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

12.2 Receiver Channel Equalization 673
Generally, there are two approaches to the problem of channel input recovery (i.e., equalization)
under ISi channels. The first approach is to determine the optimum receiver based
on channel and noise models. This approach leads to maximum likelihood sequence estimation
(MLSE), which is computationally demanding. A low-cost alternative is to design filters
known as channel equalizers to compensate for the channel distortion. In what follows, we
first describe the essence of the MLSE method for symbol recovery. By illustrating its typically
high computational complexity, we provide the necessary motivation for the subsequent
discussions on various complexity channel equalizers.
12.2.2 Maximum Likelihood Sequence Estimation (MLSE)
The receiver output samples {z[n] } depend on the unknown input QAM symbols {sn} according
to the relationship of Eq. (12. 17). The optimum (MAP) detection of {s n ) from {z[n] } requires
the maximization of joint conditional probability [Eq. (10.81 )] :
max p( ..., S n- I , S n , S n+I, ..· 1 ·.., z[n - l] , z[n], z[n + 1], ...)
{sn)
(12.18)
Unlike the optimum symbol-by-symbol detection for A WGN channels derived and analyzed
in Sec. 10.6, the interdependent relationship in Eq. (12.17) means that the optimum receiver
must detect the entire sequence {sn} from a sequence of received signal samples {z[n] }.
To simplify this optimum receiver, we first note that in most communication systems and
applications, each QAM symbol S n is randomly selected from its constellation A with equal
probability. Thus, the MAP detector can be translated into a maximum likelihood sequence
estimation (MLSE):
max p ( . .. , z[n - l] , z[n], z[n + I ] . .. · 1 · .. , S n- I , S n , Sn+! , ... )
Is,, J
(12.19)
If the original channel noise ne (t) is white Gaussian, then the discrete noise w[n] is also
Gaussian because Eq. (12.16) shows that w(t) is filtered output of ne(t). In fact, we can define
the power spectral density of the white noise n,Jt) as
N
Sne if) = 2
Then the power spectral density of the filtered noise w (t) is
Sw if) = IP(J) l 2 Sne if) = : IP (J) l 2 ( 12.20)
From this information, we can observe that the autocorrelation function between the noise
samples is
R w [i] = w[i + n]w* [n]
= w(eT + nT)w* (nT)
= L: Sw (f)e-j2rrf CT df
= N oo
J
IP(J) l 2 e - j2rrf CT df
2
- oo (12.21)