to-noise ratio E/No All ST /No and the ratio of band-pass Pseudonoise Code Tracking in Direct-Sequence Receivers Table 2.1 Evaluation of M2 for one- and two-pole Butterworth filters. (Signal X Signal Distortion)¹², M2; exp(-2B/Rs)cos(2B/Rs)- - sin(2BH/Rs)]} Manchester Code; Sm(f) = Ts 1 [3 exp(-BH/Rs) + exp(-2BH/Rs)] 1 exp(-BH/Rs)[cos(BH/Rs)- sin(BH/Rs)] + exp(-2B/R)cos(2B/Rs) - sin(2BH/Rs)]} filter bandwidth to data rate Rs, we can write S X S (2.30) where we have also made use of the definition of n in (2.16). It should be noted that (2.30) is quite similar to the expression for squaring loss of a Costas loop used for carrier synchronization of a BPSK signal (see Equation (46) of [17]). Furthermore, one can conclude from (2.30) that the squaring loss of a DLL with a correlator spacing 8 = TIN is equal to the squaring loss of a DLL with a correlator spacing T12 (a "one-delta" loop) and a data rate of 4[N - 1)/N]2 Equivalently, then