Spectrum Efficient, Localized, Orthogonal Waveforms: Closing the Gap With the Balian-Low Theorem

Abstract

The Balian-Low theorem (BLT) states the fundamental impossibility to design waveforms for L 2 (R), which 1) form an orthogonal set, 2) are time-frequency localized, and 3) attain a critical waveform density such that they form an orthogonal basis. This article closes the gap between existing waveform designs and the BLT. The main contribution is the design of orthogonal, time-frequency localized, spectrum efficient waveforms for hexagonal lattices. The waveform design is adaptive by a single design parameter, which tradesoff time-frequency localization with the waveform density. As the orthogonalization procedure is based on employing the minimum number of most time-frequency localized waveforms (Hermite functions) it is argued that the results may be optimal in terms of combined spectrum efficiency and time-frequency localization. An example is provided for waveforms for a hexagonal lattice, which are quasi-orthogonal, time-frequency localized, and up to 99% of the critical waveform density. Although the designed waveforms are not strictly orthogonal, their cross-correlation can be made arbitrarily small. The robustness in doubly dispersive channels and the efficiency for multiuser scenarios are discussed and compared to conventional orthogonal frequency division multiplexing (OFDM).

An Intrinsic Interference Mitigation Scheme for FBMC-QAM Systems

“…Without doubt, intersub-channel interference, introduced by the receiver filter of a given subcarrier overlaps with the transmitter filters of the neighboring sub-carriers in the frequency-domain due to sub-carrier nonorthogonality, is one of the important factors which impairs BER of a FBMC-QAM system. In [6], intersub-channel interference is defined as G(k, n) = H 0 ( k K )F 0 ( k K − n) for the intersub-channel k of a given sub-carrier and neighboring sub-carrier n where n = ±1, which can be extended to (16).…”

“…Prototype filters in the FBMC-QAM systems have profound impact on the OOB radiation, spectral efficiency, BER and demodulator complexity. However, according to Balian-Low theorem, it is not possible to design well-localized pulseshaping waveforms which also maintain subcarrier orthogonality and satisfy Nyquist criterion at the same time [16]. Thus, prototype filter designs found in the literature are mostly optimized for energy concentration (i.e., localization in time and frequency domain) or rapid sidelobes decay [16]- [20], [23].…”

“…However, according to Balian-Low theorem, it is not possible to design well-localized pulseshaping waveforms which also maintain subcarrier orthogonality and satisfy Nyquist criterion at the same time [16]. Thus, prototype filter designs found in the literature are mostly optimized for energy concentration (i.e., localization in time and frequency domain) or rapid sidelobes decay [16]- [20], [23]. Among them, SRRC, Mirrabasi-Martin (same as suggested in the PHY-DYAS project) and IOTA filters are often considered for FBMC-QAM systems, whose frequency responses are restated below.…”

Filtered Multicarrier Waveforms Classification: A Deep Learning-Based Approach

“…where z [n] is an additive Gaussian noise with variance σ 2 z . It is well known from the Balian-low theorem (BLT) that there exists no prototype g tx [n] that achieves orthogonality, and which is well localized both in time and frequency, and achieves critical density [12], [29], [30]. Therefore, it has been This work is licensed under a Creative Commons Attribution 4.0 License.…”

Transceiver Design for GFDM with Hexagonal Time–Frequency Allocation Using the Polyphase Decomposition

“…Accordingly, hexagonal FBMC is more robust to frequency synchronization errors and offers fewer OOB emissions concerning rectangular FBMC. Moreover, hexagonal Hermite waveforms have many potential benefits for the unsynchronized user at multi-user communications under doubly dispersive channel conditions [18]. On the other hand, fractional Fourier transform, which has equivalent computational complexity with Fourier transform (FT), is used to design a hexagonal frequency lattice structure to provide robustness on noise [19].…”