Optimum "L" filter

Legendre filters, also known as Optimum- L- filters, continuous frequency filter whose transfer function is based on the eponymous Legendre polynomials. Legendre filters were introduced in 1958 by the Greek mathematician Athanasios Papoulis.

Legendre filters are a compromise between the Butterworth filter and Chebyshev filter is: The sum frequency curve is steeper than that of Butterworth filter and has unlike the Chebyshev filter in the barrier and in the passband of a monotonic profile.

Transfer function

The squared magnitude frequency response for the filter order is given by

With the modified -th optimal polynomial, which is characterized by the fulfillment of several specific criteria that ensure the desired properties of monotonicity of the transfer function and the same maximum slope in the stop band. These are the constraints

Derivation

For linearly independent polynomials of degree, in the simplest case, can be indirect fulfillment of an approach for the searched optimal polynomial form:

With unknown coefficients. Since the integrand is an even polynomial is odd. In order to obtain a straight, to the following offers:

Both approaches automatically satisfy the condition, and, as in always is positive. For the selected Basispolynome can be canceled by example and convert it into

This is a quadratic equation in the coefficients, which can be by a coefficient easiest after dissolved. Used in remain unknown coefficients that can be solved in the nonlinear equations from the partial derivatives of. With the straight approach is to proceed analogously.

For general polynomials, the resulting system of equations to be solved for only difficult analytically. The approach, however, suggests to use the Legendre polynomials of the first kind as a basis, in the expectation that many partial integrals vanish and simplifies the derivation. This provided Papoulis 1958 in his first work before. To this end, however, the integral limits must be adapted to the properties of the Legendre polynomials and scaled so that the following equation results:

This simplifies the, respectively, significantly

For one thus obtains

Determining the maximum in the partial derivative of required after the unknown coefficients, comprising:

Note: For the inner derivative yields only the summand with index a contribution, because all the other summands of are independent. is identical to the expression in root, but as a constant parameter is carried for ease of presentation in addition to which the solution is to relate the unknown. Is then determined so that or are met.

In the formation of the left-hand side of the following knowledge is important. For all and there is the identity:

This will be

A necessary condition for a maximum is that all partial derivatives of the left-hand side of after the unknown coefficients are zero. It should be noted, that also depends on all pursuant and

Note: Only the two summands and depend on.

The total is only zero if and all are, but this is ruled out, as would be and also injured. So the term in brackets must be zero and contain the solution

Used in results

Or

With results for all coefficients

For straight Papoulis published an analog solution. After scaling to the more appropriate Intervalgrenzen then applies

Analogous to the helpful from identity applies to just

The coefficients are:

Conclusion

As a basis for the optimal polynomial using the eponymous Legendre polynomials is not absolutely necessary. Any other linearly independent polynomial basis leads to the same result, the analytical derivation is, however, much more difficult, if not impossible. Order of and to simplify the already tedious and error-prone resolution somewhat, the denominator of the respectively can provide as factors outside the integral. Which leads to

Respectively

With

Result

For the filter order 1-6 are the optimal polynomials of the filter:

Other polynomials up to 10th order are to be found in these sources.