48 IEEE COMMUNICATIONS LETTERS, VOL. 2, NO. 2, FEBRUARY 1998 Excess Bandwidth is Necessary for ISI-Free Double-Filter Receivers Umberto Mengali, Fellow, IEEE, Giorgio M. Vitetta, Member, IEEE, and Desmond P. Taylor, Fellow, IEEE Abstract— Previous work has indicated that highly effective double-filter receivers can be implemented for phase-shift keying (PSK) transmissions over linearly time-selective fading channels provided that the transmitter and receiver filters have root- raised-cosine tranfer functions with unity rolloff factor. Revisiting the detection problem within the frame of maximum-likelihood (ML) sequence estimation we prove that this filter design is unique within the class of signaling pulses which are band limited to the symbol rate. It is concluded that some waste of bandwidth is inevitable for achieving ISI-free operation with double-filter receivers. I. INTRODUCTION I T HAS BEEN shown in [1] and [2] that the intrinsic diver- sity provided by linearly time-selective fading channels can be exploited by means of double-filter processing in detectors for phase-shift keying (PSK) signals. This requires suitable pulse shaping at the transmitter and the receiver. In [1], a practical solution to this problem has been indicated. The aim of this letter is: • to establish a link between the theory of maximum- likelihood (ML) detection of PSK signals transmitted on linearly time-selective channels and the choice of impulse response for the transmitter and the receiver filters; • to prove, within this frame, that the ML detection problem leads to a unique choice for the filters which coincides with that proposed in [1]. The letter is organized as follows. Section II describes the mathematical model of the fading process and provides basic notations for PSK signals. In Section III the problem of ML detection and optimal signaling is discussed. Finally, Section IV offers some conclusions. II. CHANNEL AND SIGNAL MODELS With -ary PSK signaling over frequency-flat fading chan- nels the baseband received signal has the form (2.1) where is the fading distortion, is the transmit- ter impulse response, 1 are PSK symbols taking values Manuscript received June 12, 1997. The associate editor coordinating the review of this paper and approving it for publication was Prof. H. V. Poor. U. Mengali and G. M. Vitetta are with the Department of Information Engineering, University of Pisa, Pisa, Italy. D. P. Taylor is with the Department of Electrical and Electronic Engineer- ing, University of Canterbury, Christchurch 1, New Zealand. Publisher Item Identifier S 1089-7798(98)01918-8. 1 A real pulse is assumed throughout the paper. , and is additive white Gaussian noise (AWGN) with two-sided spectral density. In [1] and [2], it has been shown that for a linearly time-selective model of the channel, (2.1) becomes (2.2) where (2.3) and (2.4) (2.5) The random variables and are complex Gaussian and independent [1], [2]. From (2.2), it is seen that a second-order diversity can be obtained by proper processing of the received signal. III. ISI-FREE PULSE SHAPING FOR DOUBLE-FILTER RECEIVERS If the random sequences and are known, the ML estimator of the transmitted symbol sequence over the observation interval looks for the minimum of the metric [1]: (3.1) with respect to the candidate sequence Minimizing in the form (3.1) is not easy, however. The problem is greatly simplified if we can choose follows from (2.3)) so as to satisfy the following conditions: (3.2) (3.3) and (3.4) 1089–7798/98$10.00  1998 IEEE