ABSTRACT
R
, and to achieve a preassigned transducer power-gain characteristic
G
(
ω
) over the entire sinusoidal frequency spectrum.
As stated in Chapter 77, the output reflection coefficient is given by
(78.1)
where
Z
(
s
) is the impedance looking into the output port when the input port is terminated in the source resistance
R
. As shown in Chapter 77, the transducer power gain
G
(
ω
) is related to the transmission and reflection coefficients by the equation
(78.2)
Recall that in computing
ρ
(
s
) from
ρ
(
s
)
ρ
(–
s
) we assign all of the LHS poles of
ρ
(
s
)
ρ
(–
s
) to
ρ
(
s
) because with resistive load
z
(
s
) =
R
,
ρ
(
s
) is devoid of poles in the RHS. For the complex load, the poles of
ρ
(
s
) include those of
z
(–
s
), which may lie in the open RHS. As a result, the assignment of poles of
ρ
(
s
)
ρ
(–
s
) is not unique. Furthermore, the nonanalyticity of
ρ
(
s
) leaves much to be desired in terms of our ability to manipulate. For these reasons, we consider the normalized reflection coefficient defined by
s Z s z s
Z s z s ( ) = ( ) − −( )( ) + ( )
G j j j jω ρ ω ρ ω ρ ω ρ ω2 21 2
2 1 1( ) = ( ) = ( ) = − ( ) = − ( )
(78.3)
where
(78.4)
is the real all-pass function defined by the open RHS poles
s
(
i
= 1, 2, …,
q
) of
z
(–
s
). An
all-pass function
is a function whose zeros are all located in the open RHS and whose poles are located at the LHS mirror image of the zeros. Therefore, it is analytic in the closed RHS and such that
(78.5)
On the
j
ω
-axis, the magnitude of
A
(
j
ω
) is unity, being flat for all sinusoidal frequencies, and we have
(78.6)
and (78.2) becomes
(78.7)
This equation together with the normalized reflection coefficient
ρ
(
s
) of (78.3) forms the cornerstone of Youla’s theory of broadband matching [14].
