TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 165

B EXAMPLE 3.5
Optimality of P ( z ) and Q ( z ). We show here how the choices of P ( z ) and Q ( z )from
(3.5) maximize the signal to noise ratio (SNR) subject to an interference cancella-
tion constraint. To this end, let

P ( z ) ¼ p 0 þp 1 z 1 þþp L z L

be an arbitrary filter having degree L , where L is the channel degree. If the decoder
correctly produces channel symbol estimates, giving d i ¼ d i , the intersymbol inter-
ference will vanish at the equalizer output provided we choose Q ( z ) according to

Q ( z ) ¼ P ( z ) H ( z ) ^rz L

in which

r ¼ X

L

h k p Lk

k¼ 0

is the central term of the impulse response of P ( z ) H ( z ). The equalizer output then
reduces to

v i ¼ ^ rd iL þ X

L

p k b ik

k¼ 0

in which the first term is a scaled desired signal (delayed by L samples), and the
second term is the filtered noise. If the noise sequence ( b i ) is white, then the
signal to (filtered) noise ratio becomes

r 2
P k p k :

E ( d i L )
E ( b i )

SNR ¼

r ¼ P k h k p Lk ,

Since the Cauchy-Schwarz inequality may be invoked to

show that

! 2 ! X !

¼ X X

L L L

r 2

h k p k

h k p Lk

k¼ 0 k¼ 0 k¼ 0

in which equality holds if and only if h k ¼ bp L 2 k for some constant b . The signal
to filtered noise ratio is thus upper bounded as

!
:

X

L

E ( d i L )
E ( b i )

h k

SNR

k¼ 1

The right-hand side coincides with the SNR seen from the channel output. Thus,
the choice p k ¼ h L 2 k , corresponding to P ( z ) ¼ z 2 L H ( z 2 1 ), maximizes the SNR
at the equalizer output, subject to an interference cancellation constraint.