PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 349

R.4.83 The initial value (t = 0 ) and the fi nal value (t = ∞) of a given function f(t) can be

evaluated from its LT F(s), by the relations stated as follows, known as the initial

and fi nal value theorems, respectively.

a. The initial value theorem is given by

fs( ) lims [ Fs( )]0


→ ⋅

(test that all the poles of sF(s) are real and negative)

b. The fi nal value theorem states that

fs( )
lims→ 0 [ ⋅Fs( )]

(test if all the poles are on the left half of the s-plane)

R.4.84 The initial and fi nal value theorems permit to calculate f(0+) and f(∞), if one exists,

directly from the transform F(s), without the need of inverting the transform.

R.4.85 For example, the initial and fi nal value for the case of f(t) = u(t) are evaluated as

follows (recall that £[u(t)] = 1 /s), then

a. u(t = 0 +) = limitss→→[ ( )]sF s 

limit [ (s 11 s)]

b. u(t = ∞) = limitss→→ 00 [ ( )]sF s 

limit [ (s^11 s)]

The results just obtained confi rm what is already known, that is, u(t = 0 +) =

u(t = ∞) = 1.

R.4.86 The fi nal value theorem evaluates f(∞), if all the singularities are in the left half of

the s-plane. A simple pole (of F(s)) at the origin is permitted, but all the remaining

poles must be in the left half of the s-plane, and the degree of the denominator must

be greater or equal to the degree of the numerator of F(s). If any of the preceding

conditions are not met, f(t) becomes unbounded as t approaches infi nity, or physi-

cally f(t) would sustain nondecaying oscillations.

R.4.87 The transfer function, or system function, is the equation that defi nes the dynamic

properties of a linear system given by

Hs()

Laplace transform of the output

Laplace transform of the input

where H(s) is in general a rational function, given as the ratio of two polynomials

Y(s) (its output) and X(s) (its system input).

R.4.88 Recall from previous chapters that the values of s that make H(s) go to zero and

infi nity are called the system zeros and poles, respectively. Hence the transfer func-

tion can be completely defi ned in terms of its poles, zeros, and a constant multiplier

referred to as gain.

R.4.89 The LT of any function, including H(s) must be accompanied by its ROC; only

then the corresponding time function h(t) is unique. If H(s) is a one-sided trans-

form then the ROC is not needed, and defi nes a unique inverse.

For obvious reasons, the ROC must exclude the system’s poles.

R.4.90 A causal system can be defi ned in the frequency domain as the one with the ROC

located to the right of the pole having the largest real part.