PSEUDONOISE CODE TRACKING IN DIRECT-SEQUENCE RECEIVERS
2.7 THE MODIFIED CODE TRACKING LOOP
Still another PN code tracking loop configuration that attempts to combat the gain imbalance problem of the DLL without sacrificing tracking per- formance, but now with hardware simplicity rivalling the TDL, is the mod- ified code tracking looop (MCTL) [10], [11] illustrated in Figure 2.25. The principal idea of this configuration is to replace the sum channel signal of theDLL with a reference signal derived from the on-time PN code, the primary advantage being the elimination of an entire processing channel.The secondary advantage is that since the on-time channel experiences less noise power than the sum channel of Figure 2.24 does, it should serve as a better demodulation reference for the difference channel signal. In fact, in view of this, the mean-squared tracking jitter of Figure 2.25 should ideally be smaller than that of the traditional DLL of Figure 2.1 or the SD DLL of Figure 2.24.
KmKm
KmKm
KmKm1yœ21t2yœ21t22.
KmKm1yœ1t2yœ 1t221yœ1t2yœ 1t22
e1t2 3KmH1p21eœ 1t2eœ1t22 4 3KmH1p21eœ 1t2eœ1t22 4 y;œ1t2
e;œ1t2^ x1t2c1ttˆt;d2e;1t2>Km
y;œ1t2 e;œ1t2
Km
2 yœ21t2Km
2 yœ21t2
e1t2 3KmH1p2eœ 1t2 42 3KmH1p2eœ 1t2 42
8Here we allow the two phase detectors to have different gains, which we shall further assume represent all the contributions to gain in the two DLL arms.
Figure 2.24.A non-coherent DLL.
Indeed the above turns out to be the case, as will be quantitatively demonstrated shortly. Before doing so, however, and also before discussing other behavioral characteristics of the MCTL and its shortcomings, we point out that the implementation of Figure 2.25 can be reconfigured to more closely resemble that of Figure 2.24 and thereby achieve a further reduc- tion in hardware required. In particular, the difference channel signal can be generated as per the dashed lines in Figure 2.25, thus eliminating one mixer. Note, however, that in this alternate mechanization, the reference sig- nal formed from the difference of the early and late PN codes is not a con- stant envelope waveform which therefore places strict linearity requirements on the single remainining mixer. This constraint also applies to the DLL of Figure 2.24, which conceptually could also be imple- mented, analogous to Figure 2.25, by forming the sum and difference sig- nals after the input mixers.
Having derived the loop model and discussed the performance of the DLL in great detail, we shall be extremely brief in the analogous presentation for the MCTL, merely presenting the key results in summary form. In fact, to highlight the similarities between the two code tracking loops and allow easy comparison of their differences, we shall use the same equation numbers as in Section 2.1 (with a prime superscript) where appropriate.
Again assuming an input as in (2.1) and (2.2), and making similar assump- tions to those leading up to (2.7), the input to the loop filter of the MCTL can be shown [11] to be given by9
(2.7) where
(2.8) is the loop discriminator characteristic and n苲e1t,et2 is the equivalent addi-
D苲1et2D苲1etnp2; n;1,;2,;3,p e
0; et 1
3252etet2; 1 6 et 1 2
2et2et2; 12 6 et12 3
252etet2; 12 6 et1
0; et 7 1
D苲1et2^ RPN1et25RPN1et2RPN1et26 Km2
n苲e1t,et2 e1t2yo1t2y¢1t2SKm2
mˆ21ttt2D苲1et2
9For simplicity of presentation, we shall pursue only the “one-delta” loop where N2.
Figure 2.25.The modified code tracking loop (MCTL).
tive noise defined by
(2.9) with
(2.6) Figure 2.26 is a comparative illustration of the discriminator characteristics of the DLL and the MCTL. We immediately observe that the non-zero region of the MCTL characteristic is only 2/3 that of the DLL and thus the modified loop has a 1/3 less pull-in capability. Furthermore, although the two discriminator characteristics have identical slopes at t0, the DLL always creates a larger error voltage for any value of timing error.
Despite these disadvantages, when compared with the DLL, the MCTL does have an improved noise performance as previously mentioned.
e
Nˆs¢1t2H/1p2 35c1ttˆtd2c1ttˆtd26Ns1t2 4. Nˆc¢1t2H/1p2 35c1ttˆtd2c1ttˆtd26Nc1t2 4 Nˆ
so1t2H/1p2 3c1ttˆt2Ns1t2 4 Nˆ
co1t2H/1p2 3c1ttˆt2Nc1t2 4 RPN1et2 4Nˆco1t26
5RPN1et2Nˆc¢1t2 3RPN1et2 2Smˆ1ttt2
n苲e1t,et2Nˆ
co1t2Nˆc¢1t2Nˆ
so1t2Nˆs¢1t2
Figure 2.26. One-delta loop discriminator characteristics for the DLL and MCTL (reprinted from [11]).
Specifically, the equivalent noise spectral density of is given by [11]:
(2.22) withKLandM4defined in (2.23) and rHin (2.25). Following steps similar to those leading up to (2.29), we find that the normalized mean-squared tim- ing error for the MCTL is given by
(2.29) where the squaring loss is now
(2.30)
Comparing (2.30) with (2.30) for N2, we observe that the NNpower of the MCTL is one-half that of the DLL, thus producing an associated reduction in mean-squared tracking jitter. To demonstrate the magnitude of
sL
苲 M22
M4KL
BH>Rs
Es>N0
. sL
苲
s苲e2 1 2rs苲L
N苲e1et22SN0cM45RPN2 1et2123RPN1et2RPN1et2426KL
rH
d n苲e1t,et2 N苲e1et2
Figure 2.27. Minimum squaring loss comparison for NRZ data modulation (reprinted from [11]).
this reduction, it is sufficient to numerically compare the minimum squar- ing loss of (2.30) with that of (2.30) as is done in Figures 2.27 and 2.28 for the case of NRZ or Manchester data and two-pole Butterworth band-pass filters. The appropriate expressions for KL,M2, and M4are obtained from Tables 2.1 and 2.2.
With regard to the acquisition behavior of the MCTL, it has been shown [11] that for equal rms tracking jitters, i.e.,sMCTLsDLL, the maximum nor- malized search rate is given by
(2.91)
as compared with the previously found for the “one-delta” DLL.
Figure 2.29 illustrates as a function of Es/N0for NRZ and Manchester data types, and two-pole Butterworth band-pass filters. Also shown is the curve
y# y# 1
y#.6sL0MCTL
sL0DLL
.6
M4KL
2BH>Rs
Es>N0
M4KL
BH>Rs
Es>N0
Figure 2.28. Minimum squaring loss comparison for Manchester data modulation (reprinted from [11]).
corresponding to ignoring the arm filter band-limiting effects, i.e.,M2M4 KL1 and BhRs. We observe that for small values of Es/N0, the MCTL indeed has a higher search rate capability than the DLL, whereas for large values of Es/N0, it asymptotically approaches a maximum decrease of 40%
in search rate capability.
Finally, when compared with the DLL, the MCTL has the advantage of being significantly less sensitive go gain imbalances at alow values of pre- detectio signal-to-noise ratio rHresulting in lower tracking bias errors. At high rH’s, the gain imbalance sensitivities of the two configurations are nearly equivalent, both being of course much less sensitive than the addi- tional DLL.