Showing papers on "Discrete sine transform published in 2014"

Sparse Discrete Fractional Fourier Transform and Its Applications

Abstract: The discrete fractional Fourier transform is a powerful signal processing tool with broad applications for nonstationary signals. In this paper, we propose a sparse discrete fractional Fourier transform (SDFrFT) algorithm to reduce the computational complexity when dealing with large data sets that are sparsely represented in the fractional Fourier domain. The proposed technique achieves multicomponent resolution in addition to its low computational complexity and robustness against noise. In addition, we apply the SDFrFT to the synchronization of high dynamic direct-sequence spread-spectrum signals. Furthermore, a sparse fractional cross ambiguity function (SFrCAF) is developed, and the application of SFrCAF to a passive coherent location system is presented. The experiment results confirm that the proposed approach can substantially reduce the computation complexity without degrading the precision.

The Hopping Discrete Fourier Transform [sp Tips&Tricks]

Abstract: The discrete Fourier transform (DFT) produces a Fourier representation for finite-duration data sequences. In addition to its theoretical importance, the DFT plays a key role in the implementation of a variety of digital signal-?processing algorithms. Several algorithms including the fast Fourier transform (FFT) and the Goertzel algorithm have been introduced for the fast implementation of the DFT [1], [2].

Fast Discrete Fourier Transform on Generalized Sparse Grids

Abstract: In this paper, we present an algorithm for trigonometric interpolation of multivariate functions on generalized sparse grids and study its application for the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. In particular, we derive estimates for the error and the cost. We construct interpolants with a computational cost complexity which is substantially lower than for the standard full grid case. The associated generalized sparse grid interpolants have the same approximation order as the standard full grid interpolants, provided that certain additional regularity assumptions on the considered functions are fulfilled. Numerical results validate our theoretical findings.

HEVC Transform and Quantization

Abstract: This chapter provides an overview of the transform and quantization design in HEVC. HEVC specifies two-dimensional transforms of various sizes from 4 × 4 to 32 × 32 that are finite precision approximations to the discrete cosine transform (DCT). In addition, HEVC also specifies an alternate 4 × 4 integer transform based on the discrete sine transform (DST) for use with 4 × 4 luma Intra prediction residual blocks. During the transform design, special care was taken to allow implementation friendliness, including limited bit depth, preservation of symmetry properties, embedded structure and basis vectors having almost equal norm. The HEVC quantizer design is similar to that of H.264/AVC where a quantization parameter (QP) in the range of 0–51 (for 8-bit video sequences) is mapped to a quantizer step size that doubles each time the QP value increases by 6. A key difference, however, is that the transform basis norm correction factors incorporated into the descaling matrices of H.264/AVC are no longer needed in HEVC simplifying the quantizer design. A QP value can be transmitted (in the form of delta QP) for a quantization group as small as 8 × 8 samples for rate control and perceptual quantization purposes. The QP predictor used for calculating the delta QP uses a combination of left, above and previous QP values. HEVC also supports frequency-dependent quantization by using quantization matrices for all transform block sizes. This chapter also provides an overview of the three special coding modes in HEVC (I_PCM mode, lossless mode, and transform skip mode) that modify the transform and quantization process by either skipping the transform or by skipping both transform and quantization.

A low energy HEVC inverse transform hardware

Abstract: In this paper, a novel energy reduction technique for High Efficiency Video Coding (HEVC) Inverse Discrete Cosine Transform (IDCT) and Inverse Discrete Sine Transform (IDST) for all transform unit (TU) sizes is proposed. The proposed technique calculates IDCT and IDST only for DC coefficient if the values of several predetermined forward transformed low frequency coefficients in a TU are smaller than a threshold. The proposed technique reduces the computational complexity of IDCT and IDST significantly. It increases the bit rate slightly for most video frames. It decreases the PSNR slightly for some video frames, and it increases the PSNR slightly for some video frames. In this paper, a low energy HEVC 2D inverse transform (IDCT and IDST) hardware for all TU sizes is also designed and implemented using Verilog HDL. In the worst case, the proposed hardware can process 48 Quad HD (3840x2160) video frames per second. The proposed technique reduced the energy consumption of this hardware up to 32%. Therefore, the proposed hardware can be used in portable consumer electronics products that require a real-time HEVC encoder.

Multi-eigenvalue communication via the nonlinear Fourier transform

Abstract: Information transmission using only the discrete component of the nonlinear Fourier transform is studied and multi-eigenvalue signal sets are presented that achieve spectral efficiencies greater than 3 bits/s/Hz.

Real-Time Estimation of Power System Frequency Using a Three-Level Discrete Fourier Transform Method

Abstract: This paper proposes a three-level discrete Fourier transform (DFT) method to provide an accurate estimate of power system frequency in real time. The first level decomposes a power system signal into two orthogonal cosine- and sine-filtered signals. The second and third levels are used to determine the amplitude ratio of the cosine- and sine-filtered signals without encountering the zero-crossing problem and with an increase in ability to suppress harmonics and inter-harmonics. The performance of the three-level DFT method is evaluated using computer-simulated signals with harmonics and inter-harmonics. The three-level DFT method is also implemented on a digital signal processor (DSP)-based hardware prototype, and its performance in the hardware implementation is evaluated using a real-time digital simulator (RTDS). The evaluation results show that the three-level DFT method can achieve real-time estimation of power system frequency with satisfactory performance.

Method for intra prediction improvements for oblique modes in video coding

Abstract: In various embodiments, a method and a decoder include identifying a directional intra prediction mode with an angle of prediction. The method also includes identifying a first and second reference neighboring samples in a block of the video along the angle of prediction; the angle of prediction intersects a pixel to be predicted. The method further includes determining which of the first and second reference samples is nearest the angle of prediction and applying a value of the nearest reference neighboring sample to the pixel as a predictor. Also, a method and a decoder include determining whether a block type of a block of the video is intra block copy. The method also includes responsive to the block type being the intra block copy, determining a transform block size of the block and, responsive to the transform block size being 4×4, applying a discrete sine transform to the block.

CORDIC-Based Unified Architectures for Computation of DCT/IDCT/DST/IDST

Abstract: In this paper, CORDIC (coordinate rotation digital computer)-based Cooley-Tukey fast Fourier transform (FFT)-like algorithms for power-of-two point discrete cosine transform/discrete sine transform/inverse discrete cosine transform/inverse discrete sine transform are proposed and their corresponding unified architectures are developed by fully reusing the unique two basic processing elements. The proposed algorithms have some distinguished advantages, such as FFT-like regular data flow, unique post-scaling factor, and arithmetic-sequence rotation angles. The developed unified architectures can compute four different transforms by simple routing the data flow according to the specific transform without feeding different transform coefficients or different transform kernels. The unfolding technique is used to overcome the problem of difficult to realize pipeline that occur in iterative CORDIC algorithms. Compared to existing unified architectures, the proposed architectures have a superior performance in terms of hardware complexity, control complexity, throughput, scalability, modularity, and pipelinability.

A finite class of orthogonal functions generated by Routh–Romanovski polynomials

Abstract: It is known that some orthogonal systems are mapped onto other orthogonal systems by the Fourier transform. In this article we introduce a finite class of orthogonal functions, which is the Fourier transform of Routh–Romanovski orthogonal polynomials, and obtain its orthogonality relation using Parseval identity.

An algorithm for fast Hilbert transform of real functions

Abstract: A simple and accurate algorithm to evaluate the Hilbert transform of a real function is proposed using interpolations with piecewise---linear functions. An appropriate matrix representation reduces the complexity of this algorithm to the complexity of matrix-vector multiplication. Since the core matrix is an antisymmetric Toeplitz matrix, the discrete trigonometric transform can be exploited to calculate the matrix---vector multiplication with a reduction of the complexity to O(N log N), with N being the dimension of the core matrix. This algorithm has been originally envisaged for self-consistent simulations of radio-frequency wave propagation and absorption in fusion plasmas.

Image and video processing using discrete fractional transforms

Abstract: The mathematical transforms such as Fourier transform, wavelet transform and fractional Fourier transform have long been influential mathematical tools in information processing. These transforms process signal from time to frequency domain or in joint time–frequency domain. In this paper, with the aim to review a concise and self-reliant course, the discrete fractional transforms have been comprehensively and systematically treated from the signal processing point of view. Beginning from the definitions of fractional transforms, discrete fractional Fourier transforms, discrete fractional Cosine transforms and discrete fractional Hartley transforms, the paper discusses their applications in image and video compression and encryption. The significant features of discrete fractional transforms benefit from their extra degree of freedom that is provided by fractional orders. Comparison of performance states that discrete fractional Fourier transform is superior in compression, while discrete fractional cosine transform is better in encryption of image and video. Mean square error and peak signal-to-noise ratio with optimum fractional order are considered quality check parameters in image and video.

Image Watermarking in the Linear Canonical Transform Domain

Abstract: The linear canonical transform, which can be looked at the generalization of the fractional Fourier transform and the Fourier transform, has received much interest and proved to be one of the most powerful tools in fractional signal processing community. A novel watermarking method associated with the linear canonical transform is proposed in this paper. Firstly, the watermark embedding and detecting techniques are proposed and discussed based on the discrete linear canonical transform. Then the Lena image has been used to test this watermarking technique. The simulation results demonstrate that the proposed schemes are robust to several signal processing methods, including addition of Gaussian noise and resizing. Furthermore, the sensitivity of the single and double parameters of the linear canonical transform is also discussed, and the results show that the watermark cannot be detected when the parameters of the linear canonical transform used in the detection are not all the same as the parameters used in the embedding progress.

Enhancing noisy speech signals using orthogonal moments

Abstract: This study describes a new approach to enhance noisy speech signals using the discrete Tchebichef transform (DTT) and the discrete Krawtchouk transform (DKT). The DTT and DKT are based on well-known orthogonal moments: the Tchebichef and Krawtchouk moments, respectively. The representations of speech signals using a limited number of moment coefficients and their behaviour in the domain of orthogonal moments are shown. The method involves removing noise from the signal using a minimum-mean-square error in the domain of the DTT or DKT. According to comparisons with traditional methods, the initial experiments yield promising results and show that orthogonal moments are applicable in the field of speech signal enhancement. The application of orthogonal moments could be extended to speech analysis, compression and recognition.

Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform

Abstract: A movie encryption scheme is proposed using a discrete multiple-parameter fractional Fourier transform and theta modulation. After being modulated by sinusoidal amplitude grating, each frame of the movie is transformed by a filtering procedure and then multiplexed into a complex signal. The complex signal is multiplied by a pixel scrambling operation and random phase mask, and then encrypted by a discrete multiple-parameter fractional Fourier transform. The movie can be retrieved by using the correct keys, such as a random phase mask, a pixel scrambling operation, the parameters in a discrete multiple-parameter fractional Fourier transform and a time sequence. Numerical simulations have been performed to demonstrate the validity and the security of the proposed method.

Sparsity-promoting optimal control of spatially-invariant systems

Abstract: We study the optimal design of sparse and block sparse feedback gains for spatially-invariant systems on a circle. For this class of systems, the state-space matrices are jointly diagonalizable via the discrete Fourier transform. We exploit this structure to develop an ADMM-based algorithm that significantly reduces the computational complexity relative to standard approaches. Specifically, the complexity of the developed algorithm scales linearly with the number of subsystems. This is in contrast to a cubic scaling when circulant structure is not exploited. Two examples are provided to illustrate the effectiveness of the developed approach.

Novel Tridiagonal Commuting Matrices for Types I, IV, V, VIII DCT and DST Matrices

Abstract: In this letter, we first propose new nearly tridiagonal commuting matrices of discrete Fourier transform (DFT) matrix and generalized DFT (GDFT) matrix. Then, using the block diagonalizations technique of circular-centrosymmetric and centrosymmetric matrices, we derive novel tridiagonal commuting matrices for each discrete cosine transform (DCT) and discrete sine transform (DST) matrices of the types I, IV, V, and VIII from the commuting matrices of the DFT and GDFT based on the relationships in matrix forms among DFT, GDFT, and various types of DCT and DST. Moreover, the novel tridiagonal commuting matrices of various types of DCT and DST do not have multiple eigenvalues. Last, with these novel commuting matrices, we can easily determine an orthonormal set of Hermite-like eigenvectors for each of their corresponding DCT or DST matrix.

Novel OFDM Based on C-Transform for Improving Multipath Transmission

Abstract: In this paper, a new orthogonal-frequency-division multiplexing using the C-transform (C-OFDM) is introduced. An exact closed-form bit-error rate (BER) is derived for the proposed C-OFDM system over multipath channels and different kinds of modulation formats. The BER performance of the proposed C-OFDM is evaluated by mathematics and simulation for different channel models, modulation formats, under zero-padding (ZP) and minimum mean-square-error (MMSE) detection. The results are then compared with those of the discrete cosine transform (DCT) and discrete Fourier transform (DFT)-based OFDM, showing that over multipath channel, the new C-OFDM has better BER performance than both the DCT-OFDM and the DFT-OFDM. The proposed scheme is also found to achieve some reduction in the peak-to-average power ratio (PAPR) in comparison with the aforementioned OFDM schemes as the block-diagonal-structure (BDS) property of the C-transform minimizes the input signal superposition. The proposed system has the merits of being resilience to multipath channel dispersion and relatively has lower PAPR.

A fast hybrid Jacket-Hadamard matrix based diagonal block-wise transform

Abstract: In this paper, based on the block (element)-wise inverse Jacket matrix, a unified fast hybrid diagonal block-wise transform (FHDBT) algorithm is proposed. A new fast diagonal block matrix decomposition is made by the matrix product of successively lower order diagonal Jacket matrix and Hadamard matrix. Using a common lower order matrix in the form of [111-1], a fast recursive structure can be developed in the FHDBT, which is able to convert a newly developed discrete cosine transform (DCT)-II, discrete sine transform (DST)-II, discrete Fourier transform (DFT), and Haar-based wavelet transform (HWT). Since these DCT-II, DST-II, DFT, and HWT are widely used in different areas of applications, the proposed FHDBT can be applied to the heterogeneous system requiring several transforms simultaneously. Comparing with pre-existing DCT-II, DST-II, DFT, and HWT, it is shown that the proposed FHDBT exhibits less the complexity as its matrix size gets larger. The proposed algorithm is also well matched to circulant channel matrix. From the numerical experiments, it is shown that a better performance can be achieved by the use of DCT/DST-II compression scheme compared with the DCT-II only compression method.

Discrete Fourier transform based pattern classifiers

Abstract: A technique for pattern classification using the Fourier tra nsform combined with the nearest neighbor classifier is proposed. The multidimensional fast Fourier transform ( FFT) is applied to the patterns in the data base. Then the magnitudes of the Fourier coefficients are sorted in desc ending order and the first P coefficients with largest magnitudes are selected, where P is a design parameter. These coefficients are then used in fur ther processing rather than the original patterns. When a noisy pattern is presente d for classification, the pattern’s P Fourier coefficients with largest magnitude are extracted. The coefficients are a rranged in a vector in the descending order of their magnitudes. The obtained vector is referred to as the signat ure vector of the corresponding pattern. Then the distance between the signature vector of the pattern to be cl assified and the signature vectors of the patterns in the data base are computed and the pattern to be classified is matc hed with a pattern in the data base whose signature vector is closest to the signature vector of the pattern bein g classified.

Performance Comparison of Hybrid Wavelet Transform Formed by Combination of Different Base Transforms with DCT on Image Compression

Abstract: In this paper image compression using hybrid wavelet transform is proposed. Hybrid wavelet transform matrix is formed using two component orthogonal transforms. One is base transform which contributes to global features of an image and another transform contributes to local features. Here base transform is varied to observe its effect on image quality at different compression ratios. Different transforms like Discrete Kekre Transform (DKT), Walsh, Real-DFT, Sine, Hartley and Slant transform are chosen as base transforms. They are combined with Discrete Cosine Transform (DCT) that contributes to local features of an image. Sizes of component orthogonal transforms are varied as 16-16, 32-8 and 64-4 to generate hybrid wavelet transform of size 256x256. Results of different combinations are compared and it has been observed that, DKT as a base transform combined with DCT gives better results for size 16x16 of both component transforms.

A novel method of filtration by the discrete heap transforms

Abstract: In this paper, we describe the method of filtering the frequency components of the signals and images, by using the discrete signal-induced heap transforms (DsiHT), which are composed by elementary rotations or Givens transformations. The transforms are fast, because of a simple form of decomposition of their matrices, and they can be applied for signals of any length. Fast algorithms of calculation of the direct and inverse heap transforms do not depend on the length of the processed signals. Due to construction of the heap transform, if the input signal contains an additive component which is similar to the generator, this component is eliminated in the transform of this signal, while preserving the remaining components of the signal. The energy of this component is preserved in the first point, only. In particular case, when such component is the wave of a given frequency, this wave is eliminated in the heap transform. Different examples of the filtration over signals and images by the DsiHT are described and compared with the known method of the Fourier transform.

Discrete fuzzy transform of higher degree

Abstract: In this paper, we reformulate the fuzzy transform of higher degree (F m -transform) proposed originally for an approximation of continuous functions to the discrete case. We introduce two types of F m -transform which components are defined using polynomials in the first case and using specific values of these polynomials in the second case. We provide an analysis of basic properties of F m -transform.

Robust Digital Image Reconstruction via the Discrete Fourier Slice Theorem

Abstract: The discrete Fourier slice theorem is an important tool for signal processing, especially in the context of the exact reconstruction of an image from its projected views. This paper presents a digital reconstruction algorithm to recover a two dimensional (2-D) image from sets of discrete one dimensional (1-D) projected views. The proposed algorithm has the same computational complexity as the 2-D fast Fourier transform and remains robust to the addition of significant levels of noise. A mapping of discrete projections is constructed to allow aperiodic projections to be converted to projections that assume periodic image boundary conditions. Each remapped projection forms a 1-D slice of the 2-D Discrete Fourier Transform (DFT) that requires no interpolation. The discrete projection angles are selected so that the set of remapped 1-D slices exactly tile the 2-D DFT space. This permits direct and mathematically exact reconstruction of the image via the inverse DFT. The reconstructions are artefact free, except for projection inconsistencies that arise from any additive and remapped noise. We also present methods to generate compact sets of rational projection angles that exactly tile the 2-D DFT space. The improvement in noise suppression that comes with the reconstruction of larger sized images needs to be balanced against the corresponding increase in computation time.

Feature Extraction with Ordered Mean Values for Content Based Image Classification

Abstract: Categorization of images into meaningful classes by efficient extraction of feature vectors from image datasets has been dependent on feature selection techniques. Traditionally, feature vector extraction has been carried out using different methods of image binarization done with selection of global, local, or mean threshold. This paper has proposed a novel technique for feature extraction based on ordered mean values. The proposed technique was combined with feature extraction using discrete sine transform (DST) for better classification results using multitechnique fusion. The novel methodology was compared to the traditional techniques used for feature extraction for content based image classification. Three benchmark datasets, namely, Wang dataset, Oliva and Torralba (OT-Scene) dataset, and Caltech dataset, were used for evaluation purpose. Performance measure after evaluation has evidently revealed the superiority of the proposed fusion technique with ordered mean values and discrete sine transform over the popular approaches of single view feature extraction methodologies for classification.

Joint estimation algorithm for multi-targets' motion parameters

Abstract: When multiple targets are within the same radar antenna beam and cannot be separated in the range dimension, the conventional imaging methods cannot be directly used to obtain a focused radar image. In this study, a new joint estimation algorithm for multi-targets' motion parameters is proposed. In this method, the first-order Keystone transform is first applied to correct the range walk of multiple targets simultaneously, and then the Lv's transform is used to estimate the motion parameters of targets including velocity and acceleration. The signal-to-noise ratio threshold for the proposed method is also given. The proposed method is fast and can obtain the accurate parameter estimation without knowing the number of targets and their motion information. Experimental results demonstrate the performance of the proposed algorithm. Comparisons between the proposed method and other methods, the maximum-likelihood method, fractional Fourier transform and discrete polynomial transform, are performed, which show that the proposed method can efficiently obtain the accurate parameter estimation with low computational burden.