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).
The product, convolution, correlation, Wigner distribution function (WDF) and ambiguity function (AF) of two Hermite functions of arbitrary order and are derived and expressed as a bounded, weighted sum of Hermite functions. It was already known that these mathematical operations performed on Gaussians (Hermite functions of the zeroth-order) lead to a result which can be expressed as a Gaussian function again. We generalize this reciprocity to Hermite functions of arbitrary order. The product, convolution, correlation, WDF, and AF operations performed on two Hermite functions of arbitrary order lead to remarkably similar closed-form expressions, where the difference between the operations is primarily determined by distinct phase changes of the weights of the Hermite functions in the result. The closed-form expressions are generalized to the class of square-integrable functions. A key insight from the closed-form expressions is applied to the design of orthogonal, time-frequency localized communication signals which are characterized by an AF with rotational symmetry. In addition to this application, the theoretical expressions may prove useful for signal analysis in fields ranging from communications, radar and image processing to quantum mechanics.
Synchronization and matched filtering of signals in time dispersive, frequency dispersive and time-frequency dispersive channels are addressed in this paper. The 'eigenfunctions' of these channels form the signal sets under investigation. While using channel-eigenfunctions is a first requirement for undistorted data transmission, a second necessity is to achieve good synchronization over the domains of time and frequency. The synchronization problem in time-frequency for non-stationary signals is discussed. A spiral correlation method is proposed to achieve synchronization and matched filtering in time-frequency. Spiral correlation, using the pattern of a sunflower, is simulated and evaluated. It is argued that partial spiral correlation can lead to a significant reduction in computational complexity necessary for synchronization. Generalizations and identities based on the fractional Fourier transform are provided which omit the need for fractional delay filters.
This paper proposes an efficient implementation of the H.264/AVC motion estimation algorithm in hardware and software. Furthermore, a complete codesign trajectory from the HW/SW partitioning to the actual implementation on two different targets is shown. A Leon 3 + FPGA and an ARM + Montium implementation have been successfully realized. The FPGA implementation shows a speed-up of 43.6× whereas the Montium implementation shows a speedup of 22.0×, both compared to a software-only implementation. Power consumption is 42.0 mW for the FPGA and 60.2 mW for the Montium. A co-simulation tool, CosiMate, is used to achieve both on target implementations in just five weeks.
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