Fourier and Laplace
331
Observe the notation used. The same letter is used for f(t) and its transform F(w),
where the lowercase denotes a time function and the uppercase is used to defi ne
their transforms (Fourier and later in this chapter Laplace). Thus, the transform
of v(t) would be V(w) for a voltage, and the transform of i(t) would be I(w) for a
current.
Note also that the FT can be considered a limiting case of an FS, as the period
T is extended to infi nity.
R.4.37 The notation used to indicate the (direct) FT is given by
ℑ[ ( )]
( )
f t
F w
ϭ
whereas
ℑ
Ϫ1 [ ( )]
( )
F w
f t
ϭ
denotes the inverse FT of F(w).
R.4.38 The FT of f(t) exists, if the following condition is satisfi ed:
f t e
dt k
jwt
( )
Ϫ
Ϫ
ϩ
ϱ
ϱ
Ͻ Ͻ ϱ
∫
where k is a fi nite value constant.
R.4.39 The existence of the FT of f(t) denoted by F(w) is guaranteed if the Dirichlet’s conditions are satisfi ed. The Dirichlet’s conditions state (similar to the FS case)
a. f(t) may have a fi nite number of maxima and minima and a countable number of
fi nite discontinuities within a given time interval
b. f(t) must be absolutely integrable, that is,
f t dt
( ) Ͻ ϱ
Ϫ∞
∞
∫
ϩ
Note that, strictly speaking, a periodic function does not have a transform, but if
f t dt
T
T
( )
Ϫ ր
ϩ ր
Ͻ ϱ
2
2
∫
then in the limit, as T approaches infi nity, the FT exists. The preceding signals are
referred to as power signals and, therefore, satisfy the relation given by
lim
( )
T
T
T
T
f t dt
→
∫
ϱ
Ϫ ր
ϩ ր
Ͻ ϱ
1
2
2
2
R.4.40 On the contrary, if
f t dt
T
T
( )
Ϫ ր
ϩ ր
Ͻ ϱ
2
2
∫
then f(t) is referred to as a fi nite energy signal. The FT exists for fi nite energy signals
and its evaluation is an exercise in calculus.
CRC_47760_CH004.indd 331
CRC_47760_CH004.indd 331
7/28/2008 12:25:52 PM
7/28/2008 12:25:52 PM
331
Observe the notation used. The same letter is used for f(t) and its transform F(w),
where the lowercase denotes a time function and the uppercase is used to defi ne
their transforms (Fourier and later in this chapter Laplace). Thus, the transform
of v(t) would be V(w) for a voltage, and the transform of i(t) would be I(w) for a
current.
Note also that the FT can be considered a limiting case of an FS, as the period
T is extended to infi nity.
R.4.37 The notation used to indicate the (direct) FT is given by
ℑ[ ( )]
( )
f t
F w
ϭ
whereas
ℑ
Ϫ1 [ ( )]
( )
F w
f t
ϭ
denotes the inverse FT of F(w).
R.4.38 The FT of f(t) exists, if the following condition is satisfi ed:
f t e
dt k
jwt
( )
Ϫ
Ϫ
ϩ
ϱ
ϱ
Ͻ Ͻ ϱ
∫
where k is a fi nite value constant.
R.4.39 The existence of the FT of f(t) denoted by F(w) is guaranteed if the Dirichlet’s conditions are satisfi ed. The Dirichlet’s conditions state (similar to the FS case)
a. f(t) may have a fi nite number of maxima and minima and a countable number of
fi nite discontinuities within a given time interval
b. f(t) must be absolutely integrable, that is,
f t dt
( ) Ͻ ϱ
Ϫ∞
∞
∫
ϩ
Note that, strictly speaking, a periodic function does not have a transform, but if
f t dt
T
T
( )
Ϫ ր
ϩ ր
Ͻ ϱ
2
2
∫
then in the limit, as T approaches infi nity, the FT exists. The preceding signals are
referred to as power signals and, therefore, satisfy the relation given by
lim
( )
T
T
T
T
f t dt
→
∫
ϱ
Ϫ ր
ϩ ր
Ͻ ϱ
1
2
2
2
R.4.40 On the contrary, if
f t dt
T
T
( )
Ϫ ր
ϩ ր
Ͻ ϱ
2
2
∫
then f(t) is referred to as a fi nite energy signal. The FT exists for fi nite energy signals
and its evaluation is an exercise in calculus.
CRC_47760_CH004.indd 331
CRC_47760_CH004.indd 331
7/28/2008 12:25:52 PM
7/28/2008 12:25:52 PM