Page 387 - Mechanical Engineers' Handbook (Volume 2)

378 Mathematical Models of Dynamic Physical Systems
F[y(t)] Y( ) j t

y(t)e dt
Note that if y(t) 0 for t 0, and if the region of convergence for the Laplace transform
includes the imaginary axis, then the Fourier transform can be obtained from the Laplace
transform as
Y( ) [Y(s)] s j
For cases where it is impossible to determine the Fourier transform analytically, such as
when the signal is described graphically or by a table, numerical solution based on the fast
Fourier transform (FFT) algorithm is usually satisfactory.
In general, the condition T / cannot be satisfied exactly, since most physical
u
signals have no finite upper frequency . A useful approximation is to define the upper
u
frequency as the frequency for which 99% of the signal ‘‘energy’’ lies in the frequency
spectrum 0 . This approximation is found from the relation
u
u 2 2
0
0 Y( ) d 0.99 Y( ) d
where the square of the amplitude of the Fourier transform Y( ) is said to be the power
2
spectrum and its integral over the entire frequency spectrum is referred to as the ‘‘energy’’
of the signal. Using a sampling frequency 2–10 times this approximate upper frequency
(depending on the required factor of safety) and inserting a low-pass filter (called a guard
filter) before the sampler to eliminate frequencies above the Nyquist frequency /T usually
lead to satisfactory results. 4

The z-Transform
The z-transform permits the development and application of transfer functions for discrete-
time systems, in a manner analogous to continuous-time transfer functions based on the
Laplace transform. A discrete signal may be represented as a series of impulses

y*(t) y(0) (t) y(1) (t T) y(2) (t 2T)
y(k) (t kT)
N
k 0
where y(k) y*(t kT) are the values of the discrete signal, (t) is the unit impulse
function, and N is the number of samples of the discrete signal. The Laplace transform of
the series is
Y*(s) y(k)e ksT
N
k 0
where the shifting property of the Laplace transform has been applied to the pulses. Defining
st
the shift or advance operator as z e , Y*(s) may now be written as a function of z:
Y*(z) y(k) ZZ [y(t)]
N
k 0 z k
where the transformed variable Y*(z) is called the z-transform of the function y*(t). The
inverse of the shift operator 1/z is called the delay operator and corresponds to a time delay
of T.