Page 872 - The Mechatronics Handbook

0066-frame-C29 Page 2 Wednesday, January 9, 2002 7:23 PM

Continuous-Time Signals
Laplace transforms are used for system analysis of continuous-time systems, solving for system response,
and control design. The single-sided Laplace transform of a continuous-time signal, x(t), is given by

Xs() = ∫ ∞ xt()e – st t d
0
A transfer function of a linear system, H(s), can be found as the ratio of the Laplace transforms of the
output over that of the input (with zero initial conditions).
The Fourier transform is used to determine the frequency content of a signal. The Fourier transform
of x(t) is given by

∞
X w() = ∫ −∞ xt()e – jwt t d (29.1)

where ω is in units of radians per second. Notice that when x(t) = 0 for t ≤ 0, the Laplace transform is
equivalent to the Fourier transform by setting s = jω. (It should be noted that there are some additional
convergence considerations for the Fourier transform.) The frequency response of a system is defined as
the ratio of the Fourier transforms of the output over that of the input. Equivalently, it can be found
from the transfer function as H(ω) ≡ H(jω) = H(s)| s=jω . For simplicity of notation, the j is usually not
shown in the argument list, giving rise to the notation H(ω) to represent the frequency response. The
bandwidth of a system is defined as the frequency at which H(ω) = 0.707H(0).

Discrete-Time Signals
The z-transform is useful for solving a difference equation and for performing system analysis. The
z-transform of a discrete-time signal, x[n], is defined as
∞
Xz() = ∑ xn[]z −n
n=−∞
The discrete-time Fourier transform (DTFT) is used to determine the frequency content of a signal. The
DTFT and the inverse DTFT of a signal are defined by
∞
X Ω() = ∑ xn[]e −jΩn (29.2)
n=−∞
and

1
xn[] = ------ ∫ p X Ω()e jΩn d Ω (29.3)
2p −p
jΩ
Note that the DTFT can be derived from the z-transform by setting z = e . (Again, there are some
assumptions on convergence in this derivation.) Since the DTFT is periodic with period 2π, it is typically
displayed over the range [−π, π] or [0, 2π], where the frequencies of general interest are from Ω = 0 (low
frequency) to Ω = π (high frequency). The frequency response of a discrete-time system can be found
as the ratio of the DTFT of the output signal over that of the input signal. Alternatively, it can be found
jΩ
jΩ
from the transfer function as H(Ω) ≡ H(e ) = H(z)| jΩ . The notation H(Ω) is preferred over H(e )
z=e
for its simplicity. As in the continuous-time case, the bandwidth is defined as the frequency at which
H(Ω) = 0.707H(0).
©2002 CRC Press LLC