Digital Signal Processing Reference
In-Depth Information
this approach is that digital information can be taken to a higher level of
abstraction. Since digital data can be handled as numerical symbols, it be-
comes very convenient to perform calculations on it. For example, it is pos-
sible to add some amount of redundant information to a string of digital data,
which can be used later on in the transmission chain to recover the original
information in case of transmission errors in the channel. Also, digital infor-
mation is a lot more flexible to deal with than an analog-valued signal, in this
respect that digital bits can be easily buffered and restacked in order to form
new
symbols
. These data symbols, in their turn, are converted into a new analog
domain representation which is adopted to the oddities and problems that are
faced during the transmission of analog signals over the wireless (or wireline)
channel.
In Chapter 2, it was observed that the underlying mechanisms of error coding
are based on the representation of information using
sequences
of symbols.
Rather than using individual symbols to encode information bits, multiple bits
are encoded at once in a sequence of symbols. Redundant information is em-
bedded in the transmitted symbol stream by only allowing a certain
subset
of
allowed sequences out of the total set of all possible symbol sequences. Using
a clever subset of allowed sequences, the
distance
between any two possible
sequences in the allowed subset can be effectively increased. Of course, the re-
ceiver is fully aware of this fact, and will check the received sequence against
the current subset of allowed sequences. If a transmission error occurs and
some symbols get corrupted during transmission over the channel, the receiver
can recover the correct sequence by searching for the allowed sequence with
the shortest distance from the received symbol sequence.
It was also found that this
distance
between sequences should be further spec-
ified: when data is transferred over the channel in an analog shape, certain
errors will occur more frequently than others. For example, it is more likely
for a receiver to confuse two neighbouring constellation points of a symbol
than two more distant points (either in amplitude or in phase) in the constel-
lation plane. An error correction mechanism which does not take into account
the
Euclidean distance
between symbols, will waste much effort and signal en-
ergy in adding redundant information to distant constellation points, as a result
of which the overall throughput of the system shifts further away from the the-
oretical throughput capabilities of the channel. This observation has lead to the
concept of
modulation-aware coding
, in which the error coding mechanism is
aware of the way in which the analog signal is injected in the channel and also
knows exactly how the channel affects the transmitted signal in terms of noise
and interference.
