A New Approach to Optimum Pulse Shaping in Sampled Systems Using Time-Domain Filtering

The joint optimization of functions in both time and frequency domain is a classical problem in communication theory. Hilberg and Rothe 1 have recently found the lowest possible product of pulse and one-sided spectral widths and have numerically evaluated the impulse and frequency response-which is not G a u s s i a n - t h a t corresponds to this minimum. Landau, Pollak, and Slepian 2 - 4 in their classical papers have derived the pulse-form of given duration t h a t has a maximum of its energy concentrated below a certain frequency and vice versa; the solutions to this problem are given by the now well-known prolate spheroidal wave functions. Additional comments on this problem have recently been given b}' Hilberg. 5 A widespread opinion is t h a t pulses with minimum energy at high frequencies should have a rounded form with m a n y continuous derivatives. This is not true; in fact, the optimum 723 724 T H E B E L L SYSTEM T E C H N I C A L J O U R N A L , M A Y - J U N E 1 9 7 3 pulses based on the prolate spheroidal wave functions are usually not continuous at the limits of their truncation interval. Hilberg B has shown t h a t constraints of continuous derivatives tend to increase substantially the total out-of-band energy. Steep spectral roll-off above the Nyquist frequency and small residual out-of-band energy are desirable properties for signals in d a t a transmission systems to achieve maximum signaling rate over bandlimited channels and to avoid fold-over distortion in modulation and demodulation.