us to Figure 2.10) illustrated in Figure 2.23. The input to the loop filter is given by e(t) = (2.86) If both band-pass detectors of the DDL are identical, then in Figure 2.23, y+1(t) = y+2(t) = y+(t) = and y-1(t) = y-2(t) = y_(t) in which case (2.86) sim- plifies to (2.87) since again q1(t) q2(t) = 1. Note that (2.87) is identical to (2.7) and thus under ideal balanced conditions, the DDL has the same performance as the DLL. This conclusion can also be obtained by interpreting Figure 2.21 in the following manner. Note that while the TDL of Figure 2.9 time shares the early and late PN codes over a single channel (thus the loop error signal con- tains only the correlation of the input PN code with either the early or late local PN code), the DDL, by using two channels, allows both the early and late codes to be present for correlation in each half dither interval. Thus, in effect, the DDL error signal can be thought of as being generated by alter- nately switching between two DLLs which differ only in that the band-pass detectors in the two channels of one loop are in the reversed position in the other loop. Clearly, if both channels of each loop contain identical band-pass detectors, then the reversal produces no change and the effective alternate switching between the two error signals has no effect. On the other hand, if the band-pass detectors of the DDL are not identical (perhaps different gains and dc offsets), then the effective switching between the two hypo- thetical DLLs causes the dc offsets to cancel in the resulting time-averaged error signal. Also this error signal will have a gain proportional to the aver- age of the two channel gains in the DDL. 2.6 THE PRODUCT OF SUM AND DIFFERENCE DLL The product of sum and difference DLL ( DLL) [9] is another configu- ration that purports to combat the gain imbalance problem of the DLL with- out sacrificing its tracking performance. The vehicle by which this is accomplished is the replacement of the "difference of squares" operation of the DLL with a "sum and di