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 byℑ[()]ftFw( )
whereasℑ (^1) [( )]Fwf t()
denotes the inverse FT of F(w).
R.4.38 The FT of f(t) exists, if the following condition is satisfi ed:
fte()jwtdt k
∫
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 condi-
tions 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,
∫∞∞ft dt()
Note that, strictly speaking, a periodic function does not have a transform, but if
ftdt
T
T
()
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
limT ( )
T
T
T
→ ∫ ft dt
(^12)
2
2
R.4.40 On the contrary, if
ftdt
T
T
()
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.