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A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable s. Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function ...
where H is the transfer function, s is the Laplace transform variable (complex angular frequency), τ is the filter time constant, is the cutoff frequency, and K is the gain of the filter in the passband. The cutoff frequency is related to the time constant by: =
The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t). The entries of the table that involve a time delay τ are required to be causal (meaning that τ > 0).
This relationship is used in the Laplace transfer function of any analog filter or the digital infinite impulse response (IIR) filter T(z) of the analog filter. The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, ()
The impulse response for each voltage is the inverse Laplace transform of the corresponding transfer function. It represents the response of the circuit to an input voltage consisting of an impulse or Dirac delta function. The impulse response for the capacitor voltage is
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms are closely related to the Fourier transform , the Mellin transform , the Z-transform and the ordinary or one-sided Laplace transform .
The transfer function of a two-port electronic circuit, such as an amplifier, might be a two-dimensional graph of the scalar voltage at the output as a function of the scalar voltage applied to the input; the transfer function of an electromechanical actuator might be the mechanical displacement of the movable arm as a function of electric ...
The ideal integrator's transfer function corresponds to the time-domain integration property of the Laplace transform. Since its denominator is just s {\displaystyle s} , the transfer function has a pole frequency at f = 0 {\displaystyle f{=}0} .