Watermarking methods require robustness against various attacks. Conventional watermarking methods use error-correcting codes or spread spectrum to correct watermarking errors. Errors can also be reduced by embedding the watermark into the frequency domain and by using SIFT feature points. If the type and strength of the attack can be estimated, the errors can be further reduced. There are several types of attacks, such as scaling, rotation, and cropping, and it is necessary to aim for robustness against all of them. Focusing on the scaling tolerance of watermarks, we propose a watermarking method using SIFT feature points and DFT, and introduce a pilot signal. The proposed method estimates the scaling rate using the pilot signal in the form of a grid. When a stego-image is scaled, the grid interval of the pilot signal also changes, and the scaling rate can be estimated from the amount of change. The accuracy of estimating the scaling rate by the proposed method was evaluated in terms of the relative error of the scaling rate. The results show that the proposed method could reduce errors in the watermark by using the estimated scaling rate.

The present study proposes a scheme in which variable-length orthogonal codes generated by combining inverse discrete Fourier transform matrices over a finite field multiplex user data into a multiplexed sequence and its sequence forms one or a plural number of codewords for Reed-Solomon coding. The proposed scheme realizes data multiplexing, error correction coding, and multi-rate transmitting at the same time. This study also shows a design example and its performance analysis of the proposed scheme.

The present paper proposes orthogonal variable spreading factor codes over finite fields for multi-rate communications. The proposed codes have layered structures that combine sequences generated by discrete Fourier transforms over finite fields, and have various code lengths. The design method for the proposed codes and examples of the codes are shown.

Expectation propagation (EP) decoding is proposed for sparse superposition coding in orthogonal frequency division multiplexing (OFDM) systems. When a randomized discrete Fourier transform (DFT) dictionary matrix is used, the EP decoding has the same complexity as approximate message-passing (AMP) decoding, which is a low-complexity and powerful decoding algorithm for the additive white Gaussian noise (AWGN) channel. Numerical simulations show that the EP decoding achieves comparable performance to AMP decoding for the AWGN channel. For OFDM systems, on the other hand, the EP decoding is much superior to the AMP decoding while the AMP decoding has an error-floor in high signal-to-noise ratio regime.

In this letter, an efficient hybrid direction-of-arrival (DOA) estimation scheme is devised for massive uniform rectangular array. In this scheme, the DOA estimator based on a two-dimensional (2D) discrete Fourier transform is first applied to acquire coarse initial DOA estimates for single data snapshot. Then, the fine DOA is accurately estimated through using the iterative search estimator within a very small region. Meanwhile, a Nyström-based method is utilized to correctly compute the required noise-subspace projection matrix, avoiding the direct computation of full-dimensional sample correlation matrix and its eigenvalue decomposition. Therefore, the proposed scheme not only can estimate DOA, but also save computational cost, especially in massive antenna arrays scenarios. Simulation results are included to demonstrate the effectiveness of the proposed hybrid estimate scheme.

A light field, which is equivalent to a dense set of multi-view images, has various applications such as depth estimation and 3D display. One of the essential problems in light field applications is light field interpolation, i.e., view interpolation. The interpolation accuracy is enhanced by exploiting an inherent property of a light field. One example is that an epipolar plane image (EPI), which is a 2D subset of the 4D light field, consists of many lines, and these lines have almost the same slope in a local region. This structure induces a sparse representation in the frequency domain, where most of the energy resides on a line passing through the origin. On the basis of this observation, we propose a group sparsity prior suitable for light fields to exploit their line structure fully for interpolation. Specifically, we designed the directional groups in the discrete Fourier transform (DFT) domain so that the groups can represent the concentration of the energy, and we thereby formulated an LF interpolation problem as an overlapping group lasso. We also introduce several techniques to improve the interpolation accuracy such as applying a window function, determining group weights, expanding processing blocks, and merging blocks. Our experimental results show that the proposed method can achieve better or comparable quality as compared to state-of-the-art LF interpolation methods such as convolutional neural network (CNN)-based methods.

This letter presents an improved hybrid direction of arrival (DOA) estimation scheme with computational efficiency for massive uniform linear array. In order to enhance the resolution of DOA estimation, the initial estimator based on the discrete Fourier transform is applied to obtain coarse DOA estimates by a virtual array extension for one snapshot. Then, by means of a first-order Taylor series approximation to the direction vector with the one initially estimated in a very small region, the iterative fine estimator can find a new direction vector which raises the searching efficiency. Simulation results are provided to demonstrate the effectiveness of the proposed scheme.

The windowed interpolation DFT methods have been utilized to estimate the parameters of a single frequency and multi-frequency signal. Nevertheless, they do not work well for the real-valued sinusoids with closely spaced positive- and negative- frequency. In this paper, we describe a novel three-point windowed interpolation DFT method for frequency measurement of real-valued sinusoid signal. The exact representation of the windowed DFT with maximum sidelobe decay window (MSDW) is constructed. The spectral superposition of positive- and negative-frequency is considered and calculated to improve the estimation performance. The simulation results match with the theoretical values well. In addition, computer simulations demonstrate that the proposed algorithm provides high estimation accuracy and good noise suppression capability.

This letter presents an efficient hybrid direction of arrival (DOA) estimation scheme for massive uniform linear array. In this scheme, the DOA estimator based on a discrete Fourier transform (DFT) is first applied to acquire coarse initial DOA estimates for single data snapshot. And then, the fine DOA is accurately estimated through using the iterative search estimator within a very small region. It iteratively searches for correct DOA vector by minimizing the objective function using a Taylor series approximation of the DOA vector with the one initially estimated. Since the proposed scheme does not need to perform eigen-decomposition and spectrum search while maintaining better DOA estimates, it also has low complexity and real-time capability. Simulation results are presented to demonstrate the efficiency of the proposed scheme.

The Discrete Fourier Transform Test (DFTT) is a randomness test in NIST SP800-22. However, to date, the theoretical reference distribution of the DFTT statistic has not been derived, which is problematic. We propose a new test using power spectrum variance as the test statistic whose reference distribution can be derived theoretically. Note that the purpose of both the DFTT and the proposed test is to detect periodic features. Experimental results demonstrate that the proposed test has stronger detection power than the DFTT and that it test can be used even for short sequences.

In this paper, we propose a method of embedding and detecting data in printed images with several formats, such as different resolutions and numbers of blocks, using the camera of a tablet device. To specify the resolution of an image and the number of blocks, invisible markers that are embedded in the amplitude domain of the discrete Fourier transform of the target image are used. The proposed method can increase the variety of images suitable for data embedding.

In general, in-focus images are used in visual object detection because image blur is considered as a factor reducing detection accuracy. However, in-focus images make it difficult to separate target objects from background images, because of that, visual object detection becomes a hard task. Background subtraction and inter-frame difference are famous schemes for separating target objects from background but they have a critical disadvantage that they cannot be used if illumination changes or the point of view moves. Considering these problems, the authors aim to improve detection accuracy by using images with out-of-focus blur obtained from a camera with a shallow depth of field. In these images, it is expected that target objects become in-focus and other regions are blurred. To enable visual object detection based on such image blur, this paper proposes a novel scheme using DFT-based feature extraction. The experimental results using synthetic images including, circle, star, and square objects as targets showed that a classifier constructed by the proposed scheme showed 2.40% miss rate at 0.1 FPPI and perfect detection has been achieved for detection of star and square objects. In addition, the proposed scheme achieved perfect detection of humans in natural images when the upper half of the human body was trained. The accuracy of the proposed scheme is better than the Filtered Channel Features, one of the state-of-the-art schemes for visual object detection. Analyzing the result, it is convincing that the proposed scheme is very feasible for visual object detection based on image blur.

In this paper, a convolution theorem which is analogous to the theorem for Fourier transform is shown among a certain type of polynomials. We establish a fast method of the multiplication in a special class of quotient rings of multivariate polynomials over q-element finite field GF(q). The polynomial which we treat is one of expressing forms of the multiple-valued logic function from the product of the semigroups in GF(q) to GF(q). Our results can be applied to the speedup of both software and hardware concerning multiple-valued Boolean logic.

A projective Reed-Muller (PRM) code, obtained by modifying a Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and the dual code of a PRM code are known, and some decoding examples have been presented for low-dimensional projective spaces. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of correctable errors of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of the minimum distance decoding.

The authors proposed an algorithm for calculation of new lower bound (rank bounded distance) using the discrete Fourier transform in 2010. Afterward, we considered some algorithms to improve the original algorithm with moving the row or column. In this paper, we discuss the calculation method of the rank bounded distance by conjugate elements for cyclic codes.

Least squares error (LSE) method adopted recursively can be used to track the frequency and amplitude of signals in steady states and kinds of non-steady ones in power system. Taylor expansion is used to give another version of this recursive LSE method. Aided by variable-windowed short-time discrete Fourier transform, recursive LSEs with and without Taylor expansion converge faster than the original ones in the circumstance of off-nominal input singles. Different versions of recursive LSE were analyzed under various states, such as signals of off-nominal frequency with harmonics, signals with step changes, signals modulated by a sine signal, signals with decaying DC offset and additive Gaussian white noise. Sampling rate and data window size are two main factors influencing the performance of method recursive LSE in transient states. Recursive LSE is sensitive to step changes of signals, but it is in-sensitive to signals' modulation and singles with decaying DC offset and noise.

The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. It is necessary to construct the maximum value of the independent set. However, its computational complexity is very large. In this paper, we consider cyclic codes defined by their defining set, and a new method to calculate the lower bound of the minimum distance using the discrete Fourier transform (DFT) is shown. The computational complexity of this method is compared with the shift bound's one. Moreover construction of independent set is shown.

In this paper, we propose a normalization method dividing the gradient vector by the sum of the diagonal and two adjoining elements of the matrix expressing the correlation between the components of the discrete Fourier transform (DFT) of the reference signal used for the identification of unknown system. The proposed method can thereby improve the estimation speed of coefficients of adaptive filter.

In this letter, an area-efficient architecture for the hardware implementation of the real-time prime factor Fourier transform (PFFT) is presented. In the proposed architecture, a prime length DFT module with the one-point-per-cycle (OPPC) property is implemented by the parallel distributed arithmetic (DA), and a cyclic convolution feature is exploited to simplify the structure of the DA cells. Based on the proposed architecture, a real-time 65-point PFFT processor is designed, and the synthesis results show that it saves over 8% gates compared to the existing real-time 64-point DFT designs.

For cyclic codes some well-known lower bounds and some decoding methods up to the half of the bounds are suggested. Particularly, the shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. In this paper we consider cyclic codes defined by their defining set, and new simple derivation of the shift bound using the discrete Fourier transform with unknown elements and the Blahut theorem is shown. Moreover two examples of binary cyclic codes are given.