Signal Interpolation Research Articles

Convolution-Based Trigonometric Interpolation of Band-Limited Signals

This paper presents a method to obtain a trigonometric polynomial that accurately interpolates a given band-limited signal from a finite sequence of samples. The polynomial delivers accurate approximations in the range covered by the sequence, except for a short frame close to the range limits. Besides, its accuracy increases exponentially with the frame width. The method is based on using a band-limited window in order to reduce the truncation error of a convolution series. It is shown that the polynomial can be efficiently constructed and evaluated using algorithms designed for the discrete Fourier transform (DFT). Specifically, two basic procedures are presented, one based on the fast Fourier transform (FFT), and another based on a recursive update algorithm for the short-time FFT. The paper contains three applications. The first is a variable fractional delay (VFD) filter, which consists of a short-time FFT combined with the evaluation of a trigonometric polynomial. This filter has low complexity and can be implemented using CORDIC rotations. The second is the interpolation of nonuniform Fourier summations, where the proposed method eliminates the need to interpolate any kernel sample. Finally, the third can be viewed as a generalization of the FFT convolution algorithm and makes it possible to interpolate the output of an finite-impulse-response (FIR) filter efficiently.

Universal Sampling and Interpolation of Band-Limited Signals

We ask if there exist discrete “universal” sets \(\bigwedge\) of given finite density such that every signal f with bounded spectrum of small measure can be recovered from the samples \(f(\lambda),\,\lambda \in \bigwedge\). We prove that uniqueness in this problem can be achieved in general situation. On the other hand, for stable reconstruction it is crucial whether the spectrum is compact or dense.

There Exists No Globally Uniformly Convergent Reconstruction for the Paley–Wiener Space ${{\cal PW}}_{\pi}^{1}$ of Bandlimited Functions Sampled at Nyquist Rate

The bandlimited interpolation of signals and the convergence behavior of the Shannon sampling series are discussed in order to show that it is desirable to have a uniformly convergent reconstruction, for as large a space of signals as possible. In this paper general sampling series are analyzed for the frequently utilized Paley-Wiener space <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PW</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pi</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> , which is the largest space in the scale of Paley-Wiener spaces. The analysis is done not only for the Shannon sampling series, but for a whole class of axiomatically defined reconstruction processes. It is shown that for this very general class, which contains all common sampling series including the Shannon sampling series, a uniformly convergent reconstruction is not possible for the space <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PW</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pi</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> . Moreover, a universal signal is identified that causes the divergence behavior for all sampling series. Finally, a lower and an upper bound are derived and used to describe the asymptotic behavior of the peak value of the finite sampling series.

Increasing the sine-cosine transformation accuracy in signal approximation and interpolation

Sine-cosine transformations on the interval −0.5T ≤ t ≤ 0.5T, which are equivalent to a cosine transformation on the interval 0 ≤ t ≤ T, are considered. Relationships for the error variance of signal reconstruction are obtained. Depending on the a priori information on the signal, the relationships make it possible to choose for its representation the most suitable orthogonal expansion. A stationary random signal model is used for comparing the expansion with a conventional sine-cosine transformation.

A method for estimating the accuracy of quadrature output sensors

The errors of quadrature signal interpolation are analyzed. Quantitative assessment of the noise effect and the quadrature signal distortion is given. A method for estimating the accuracy of quadrature output sensors is proposed.

Single-Bit Oversampled A/D Conversion With Exponential Accuracy in the Bit Rate

A scheme for simple oversampled analog-to-digital (A/D) conversion using single-bit quantization is presented. The scheme is based on recording positions of zero-crossings of the input signal added to a deterministic dither function. This information can be represented in a manner such that the bit rate increases only logarithmically with the oversampling factor r. The input band-limited signal can be reconstructed from this information locally with O(1/r) pointwise error, resulting in an exponentially decaying distortion-rate characteristic. In the course of studying the accuracy of the proposed A/D conversion scheme, some new results are established about reconstruction of band-limited signals from irregular samples using linear combination of functions with fast decay. Schemes for local interpolation of band-limited signals from quantized irregular samples are also proposed.

Nonuniform Interpolation of Noisy Signals Using Support Vector Machines

The problem of signal interpolation has been intensively studied in the information theory literature, in conditions such as unlimited band, nonuniform sampling, and presence of noise. During the last decade, support vector machines (SVM) have been widely used for approximation problems, including function and signal interpolation. However, the signal structure has not always been taken into account in SVM interpolation. We propose the statement of two novel SVM algorithms for signal interpolation, specifically, the primal and the dual signal model based algorithms. Shift-invariant Mercer's kernels are used as building blocks, according to the requirement of bandlimited signal. The sine kernel, which has received little attention in the SVM literature, is used for bandlimited reconstruction. Well-known properties of general SVM algorithms (sparseness of the solution, robustness, and regularization) are explored with simulation examples, yielding improved results with respect to standard algorithms, and revealing good characteristics in nonuniform interpolation of noisy signals.

Optimal signal interpolation using the values of signal and its derivative

The effect of the bias in a sequence of signal samples and sequence of signal derivative samples on the signal reconstruction accuracy is considered for a bandwidth-limited signal.

Interval Timer Control of Puberty in Photoinhibited Siberian Hamsters

Puberty, which is markedly delayed in male Siberian hamsters (Phodopus sungorus) born into short day lengths, is controlled by an interval timer regulated by the duration of nocturnal melatonin secretion. Properties of the interval timer were assessed by perturbing normal patterns of melatonin secretion in males gestated and maintained thereafter in 1 of 2 short day lengths, 10 h light/day (10 L) or 12L. Melatonin secretion of short-day hamsters was suppressed by constant light treatment or modified by daily injection of propranolol to mimic nocturnal melatonin durations typical of long-day hamsters. Constant light treatment during weeks 3 to 5 induced early incomplete gonadal growth in 12L but not 10 L hamsters but did not affect late onset of gonadal development indicative of puberty in either photoperiod. Propranolol treatment during postnatal weeks 3 to 5 induced transient growth of the testes and ultimately delayed the timing of puberty by 3 weeks. Similar treatments between weeks 5 and 7 or on alternate weeks for 24 weeks did not affect the interval timer. The first 2 weeks after weaning may constitute a critical period during which the interval timer is highly responsive to photoperiod. Alternatively, the hamsters' photoperiodic history rather than age or developmental stage may be the critical variable. The interpolation of long-day melatonin signals at the time of weaning does not appear to reset the interval timer to its zero position but may reduce timer responsiveness to long-day melatonin signals several weeks later.

The Calculation of Dynamic Errors in Signal Transformation Circuits of Analog-to-Digital Converters for Mechatronic Systems

Precision and wide-band analog-to-digital converters (ADC) are used for information processing in mechatronic systems. The analog signal transformation circuits, such as sample, and hold circuit (SHC) and an analog signal interpolation circuit (ASIC), are applied with the aim of increasing the precision of converters. These circuits allow for reducing the quantity of comparators in the converter and increase their dynamic stability. The models of SHC and ASIC as well as the results of the calculation precision and dynamic parameters of such circuits are presented in this paper. Equations for the calculation of the aperture error in the Gaussian and a sinusoidal input signal were derived. The structural model, proposed for SHC, evaluates the nonlinearity of a transfer characteristic, and the influence of noise in the signal and strobe channels. The results of the theoretical research and analytical equations for the evaluation of the number of signal interpolation block differential amplifiers, which depends on the analog signal’s maximal frequency and the number of segments of the converter, are presented. The results of this work allow one to estimate the main precision and dynamic parameters of ADC transformation circuits.

Experimental calibration of a cost-effective X-band weather radar for climate ecological studies in southern Ecuador

In this paper setup, operational problems and a straightforward calibration approach for a cost-effective X-Band radar are presented. The LAWR (Local Area Weather Radar) system is based on conventional ship radar technology which is adapted to register rainfall within a range of about 60 km with a spatial resolution of 500 m per pixel. The instrument offers neither Doppler processing nor vertical scan capabilities but uses 20° wide (vertical) beam. The calibration suffers from an unfavorably distributed and very sparse rain gauge network, heavy clutter contamination of the signal and obstructions by surrounding terrain. A specific scaling approach is developed, that includes satellite data on cloud frequency and distribution, to overcome these limitations. Observed clutter is removed and missing values are replaced by bilinear interpolation of the undisturbed signals. A temporal and spatial bias of the radar signal is corrected using an omni-directional spatial distribution hypothesis. This is possible because of the location of the radar site in the transition zone between high rainfall on the eastern Andean slopes and low rainfall on the leeward side. A further limitation of the system is that the LAWR does not provide information on the measured reflectivity Z but dimensionless counts (8 bit resolution). Calibration is performed assuming a linear relation between radar output and rainfall as recommended by the systems manufacturer. The intercomparison of rain gauge and scatterometer data with calibrated radar rainfall reveals a good performance of the developed calibration approach.

Generalized smoothing splines and the optimal discretization of the Wiener filter

We introduce an extended class of cardinal L/sup */L-splines, where L is a pseudo-differential operator satisfying some admissibility conditions. We show that the L/sup */L-spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional /spl par/Ls/spl par//sub L2//sup 2/, subject to the interpolation constraint. Next, we consider the corresponding regularized least squares estimation problem, which is more appropriate for dealing with noisy data. The criterion to be minimized is the sum of a quadratic data term, which forces the solution to be close to the input samples, and a "smoothness" term that privileges solutions with small spline energies. Here, too, we find that the optimal solution, among all possible functions, is a cardinal L/sup */L-spline. We show that this smoothing spline estimator has a stable representation in a B-spline-like basis and that its coefficients can be computed by digital filtering of the input signal. We describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of L is rational (which corresponds to the case of exponential splines). We justify these algorithms statistically by establishing an equivalence between L/sup */L smoothing splines and the minimum mean square error (MMSE) estimation of a stationary signal corrupted by white Gaussian noise. In this model-based formulation, the optimum operator L is the whitening filter of the process, and the regularization parameter is proportional to the noise variance. Thus, the proposed formalism yields the optimal discretization of the classical Wiener filter, together with a fast recursive algorithm. It extends the standard Wiener solution by providing the optimal interpolation space. We also present a Bayesian interpretation of the algorithm.

Adaptive online correction and interpolation of quadrature encoder signals using radial basis functions

This paper considers the development of an adaptive online approach for the correction and interpolation of quadrature encoder signals, suitable for application to precision motion control systems. It is based on the use of a two-stage double-layered radial basis function (RBF) neural network. The first RBF stage is used to adaptively correct for the imperfections in the encoder signals such as mean, phase offsets, amplitude deviation and waveform distortion. The second RBF stage serves as the inferencing machine to adaptively map the quadrature encoder signals to higher order sinusoids, thus, enabling intermediate positions to be derived. Experimental and simulation results are provided to verify the effectiveness of the RBF approach.

Effect of interpolation on PWVD computation and instantaneous frequency estimation

The peak of the polynomial Wigner–Ville distribution (PWVD) can be used for estimating the instantaneous frequency (IF) of monocomponent polynomial phase signals. However, the PWVD kernel, optimized to yield a time–frequency distribution (TFD) localized along the IF, comprises of fractional-time-sampled signals. When implemented in a discrete-time scenario, this calls for signal interpolation. We study three interpolation schemes—linear, cubic polynomial and sinc and derive expressions for the variance of the interpolated samples in the presence of noise. In representing nonstationary signals using the PWVD, the instantaneous energy content of noise auto-terms and signal-noise cross-terms is found to be the least for linear interpolation scheme. For polynomial IF estimation using the peak of the PWVD, it was found that linear interpolation is a computationally efficient way of obtaining reasonably good estimates at low signal-to-noise ratios (SNRs). For high SNRs, sinc interpolation outperforms the other two schemes. Similar results were found when the experiment was extended to sinusoidal IF signals also.

Fractional biorthogonal partners in channel equalization and signal interpolation

The concept of biorthogonal partners has been introduced recently by the authors. The work presented here is an extension of some of these results to the case where the upsampling and downsampling ratios are not integers but rational numbers, hence, the name fractional biorthogonal partners. The conditions for the existence of stable and of finite impulse response (FIR) fractional biorthogonal partners are derived. It is also shown that the FIR solutions (when they exist) are not unique. This property is further explored in one of the applications of fractional biorthogonal partners, namely, the fractionally spaced equalization in digital communications. The goal is to construct zero-forcing equalizers (ZFEs) that also combat the channel noise. The performance of these equalizers is assessed through computer simulations. Another application considered is the all-FIR interpolation technique with the minimum amount of oversampling required in the input signal. We also consider the extension of the least squares approximation problem to the setting of fractional biorthogonal partners.

A centrosymmetric matrix based technique for the interpolation of a Hermitian signal

AbstractThis work is concerned with the signal interpolation problem, i.e. given only samples of a signal, a method is derived for evaluating its samples on finer grids. The derivation is based on a discrete‐time decimation formula. In the special case where the known samples have the Hermitian property, two schemes are presented and mathematically proved to result in interpolated points having the same property. The first scheme does not utilize the known sample at the origin and results in a square system of equations to be solved for the unknown interpolated points of the signal. The second scheme has the merit of utilizing all the known samples, but it results in an overdetermined system of equations to be solved by the least squares method. The exploitation of the elegant properties of the involved centrosymmetric matrices is central to the treatment presented here. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Reduction of electronic delay in active noise control systems— A multirate signal processing approach

Electronic delay has been a critical problem in active noise control (ANC) systems. This is true whether a feedforward structure or a feedback structure is adopted. In particular, excessive delays would create a causality problem in a feedforward ANC system of a finite-length duct. This paper suggests a multirate signal-processing approach for minimizing the electronic delay in the control loop. In this approach, digital controllers are required in decimation and interpolation of discrete-time signals. The computation efficiency is further enhanced by a polyphase method, where the phases of low-pass finite impulse response (FIR) filters must be carefully designed to avoid unnecessary delays. Frequency domain optimization procedures based on H1, H2, and Hinfinity norms, respectively, are utilized in the FIR filter design. The proposed method was implemented by using a floating-point digital signal processor. Experimental results showed that the multirate approach remains effective for suppressing a broadband (200-600 Hz) noise in a duct with a minimum upstream measurement microphone placement of 20 cm.

Interpolation of Signals for Colorimetric Computation

Interpolation of Signals for Colorimetric Computation

Multiplierless interpolator for a delta-sigma digital to analog converter

A simplified algorithm for digital signal interpolation and a novel architecture to implement the algorithm in an integrated circuit ('IC') with significant space constraints are presented. According to the embodiments of the present invention, the interpolator is divided into two parts. The first part of the interpolator increases the sample rate by a factor of two and smoothes the signal using a half-band Infinite Impulse Response ('IIR') filter. The second part of the interpolator increases the sample rate of the signal by a factor of thirty-two using a zero-order-hold ('ZOH') circuit. In one embodiment, the half-band IIR filter is implemented using an all-pass lattice structure to minimize quantization effects. The lattice coefficients are chosen such that the structure can achieve all filter design requirements, yet is capable of being implemented with a small number of shifters and adders, and no multipliers.

Precise interpolation for band‐limited digital signals based on sampling theorem incorporated with window functions

AbstractThis paper proposes an interpolation method called the “window function method” in which the sampling theorem and the window functions are combined. The cause of the generation of the interpolation error is the folding distortion due to the multiplication of the time‐limited window to the input signal. The conditions to enable precise interpolation are that the functions are continuous, the windows with large spectral decays in the side lobes are normalized such that the window function values are unity at the interpolation points, and the main lobe width is set to an optimum. When the interpolation precision is compared with those by other interpolation methods, the present one is about the same as that in the “filter method” (the conventional method using the filter to eliminate the unwanted image components) but is somewhat poorer than that in the “least RMS error method” (theoretical limit) if Kaiser–Bessel windows are used. However, the number of input signal data to obtain the same interpolation precision is larger only by 10%. Hence, the operating time to obtain the required interpolation precision is close to optimum. The calculation of the coefficients needed for interpolation is basically a simple substitution and can avoid a design of higher‐order filters (in the filter method) and solution of the ill‐conditioned simulation equations (in the least RMS error method). © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 84(7): 63–77, 2001