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## 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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