Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in the mathematics literature, the PSF is commonly stated and proven in the pointwise sense for various types of $L _{ 1 }$ signals. This $L _{ 1 }$ assumption is, however, too restrictive for many signal-processing tasks that demand the sampling of possibly growing signals. In this paper, we present two generalized versions of the PSF for d-dimensional signals of polynomial growth. In the first generalization, we show that the PSF holds in the space of tempered distributions for every continuous and polynomially growing signal. In the second generalization, the PSF holds in a particular negative-order Sobolev space if we further require that d∕2 + ε derivatives of the signal are bounded by some polynomial in the $L _{ 2 }$ sense.

Generalized Poisson Summation Formula for Tempered Distributions

Quy Ha Nguyen, Michaël Unser

The Poisson summation formula (PSF), which relates the sampling of an analog signal with the periodization of its Fourier transform, plays a key role in the classical sampling theory. In its current forms, the formula is only applicable to a limited class of signals in

L _{ 1 }

. However, this assumption on the signals is too strict for many applications in signal processing that require sampling of non-decaying signals. In this paper we generalize the PSF for functions living in weighted Sobolev spaces that do not impose any decay on the functions. The only requirement is that the signal to be sampled and its weak derivatives up to order 1∕2 + ε for arbitrarily small ε > 0, grow slower than a polynomial in the

L _{ 2 }

sense. The generalized PSF will be interpreted in the language of distributions.

IEEE2015

Signal Processing in Space and Time

Francisco Pereira Correia Pinto

Sound waves propagate through space and time by transference of energy between the particles in the medium, which vibrate according to the oscillation patterns of the waves. These vibrations can be captured by a microphone and translated into a digital signal, representing the amplitude of the sound pressure as a function of time. The signal obtained by the microphone characterizes the time-domain behavior of the acoustic wave field, but has no information related to the spatial domain. The spatial information can be obtained by measuring the vibrations with an array of microphones distributed at multiple locations in space. This allows the amplitude of the sound pressure to be represented not only as a function of time but also as a function of space. The use of microphone arrays creates a new class of signals that is somewhat unfamiliar to Fourier analysis. Current paradigms try to circumvent the problem by treating the microphone signals as multiple "cooperating" signals, and applying the Fourier analysis to each signal individually. Conceptually, however, this is not faithful to the mathematics of the wave equation, which expresses the acoustic wave field as a single function of space and time, and not as multiple functions of time. The goal of this thesis is to provide a formulation of Fourier theory that treats the wave field as a single function of space and time, and allows it to be processed as a multidimensional signal using the theory of digital signal processing (DSP). We base this on a physical principle known as the Huygens principle, which essentially says that the wave field can be sampled at the surface of a given region in space and subsequently reconstructed in the same region, using only the samples obtained at the surface. To translate this into DSP language, we show that the Huygens principle can be expressed as a linear system that is both space- and time-invariant, and can be formulated as a convolution operation. If the input signal is transformed into the spatio-temporal Fourier domain, the system can also be analyzed according to its frequency response. In the first half of the thesis, we derive theoretical results that express the 4-D Fourier transform of the wave field as a function of the parameters of the scene, such as the number of sources and their locations, the source signals, and the geometry of the microphone array. We also show that the wave field can be effectively analyzed on a small scale using what we call the space/time-frequency representation space, consisting of a Gabor representation across the spatio-temporal manifold defined by the microphone array. These results are obtained by treating the signals as continuous functions of space and time. The second half of the thesis is dedicated to processing the wave field in discrete space and time, using Nyquist sampling theory and multidimensional filter banks theory. In particular, we show examples of orthogonal filter banks that effectively represent the wave field in terms of its elementary components while satisfying the requirements of critical sampling and perfect reconstruction of the input. We discuss the architecture of such filter banks, and demonstrate their applicability in the context of real applications, such as spatial filtering and wave field coding.

EPFL2010

Hybrid continuous-discrete-time multi-bit delta-sigma A/D converters with auto-ranging algorithm

Sergio Pesenti

In wireless portable applications, a large part of the signal processing is performed in the digital domain. Digital circuits show many advantages. The power consumption and fabrication costs are low even for high levels of complexity. A well established and highly automated design flow allows one to benefit from the constant progress in CMOS technologies. Moreover, digital circuits offer robust and programmable signal processing means and need no external components. Hence, the trend in consumer electronics is to further reduce the part of analog signal processing in the receiver chain of wireless transceivers. Consequently, analog-to-digital converters with higher resolutions and bandwidths are constantly required. The ultimate goal is the direct digitization of radio frequency signals, where the conversion would be performed immediately after the front-end amplifier. ΔΣ-modulation-based converters have proved to be the most suitable to achieve the required performance. Switched-capacitor implementations have been widely used over the last two decades. However, recent publications and books have shown that continuous-time architectures can achieve the same performance with lower power consumption. Most designs found throughout the literature use a single- or few-bit internal quantizer with a high-order modulation. As a result, in order to achieve the resolutions and bandwidths required today, the sampling frequency must exceed 100MHz. This approach leads to non-negligible power consumption in the clock generation. Moreover, the presence of such fast squared signals is not suitable for a system-on-chip comprising radio frequency receivers. In this thesis we propose a low-power strategy relying on a large number of internal levels rather than on a high sampling frequency or modulation order. Besides, a hybrid continuous-discrete-time approach is used to take advantage of the accuracy of switched-capacitor circuits and the low power consumption of continuous-time implementation. The sensitivity to clock jitter brought about by the continuous-time stage is reduced by the use of a large number of levels. An auto-ranging algorithm is developed in this thesis to overcome the limitation of a large-size quantizer under low-voltage supply. Finally, the strategy is applied to a design example addressing typical specifications for a Bluetooth receiver with direct conversion.

EPFL2007