Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals - Global Navigation Satellite Systems: Signal,Theory and Applications - page 63

T
2

3T
4

c

0 T

4

0 T c 0 T c /2 T c 0 T c

0T c /2 T c

Fig. 8. Examples of SCS Waveforms

4.2 Vector representation

Any SCS waveform can be represented by a coordinate vector. Given a pair of

T and T ,

{ } ( )

we can construct a set of function ψ tk : = , , , M − 1 , where

k

1

⎧

( )

, kT ≤< + t k 1 T

= ⎩

()

s s

ψ t T (8)

c

k

0, others

As one can easily confirm, these M functions are orthogonal to each other and any SCS chip
waveform

( )

p t with the same

T and M can be written as a linear combination of SCS

{ }

( )

ψ t , that is

k

M

∑ 1

−

() ()

p t = ψ t ⋅ d (9)

SCS k k

k = 0

( )

onto ( )

where

d is the projection of

p t ψ t , i.e.

SCS k

T

() ()

∫ SCS 0

dMp t = ψ t t (10)

k k

( )

Therefore, each chip symbol p t corresponds to a coordinate vector

SCS

[ ] T

d = dd , , , d − (11)

01 M 1

in the space spanned by ( )
k

. To meet the energy normalization
condition of the SCS chip waveform, each vector d must satisfy

ψ t for k = − 0,1, , 1

1 1

2

d = d d

T

= 1 (12)

MM

Given the spreading chip rate f c and the vector d , the chip waveform p ( t ) is determined.
Borrowing from the notation of BCS signals (Hegarty et al., 2004), we call the vector d
shape vector, and use the notation p ( t ; d , f c ) to denote a SCS signal whose shape vector is d
and the chip rate is f c . If it is understood from context, we will omit f c from the notation.
It can be seen that the chip waveforms employed in most of the modulations in satellite
navigation such as BPSK-R, BOC with even order, and BCS are special cases of SCS
waveforms. Besides, almost all the auxiliary signal chip waveforms used in SC algorithms
also belong to this family. When d =1, p ( t ; d ) degenerates to the rectangular pulse, and when
[ ] T 21

d =− 1, 1, ,1, − 1 , p ( t ; d ) is the chip waveform of a sine phased BOC signal with the

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