A modified elliptic function with diminishing ripples

Abstract

A modified elliptic low-pass filter function is proposed. T h e modified elliptic function possesses progressively diminishing ripples in both passband and stopband, and improves the frequency and the time-domain characteristics a s compared with classical elliptic function. And it is realizable in the doublyterminated ladder structures for the order n even or odd, thus lending themselves amenable to high-quality active RC or switched capacitor filters through the simulation techniques. I . INTRODUCTIONElectric and electronic filters are the essential products of informational transmission equipments for signal and image processing in the communications, etc.. Butterworth, Chebyshev, inverse Chebyshev, elliptic and Bessel functions are used for filter design.The all-pole functions such as Butterworth, Chebyshev functions are required high-order function to satisfy the given set of specifications.For a given set of magnitude specifications, the elliptic functions require the lowest order n compared with the other kinds of filter functions. This optimality can be ascribed to the fact that the magnitude characteristic exhibits in both passband and stopband, equal widths of ripples having n number of peaks and valleys, and the steepest slope a t the cutoff frequency. The other characteristics such as group delay and step response, however, are rather undesirable than other filter functions ill.The magnitude characteristics of the elliptic function are the fact that for even n the function does not vanish a t s = m because both the numerator and the denominator polynomials are of order n. A consequence of the above fact is that the passive RLC ladder network realization is impossible unless the coupled coils are employed 121. The elliptic functions thus are classified three types of characteristic in the even order [31.In a case A, the magnitude of the elliptic function will have a finite value at w = a , so a passive RLC ladder network realization of the function is impossible. The case B may be realized without coupled coils as long as the source and load resistance termination does not have the same value. In a case C, the magnitude at o = O is the maximum passband value. But the attenuation of the stopband is much smaller than that of the case B.This paper proposes the modified elliptic function with diminishing ripple characteristics in both passband and stopband. Since the magnitude assumes the maximum value at w =O a.nd vanishes a t w = C O , it is possible to realize the function in a doubly-terminated ladder configuration with the terminating resistances R'1=Rz which corresponds to the maximum power transfer as well as desirable sensitivity performance [41. Moreover, the modified elliptic functions are improved the frequency and the time domain characteristics due to the lower pole-Q values.

A new type of transitional filter based on summation of polynomials

“…These comprise the well-known transitional Butterworth--Chebyshev (TBC) [18,19] and transitional ultraspherical-ultraspherical [20]. More recent approaches exploit maximally flat magnitude (Butterworth) and ripple responses to achieve lumped element realizations without transformers [21,22]. The blending of the polynomials results in different pole patterns distinct from the interpolation described in item 2 above.…”