Safran CORTEX Series - Impact of Phase Errors on Doppler Computing; Sampling Period Resolution; Error Due to Computing

HIGH DATA RATE RECEIVER
HDR-4G+ USER’S MANUAL
Ref. DTU 100782
Is.Rev 3.5
Date: June 1, 2021
© Safran Data Systems – IMP000074 e14r1
This document is the property of Safran Data Systems.
It cannot be duplicated or distributed without expressed written consent.
Page 117
A typical second-order Doppler variation of 10 Hz.s
-2
and third-order Doppler variation of 10
-4
Hz.s
-3
give
a phase error lower than (4.1610
-3
+ 6.2510
-9
) Hz for a sampling period Ts of 0.1 s, and lower than
(41.6 + 6.2510
-3
) Hz for a sampling period Ts of 10 s.
Impact of Phase Errors on Doppler Computing 3.3.10.8
If 
err
is the error on the accumulated phase, the error on the computed Doppler is lower than
Ts
err  2
2
1
32

, where Ts is the sampling period.
So, the maximum error on the Doppler computing due to the linear interpolation on the second-order
phase variation is 


2
2
T
Ts
DR

.
For example, with a Doppler rate of 1000 Hz/s, the maximum error will be 0.43 Hz for a sampling period
of 0.1 s and 0.0043 Hz for a sampling period of 10 s.
The relation between range rate and Doppler is:
c
v
f
f
r2


, where f is the satellite carrier frequency, v
r
the radial velocity (range rate) and c the light celerity.
For a 2.2-GHz beacon, the corresponding range rate error will be 2.93 cm/s for a sampling period
of 0.1 s and 0.293 mm/s for a sampling period of 10s.
Sampling Period Resolution 3.3.10.9
Error Due to Computing 3.3.10.9.1
The accumulated phase and Doppler sampling is processed owing to an internal 64 bits- NCO running
at T = 10 ms. The effective sampling period is:
T
incr
Tseff 
64
2
where incr is the NCO phase increment.
So the resolution on the period is


T
Tse
incr
T
e
eincrincr
T







64
2
2
64
64
2
2
11
2
where e is the error made on
the phase increment computing (in this computing Ts
eff
is approximated by Ts, and e is consided as
<<< incr).
Due to double-float format limitations, e = 2
14
is higher than the possible error (11 bits come from the
difference between the 64 bits and the 53 bits mantissa of the double-float representation, on which up
to 3 bits may be lost in the computing process).
When Ts = 10 s (worst case), the resolution is 6.810
-12
s.
This resolution is effective when the programmed sampling period is a value for which the floating
representation does not generate any error (examples: 10 s, 1 s or 0.125 s).
An error  made on the sampling period generates an error on the Doppler estimation:
 
Ts
Ts
tf
TsTs
Tstt
TsTs
Tstt
Ts
Tstt
Ts
Tstt
















2
)()(
)(
)()(
)()()()(
,
where f is the current IF frequency.
With a typical value of 720 MHz, the Doppler computing error will be 39 mHz with Ts = 0.125 s.