Pseudo-Noise (PN) Ranging Systems - CCSDS

CCSDS INFORMATIONAL REPORT CONCERNING PSEUDO-NOISE RANGING SYSTEMS
2.7.2.4 Square-Square Matched Case
In this case the clock is a square wave signal, so the expression of 2.7.2.2 becomes:
[ + ⋅ ( + ) ]
m t ω
ω Sin
( ) = ( ) + ( ) = 2 sin 0
t +
1444442444443
t
C t n t s t x θ
s(
t)
[ ( ω + ω ) t]
+ 2n
( t)
sin[
( ω + ) t]
+ 2nc
( t)
cos 0 t
s
0 ωt
1444444442444444443
n(
t )
The signal y(t) at the carrier demodulator output is given by:
( ω t + θ ) + n ( t)
cos(
ω t)
n ( t)
sin(
ω t)
y( t)
= C sin( m)
⋅Sin
t
c
t + s
t
Developing the above expressions one can find the in-phase and quadrature signal
components at the integrator output (see figure 2-18):
W
W
I
Q
= ∫
T
2C sin(
m)
⋅Sin
( ωtt
+ θ ) ⋅Sin(
ωtt)
dt =
⎡ 2θ
⎤
C sin(
m)
⋅T
⎢
1−
⎣ π ⎥
⎦
= ∫ 2C sin(
m)
⋅Sin
( ωtt
+ θ ) ⋅Cos(
ωtt)
dt =
⎡ 2θ
⎤
C sin(
m)
⋅T
⎢
−
⎣ π ⎥
⎦
T
For the noise, the expressions for the rms value at the integrator output are still valid yielding
for the ranging delay τ (θ = ωtτ):
and
where:
π −WQ
τ =
2 ⋅ω
W −W
σ τ
⎡
π ⎛
≈ ⎢⎜
2ω
⎢⎜
t
⎣⎝
t
I
Q
W
⎞
⎟
⎠
2
N
⋅
0
⎛
T + ⎜⋅
⎝
2 ( W −W
) ⎟ 2 ⎜ ( W −W
)
I
Q
Q
⎤
N0
⋅ T ⎥
2 ⎥
⎦
CCSDS 414.0-G-1 Page 2-49 March 2010
−W
2
⎡
π N ⎛ W ⎞ ⎛
0 ⎢⎜
Q ⎟ ⎜ −W
I
= T
+ ⋅
2ω 2 ⎢⎜
t
⎣⎝
I Q ⎠ ⎝ I Q
2
π N W
0 I + W
= T
2ωt 2 I Q
2 ( W −W
) ⎟ ⎜ ( W −W
)
2
Q
( ) 2
W −W
I
I
Q
2
2
⎞
⎟
⎟
⎠
2
⎞
⎟
⎟
⎠
2
⎤
⎥
⎥
⎦
1 2
=
1 2
=