Detection, coding, and decoding
The previous chapter showed how to characterize noise as a random process and this chapter
uses that characterization to retrieve the signal from the noise corrupted received waveform.
As one might guess, this is not possible without occasional errors when the noise is unusually
large. The objective then, is to retrieve the data while minimizing the eﬀect of these errors.
This process of retrieving data from a noise corrupted version is known as
Detection, decision making, hypothesis testing, and decoding are synonyms. The word
refers to the eﬀort to detect whether some phenomenon is present or not on the basis of obser-
vations. For example, a radar system uses the observations to detect whether or not a target is
present; a quality control system attempts to
whether a unit is defective; a medical test
whether a given disease is present. The meaning of detection has been extended in the
digital communication ﬁeld from a yes/no decision to a decision at the receiver from a ﬁnite set
of possible transmitted signals. Such a decision from a set of possible transmitted signals is also
, but here the possible set is usually regarded as the codewords in a code rather
than the signals in a signal set.
is, again, the process of deciding between a
number of mutually exclusive alternatives.
is the same, and here the mutually
exclusive alternatives are called hypotheses. We use the word hypotheses for the possible choices
in what follows, since the word conjures up the appropriate intuitive image of making a choice
between a set of alternatives, where only one alternative is correct and there is a possibility of
These problems will be studied initially in a purely probabilistic setting. That is, there is a
probability model within which each hypothesis is an event. These events are mutually exclusive
and collectively exhaustive, i.e., the sample outcome of the experiment lies in one and only one
of these events, which means that in each performance of the experiment, one and only one
hypothesis is correct. Assume there are
. The sample
outcome of the experiment will lie in one of these
events. This deﬁnes a random symbol
As explained more fully later, there is no fundamental diﬀerence between a code and a signal set.
The principles here apply essentially without change for a countably inﬁnite set of hypotheses; for an uncount-
ably inﬁnite set of hypotheses, the process of choosing an hypothesis from an observation is called
Typically, the probability of choosing correctly in this case is 0 and the emphasis is on making an estimate that
is close in some sense to the correct hypothesis.