IEEE - Aerospace and Electronic Systems - September 2019 - 53

Angkasa et al.

APPENDIX

parameters given by Berner et al. [11]

This appendix summarizes the analysis leading to the
results that appear in Figures 1 and 2.

DISTRIBUTION OF DOWNLINK POWER

fRS ¼ phase deviation of carrier by ranging signal; rad rms

fRN ¼ phase deviation of carrier by noise; rad rms
fT ¼ phase deviation of carrier by telemetry; rad:
It is necessary to relate fRS and fRN to the following
downlink modulation parameter:
fR ¼ phase deviation of carrier due to signal plus
noise in the transponder's ranging channel; rad rms:
The parameter fR is implemented in the transponder
and should be known to the link designer. The modulation
parameters fRS and fRN are related to fR by
fRS ¼

fR ; qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
rTR
fR Á 1þr
;

regenerative
nonregenerative

0;
1
ﬃ;
fR Á pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

regenerative
nonregenerative

fRN ¼

1þrTR

where rTR is the SNR in the transponder's ranging
channel. For nonregenerative ranging, the ranging
channel of the transponder is wideband (typically 1.5
MHz, in order to accommodate a 1-MHz range clock)
and rTR ( 1; this causes fRS ( fR . For regenerative
ranging, the code-tracking loop in the transponder will
have a narrow bandwidth (likely about 1 Hz)
so that rTR ) 1. A rTR of À25 dB has been assumed
for nonregenerative ranging, and this is a typical value
for deep-space ranging. For regenerative ranging with
a code-tracking loop having a bandwidth that is six
orders of magnitude smaller than the bandwidth of
the nonregenerative transponder's wideband ranging
channel, rTR would be, for the same link parameters,
þ35 dB or more. For regenerative ranging, the
noise in the ranging channel is negligible and so
there is effectively no noise modulated onto the downlink carrier.
The available power PR in the ranging sidebands is
related to the total downlink power PT and the modulation
SEPTEMBER 2019

PD ¼ J02

pﬃﬃﬃ

2
2 fRS Á eÀfRN Á sin2 ðfT Þ Á PT :

The power PC in the residual carrier is related to PT
and the modulation parameters given by Berner et al. [11]
pﬃﬃﬃ

2
PC ¼ J02 2 fRS Á eÀfRN Á cos2 ðfT Þ Á PT :
Because the ranging signal has half-cycle sinusoidal
pulse shapes and is phase modulated onto the carrier, the
sideband levels are governed by the Bessel functions
J0 ðÁÞ and J1 ðÁÞ of the first kind and of order 0 and 1. The
power in each signal component increases as the corresponding phase deviation increases; for example, PR
increases as fRS increases (with the constraint
0 fRS fR ). Also, PR decreases as fRN increases
(with the constraint 0 fRN fR ), which can be understood as an increasing amount of downlink power being
wasted in noise sidebands, leaving less power for the ranging sidebands.

RANGE CODE

and
(

pﬃﬃﬃ

2
2 fRS Á eÀfRN Á cos2 ðfT Þ Á PT :

The power PD in the telemetry sidebands is related to PT
and the modulation parameters given by Berner et al. [11]

To proceed with this analysis, the distribution of power
on the downlink phase-modulated carrier must be calculated. When both ranging and telemetry are present
on the downlink, the relative distribution of power on
the downlink depends on the following modulation
parameters:

(

PR ¼ 2 J12

The CCSDS T4B range code was designed for a range measurement at planetary distances [7]. This range code is built
from a set of six component codes, the first of which is the
range clock. As part of the range measurement, the DSN
measures the phase delay experienced by the range clock in
passing from the DSN transmitter to the spacecraft and
then to the DSN receiver.11 (In principle, the time delay
might be measured directly, but with the poor SNRs available on deep-space links, this would lead to poor results. A
phase measurement, from which time delay and range may
be inferred, gives much better accuracy.)
The phase measurement on the range code has a
phase ambiguity. This ambiguity may be understood
by recognizing that the total phase delay is a large
integer number of cycles of the range clock plus a
fraction of one cycle of the range clock. The measurement of the range-clock phase delay can only identify
the fractional part. This fractional part gives the range
measurement its precision (s R ). The resolution of this
ambiguity requires that the integer number of rangeclock cycles be determined as well. It is the purpose
of the other five code components to aid the ambiguity
resolution. The probability of acquisition Pacq is the
probability that the ambiguity resolution is successful.
Both s R and Pacq are essential performance parameters
for the range measurement.

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