This paper presents an algorithm for deriving uniformly spaced samples from nonuniformly spaced samples of a signal (and visa versa). The nonuniformity must be a function of linear sampling i.e. to derive uniformly spaced samples from an exponentially sampled signal. One place this situation occurs is in the calculation of some integral transforms like the Hankel transform. The algorithm runs quickly, in linear time and can thus be used with the FFT to create fast transform algorithms and provide a superior alternative to linear interpolation.
We describe the implementation of a novel adaptive wireless communications waveform for interference avoidance (IA) in re-configurable logic devices. While other transform domain-based IA waveforms have been suggested, the wavelet packet modulation (WPM) system described here is unique in its multiplexing of complex quadrature amplitude modulation symbols onto orthogonal wavelet packets for unrivaled time-frequency agility. We examine the realization aspects of dynamically instantiating the transmit side inverse discrete wavelet packet transform (DWPT) and receive side DWPT filter bank structures, and the WPM symbol timing recovery, in Field Programmable Gate Array (FPGA) devices. This work applies Trenas' re-configurable wavelet packet transform (WPT) architecture to a wireless communications system, draws upon Jones' theoretical foundation for orthogonally multiplexed communications, and utilizes Lindsey's WPM supersymbol tuning and Kjeldsen's WPM symbol synchronization algorithms.
Circular Trellis Coded Modulation (CTCM) defines a family of (block) trellis codes which use a unique algebraic constraint, imposed on the start state, to produce a strong tail-biting property without the inefficiency of driving the encoder state to zero by using a sequence of input zeroes. From the beginning of CTCM, elements of the Galois field, GF(pm), have served dual roles, labeling both s ystem trellis nodes and valid input symbols. This dual use of field elements facilitates exploitation of the algebraic structure of GF(pm). The system trellis always take a particularly simple and advantageous form (called the pn-fly form) whenever the alphabet of valid input symbols is chosen to be (a coset of) any additive subgroup of the additive group structure of GF(pm). This paperproposes a family of signal mappings that complete the definition of the CTCM system by providing structurally consistent output labels for the trellis edges. The completion of a structural definition greatly facilitates system analysis, especially the (future) geometrically precise construction of a related signal constellation. At the same time, it preserves the possibility of an advantegeous receiver structure.
KEYWORDS: Global Positioning System, Sensors, Signal detection, Interference (communication), Sensor performance, Data communications, Satellites, Error analysis, Modulation, Signal to noise ratio
Multi-user detection (MUD) techniques are applied to the GPS interference suppression/rejection problem, such that recovery of GPS satellite signals out of an environment contaminated with one or more correlated interference sources is afforded. In modeling the narrowband digital interference as a set of multiple users it is shown that recovery is possible via any of several MUD algorithms, including the conventional detector, the decorrelating detector, the optimum linear detector, and the minimum mean square error (MMSE) detector. The relationship between the underlying structure of the GPS symbol (C/A code) and the performance of these detectors is studied such that bit error rate (BER) and signal-to-interference ratio (SIR) are developed analytically.
The novel use of a Maximum Likelihood Decision-Directed (MLDD) synchronization scheme for a Wavelet Packet Modulation (WPM) System is discussed. MLDD synchronization allows symbol synchronization without edge detection. Multi-Carrier Modulation (MCM) techniques are being increasingly employed in military communication networks to combat time-dispersive and time-variant channel effects. One MCM method, Coded Orthogonal Frequency Division Multiplexing, has been particularly effective against multipath fading environments. Wavelet packet modulation has recently been introduced to address the need for improved transmission performance over channels with joint time and frequency interference components. Wavelet Packet Modulation's strength rests in a flexible, custom mapping of the desired signal on the communications channel at the transmitter to avoid a variety of a priori noise/interference patterns. The use of wavelet packet modulation on wireless channels has been hindered due to symbol synchronization not being achievable with conventional edge detection techniques. A MLDD receiver symbol timing recovery algorithm is presented here that should help expedite the adoption of WPM on wireless channels.
We give an explicit expression for the transform of a signal in an arbitrary representation which has first been filtered in another representation. Using this formula we connect the work of Cohen for obtaining convolution and correlation theorems in arbitrary representations with the work of Lindsey and Suter for partitioning the space of integral transforms.
Circular Trellis Coded Modulation (CTCM) defines a family of (block) trellis codes which use a unique algebraic constraint, imposed on the start state, to produce a strong tail-biting property without the inefficiency of driving the encoder state to zero, using a sequence of input zeroes. Previous papers have investigated specific practical results in the case of binary and 4-ary signaling, using the elements of the Galois field GF(2m) to label the trellis. The `4-fly' has proven to be the central feature of the trellis structure, allowing exceptional performance. The present paper generalizes the 4-fly structure, naturally related to GF(2m), to the case of pj-flies, which support efficient pj-ary signaling using trellises labeled by the field GF(pm), valid both for p equals 2 and for odd characteristic. This opens numerous practical system design options, allows greater flexibility for the transmitter design, and lays the mathematical groundwork required to support a much more systematic and general analysis of the CTCM trellis structure. This paper gives the general definitions and records the first elements of the structural analysis. Detailed knowledge of the trellis structure is key to the minimization of decoder complexity.
Subspace projection techniques are effective in excising FM interferers in GPS receivers. The FM jammers are instantaneously narrowband and have clear time-frequency (t- f) signatures that are distinct from the GPA C/A spreading codes. The instantaneous frequency (IF) estimate, provided by the time-frequency distribution, or any other estimator, is used to form the jammer subspace. The interference rejection is implemented by projecting the received data onto the orthogonal subspace of the jammer subspace. Errors in IF estimations, however, perturb the projection matrix and allow some of the jammer power to escape the projection operation. This in turn leads to degraded receiver performance and lower SINR. This paper derives the signal- to-interference-and-noise ratio (SINR) of the GPS receivers in the presence of Gaussian IF estimation errors. It is shown that IF estimation errors can substantially impede interference rejection in GPS using subspace projection techniques.
Lindsey and Suter have shown that for many transforms the transform of the convolution of two functions have the same functional form. We explain the origin of this result and derive the condition on the transformation kernel for when this should be the case. In addition we consider the general transform of the inverse and direct scale transform and obtain condition on the kernel so that the transform gives similar functional forms.
Investigating a number of different integral transforms uncovers distinct patterns in the type of scale-based convolution theorems afforded by each. It is shown that scaling convolutions behave in quite a similar fashion to translational convolution in the transform domain, such that the many diverse transforms have only a few different forms for convolution theorems. The hypothesis is put forth that the space of integral transforms is partitionable based on these forms.
Investigating a number of different integral transforms uncovers distinct patterns in the type of translation convolution theorems afforded by each. It is shown that transforms based on separable kernels (aka Fourier, Laplace and their relatives) have a form of the convolution theorem providing for a transform domain product of the convolved functions. However, transforms based on kernels not separable in the function and transform variables mandate a convolution theorem of a different type; namely in the transform domain the convolution becomes another convolution--one function with the transform of the other.
KEYWORDS: Wavelets, Filtering (signal processing), Transceivers, Modulation, Analog electronics, Signal processing, Quadrature amplitude modulation, Digital signal processing, Image filtering, Binary data
Current transceiver designs for wavelet-based communication systems are typically reliant on analog waveform synthesis, however, digital processing is an important part of the eventual success of these techniques. In this paper, a transceiver implementation is introduced for the recently introduced wavelet packet modulation scheme which moves the analog processing as far as possible toward the antenna. The transceiver is based on the discrete wavelet packet transform which incorporates level and node parameters for generalized computation of wavelet packets. In this transform no particular structure is imposed on the filter bank save dyadic branching, and a maximum level which is specified a priori and dependent mainly on speed and/or cost considerations. The transmitter/receiver structure takes a binary sequence as input and, based on the desired time- frequency partitioning, processes the signal through demultiplexing, synthesis, analysis, multiplexing and data determination completely in the digital domain - with exception of conversion in and out of the analog domain for transmission.
Within the past few years, wavelet transforms and filter banks have received considerable attention in the technical literature, prompting applications in a variety of disciplines including applied mathematics, speech and image processing and compression, medical imaging, geophysics, signal processing, and information theory. More recently, several researchers in the field of communications have developed theoretical foundations for applications of wavelets as well. The objective of this paper is to survey the connections of wavelets and filter banks to communication theory and summarize current research efforts.
This work presents a generalized signaling strategy for orthogonally multiplexed communication. Wavelet packet modulation (WPM) employs the basis functions from an arbitrary pruning of a full dyadic tree structured filter bank as orthogonal pulse shapes for conventional QAM symbols. The multi-scale modulation (MSM) and M-band wavelet modulation (MWM) schemes which have been recently introduced are handled as special cases, with the added benefit of an entire library of potentially superior sets of basis functions. The figures of merit are derived and it is shown that the power spectral density is equivalent to that for QAM (in fact, QAM is another special case) and hence directly applicable in existing systems employing this standard modulation. Two key advantages of this method are increased flexibility in time-frequency partitioning and an efficient all-digital filter bank implementation, making the WPM scheme more robust to a larger set of interferences (both temporal and sinusoidal) and computationally attractive as well.
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