Page 137 - Fundamentals of Communications Systems

Complex Baseband Representation of Bandpass Signals 4.13

4.4.2 Bandwidth of Bandpass Signals
The ideas of bandwidth of a signal extend in an obvious way to bandpass signals.
Recall engineers define bandwidth as being the amount of positive spectrum
that a signal occupies. For bandpass energy signals we have the following two
definitions that are analogous to the bandwidth definitions in Chapter 2.
(f ) then B X is determined as
Definition 4.2 If a signal x c (t) has an energy spectrum G x c

(f 1 )) (4.16)
10 log max G x c (f ) = X + 10 log(G x c
f
(f ) for 0 < f < f 1 and
where G x c (f 1 ) > G x c

(f 2 )) (4.17)
10 log max G x c (f ) = X + 10 log(G x c
f
(f ) for f > f 2 where f 2 − f 1 = B X .
where G x c (f 2 ) > G x c
(f ) then B P = min(f 2 − f 1 )
Definition 4.3 If a signal x c (t) has an energy spectrum G x c
such that
f 2
2 G x c (f ) df
f 1
P = (4.18)
E x c
where f 2 > f 1 .
Note the reason for the factor of 2 in Eq. (4.21) is that half of the energy of the
bandpass signal is associated with positive frequencies and half of the energy
is associated with negative frequencies.
(f ) being re-
Again, for bandpass power signals similar ideas hold with G x c
(f , T ).
placed with S x c
(f , T m ) (see
Definition 4.4 If a signal x c (t) has a sampled power spectral density S x c
Eq. (2.56)) then B X is determined as

(f 1 , T m )) (4.19)
10 log max S x c (f , T m ) = X + 10 log(S x c
f
(f , T m ) for 0 < f < f 1 and
where S x c (f 1 , T m ) > S x c

(f 2 , T m )) (4.20)
10 log max S x c (f , T m ) = X + 10 log(S x c
f
(f , T m ) for f > f 2 where f 2 − f 1 = B X .
where S x c (f 2 , T m ) > S x c
(f , T m ) then B P = min(f 2 −f 1 )
Definition 4.5 If a signal x c (t) has an power spectrum S x c
such that
f 2
2 S x c (f , T m ) df
f 1
P = (4.21)
(T m )
P x c
(T m ) is defined in Eq. (2.57).
where f 2 > f 1 and P x c