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A novel method is proposed to construct arbitrary length perfect integer sequences based on all-pass polynomial. It uses a fact that associative sequence of an all-pass polynomial is guaranteed to be perfect. In addition, geometric series method is our special case. Moreover, perfect Gaussian integer sequences can also be constructed. Furthermore, this approach can be applied to symmetric perfect sequences design. Finally, perfect sequence with low degree of freedom can be achieved by the difference set. Concrete examples are illustrated.

In this letter, a novel method is proposed to generate discrete approximations of continuous Hermite–Gaussian functions with noninteger shift and dilation. Moreover, the approximations preserve orthogonality. In addition, the discrete Hermite–Gaussian functions reach the minimum time-bandwidth product under discrete version of Gabor uncertainty principle. Two applications are demonstrated. In fractional delay application, the proposed method is compared with the windowing method and proved to have smaller delay error. Another one is the signal expansion based on the proposed complete orthogonal basis.

In this letter, a closed-form orthogonal Ramanujan integer basis is proposed and obtained by performing Gram–Schmidt process from the Ramanujan sum and its circular shift. It has a surprisingly simple and sparse form, which is better than the original complete Ramanujan basis. The relationship between the original basis and proposed basis is also clearly illustrated. Therefore, the original basis can be replaced easily.

A novel method to generate integer 2-D discrete Fourier transform (DFT) pairs and eigenvectors was proposed. Using projection slice theorem and Ramanujan’s sum, the 2-D spatial signal is decomposed into 2-D gcd-delta functions which contain only zeroes and ones. The 2-D DFT of 2-D gcd-delta functions are also integers. The integer 2-D DFT pairs can be applied to obtain integer 2-D DFT eigenvectors and 2-D period detection. The connection between 2-D gcd-delta function and multidimensional Ramanujan’s Sum is also illustrated with two numerical examples.

In this letter we focus on discrete integer-periodic signals, whose periodicity are different from the periodicity of continuous periodic signals in many aspects. We introduce a class of discrete periodic signals called intrinsic integer-periodic function (IIPF). An IIPF contains only a single period in terms of downsampling, which leads to some interesting properties for analyzing periodic components from a discrete signal. We show that one can use Ramanujan sum to decompose discrete periodic signals into IIPF components. Finally, we also propose an integer periodic spectrum rather than frequency spectrum. Our results show that the proposed integer periodic spectrum outperforms the conventional Ramanujan Fourier transform.

In this work, we describe some methods to generate zero autocorrelation Gauss integer sequences of arbitrary length. These methods can be combined with each other so that more choices can be made.

We propose a system to enhance the visual quality of displays under low backlight. Our system can not only prolong the battery life but also enhance visual quality. First, we propose a constrained L0 gradient image decomposition to separate base layer and detail layer. Afterwards, we design a median-preserving gamma correction for compensation and a JND-based boosting for detail enhancement. Experimental results show that our system outperforms than the compared systems.

In this letter, a novel tunable frequency response is proposed for variable fractional delay filter with careful consideration on conjugate symmetry. This proposed fractional delay method can be efficiently realized by FFT using the tunable fractional delay parameter. Its equivalent closed form to windowing method is derived and presented. Several comparisons to other windowing methods are also demonstrated, such as the magnitude response, group delay, and approximation error between ideal frequency response.

In this letter, a new commuting matrix with random discrete Fourier transform (DFT) eigenvectors is first constructed. A random discrete fractional Fourier transform (RDFRFT) kernel matrix with random DFT eigenvectors and eigenvalues is then proposed. The RDFRFT has an important feature that the magnitude and phase of its transform output are both random. As an application example, a security-enhanced image encryption scheme based on the RDFRFT is illustrated.

In this letter, an iterative weighted least-squares (LS) approach is proposed for the design of variable fractional-delay (VFD) finite impulse response (FIR) digital filters in minimax sense. For deriving the elements of relevant matrices, there is no need for numerical integration or closed-form formulation and they obey given specifications. Comparing with the existing method, the proposed method does not lead to a misleading problem, and the performance of convergence is also better. Especially, it is found that the used LS weighting function is only dependent on the frequency, not on the tunable parameter, which makes it easier for the minimax design of VFD FIR digital filters.

A special class of odd-symmetric length-4N periodic signals is studied, and it is shown how the odd Ramanujan sums are used as weighting coefficients to compute their pure imaginary discrete Fourier transform (DFT) integer-valued coefficients. The odd Ramanujan sum, being the sums of complex roots of unity, can be calculated either using closed-form formulas or computed recursively through the impulse response of a derived infinite impulse response (IIR) filter. This special class of odd-symmetric signals and odd Ramanujan sums can be combined together with the previous even-symmetric special class signals and the well-known even Ramanujan sums as a useful tool for signal processing

Eight types of maximally flat finite impulse response (FIR) fractional delay filters are proposed. They are designed by separately approximating the real part and the imaginary part of the frequency response. Among these eight types of filters, two of them are equivalent to the existing design results in the open literature. Therefore, a more general closed-form design of the maximally flat FIR fractional delay filters is described in this letter, and some design examples of FIR fractional delay filters will be illustrated

The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security.

The three-dimensional (3-D) Fourier transform of the nonaccelerating object in an image sequence lies on a plane passing through the origin of the 3-D frequency domain. Instead of using a 3-D frequency planar filter, we propose a new filter design technique for object tracking using a one-dimensional trajectory filter-in the spatiotemporal domain. This filter can track a moving object with linear or nonlinear trajectory without prior knowledge of the target trajectory. Also, it is easy to design and has a very low computational complexity. More importantly, it can be implemented in real time.

In this letter, we present a novel robust method for embedding watermarks into dithered halftone images. The method is named paired sub-image matching ordered dithering (PSMOD), of which the decoder is provided with a priori information of the original watermark. The method utilizes the bit and sub-subimage interleaving preprocesses. The experiments show that the technique is sufficiently robust to guard against the cropping, tampering, and printed-and-scanned degradation processes, in either B/W or color dithered images. This technique is also sufficiently flexible for various levels of embedded capacities.

Most of the existing methods designed to implement fractional interpolation and decimation are limited by rational scaling factors such as L/M, where L and M are positive integers. The general procedure is usually done with up-sampling by L first, and then followed by down-sampling by M. This scheme, however, requires two steps and higher sampling rates to fulfill and is not capable in dealing with irrational scaling factors, which cannot be represented by L/M. We propose a new method to solve the above two difficulties by a single step. Furthermore, a new structure of a two-channel nonuniform perfect reconstruction filter bank is derived using this new method, and the experimental results are presented.

In this letter, we propose a novel all-pass (AP) fractional delay filter whose denominator polynomial is obtained by truncating the power series of a certain function. This function is derived from the frequency response of the denominator whose magnitude response is related to the desired phase response through the Hilbert transform since the denominator of a stable AP filter is of minimum phase. The target function and corresponding power series are calculated analytically and expressed in closed form. The closed-form expressions facilitate the analysis of stability. According to the properties for the coefficients of the denominator polynomial, we show that the proposed AP filter is stable for positive delay. Numerical examples indicate that the phase delays of the proposed filters are flat around DC.

This letter presents an efficient split vector-radix-2/8 fast Fourier transform (FFT) algorithm. The split vector-radix-2/8 FFT algorithm saves 14% real multiplications and has much lower arithmetic complexity than the split vector-radix-2/4 FFT algorithm. Moreover, this algorithm reduces 25% data loads and stores compared with the split vector-radix-2/4 FFT algorithm.

In this paper, the Taylor series expansion is used to transform the design problem of a fractional delay filter into the one of a first-order differentiator such that the conventional finite-impulse response and infinite-impulse response differentiators can be applied to design a fractional delay filter directly. The proposed structure is more efficient than the well-known Farrow structure in terms of filter coefficient storage because only one first-order differentiator needs to be designed and implemented. Moreover, one design example is demonstrated to illustrate the effectiveness of this new design approach.

In this paper, we propose a design of two-dimensional (2-D) linear-phase, diamond-shaped (DS) finite impulse response (FIR) filters by using Bernstein polynomials. Although the one-dimensional (1-D) FIR filter designed by the Bernstein polynomials has been well investigated, this approach has not been broadly applied to 2-D filter design yet. We present a novel method of designing the 2-D FIR DS filter. In order to be approximated by a 2-D Bernstein polynomial, the 2-D symmetrical frequency response is transformed into a new domain. The key observation is that the region of support of the transformed frequency response is not diamond-shaped. The boundary of the new region of support represents an ellipse, a circle, or a line, and is analytically derived. The resultant magnitude responses are flat in the passband and stopband.

In this letter, a closed-form expression for the impulse response of the generalized half-band (HB) maximally flat (MF) FIR filters is obtained by solving the linear equations of the MF conditions. Based on the resultant impulse responses, an efficient implementation structure is derived. The dynamic range of the multipliers of the new structure is shown to be greatly reduced in comparison to the one of the direct form impulse response.