Performance of Stand-Alone GPS - Understanding GPS: Principles and Applications - page 309

the propagation velocity v p of the signal's carrier phase differs from the velocity v g
associated with the waves carrying the signal information. The information-
carrying aspect can be thought of as a group of waves traveling at slightly different
frequencies.
To clarify the concepts of group and phase velocities, consider two components,
S 1 and S 2 , of an electromagnetic wave with frequencies f 1 and f 2 (or ω 2 ) and

phase velocities v 1 and v 2 , traveling in the x -direction. The sum S of these signals is

ω 1 and

t x
v

x
v

SS S =+ = sin ω − + sin ω t −

1 2 1 2

1 2

Using the trigonometric identity,

1
2

1
2

( ) ( )

sin α + sin β = 2 cos αβ − ⋅ sin αβ +

we find that

ωω ωω

1
2

1
2

1
2

1
2

( ) ( )

S = 2 cos ωω − t −

vv x

1

−

2

×

sin ωω + t −

vv x

1

+

2

1 2 1 2

1 2 1 2

1
2

x 1

2

1
2

ωω

( ) ( )

1 2

= 2 cos ωω − t − × sin ωω + t −

vv x

+

1 2 1 2

1
2
1
2

( )

ωω −

1 2

1 2

ωω

1 2

−

vv

1 2

The cosine part is a wave group (the modulation imposed on the sinusoid—that
part of the wave that carries the information) that moves with velocity

vv

1
2
1
2

1

−

2

( )

ωω −

( )

2 π f − f

f − f λλ

1 2

1 2 1 2

v = = =

1 2

=

g

11

ωω f

v

f
v

11

−

1

−

2

2 π

1

−

2

−

λλ

vv λλ 1 2

(7.6)

1 2 1 2 1 2

v vvv

1

−

1

+−

1 2

λ λ λ λ

vv −

1 2 2 2

2 1

= =− v λ

1 1

λ−λ

11

2 1

−

λλ

1 2

where λ 2 are the corresponding signal wavelengths.

For signals with narrow bandwidths relative to the carrier frequency, such as
the GPS signals, we can replace v 2 −

λ 1 and

v 1 by the differential dv , λ 2 − λ 1 by the differential

d λ , and λ 1 by λ 2 , and add the subscript p to v to denote phase velocity explicitly to get