Showing papers on "Discrete-time Fourier transform published in 1996"

Digital computation of the fractional Fourier transform

Abstract: An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.

Fourier transforms of colour images using quaternion or hypercomplex, numbers

Abstract: The 2D quaternion, or hypercomplex, Fourier transform is introduced. This transform makes possible the handling of colour images in the frequency domain in a holistic manner, without separate handling of the colour components, and it thus makes possible very wide generalisation of monochrome frequency domain techniques to colour images.

On the relationship between the Fourier and fractional Fourier transforms

Abstract: In recent years, the fractional Fourier transform has been the focus of many research papers. In this letter, we show that the fractional Fourier transform is nothing more than a variation of the standard Fourier transform and, as such, many of its properties, such as its inversion formula and sampling theorems, can be deduced from those of the Fourier transform by a simple change of variable.

Rapid computation of the discrete Fourier transform

Abstract: Algorithms for the rapid computation of the forward and inverse discrete Fourier transform for points which are nonequispaced or whose number is unrestricted are presented. The computational procedure is based on approximation using a local Taylor series expansion and the fast Fourier transform (FFT). The forward transform for nonequispaced points is computed as the solution of a linear system involving the inverse Fourier transform. This latter system is solved using the iterative method GMRES with preconditioning. Numerical results are given to confirm the efficiency of the algorithms.

Calculation of Demagnetizing Field Distribution Based on Fast Fourier Transform of Convolution

Abstract: It is confirmed that the calculation of the demagnetizing field in micromagnetic simulations can be accelerated significantly by using the discrete convolution theorem and the fast Fourier transform (FFT). When the magnetization distribution is periodic, application of the theorem to the demagnetizing field calculation is straightforward. Unlike the previously reported FFT method which is based on the continuous Fourier transform of the demagnetizing field, the method can also be used in the case of non-periodic magnetization structures. It is also confirmed that the result obtained using the new FFT method coincides with that of the conventional direct method, as expected. The principle of calculation and the results of one- and two-dimensional calculations which show the validity and effectiveness of the developed method are presented.

Computer Explorations in Signals and Systems Using MATLAB

Abstract: 1. Signals and Systems. Tutorial: Basic MATLAB Functions for Representing Signals. Discrete-Time Sinusoidal Signals. Transformations of the Time Index for Discrete-Time Signals. Properties of Discrete-Time Systems. Implementing a First-Order Difference Equation. Continuous-Time Complex Exponential Signals. Transformations of the Time Index for Continuous-Time Signals. Energy and Power for Continuous-Time Signals. 2. Linear Time-Invariant Systems. Tutorial: conv. Tutorial: filter. Tutorial: lsim with Differential Equations. Properties of Discrete-Time LTI Systems. Linearity and Time-Invariance. Noncausal Finite Impulse Response Filters. Discrete-Time Convolution. Numerical Approximations of Continuous-Time Convolution. The Pulse Response of Continuous-Time LTI Systems. Echo Cancellation via Inverse Filtering. 3. Fourier Series Representation of Periodic Signals. Tutorial: Computing the Discrete-Time Fourier Series with fft. Tutorial: freqz. Tutorial: lsim with System Functions. Eigenfunctions of Discrete-Time LTI Systems. Synthesizing Signals with the Discrete-Time Fourier Series. Properties of the Continuous-Time Fourier Series. Energy Relations in the Continuous-Time Fourier Series. First-Order Recursive Discrete-Time Filters. Frequency Response of a Continuous-Time System. Computing the Discrete-Time Fourier Series. Synthesizing Continuous-Time Signals with the Fourier Series. The Fourier Representation of Square and Triangle Waves. Continuous-Time Filtering. 4. The Continuous-Time Fourier Transform. Tutorial: freqs. Numerical Approximation to the Continuous-Time Fourier Transform. Properties of the Continuous-Time Fourier Transform. Time- and Frequency-Domain Characterizations of Systems. Impulse Responses of Differential Equations by Partial Fraction Expansion. Amplitude Modulation and the Continuous-Time Fourier Transform. Symbolic Computation of the Continuous-Time Fourier Transform. 5. The Discrete-Time Fourier Transform. Computing Samples of the DTFT. Telephone Touch-Tone. Discrete-Time All-Pass Systems. Frequency Sampling: DTFT-Based Filter Design. System Identification. Partial Faction Expansion for Discrete-Time Systems. 6. Time and Frequency Analysis of Signals and Systems. A Second-Order Shock Absorber. Image Processing with One-Dimensional Filters. Filter Design by Transformation. Phase Effects for Lowpass Filters. Frequency Division Multiple-Access. Linear Prediction on the Stock Market. 7. Sampling. Aliasing due to Undersampling. Signal Reconstruction from Samples. Upsampling and Downsampling. Bandpass Sampling. Half-Sample Delay. Discrete-Time Differentiation. 8. Communications Systems. The Hilbert Transform and Single-Sideband AM. Vector Analysis of Amplitude Modulation with Carrier. Amplitude Demodulation and Receiver Synchronization. Intersymbol Interference in PAM Systems. Frequency Modulation. 9. The Laplace Transform. Tutorial: Making Continuous-Time Pole-Zero Diagrams. Pole Locations for Second-Order Systems. Butterworth Filters. Surface Plots of Laplace Transforms. Implementing Noncausal Continuous-Time Filters. 10. The z-Transform. Tutorial: Making Discrete-Time Pole-Zero Diagrams. Geometric Interpretation of the Discrete-Time Frequency Response. Quantization Effects in Discrete-Time Filter Structures. Designing Discrete-Time Filters with Euler Approximations. Discrete-Time Butterworth Filter Design Using the Bilinear Transformation. 11. Feedback Systems. Feedback Stabilization: Stick Balancing. Stabilization of Unstable Systems. Using Feedback to Increase the Bandwidth of an Amplifier. Bibliography. Index.

The discrete periodic Radon transform

Abstract: In this correspondence, a discrete periodic Radon transform and its inversion are developed. The new discrete periodic Radon transform possesses many properties similar to the continuous Radon transform such as the Fourier slice theorem and the convolution property, etc. With the convolution property, a 2-D circular convolution can be decomposed into 1-D circular convolutions, hence improving the computational efficiency. Based on the proposed discrete periodic Radon transform, we further develop the inversion formula using the discrete Fourier slice theorem. It is interesting to note that the inverse transform is multiplication free. This important characteristic not only enables fast inversion but also eliminates the finite-word-length error that may be generated in performing the multiplications.

Signal processor with improved efficiency

Abstract: A signal processor arranged and constructed to provide a discrete time Fourier transform (DTFT) corresponding to a sequence of samples, including a current, preferably, last sample, of a signal, the signal processor including a recursive structure (303), coupled to the signal and operating on the sequence of samples, for providing an output (309) and a first previous output (310), the output proportional to a combination of the current sample of the signal, and the first previous output weighted by a sinusoidal function, less a second previous output (316), the sinusoidal function having an argument corresponding to an arbitrary frequency, and a combiner (330) coupled to the output and the first previous output for providing a DTFT signal proportional to a DTFT evaluated at the arbitrary frequency for the sequence of samples.

Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class

Abstract: We consider the Cohen (1989) class of time-frequency distributions, which can be obtained from the Wigner distribution by convolving it with a kernel characterizing that distribution. We show that the time-frequency distribution of the fractional Fourier transform of a function is a rotated version of the distribution of the original function, if the kernel is rotationally symmetric. Thus, the fractional Fourier transform corresponds to rotation of a relatively large class of time-frequency representations (phase-space representations), confirming the important role this transform plays in the study of such representations.

Discrete fractional Fourier transform

Abstract: The continuous fractional Fourier transform (FRFT) represents a rotation of signal in time-frequency plane, and it has become an important tool for signal analysis. A discrete version of fractional Fourier transform has been developed but its results do not match those of continuous case. In this paper, we propose a new version of discrete fractional Fourier transform (DFRFT). This new DFRFT will provide similar transforms as those of continuous fractional Fourier transform and also hold the rotation properties.

Discrete Fourier Transform

Abstract: In chapter 6, we investigated the definition and properties of the discrete-time Fourier transform X(e jω ), with ω being a continuous frequency variable, and found it to be very useful for analyzing a wide variety of signals and systems of theoretical interest. However, much of the practice of digital signal processing is done in computers where we cannot evaluate a continuum of frequencies ω, nor can we input and store an infinite-duration sequence x(n). Hence, for actual data sequences, as opposed to theoretically defined signals, we cannot compute the Fourier transform, in general.

The discrete fractional Fourier transformation

Abstract: Based on the fractional Fourier transformation of sampled periodic functions, the discrete form of the fractional Fourier transformation is obtained. It is found that for a certain dense set of fractional orders it is possible to define a discrete transformation. Also, for its efficient computation a fast algorithm, which has the same complexity as the FFT, is given.

Constrained imaging: overcoming the limitations of the Fourier series

Abstract: Magnetic resonance imaging (MRI) is usually implemented as a Fourier transform-based technique. During data acquisition, spatially resolved information relating to spin density, relaxation rates, chemical shifts, and other parameters is phase and frequency encoded in the measured data. Image reconstruction is accomplished through the use of the Fourier series model, which can be evaluated efficiently using a fast Fourier transform (FFT) algorithm. Theoretically, the Fourier series is capable of producing perfect images if the data space (often called k-space) is sufficiently covered. In practice, several problems arise with this model due to finite sampling. Specifically, finite sampling leads to a truncation or the Fourier series, which results in image blurring and ringing. Image blurring is attributed to a loss of spatial resolution. In fact, with the Fourier series model, the resulting image resolution is limited to roughly the reciprocal of the frequency interval over which the data are sampled. The ringing artifact is due to the well-known Gibbs phenomenon, which is more pronounced for images with sharp edges. In order to overcome these limitations associated with the direct application of the Fourier series model, many alternatives have been proposed in the past decade to incorporate a priori information into the imaging process. This article discusses the constrained imaging concept. Specifically, the authors review 3 model-based imaging techniques that the authors have developed in the past few years. An essential feature of these methods is that a parametric model in the form of a generalized series is superimposed on the underlying measured data or image.

The errors in FFT estimation of the Fourier transform

Abstract: The problem of determining the error in approximating the Fourier transform by the discrete Fourier transform is studied. Exact formulas for the relative error are established for classes of functions, called canonical-k (k/spl ges/0), and asymptotic error formulas are established for a much wider class of functions, called order-k. The formulas are dependent only on the class and not on the function in the class whose Fourier transform is being approximated, and this facilitates the application of the results.

Convolution using the undecimated discrete wavelet transform

Abstract: Convolution is one of the most widely used digital signal processing operations. It can be implemented using the fast Fourier transform (FFT) with a computational complexity of O (N log N). The undecimated discrete wavelet transform (UDWT) is linear and shift invariant, so it can also be used to implement convolution. We propose a scheme to implement the convolution using the UDWT, and study its advantages and limitations.

Analysis of algorithms for nonuniform-time discrete Fourier transform

Abstract: In order to perform spectral analysis of nonuniformly sampled time domain signals, algorithms for the Nonuniform-Time Discrete Fourier Transform (NUT-DFT) are developed and evaluated The NUT-DFT takes nonuniform time domain data and produces a uniform sampled spectrum. The proposed algorithms for NUT-DFT are based on approximating the continuous-time Fourier transform by different numerical integration algorithms. Special attention is paid to the ability of the transform to capture signals with frequency components above half the average sampling rate. The performance of the algorithms is studied experimentally by analysing both nonuniformly and uniformly sampled data.

Enhanced fast fourier transform technique on vector processor with operand routing and slot-selectable operation

Abstract: An apparatus and a method perform an N-point Fast Fourier Transform (FFT) on first and second arrays having real and imaginary input values using a processor with a multimedia extension unit (MEU), wherein N is a power of two. The invention repetitively sub-divides the N-point Fourier Transform into N/2-point Fourier Transforms until only a 2-point Fourier Transform remains. Next, it vector processes the 2-point Fourier Transform using the MEU and cumulates the results of the 2-point Fourier Transforms from each of the sub-divided N/2 Fourier Transforms to generate the result of the N-point Fourier Transform.

Numerical model estimating the capabilities and limitations of the fast Fourier transform technique in absolute interferometry.

Abstract: Anumerical model was developed to emulate the capabilities of systems performing noncontact absolute distance measurements. The model incorporates known methods to minimize signal processing and digital sampling errors and evaluates the accuracy limitations imposed by spectral peak isolation by using Hanning, Blackman, and Gaussian windows in the fast Fourier transform technique. We applied this model to the specific case of measuring the relative lengths of a compound Michelson interferometer. By processing computer-simulated data through our model, we project the ultimate precision for ideal data, and data containing AM–FM noise. The precision is shown to be limited by nonlinearities in the laser scan.

L1-approximation of Fourier series of complex-valued functions

Abstract: V. B. Stanojevic suggested in her recent paper that it would be of interest to prove a corresponding L1-convergence theorem for Fourier series with complex O-regularly varying quasimonotonc coefficients. The present paper will discuss this question and establish L1-convergence and. furthermore. L1-approximation theorems for complex-valued integrable functions.

Every Fourier optical system is equivalent to consecutive fractional-Fourier-domain filtering

Abstract: We consider optical systems composed of an arbitrary number of lenses and filters, separated by arbitrary distances, under the standard approximations of Fourier optics. We show that every such system is equivalent to (i) consecutive filtering operations in several fractional Fourier domains and (ii) consecutive filtering operations alternately in the space and the frequency domains.

A Generalization of the Theorem of Hardy: A Most General Version of the Uncertainty Principle for Fourier Integrals

Abstract: N. Wiener remarked that a non - identically vanishing real function and its Fourier transform cannot both decay “very fast”. It was Hardy who specified and proved this assertion in 1933. In the present paper Hardy's theorem will be generalized. Moreover, it will be shown that further weakening of these assumptions does not make sense.