Legendre Filters

The Legendre filter is an all pole filter that produces the steepest cutoff frequency without producing any passband ripples of oscillating slopes. The pass band curve slope varies between downward and zero only, never upwards. Legendre filters are sometimes referred to as Monotonic Class L filters.

Legendre filters are useful in applications which require a steep cutoff at the pass band edge without pass band ripples, or in cases where a Chebyshev I filter produces too much group delay at the pass band edge.

 

Derivation

The filter is characteristic function is derived from the Legendre polynomials in which the filter is named after.

The transfer function is:

Where L(S) is a squared characteristic polynomial derived from the Legendre polynomials.

The -3dB bandwidth is set with the variable e. There are no provisions for stop band zeros or an asymmetrical band pass with Legendre filters.

 

Inverse Legendre Filters

The Legendre function may apply to the stop band too, producing the most rapid monotonic attenuation in the stop band near the cutoff frequency. However, this requires complex stop band zeros, making the function unrealizable with ideal inductors and capacitors. FilterSolutions supports Inverse Legendre filters for active, switched capacitor and digital filters.