2021-Brubaker, Phil - Optimum Matched Filter Design (Transfer Function)
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Abstract
Optimum Matched Filter (Transfer Function)
(Nested Processes ... Each Process controlled by a Solver)
Problem Description
The transfer function H(s) is the Laplace transform of the output signal Yout(s)* divided by the Laplace transform of the input signal Yin(s)*: that is H(s)=
where each signal's transform is assumed to be a ratio of polynomials. Thus, H(s) can likewise be stated in the form:
Equation 2.1 Generalized H(s)
Assuming the numerator and denominator can be factored, yields H(s) in the general form
Equation 2.2 Factored Transfer Function
where each Zi is known as a "zero" and the Pi as a "pole" of the transfer function. Zi and Pi are complex points in the Laplace domain.
A realizable transfer function must have poles and zeros with their conjugate points. That is, poles and zeros come in pairs. If a pole or zero is located at the complex point si + jwi, then its conjugate is located at si - jwi. Thus, a generalized transfer function is stated as
Equation 2.3 Generalized Transfer Function H(s)
Given n-data points from a Bode plot (see Figure 2.1 below) that define the mainlobe of the desired transfer function, find the optimal Pole/Zero constellation such that H(s) has equal sidelobe peak amplitudes in a Bode plot and curve fits the given data in the mainlobe.
Bode Plot: Mainlobe with 3 Sidelobes
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