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.
Interpolation Process 3.3.10.6
In some cases, when the sampling rate and the generation rate are asynchronous, a linear
interpolation is applied to provide the accumulated phases at the required sampling frequency.
Such an interpolation generates an error. Let’s call
inter
the interpolated phase,
err
the phase error, T
the 10-ms period and the time between 0 and T when the phase is interpolated.
)()()()(int tTt
T
tter
The Taylor expression of the phase at t+ is:
)(
2
)(')()()( 2
2
tRtDtDFtt
, where D(t) is the input Doppler and R
2
(t) the rest such that:
)''(sup
!3
)(
,
3
2 xDtR
ttx
(Lagrange overestimation).
)()(
2
)'(
)()()()()()()( 22int
tRTtR
T
T
tD
tttTt
T
ttt ererr
so
)''(sup
3
)'(sup
8
)(
,
3
,
2
xD
T
tD
T
t
TttxTttx
err
For instance, a Doppler rate of 1000 Hz/s with a second-order Doppler variation of 10 Hz.s
-2
give a
phase error lower than (0.0125 + 3.3310
-6
) cycle.
IN MOST CASES, USER WOULD NOT USE CRANKY VALUES FOR THE SAMPLING RATE. THUS, NO INTERPOLATION WOULD BE APPLIED,
BECAUSE ACCUMULATED PHASE ARE GENERATED SYNCHRONOUSLY WITH THE SAMPLING RATE DATES.
Doppler Computing 3.3.10.7
The Doppler measurements are computed owing to the accumulated phase interpolated every Ts
seconds according to the formula
IFRf
Ts
tTst
Ts
tf
)()(
2
1
2
32
, where f is the computed
Doppler, f
IFR
is the IF Receiver center frequency and the accumulated phase with a resolution of 2
-32
cycle.
The Doppler error is
2
'
)()(
2
Ts
t
Ts
tTst
Ts
tferr
With
)()(
6
)''(
2
)'()()( 3
)3(
3
2
TstRt
Ts
t
Ts
tTstTst
where
)(sup
!4
)(
)4(
,
4
3 x
Ts
TstR
Tsttx
,
and
2
')(
2
2
)''(
2
)'(
2
' 2
)3(
2
Ts
tRt
Ts
t
Ts
t
Ts
t
where
)(sup
!3
2
2
'
)4(
,
3
2 x
Ts
Ts
tR
Tsttx
.
So
)(sup
16
)(sup
24
2
)3(
,
3
)2(
,
2
xD
Ts
xD
Ts
Ts
tf
TsttxTsttx
err
where D(t) is the input Doppler.
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