It is shown that a ranging receiver with a sufficient and reasonable number of correlators is competitive with the current sequential component ranging system by some 1.5 to 2.5 dB. The optimum transmitter code, the optimum receiver, and a near-maximum-lilelihood range-estimation algorithm are presented

Tausworthe, R. C.

English

NASA Technical Reports Server

N88-12110 
, 
TDA Progress Report 42-91 
Tau Ranging Revisited 
R .  C. Tausworthe 
Information Systems Division 
The advent o f  reasonably fast and inexpensive Very Large Scale Integration (VLSI) 
components offers an opportunity to reconsider the merits of composite-code-uplink 
ranging systems, abandoned years ago. It is shown in this article that a ranging receiver 
with a sufficient and reasonable number o f  correlators is competitive with the current 
sequential component ranging system and may outperform that system by  some 1.5- 
2.5 dB. The optimum transmitter code, the optimum receiver, and a near-maximum- 
likelihood range-estimation algorithm are presented. 
July-September 1987 
1. Introduction and Background 
Prior t o  1973, planetary and lunar spacecraft ranging sys- 
tems at  JPL utilized a transmitted uplink code made by com- 
bining binary clock and pseudonoise sequences in a majority- 
vote logic [ 1 ] .l The “composite-code-uplink’’ ranging receiver 
consisted of one or two channels that correlated the trans- 
ponded signal with combinations of the clock and each sepa- 
rate component sequentially through each successive symbol- 
delay t o  determine the precise delay on each of the compo- 
nents. Since transmitted power was distributed among the 
various clock and pseudonoise code components but the 
receiver was sensitive t o  only one component at one phase at  
a time, acquisition time was longer by about a factor of 
16 than if the receiver could have processed all the received 
power during the acquisition time. 
Consequently, once sufficient analysis and precautions had 
been taken to ensure that uplink ranging sidebands would not 
interfere with the spacecraft command system, a “sequential- 
component-uplink” ranging method [2] was devised that is 
still being used today. The term “sequential,” in this case, 
refers t o  the transmitted code, which is a time series of square 
waves of successively shorter wavelengths. The receiver, still a 
fewcorrelator device, is programmed to acquire the sequen- 
tially transmitted components one by one. The newer method 
had a 16:l acquisition-time advantage over the older scheme 
because it could utilize all the transponded power for each 
component during the acquisition time. The necessity t o  pro- 
gram the uplink code and to  program the receiver t o  switch 
components at the proper round-trip-light-time interval was 
a disadvantage compensated by the signal-to-noise advantage. 
‘R. W. Tappan and R. C. Tausworthe, “DSIF Technical Description, 
Planetary Ranging Equipment, Mariner-Mars 1971 Configuration,” 
JPL Document TD505943A (internal document), Jet Propulsion 
Laboratory, Pasadena, California, February 4 ,  1972. 
During a transition period, both ranging techniques were 
used, The older, composite transmitted code method was 
referred t o  as the ‘‘T” system, and the newer, sequential com- 
ponent code method was called the ‘‘p” system.’These names 
were dubbed by  Robertson Stevens, now the Chief Engineer 
I 31 8 
https://ntrs.nasa.gov/search.jsp?R=19880003328 2020-03-20T09:15:52+00:00Z
of the Deep Space Network, one day while strategizing at the 
blackboard in a design meeting, groping for a notation to dis- 
tinguish the two. They derive from the initials of the then- 
purveyors of the two systems: the author (guess which one) 
and Warren Martin. Usage of these designations has decayed 
over the years because the 7 system is no longer extant. The 
designations are reinstituted in this article for brevity in refer- 
encing the two schemes. 
The 7 planetary composite transmitter code was generated 
by combining a clock square wave with a majority-vote logic 
of 5 pseudonoise sequences in an exclusive-OR fashion. The 
components had symbol-periods of 2, 7, 11, 15 ,  19, and 23, 
for a total code period of N = 1,009,470. This, when clocked 
at a symbol-period to of about 1 psec, gave a repetition per- 
iod of about 1 sec, yielding a 2-way range ambiguity interval 
of approximately 150,000 km. 
The majority-vote combining logic in the old-7 system was 
chosen [3] , [4] because it evenly (and optimally, for that strat- 
egy) distributed the power among all components for sequen- 
tial detection. All component delay measurements were thus 
made with times approximately proportional to the compo- 
nent periods. The clock-code (period 2) phase was acquired 
first, and then the other 75 code phase correlations were made 
sequentially. Two correlator channels were time-shared during 
range acquisition in a way that balanced the channel gains and 
removed any residual dc bias voltage in the baseband detection 
process. 
The reason for utilizing only a few correlators in ranging 
receivers until now has been that each correlator channel con- 
sisted of relatively expensive analog and unit-logic digital 
hardware. However, with the advent of reasonably high speed 
digital analog-to-digital devices and very large scale integrated 
(VLSI) digital devices, correlators may now be made at much 
more modest cost [7].  
It is now economically feasible under VLSI technology to 
put the needed number of correlators into a receiver to build a 
detector for each of the components at each symbol-delay of 
the composite code. (It still may be impractical to build a full 
matched filter for the overall transmitted code, however.) It 
is therefore appropriate to reevaluate the relative merits of the 
two ranging methods under the removed constraint that pre- 
viously fixed the number of correlator channels. 
This article shows that a new-7 composite-component up- 
link code (or “ ~ 7 , ”  for brevity) which utilizes a new combining 
logic for the transmitter code and a 77-correlator receiver is 
again favorable in performance. In fact, it is shown that the 
vr method is only about 0.25 dB below the performance of a 
matched filter for the optimal transmitter code. As the /J sys- 
tem is now configured, about half of the range-measurement 
time is spent in correlating with the highest-frequency compo- 
nent (the clock), the other half being spent determining the 
range-cells of the lower-frequency components. The w system 
thus outperforms it by some 2.5 dB in signal-to-noise ratio. 
A companion paper by the author and J. R. Smith [7] dis- 
cusses the requirements and conceptual design of the correla- 
tor channels and VLSI devices required. 
II. The Transmitted Code 
The code component periods remain the same as in the 
earlier 7 system: 2, 7, 11, 15, 19, and 23. The combining logic 
for the transmitted code, however, is now taken to be 
x(c)=  c1 XOR AND(c) (1) 
in which c = (cl, . . . , c6),  c1 is the clock (period 2) sequence, 
XOR is the exclusive-OR function, and AND ( ) is the logi- 
cal-AND of all 6 component sequences. The logical-AND, or 
“unanimous-vote” logic, is the limiting case of the majority- 
vote logic used previously. 
Since AND (c) contains only one “1” in its truth table of 
64 entries, the in-phase cross-correlation of x(c) with c1 will 
be [31 
The in-phase cross-correlation of x(c) with a code of the form 
(cl XOR ci), for i = 2, . . . , 6 ,  is only 
1 + 1  
64 Rxi = -= 0.031 (3) 
(These figures are only approximate, being influenced slightly 
by the sense of imbalances between 0’s and 1’s in each of the 
pseudonoise sequences. These imbalances can be chosen to 
further optimize the reception, but this is left as an exercise 
for the implementer.) 
The cross-correlations as functions of ranging delay are 
shown in Fig. 1. 
111. Performance 
The v~ receiver achieves range measurement precision by 
clock-component correlation, just as did both predectssors. 
But since the clock component of the transmitted code con- 
tains 94 percent (Le., 0.972) of the total ranging power, there 
31 9 
is only a 0.27 dB degradation in acquisition time from trans- 
mitting the clock component alone (such a code would not ,  
however, remove the ambiguity of the range). 
The requirements for range accuracy demand that the stan- 
dard deviation of the range measurement due to  noise be 
about l/lOOO of the “chip” time. to ,  or symbol-rate of the 
clock. The relative clock variance ui of A = 7/ro is thus about 
In order t o  achieve this accuracy, the received signal- 
energy/noise-density ratio must accordingly be high [5] . 
1 - 
= 6.7 x lo4 - 
ST 
N o  16R:,u; (4) 
where S is the total signal power, T is the correlator integra- 
tion time, and No is the received noise (single-sided) spectral 
density. 
Detection of the pseudonoise component phase is accom- 
plished by correlating the received signal with each of the 
separate phases of ( c ,  XOR c,) .  The power in each component 
is only about 0.001 = 0.312 of the total, so the component 
detection-energy/noise-density ratio is about 
S,T R: iST  
= 0.001 X 6.7 X lo4 = 67 (5) - - -- 
NO NO 
The maximum required pseudonoise component-energy/ 
noise-density ratio for an error probability of 0.01 was about 
10 for the old-7 system. Since detection of the pseudonoise 
range cell (see below) involves summing channel values for 
pairs of correlators, the additional noise may degrade the 
required SjT/No t o  about 20 (a full analysis has not yet been 
made). The better-than-a-factor-of-three margin ensures, how- 
ever, that unerring full-range acquisition is almost certain. 
IV. Maximum-Li keli hood Receiver 
The maximum-likelihood estimator of the clock compo- 
nent phase is well known and will not be repeated here. The 
remaining pseudonoise code delays, however, are t o  be esti- 
mated from measurements made in parallel with the clock 
delay determination. This approach differs significantly from 
the old-7 method, where the receiver pseudonoise sequences 
were acquired after the clock so that the receiver codes could 
be adjusted t o  then be in step with the received signal. Thus, 
whereas the old-7 system enjoyed full component correlation 
in only one integration bin per component, the UT scheme 
must make do with partial component correlation in two adja- 
cent correlation channels for each component. 
The derivation of the maximum-likelihood detector is 
straightforward: We presume that we receive the transmitted 
binary ranging signal x( t )  = x(c[ t ] ) ,  normalized here t o  unit 
power, immersed in wideband Gaussian noise n( t ) ,  as 
y ( t )  = m(t - 7) t n ( t )  (6) 
where (Y = S1/*. The time delay 7 is to  be estimated as that 
value ? maximizing the conditional probability (density) 
function 
Under the usual assumption that 7 is uniformly distributed 
over the unambiguous-range interval, the likelihood ratio, by 
Bayes’ rule, is 
where the interval (0, T )  dependency has been suppressed for 
notational convenience. 
The probability of receivingy(t), given 7, is the probability 
that the noise in n(r) = y ( r )  - m(r - 7). Because of the wide- 
band Gaussian character of the noise, the likelihood ratio 
becomes [6] 
exp (-$ lT [y ( t )  - m(t 
exp (-& lT [y(r )  - m(t A =  
where exp(x) is the exponential function ex. 
By noting x 2 ( t )  = 1, canceling like terms in the numerator 
and denominator, and taking logarithms, we find that the con- 
dition on ?is that 
That is, ? will be the maximum-likelihood estimator of  7 
provided that it maximizes the correlation between the ob- 
served y ( t )  and the delayed transmitted code. However, a 
continuum of correlators is infeasible, so we must infer? from 
the finite number of measurements we d o  make. It has been 
shown [3] that the maximum-likelihood value can be inferred 
using correlations of the incoming signal and various delays of 
320 
the transmitted codes, in the form x1 = c1 and xi = c1  XOR 
c i f o r i = 2 ,  . . . ,  6. 
To decrease ranging inaccuracy caused by waveform distor- 
tion within the communication system, estimation of the 
clock component phase in the current p system is performed 
by maximum-likelihood methods applied only to the funda- 
mental harmonic of the received clock component. This 
results in a modest, justifiable increase in required integration 
time. The UT method would presumably have the same require- 
ment for this clock estimation scheme. 
Since almost all of the transmitted power is in the clock 
component of the code, the contribution of the 75 other cor- 
relators in improving the accuracy of the clock phase estimate 
will be negligible. Hence, the value of (?mod 2t0) may be 
determined from the clock-channel correlators alone. (Com- 
bined, weighted estimation of the clock phase from all chan- 
nels can be done, however, with only slightly more complexity 
in the estimation program, if desired.) Since 94 percent of the 
transmitted power is in the clock component, and since 
maximum-likelihood estimation is performed on this compo- 
nent, the clock phase estimate is very nearly the same as the 
maximum-likelihood estimate of a pure clock signal. 
Thus, any method that with high likelihood selects the 
proper range-cell delays of the remaining components will 
measure the range within 0.27 dB of the performance of a 
maximum-likelihood device, insofar as ranging accuracy is 
concerned. 
The clock-channel measurement of (; mod 2 to) is required 
to be very close to the actual value of (T  mod 2t0)  for system 
accuracy. From this value, (?mod f,) may be determined, as 
well as the + 1  sense of the in-step correlation (Fig. 1). There- 
fore, determination of (? mod Nt,) additionally requires only 
the estimation of the integer values k ,  such that 
for each of the remaining pseudonoise components (period 
Nito>. We may estimate these values from the correlator out- 
puts of each pseudonoise component and combine them to 
form the overall range using the Chinese remainder theorem, 
just as did the previous T system. 
Only two of the correlator integration valuesl., j =  1 ,  . . . , 
Ni for the i th  code component may derive from partial corre- 
lation with the true delay. The other Iii values correspond to 
11 
out-of-phase correlation levels. Let zii denote the normalized 
sum of adjacent correlators at the j t h  position of the i th code 
component: 
= Rxi(7 - ifo) t Rx i ( ~  - ( j  t l)to) t nii 
for an appropriately defined noise term nii. The signal portion 
of zii, denoted Zij in the equation above (see Fig. 2), 
- 
2.. = 
11 
RxiX  1 t - - - i t  1 i f ( j -  l ) toGT<jto i ilk l 
if ito Q T < ( j  t 1) to 
0 elsewhere 
(14) 
If k is the correct range cell, i.e., kt, Q T < (k t ]) to, then 
the geometry of the correlation function (Fig. 1) leads to the 
estimator 
The right-hand side of this equation should be ei mod to ) ,  
as already estimated accurately by clock-channel computa- 
tions, within the expected noise deviation. 
If there were no noise, the value of j yielding the maximum 
zii would be the correct range cell. But with noise, the cells on 
either side must also be scrutinized, as the spillover correlation 
and noise may cause us otherwise to choose the index of the 
wrong range cell. 
n 
HenceLlet ki $e thaij-index for which zii is maximum, and 
let k ,  be k,  - 1 ,  k,, or k,  t 1, whichever minimizes the differ- 
ence between ?, and ?l .  This k,  is a high-likelihood range- 
cell estimate for the ith component because the maximum- 
likelihood range celI is certainly one of the three candidates, 
and the comparison of the symbol-fraction range offset, calcu- 
lated as above, against the accurate determination of the clock 
321 
symbol fraction, rules out the other two candidates with high 
probability. 
Analysis indicates that the probability of making the wrong 
choice would be very remote. As may be seen in the geometry 
of the cross-correlation function, shown in Fig. 1, the wrong 
pair of correlator values inserted into the symbol fraction 
formula above produces a significantly different estimate for 
I\ than for 
V. Range Calculation Algorithm 
The procedure used by the VT receiver to reconstruct the 
range is therefore basically the same as in the old-7 system, 
except for the way the component range cell determinations 
are made. Having read the 77 correlation values all at once 
after an integration interval T,  the normal calculations on the 
two clock values determine the clock component range delay 
(mod 24,) with high accuracy. The remaining 75 correlation 
values are grouped by componens and then a value zi is 
chosen for each component. This ki is the range-cell index j 
that maximizes the sum of adjacent correlazon valu_es IiJ 
+ Ii,i+l. Each of the three range-cell indices ki - 1, k i ,  and 
ki + 1 in turn is hypothesized to be the correct range-cell in- 
dex. The appropriate correlator values for each of these indices 
are then used to compute a symbol-fraction offset using the 
estimator formula above. The candidate that comes closest to 
the clock-channel measurement wins. 
.% 
Acknowledgments 
The author gratefully acknowledges the encouragement and assistance of several col- 
leagues. Paul Headley and Mike Rodriguez were instrumental in setting up the first meet- 
ing to discuss the possibilities of reconsidering multicomponent-uplink ranging. Shirley 
Peak and Bob Bunker performed a custom-VLSI feasibility study to determine whether 
conventional correlator channels could be built economically. John R. Smith performed a 
conceptual design of the VLSI correlators and spent a significant amount of time with the 
author and others discussing the signal detection and correlation method. Harold Baugh 
answered many questions concerning the current p ranging system and its planned up- 
grade. John Smith also proposed the logic simplification discussed in the companion 
article [7]. 
322 
References 
[ 11 S .  W. Golomb, et al., Digital Communications with Space Applications, Englewood 
Cliffs, New Jersey: Prentice-Hall, 1964. 
[2] J. H. Yuen (ed.), Deep Space Telecommunications Systems Engineering, JPL Pub- 
lication 82-76, Jet Propulsion Laboratory, Pasadena, California, pp.123-178, July 
1982. 
[3] R. C. Tausworthe, “Optimal Ranging Codes,” Transactions of the IEEE PTG-SET, 
vol. 10, no. 1,  pp. 19-30, March 1964. 
[4] R. C. Tausworthe, “Ranging the 1967 Mariner to Venus,’’ in Proc. of IEEE National 
Convention, pp. 294-295, New York, March 1967. 
[SI J .  H. Yuen (ed.), Deep Space Telecommunications Systems Engineering, JPL Publica- 
tion 82-76, Jet Propulsion Laboratory, Pasadena, California, pp. 160-161, July 
1982. 
[6] W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals 
[7] R. C. Tausworthe and J .  R. Smith, “A Simplified, General-Purpose Deep-Space Rang- 
ing Correlator Design,” TDA Progress Report 42-91, vol. July-September, Jet Pro- 
pulsion Laboratory, Pasadena, California, November 15, 1987. 
andNoise, New York: McGraw-Hill, 1958. 
323 
w 
3 
> 
z 
+ 
2 
0 
4 
w oc 
LT 
0 
U 
TIME DELAY 
e 
Fig. 1. Composite-code cross-correlation function RXi ( T )  
fl 
2 + 
3 
0 
oc 
!5 W 
oc 
oc 
8 + z 
W 
V 
2 
6 
0 
I 
3 
m 
U 
RANGE DELAY 
Fig. 2. Sum of adjacent correlator outputs versus range delay 
324 


Tau ranging revisited

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