Digital Signal Processing Reference
In-Depth Information
and this is how the power constraint ultimately affects themaximumachiev-
able bitrate. Note that the above derivation has been carried out with very
specific hypotheses on both the signaling alphabet and on the decoding al-
gorithm (the hard slicing); the upper bound on the achievable rate on the
channel is actually a classic result of information theory and is known un-
der the name of Shannon's capacity formula. Shannon's formula reads
C
=
B
log
2
1
+
l
g
r
,
y
i
d
.
,
©
,
L
s
S
N
where
C
is the absolute maximum capacity in bits per second,
B
is the avail-
able bandwidth in Hertz and
S
/
N
is the signal to noise ratio.
Design Example Revisited.
Let us resume the example on page 341 by
assuming that the power constraint on the telephone line limits the max-
imum achievable SNR to 22 dB. If the acceptable bit error probability is
p
e
=
10
−
6
, Equation (12.20) gives us a maximum integer value of
M
4bits
per symbol. We can therefore use a regular 16-point square constellation;
recall we had designed a system with a baud rate of 2400 symbols per sec-
ond and therefore the final reliable bitrate is
R
=
9600 bits per second. This
is actually one of the operating modes of the V.32 ITU-T modem standard.
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=
12.3 Modem Design: the Receiver
The analog signal
s
created at the transmitter is sent over the telephone
channel and arrives at the receiver as a distorted and noise-corrupted signal
s
(
t
)
. Again, since we are designing a purely digital communication system,
the receiver's input interface is an A
(
t
)
D converter which, for simplicity, we
assume, is operating at the same frequency
F
s
as the transmitter's D
/
Acon-
verter. The receiver tries to undo the impairments introduced by the chan-
nel and to demodulate the received signal; its output is a binary sequence
which, in the absence of decoding errors, is identical to the sequence in-
jected into the transmitter; an abstract view of the receiver is shown in Fig-
ure 12.16.
/
ˆ
s
[
n
]
..01100
01010...
ˆ
(
)
s
t
RX
F
s
Figure 12.16
Abstract view of a digital receiver.
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ITU-T is the Standardization Bureau of the International Telecommunication Union.
