346 Practical MATLAB® Applications for Engineers
R.4.75 The LT can be viewed as a generalized case of the FT.
Recall that the FT was defi ned by
Fw() f te()jwtdt
∫Observe that in the LT, f(t) is expressed in terms of e−st, whereas the FT represents
f(t) in terms of e−jwt. The FT is then a special case of the LT in which s = jw. In gen-
eral, the Laplace variable s is complex, defi ned as s = σ + jw, where σ and w are
real.
R.4.76 F(s) is referred as the direct LT of f(t), denoted by
F(s) = £ [f(t)]
R.4.77 The process of obtaining the time function f(t) from the LT F(s) is referred to as the
ILT. Assuming its existence, ILT is denoted by
f(t) = £−^1 [F(s)]
Mathematically, the variable s represents a complex frequency; however, it is not
necessary to pursue this interpretation to make use of the transformation.
R.4.78 The suffi cient conditions for the existence of the LT are that f(t) must be sectionally
continuous in every fi nite interval 0 ≤ t ≤ M, and of exponential order a for t > M,
then the LT F(s) as well as the ILT exist and are unique over the range s > a. Unique-
ness will always be assumed unless otherwise stated (Lerch’s theorem).
R.4.79 The region of convergence (ROC) of the LT is the region where the transform exists,
and is unique. Recall that causal signals are defi ned by ƒ(t) = 0, for t < 0 and causal
systems are defi ned by h(t) = 0, for t < 0. In either case the ROC is in the right half
of the complex plane, and the transform used is called, for obvious reasons, the
unilateral LT, given as follows:
£[ ( )]ftFs( ) fte ds( )st0∫
R.4.80 As mentioned earlier, the notation used to defi ne f(t) from its ILT is given by
f(t) = £−^1 [F(s)]
where
ft
jFse dsst
jwjw
() ()1
2
∫R.4.81 Table 4.2 summarizes some of the standard time–frequency (assuming causality)
LT pa i r s.