Safran CORTEX CRT Quantum, User Manual

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COMMAND RANGING & TELEMETRY UNIT CORTEX
CRT QUANTUM USER'S MANUAL
Ref. DTU 100042
Is.Rev. 5.17
Date: Dec.
03, 2021Sept. 30, 2021
© Safran Data Systems
This document is the property of
Safran Data Systems
.
It cannot be duplicated or distributed without expressed written consent.
Page 192
3.6.13.3.
Ambiguity
ollows:
𝜑
𝑖
= 〈𝑓
𝑖
𝜏〉
1
𝜑
𝑀
= 〈𝑓
𝑀
𝜏〉
1
the number of turns of the major tone that is compatible with the phase measurements on minor tones:
𝜏 ≅
𝑘 + 𝜑
𝑀
𝑓
𝑀
, 𝜑
𝑖
≅ 〈𝑓
𝑖
𝑘 + 𝜑
𝑀
𝑓
𝑀
〉
1
There is a multitude values of K added to k that leads to the same phase measurements
〈𝑓
𝑖
𝑘 + 𝜑
𝑀
𝑓
𝑀
〉
1
= 〈𝑓
𝑖
𝑘 + 𝐾 + 𝜑
𝑀
𝑓
𝑀
〉
1
All the K are integer multiple of the lowest one: K = n * K
0
K
0
is called the maximum ambiguity. Let’s estimate K
0
.
〈𝑓
𝑖
𝑘 + 𝜑
𝑀
𝑓
𝑀
〉
1
= 〈𝑓
𝑖
𝑘 + 𝐾
0
+ 𝜑
𝑀
𝑓
𝑀
〉
1
The same equation can be rewritten with the integer ratios r
i
, r
M
instead of the frequencies f
i
, f
M
:
〈𝑟
𝑖
𝑘 + 𝜑
𝑀
𝑟
𝑀
〉
1
= 〈𝑟
𝑖
𝑘 + 𝐾
0
+ 𝜑
𝑀
𝑟
𝑀
〉
1
In other words
0 = 〈𝑟
𝑖
𝐾
0
𝑟
𝑀
〉
1
⟺ ∃𝑘
𝑖
∈ ℕ, 𝑟
𝑖
𝐾
0
= 𝑘
𝑖
𝑟
𝑀
Using relatively prime versions p
i
, p
M
of r
i
, r
M
𝑝
𝑖
=
𝑟
𝑖
gcd(𝑟
𝑖
, 𝑟
𝑀
)
=
𝑙𝑐𝑚(𝑟
𝑖
, 𝑟
𝑀
)
𝑟
𝑀
, 𝑞
𝑖
=
𝑟
𝑀
gcd(𝑟
𝑖
, 𝑟
𝑀
)
=
𝑙𝑐𝑚(𝑟
𝑖
, 𝑟
𝑀
)
𝑟
𝑖
Consequently, since p
i
and q
i
are relative prime numbers,
∃𝑀
𝑖
∈ ℕ, 𝐾
0
= 𝑀
𝑖
𝑞
𝑖