The origins and uses of complex signals - Statistical Signal Processing of Complex-Valued Data - page 7

Im

2 ( A + + A − )

q + − q −
2

Re

2 A + −

A −

Figure 1.2 A typical polarization ellipse.

component of the form A u cos( ω 0 t + θ u )inthe u -direction and another of the form

A v sin( -direction.

But there is more. Euler's identity may be used to write the electric field vector as

ω 0 t + θ v )inthe v

1

2 A u e j θ u e j ω 0 t

1

2 A u e − j θ u e − j ω 0 t

1

2 j A v e j θ v e j ω 0 t

1

2 j A v e − j θ v e − j ω 0 t

x ( t ) = + + +

2 A u e j θ u

j A v e j θ v

2 A u e − j θ u

j A v e − j θ v

e j ω 0 t e − j ω 0 t

1 1

= + + + . (1.8)

A + e j θ + A − e − j θ −

This representation of the two-dimensional electric field shows it to be the superpo-
sition of a two-dimensional, circularly polarized, component of the form A + e j θ + e j ω 0 t
and another of the form A − e − j θ − e − j ω 0 t . The first rotates counterclockwise (CCW)
and is said to be left-circularly polarized . The second rotates clockwise (CW) and
is said to be right-circularly polarized . In this representation, the complex constants
A + e j θ + and A − e − j θ − fix the amplitude and phase of their respective circularly polarized
components.
The circular representation of the ellipse makes it easy to determine the orientation
of the ellipse and the lengths of the major and minor axes. In fact, by noting that the
magnitude-squared of x ( t )is

2

A 2 A 2

−

| x ( t ) | =

+ +

2 A + A − cos( θ + + θ − + 2 ω 0 t ) + ,itis

2 has a maximum value of ( A + +

A − ) 2 at

easy to see that | x ( t ) | θ + + θ − + 2 ω 0 t = 2 k π ,

A − ) 2 and a minimum value of ( A + − at θ + + θ − + 2 ω 0 t = (2 k + 1) π . This orients the

major axis of the ellipse at angle ( θ + − θ − ) / 2 and fixes the major and minor axis lengths

at 2( A + + A − ) and 2 | A + − A − | . A typical polarization ellipse is illustrated in Fig. 1.2 .

Jones calculus
It is clear that the polarization ellipse x ( t ) may be parameterized either by four real
parameters ( A u ,

A v ,θ u ,θ v ) or by two complex parameters ( A + e j θ + , A − e − j θ − ). In the

ω 0 t ) , ω 0 t )), and in the second case, we

first case, we modulate the real basis (cos( sin(

modulate the complex basis (e j ω 0 t e − j ω 0 t ). If we are interested only in the path that the

electric field vector describes, and do not need to evaluate x ( t 0 ) at a particular time t 0 ,
knowing the phase differences

,

θ + − θ − rather than the phases themselves
is sufficient. The choice of parameterization - whether real or complex - is somewhat
arbitrary, but it is common to use the Jones vector [ A u ,

θ u − θ v or

A v e j( θ u − θ v ) ] to describe the state
of polarization. This is illustrated in the following example.