Interpolation Process; Doppler Computing - Safran CORTEX Series User Manual

3.3.10.6

Interpolation Process

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 
the 10-ms period and  the time between 0 and T when the phase is interpolated.
 
   
   
( t ) ) ( t ( t T
int er
T
The Taylor expression of the phase at t+ is:
 
   
    
( t ) ) ( t F D ) ( t

3

 
R ( t ) sup D ''( x
2
! 3  
  
x t t ,
    
   
( t ) ( t ) ( t
err int er
T 2
 
  
( t ) sup
so
err
8 
 
x t t , T
For instance, a Doppler rate of 1000 Hz/s with a second-order Doppler variation of 10 Hz.s
phase error lower than (0.0125 + 3.3310
I ,
N MOST CASES USER WOULD NOT USE CRANKY VALUES FOR THE SAMPLING RATE
BECAUSE ACCUMULATED PHASE ARE GENERATED SYNCHRONOUSLY WITH THE SAMPLING RATE DATES
3.3.10.7

Doppler Computing

The Doppler measurements are computed owing to the accumulated phase interpolated every Ts
seconds according to the formula
is the IF Receiver center frequency and  the accumulated phase with a resolution of 2
Doppler, f
IFR
cycle.
The Doppler error is
f
err
 
   
( t Ts ) ) ( t Ts
With
Ts 4

 
R ( t Ts )
where
3
! 4
 
 
   Ts
Ts
' t '( t )
and
2
2
 
Ts
 

 
Ts
R ' t
where
2
2
 
Ts 2
  
Ts
f t sup
So
err
2
24

x
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


) ) ( t

2

   
D ) ( ' t R ( t )
, where D(t) is the input Doppler and R
2
2
)
(Lagrange overestimation).
   
   
    
) ( t T ) ) ( t
T
T 3
 
D '( t ) sup D ''( x )
 3  
 
x t t , T
-6
) cycle.
 
  
Ts
t f
2
 
 
 
( t Ts ) ) ( t
 
Ts
t
2
Ts
Ts 2 Ts 3
  
   
(
'( t ) ''( t )
2 6


( ) 4
sup ( x )
,
 
 
x t t , Ts
 
2
Ts
2
 
   
( ) 3
''( t ) ) ( t R
2
3
2


( ) 4
sup ( x )
.
! 3  
 
x t t , Ts
Ts 3
 
( ) 2
D ( x ) sup
  16 
  
t t , Ts x t t , Ts
the interpolated phase, 
inter


D '( t )
   
   
( t ) ) ( t T
2
. T
HUS
 
 
( t Ts ) ) ( t
1  
f
IFR
2 32 Ts
 

 
Ts
' t
2

 
) 3
) ( t R ( t Ts )
3
 


Ts
' t
2
2
( ) 3
D ( x )
where D(t) is the input Doppler.

Safran Data Systems
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It cannot be duplicated or distributed without expressed written consent.
Ref. DTU 100782
Is.Rev 3.5
Date: June 1, 2021
the phase error, T
err
(t) the rest such that:
2



    
R ( t T ) R ( t )
2 2
T
-2
,
NO INTERPOLATION WOULD BE APPLIED
.
, where f is the computed
.
give a
,
-32
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