Discrete-time representations - Signal Processing and Optimization for Transceiver Systems - page 130

We know that the transmitted symbol s ( n ) belongs to a constellation such as,
for example, the PAM or QAM constellation (Sec. 2.2). The detector shown in
the figure obtains an estimate s est ( n ) of this constellation symbol based on the
noisy received symbol

s ( n ).

4.7 Oversampling the received signal

Consider Fig. 4.18(a), where the received and filtered signals are sampled with
sample spacing T/L ,where L is an integer. The sampling rate at the receiver
is L times higher than the symbol rate 1 /T. In this case the equivalent digital
model is different from Fig. 4.5. We will see in Sec. 4.8 that the digital model
is an interpolation filter model (see Fig. 4.21 for a preview). One advantage of
oversampling at the receiver is that the digital equalizer, indicated as G d ( e jω )
in Fig. 4.18(a), becomes simpler. For example, FIR channels can be equalized
with FIR equalizers, as we shall see. The equalizers are therefore guaranteed to
be stable. However, additional bandwidth is required in order to take advantage
of oversampling, as we shall explain (Sec. 4.8.5).
One remark is in order here. In principle, a receiver with oversampling can
always be replaced with a receiver with no oversampling. To show this we use the
so-called “noble identity” shown in Fig. 4.19(a) (proved in Sec. G.2 of Appendix
G). Using this we see that the digital equalizer G d ( e jω ) can be moved to the left
of the C/D block, as shown in Fig. 4.18(b). Defining

G ( jω )= G ( jω ) G d ( e jωT/L ) ,

and using the simple identity shown in Fig. 4.19(b), we can redraw this as in
Fig. 4.18(c). Thus the digital equalizer G d ( e jω ) has been absorbed into the
equalizer G ( jω ) . This shows that a receiver with oversampling can be replaced
with an equivalent receiver with no oversampling.
But there are some practical advantages to oversampling. For example, we
will see that FIR equalizers can be used to equalize FIR channels (Sec. 4.8).
There are also important advantages when we take into account errors in tim-
ing phase at the receiver; see Gitlin and Weinstein [1981], Proakis [1995], and
references therein.