I. Introduction
Inter-Symbol interference (ISI) induced by the channel often significantly impairs simple receiver performance. To alleviate this effect, block transmissions are widely adopted [1], [19]. In such a transmission scheme, the transmitted data stream is divided into consecutive equal size blocks and redundancy is added between blocks. Given proper selection of redundancy length, the inter-block interference (IBI) can be completely removed. Examples of this block transmission include discrete multitone (DMT) [16], [18] and orthogonal frequency division multiplexing (OFDM) modulations [4], [15], [17], which have been adopted in standards [14] and [5]. Previous studies have demonstrated that many existing modulations can be formulated within a unifying multirate filterbank transceiver model [2], [7], [9], [32]. Building on this framework, the FIR filterbanks used in the transmitter/receiver are usually known as FIR filterbanks precoder/equalizer (or are simply termed precoder/equalizer). Particularly, the precoder/equalizer with the property of perfect reconstruction (PR) is a zero-forcing (ZF) precoder/equalizer [2], [23], [26]. A situation in which the length of the FIR fitlerbanks does not exceed block-plus-redundancy/block size is referred to as a zero-order (or a block-based) precoder/equalizer. To facilitate discussion, the nonblock-based transceiver generally represents that both the precoder and the equalizer are not restricted to being zero-order. Relaxing the PR constraint reveals the potential of filtered multitone (FMT) modulation in many applications [24]. However, this study restricts itself to the problem of PR with a (FIR filterbanks) ZF equalizer, especially for a block-based precoder over physical single-input–single-output (SISO) channels.