Classical transceiver techniques - Signal Processing and Optimization for Transceiver Systems - page 205

q ( n )

s ( n )
est

y ( n )

+

x ( n ) r ( n )

s ( n )

H ( z )

+ + detector

FIR channel

−

B ( z )

pre-equalizer 1/ H ( z )

Figure 5.22 . An IIR pre-equalizer for the FIR channel H ( z ), inserted at the trans-
mitter end.

Note that the receiver uses only a detector, and no equalizers. In Sec. 5.7
we explained that if 1 /H ( z ) is used at the receiver then the channel noise can
get severely amplified. This problem is no longer present in the pre-equalized
transceiver. However, a dual problem arises at the transmitter. Namely, the
amplitudes of the samples x ( n ) which enter the channel can get very large (es-
pecially if 1 /H ( z ) has poles close to or outside the unit circle), increasing the
channel input power. In the case of equalization at the receiver we overcame
this problem by moving the detector, so that it sits inside the feedback loop.
There is a somewhat similar trick we can perform for the pre-equalizer. Namely,
we insert an amplitude-limiting operator inside the feedback loop, indicated as
“mod V ” (read as modulo V ) in Fig. 5.23. The description of this operator is
given below; we will see that it is a nonlinear device, and this makes the pre-
equalizer a nonlinear equalizer. The mod V operator is used at the receiver also,
just prior to detection. We will show that, in spite of the presence of nonlinearly,
the pre-equalization operation perfectly equalizes the channel (i.e., s ( n )= s ( n )

in the absence of noise).
The mod V operator is a memoryless device with input-output characteristics
described by Fig. 5.24. Given any real number u , it produces a number v in the
range 5

− 0 . 5 V ≤ v< 0 . 5 V (5 . 91)

such that u v is a multiple of V ). That is, given an

arbitrary real u , the operator simply subtracts a multiple of V from it so that the
result is in the range [

− v =0mod V (i.e., u −

0 . 5 V, 0 . 5 V ) . Thus, imagine that the real axis is divided
into intervals of length V .Ifanumber u does not fall in the fundamental period
given by (5.91), it is simply brought back to the appropriate position within this
period. This is demonstrated in Fig. 5.25. The numbers marked

−

× , which are

outside the range [ 0 . 5 V, 0 . 5 V ) , are mapped into the numbers marked by little

circles falling within the desired range [

−

− 0 . 5 V, 0 . 5 V ) .

5 For complex constellations such as QAM, the definition of the modulo V operator can be
appropriately extended.