A class of magnitude complementary loudspeaker crossovers

Necessary and sufficient conditions are developed for a pair of transfer functions to have magnitude responses which sum to a constant. If, in addition, the transfer function pair is constrained to be all-pass complementary, the transfer function pair so obtained is well suited for loudspeaker crossover applications. Such crossover systems are characterized by a pair of transfer functions which exhibit the same phase angle at all frequencies. This property has twofold implications: i) in active crossover biamplified audio systems, whereby the low frequencies and high frequencies are reproduced using separate power amplifiers, in-phase crossover transfer functions require less power in subsequent amplifier stages than other designs to achieve a given acoustic sound pressure level; and ii) the summed acoustic magnitude response is least sensitive to noncoincidental mounting of the low- and high-frequency transducers when the crossover transfer functions exhibit the same phase angle at all frequencies. The class of transfer functions realizable is quite wide, and includes squared versions of Butterworth, Chebyshev, and elliptic transfer functions of all orders.