B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

12.3 Linear T-Spaced Equalization (TSE) 679
Example 12. 1 Consider a first-order channel
H(z) = l +z- 2
Determine the noise amplification effect on the ZF equalizer for a BPSK transmission.
Because H (ei 2 nf) = 0 whenf = ±1/4, it is clear that H(z) has spectral nulls. By applying
the ZF equalizer, we have
Therefore,
1
f [ij - z d7
f i - 1-u
- 2nj 1 +z- 2
=
{ 0 i < u
( - l) i -u i :':: U
L ll [i]l 2 = L Cl) = 00
i=O
i=u
CX)
CX)
This means that the BER of the BPSK transmission equals
Pb = Q (0) = 0.5
The noise amplification is so severe that the detection is completely random.
Example 12. 1 clearly shows the significant impact of noise amplification due to ZF
equalization. The noise amplification effect strongly motivates other design methodologies
for equalizers. One practical solution is the minimum mean square error (MMSE) design.
12.3.2 TSE Design Based on MMSE
Because of the noise amplification effect in ZF equalization, we must not try to eliminate the
ISi without considering the negative impact from the noise term. In fact, we can observe the
equalizer output in Eq. (12.30) and quantify the overall distortion in d[n] by considering the
difference (or error)
CXJ
d[n] - Sn-u = L C;S11-i - Sn-u + L f[i]w[n - i]
CXJ
(12.34)
To reduce the number of decision errors when
dee (d [n]) -/- Sn-u
it would be sensible to design an equalizer that would minimize the mean square error between
d[n] and Sn- u • In other words, the MMSE equalizer design should minimize
ld[n] - Sn-u l 2 (12.35)