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

12.2 Receiver Channel Equalization 675
Indeed, Eq. (12.24) tells us that z[n-i] is Gaussian with equal varianceN/2 and mean value of
Therefore, the MLSE optimum receiver under Gaussian channel noise and root-raised-cosine
pulse shape P rrc Ct) [Eq. (12.23)],
max ln [np (z[n - i] 1 . .. , Sn- 1 , Sn , S n+! , ... )]
{>11)
°!:
ax
I-! L lz[n - i] - L h[k ]sn-i-k 1
{ nl
i
k
2
]
(12.26a)
Thus, MLSE is equivalent to
min L lz[n - i] - L h[k]sn-i-k l
{sn)
i k
2
(12.26b)
For a vast majority of communication channels, the impulse response h[k] can be closely
approximated as a finite impulse response (FIR) filter of some finite order. If the maximum
channel order is L such that
then the MLSE receiver needs to solve
L
H(z) = L h[k] z - k
k=O
min L lz[n - i] - t h[k]sn-i-kl
{sn)
i k=O
2
(12.27)
We note that the MLSE algorithm requires that the receiver possess the knowledge of the discrete
channel coefficients {h[k ] } . When exact channel knowledge is not available, the receiver
must first complete the important task of channel estimation.
MLSE Complexity and Practical Implementations
Despite the apparent high complexity of the MLSE algorithm [Eq. (12.27)], there exists a
much more efficient solution given by Viterbi 2 based on the d y namic programming principle of
Bellman. 3 This algorithm, often known as the Viterbi algorithm, does not have an exponentially
growing complexity as the data length grows. Instead, if the QAM constellation size is M,
then the complexity of the Viterbi algorithm grows according to M L . The Viterbi algorithm is a
very powerful tool, particularly when the channel order L is not very long and the constellation
size M is not huge. The details of the Viterbi algorithm will be explained in Chapter 14 when
we present the decoding of convolutional codes.
MLSE is very common in practical applications. Most notably, many GSM cellular
receivers perform the MLSE detection described here against multipath distortions. Because
GSM uses binary constellations in voice transmission, the complexity of the MLSE receivers
is reasonably low for common cellular channels that can be approximated as FIR responses of
order 3 to 8.