Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals - Global Navigation Satellite Systems: Signal,Theory and Applications - page 73

{ }

S d = ∈ R

2

: dd

2

+

2

= 2, dd > ≥ 0 (48)

1 1 2 0 1

In the case M = (Yao, 2008), UVS can be expressed as 4

⎧ 0 ≤≤≤ ddd ⎫

⎧ ⎫⎧ ⎫

3 0 1

⎪ ⎪

⎪ ⎪⎪ ⎪

dd >≥ 0

0

dd ≤

⎪ ⎪ ⎪ ⎪ ⎪⎪

S d

4 0 3

∪

3 2

=∈ R (49)

⎨ ⎨ ⎬ ⎨ ⎬⎬ 1

dd == dddd

dddd

−≤−≤

⎪ ⎪ ⎪ ⎪ ⎪⎪

1 2 3 1 2 0

⎪ ⎪ ⎪ ⎪ ⎪⎪

−+−≠ 0

⎩ ⎭ ⎩ ⎭

⎩ ⎭

0 1 2 3

For larger M , as the degrees of freedom of d increase, the explicit expressions for the
constraints on d become complex and hard to derive. The full UVS can be obtained
through the use of numerical method. However, it is easy to verify that any element in one
of the subset of UVS

⎧ ⎫

⎧ dd

d

> ≥ 0 ⎫

⎪ ⎪

0 M − 1

S d ′ =∈ R

M

(50)

⎨ ⎨ ⎬⎬ 1

== 0, i 1,2, , M − 2

⎪ ⎩ ⎭ ⎪

⎩ ⎭

i

can make a triangular SCF for all M even.

6.4 Step 4 - Local waveform optimization

Under the energy normalization restriction (12),

d in (50) has one degree of freedom. So by

defining κ = d / d , the shape vector in S can be expressed as

M − 1 0

( )

T

d =

M

,0, ,0, κ

M

(51)

1

2 2

1 + κ 1 + κ

Utilizing (51), (19), and (38), without considering front-end filtering, we can obtain the
expression of R

( ) ( ) ( )

⎧

M 24 21 κτ −+− κ T 1

2

−
−

κ
κ

T

, τ <

c c

= ⎨
⎩

( )

( )

( )

M

2

R τκ ;

MT 1 + κ

(52)

c

0, others

the base line half width of which is

() (

)

1 − κ

T

w κ = (53)

( )

M 2 − κ

and the height of the peak is

(

21
1

− κ

)

()

(54)

h κ =

( )

M + κ

2

Figure 15 (a) and (b) show some SCFs with different κ for BOC( n , n ) and BOC(2 n , n ) signals,
respectively. Figure 16 depicts the discriminator characteristic curve of the early-minus-late
power (EMLP) loop which uses SCF instead of BOC( n , n ) ACF.