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3.3.4.5. Impact of Phase Errors on Doppler Computing
If ��
is the error on the accumulated phase, the error on the computed Doppler is lower than
������
1
��
������
×
, where Ts is the sampling period.
32
2
����
So, the maximum error on the Doppler computing due to the linear interpolation on the second-order phase
2
����
��
variation is ≤
(
)
����
2
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 10s.
The relation between range rate and Doppler is:
∆��
−2��
��
, where f is the satellite carrier frequency, v
��
��
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.
3.3.4.6. Sampling Period Resolution
3.3.4.6.1. Error Due to Computing
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:
64
2
����
=
× �� where �������� is the NCO phase increment. So the resolution on the period is
������
��������
1
1
64
2
× �� × (
) ≈ �� ×
��������
��������+��
where �� is the error made on the phase increment computing (in this computing ����
�� is considered as ≪ ��������).
Due to double-float format limitations, e = 2
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 = 10s (worst case), the resolution is 6.8x10
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:
�� ( �� ) − �� ( �� − ���� )
����
where f is the current IF frequency.
With a typical value of 70 MHz, the Doppler computing error will be 3.8 mHz with Ts = 0.125s.
© Safran Data Systems
COMMAND RANGING & TELEMETRY UNIT CORTEX
CRT QUANTUM USER'S MANUAL
.
64
2
2
��
������
=
2
64
��������
2
��
14
�� ( �� ) − �� ( �� − ���� )
= �� ( �� ) − �� ( �� − ���� ) ×
−
���� + ��
= ��(�� − ���� 2 ⁄ ) × 2
−23
This document is the property of
It cannot be duplicated or distributed without expressed written consent.
the radial velocity (range rate) and c the light celerity.
r
is higher than the possible error (11 bits come from the difference
-12
s.
���� × ( ���� + �� )
Safran Data Systems
.
Ref.
DTU 100042
Is.Rev.
5.17
Date:
Dec.
03, 2021Sept. 30, 2021
is approximated by ����, and
������
�� ( �� ) − �� ( �� − ���� )
��
≈
����
��
×
����
Page 51
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