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

12.4 Linear Fractionally Spaced Equalizers (FSE) 685
Any sampling rate of the form 1/ 1'. = m/T (m > 1) will be above the Nyquist sampling rate
and can avoid aliasing. For analysis, denote the sequence of channel output samples as
00
z(k1'.) = L snh(k1'. - nT) + w(k1'.)
n=O
00
= L snh(k1'. - nm 1',.) + w(k 1'.)
n=O
(12.46)
To simplify our notation, the oversampled channel output z(k 1'.) can be reorganized ( decimated)
into m parallel subsequences
z; [k] z(kT + i1'.)
= z(km1'. + i1'.)
00
= L snh(km1'. + i1'. - nm 1'.) + w(km1'. + i1'.).
n=O
00
= L s ,, h(kT - nT + i1'.) + w(kT + i1'.) i = l, ... , m
n=O
(12.47)
Each subsequence z;[k] is related to the original data via
z; [k] z(kT + i1'.) = sk * h(kT + i1'.) + w(kT + i1'.)
In effect, each subsequence is an output of a linear subchannel. By denoting each subchannel
response as
and the corresponding subchannel noise as
00
h; [k] h(kT + i1'.) {=::::} H; (z) = L h;[k]z - k
k=O
then the reorganized m subchannel outputs are
00
z;[k] = L s ,, h;[k - n] + w; [k]
n=O
00
= L h;[nlsn-k + w; [k] i = l, ... , m
n=O
(12.48)
Thus, these m subsequences can be viewed as stationary outputs of m discrete channels
with a common input sequence s[k] as shown in Fig. 12.7. Naturally, this represents a singleinput-multiple-output
(SIMO) system analogous to a physical receiver with m antennas. The
FSE is in fact a bank of m filters {F; (z)} that jointly attempts to minimize the channel distortion
shown in Fig. 12.7.