563489578934

Sec. 3–4 Digital Signaling 157
Eq. (3–27), would have a time span of T 0 seconds. The data rate could be computed using the
following definitions:
DEFINITION. The baud (symbol rate)is †
D = N>T 0 symbols/s
(3–28)
where N is the number of dimensions used in T 0 s.
DEFINITION. The bit rate is
R = n>T 0 bits/s
(3–29)
where n is the number of data bits sent in T 0 s.
For the case when the w k ’s have binary values, n = N, and w(t) is said to be a binary signal. When
the w k ’s are assigned more than two possible values (i.e., when they are not binary), w(t) is said
to be a multilevel signal. These two types of signaling are discussed in separate sections later.
A critical question is, If the waveform of Eq. (3–27) is transmitted over a channel and
appears at the input to the receiver, how can a receiver be built to detect the data? Because
w(t) is an orthogonal series, the formal way to detect the data is for the receiver to evaluate the
orthogonal series coefficient. That is, using Eq. (2–84), we get
T 0
w k = 1 w(t)w k
*(t)dt, k = 1, 2, Á , N
(3–30)
K k L0
where w(t) is the waveform at the receiver input and w k (t) is the known orthogonal function
that was used to generate the waveform. It can also be shown that Eq. (3–30) is the optimum
way to detect data when the received signal is corrupted by white additive noise; this procedure
is called matched filter detection using correlation processing and is described in Sec. 6–8.
Detection of data is illustrated in Examples 3–7 and 3–8.
Vector Representation
The orthogonal function space representation of Eq. (3–27) corresponds to the orthogonal
vector space represented by
N
w = a w j w j (3–31a)
j =1
where the boldface type denotes a vector representing the waveform of Eq. (3–27), w is an
N-dimensional vector in Euclidean vector space, and { w j } is an orthogonal set of N-directional
vectors that become a unit vector set if the K j ’s of Eq. (2–78) are all unity. A shorthand notation
for the vector w of Eq. (3–52a) is given by a row vector:
w = (w 1 , w 2 , w 3 , Á , w N )
(3–31b)
† In the technical literature, the term baud rate instead of baud is sometimes used, even though baud rate is a
misnomer, since the term baud, by definition, is the symbol rate (symbolss).