Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals - Global Navigation Satellite Systems: Signal,Theory and Applications - page 69

M − 1

⎧

∑
∑

⎫

2

dM
d

=

i

⎪ ⎪

i
k

= 0

⎪ ⎪

S d R = ⎨ (30)

()

i

M

⎬

∈ : − 1 > 0

L i

⎪

i = 0

⎪

dd = , k = , , ,

M

− 1

⎪ ⎪

⎩ ⎭

k Mk −− 1

2

With an appropriate weight coefficient α, all the undesired positive side peaks of

2
R can

be canceled by subtraction. From (18), the coefficient α must satisfies

M − k

( )

α r ≥ R TM / = (31)

k B c

M

for k = 1, 2, , M − 1 , or

M − k

α ≥ max

k

. (32)

M r

≠ 0

k

5.4 Step 4 - Local waveform optimization

So far the effect of thermal noise has not been considered. In fact, the coefficient α amplifies
noise components in B/ R . Under a given pre-correlation SNR, the larger α is, the lower
SNR in the SCF is. Therefore, from the viewpoint of sensitivity, it is desired that α be as
small as possible. So that with a given

d the optimum α is

M − k

α = max

k

(33)

()

Mr d

≠ 0

k L

and in UVS, the optimum

d is the one minimizing α, that is

M − k

d = arg min α = arg min max (34)

opt

M r

d ∈ S d ∈ S k ≠ 0

L L

k

It can be proved that the explicit expression of (34) is

[ ] T

d = dd , , d − (35)

opt 0 1 M 1

where

⎨ dd = =

M − 1

0 M − 1

23
1

M −

(36)

i − 1

() (

)

−

dd = = , i = , , ,

M

− 1

⎩

i Mi −+ 1 2

23 M −

and α =−. 2 M 3

min

Figure 11 (a)-(c) depict the optimum local SCS waveforms for BOC( n , n ), BOC(2 n , n ), and
BOC(3 n , n ) signals respectively. For BOC( n , n ) signals,

( ) T

α = . It
can be found that the SC method proposed in (Julien, et al., 2007) is equivalent to this case.
The local symbol is simply a rectangular pulse. For BOC(2 n , n ),

M = . So 2 d = 1,1 and 1

min

opt

( ) T

d =

9

1,,,1

11

and

opt 5 3 3

( ) T

α = . For BOC(3 n , n ), 5 d =

1

5,1, − − 1, 1,1, 5 and α = . It can be seen that the 9

min opt min

3