Fractional Cosine Research Articles

An Improved Image Compression Algorithm Using Wavelet and Fractional Cosine Transforms

An Improved Image Compression Algorithm Using Wavelet and Fractional Cosine Transforms

Fractional quaternion cosine transform and its application in color image copy-move forgery detection

In this paper, fractional quaternion cosine transforms (FrQCT) is proposed to generalize the conventional fractional cosine transforms (FrCT) to quaternion signal processing in a holistic manner. Firstly, the new transform FrQCT is defined and the proof of its inverse transform is presented. An efficient discrete implementation method of FrQCT is then proposed, in which the relationship between FrQCT and FrCT of four components is used for a quaternion signal. Finally, a new color image copy-move forgery detection algorithm based on FrQCT and modified PatchMatch matching algorithm is proposed to evaluate the performance of the proposed FrQCT. Experimental results on two public datasets (FAU dataset and GRIP dataset) demonstrate that: (a) the proposed efficient implementation method takes only half the computational time of the direct method; (b) the proposed FrQCT-based forgery detection algorithm can achieve a better performance than some state-of-the-art algorithms, especially in the additional operation case.

Existence Results of Mild Solutions for Impulsive Fractional Integrodifferential Evolution Equations With Nonlocal Conditions

Abstract In this paper, we investigate the existence of mild solutions of impulsive fractional integrodifferential evolution equations with nonlocal conditions via the fixed point theorems and fractional cosine family combined with solutions operator theorems. Our results improve and generalize some classical results. Finally, an example is given to illustrate the main results.

On generalized fractional cosine and sine transforms

Abstract In this paper, we define and investigate fractional cosine and sine transforms for functions as well as distributions. For these transforms, several properties are provided, the representation theorems are proved and the analyticity is established.

Faster-Than-Nyquist Non-Orthogonal Frequency-Division Multiplexing for Visible Light Communications

In this paper, we propose a faster-than-Nyquist non-orthogonal frequency-division multiplexing (NOFDM) scheme for visible light communications (VLCs) where the multiplexing/demultiplexing employs the inverse fractional cosine transform (IFrCT)/FrCT. Different to the common fractional Fourier transform-based NOFDM (FrFT-NOFDM) signal, FrCT-based NOFDM (FrCT-NOFDM) signal is real-valued, which can be directly applied to the VLC systems without the expensive up-conversion and thus it is more suitable for the cost-sensitive VLC systems. Under the same transmission rate, FrCT-NOFDM signal occupies smaller bandwidth compared to OFDM signal. When the bandwidth compression factor $\alpha$ is set to 0.8, 20% bandwidth saving can be obtained. Therefore, FrCT-NOFDM has higher spectral efficiency and suffers less high-frequency distortion compared to OFDM, which benefits the bandwidth-limited VLC systems. As the simulation results show, bit error rate performance of FrCT-NOFDM with $\alpha$ of 0.9 or 0.8 is better than that of OFDM. Meanwhile, FrCT-NOFDM has a superior security performance. In conclusion, FrCT-NOFDM shows the potential for application in the future VLC systems.

Securing image information using double random phase encoding and parallel compressive sensing with updated sampling processes

Recently, a new kind of image encryption approach using compressive sensing (CS) and double random phase encoding has received much attention due to the advantages such as compressibility and robustness. However, this approach is found to be vulnerable to chosen plaintext attack (CPA) if the CS measurement matrix is re-used. Therefore, designing an efficient measurement matrix updating mechanism that ensures resistance to CPA is of practical significance. In this paper, we provide a novel solution to update the CS measurement matrix by altering the secret sparse basis with the help of counter mode operation. Particularly, the secret sparse basis is implemented by a reality-preserving fractional cosine transform matrix. Compared with the conventional CS-based cryptosystem that totally generates all the random entries of measurement matrix, our scheme owns efficiency superiority while guaranteeing resistance to CPA. Experimental and analysis results show that the proposed scheme has a good security performance and has robustness against noise and occlusion.

Capacity limit for faster-than-Nyquist non-orthogonal frequency-division multiplexing signaling

Faster-than-Nyquist (FTN) signal achieves higher spectral efficiency and capacity compared to Nyquist signal due to its smaller pulse interval or narrower subcarrier spacing. Shannon limit typically defines the upper-limit capacity of Nyquist signal. To the best of our knowledge, the mathematical expression for the capacity limit of FTN non-orthogonal frequency-division multiplexing (NOFDM) signal is first demonstrated in this paper. The mathematical expression shows that FTN NOFDM signal has the potential to achieve a higher capacity limit compared to Nyquist signal. In this paper, we demonstrate the principle of FTN NOFDM by taking fractional cosine transform-based NOFDM (FrCT-NOFDM) for instance. FrCT-NOFDM is first proposed and implemented by both simulation and experiment. When the bandwidth compression factor α is set to 0.8 in FrCT-NOFDM, the subcarrier spacing is equal to 40% of the symbol rate per subcarrier, thus the transmission rate is about 25% faster than Nyquist rate. FTN NOFDM with higher capacity would be promising in the future communication systems, especially in the bandwidth-limited applications.

Pseudo-differential operators involving Fractional Fourier cosine (sine) transform

A brief introduction to the fractional Fourier cosine transform as well as fractional Fourier sine transform and their basic properties are given. Fractional Fourier cosine (fractional Fourier sine) transform of tempered distributions is studied. Pseudo-differential operators involving these transformations are investigated and discussed the continuity on certain spaces $\mathcal{S}_e $ and $\mathcal{S}_o$ .

Integral transformations applied to image encryption

In this paper we consider the application of the integral transformations for image encryption through optical systems, a mathematical algorithm under Matlab platform using fractional Fourier transform (FrFT) and Random Phase Mask (RPM) for digital images encryption is implemented. The FrFT can be related to others integral transforms, such as: Fourier transform, Sine and Cosine transforms, Radial Hilbert transform, fractional Sine transform, fractional Cosine transform, fractional Hartley transform, fractional Wavelet transform and Gyrator transform, among other transforms. The encryption scheme is based on the use of the FrFT, the joint transform correlator and two RPMs, which provide security and robustness to the implemented security system. One of the RPMs used during encryption-decryption and the fractional order of the FrFT are the keys to improve security and make the system more resistant against security attacks.

A Proposed Orthogonal Chirp Division Multiplexing (OCDM) Multicarrier Transceiver Based on the Discrete Fractional Cosine Transform

Doppler spread is a phenomenon caused by rapid changes in the channel response due to the movement of the transmitter/receiver through a multipath environment, which is also known as a doubly selective channel response. In doubly selective fading channel where the channel frequency response is regarded as rapidly time-varying, the inter carrier interference (ICI) occurs mainly because the traditional multicarrier transceivers are incapable to diagonalize the channel matrix. The proposed scenario is for an Orthogonal Chirp Division Multiplexing (OCDM) multicarrier system derived from Discrete Fractional Cosine Transform (DFrCT). The proposed DFrCT-OCDM multicarrier system is proofed to be less signal processing complexity and easier to implement, compared to the traditional Fast Fourier Transform (FFT). Simulation results reveal that the DFrCT provides superior performance reflected the significant improvement in the BitError Rate which reaches to 10 dB, better spectrum efficiency and better energy concentration properties than the conventional FFT. Finally, several equlizers are applied and they all show that the proposed DFrCT-OCDM over performs the other systems with all these equlizers.

Convolution theorem for fractional cosine‐sine transform and its application

Fractional cosine transform (FRCT) and fractional sine transform (FRST), which are closely related to the fractional Fourier transform (FRFT), are useful mathematical and optical tool for signal processing. Many properties for these transforms are well investigated, but the convolution theorems are still to be determined. In this paper, we derive convolution theorems for the fractional cosine transform (FRCT) and fractional sine transform (FRST) based on the four novel convolution operations. And then, a potential application for these two transforms on designing multiplicative filter is presented. Copyright © 2016 John Wiley & Sons, Ltd.

A Keystone-Based Cosine Transform

This paper introduces a new time-frequency transform, referred to as the keystone-based cosine transform (KCT), to process real-valued multi-component linear frequency-modulated signals. Theoretical derivation shows that the KCT is a bilinear-based transform, but has a property of asymptotic linearity with negligible cross-terms. It also has high signal energy concentration. Experimental simulations show that compared with fractional cosine transform, the KCT can provide better performance on the distribution concentration and noisy signal detection.

Image encryption scheme based on random fractional discrete cosine transform and dependent scrambling and diffusion

An image encryption scheme is proposed by combining the random fractional discrete cosine transform (RFrDCT) with the dependent scrambling and diffusion (DSD). The application of the randomization, irrational choice and vectorization of fractional orders and the randomization of generating sequences improves the key-sensitivity and thus enlarges the key space greatly. Both the locations and the values of RFrDCT transformed coefficients are changed during the stage of DSD to further enhance the security of image encryption scheme. Numerical simulations demonstrate that the proposed image encryption scheme is feasible, secure and capable of resisting common classical attacks.

Fractional order semilinear Volterra integrodifferential equations in Banach spaces

Sufficient conditions are established for the existence results of fractional order semilinear Volterra integrodifferential equations in Banach spaces. Results are obtained by using the theory of fractional cosine families and fractional powers of operators.

Peak-to-Average Power Ratio Performance Analysis for Orthogonal Chirp Division Multiplexing Multicarrier Systems Based on Discrete Fractional Cosine Transform

In doubly selective fading channels, the orthogonal frequency division multiplexing (OFDM) multicarrier system may fail. Chirp like basis (fractional Fourier transform-fractional cosine transform) may be used instead of complex exponential basis in this case to improve the system performance. However, in multicarrier transmission, the high peak to average power ratio (PAPR) of the transmitted signal is one of the difficult problems that face both the chirp and the exponential basis. In this paper, an evaluation for the PAPR performance of a multicarrier system based on the fractional cosine transform (FrCT) is introduced and then compared with DFrFT and FFT. Moreover, applying the SLAM technique over these systems is provided to understand the behaviour of these systems when applying SLAM. Simulations verify that this system obtains a better PAPR performance. Moreover, further PAPR reduction can be gained using the well-known PAPR reduction methods. Moreover, applying SLAM technique improves the performance of (dB) by 4 dB to 5 dB and all systems become as competitive to each other when SLAM is applied. Finally, BER performance comparison among OFDM, Discrete Cosine Transform MCM (DCT- MCM), Discrete Hartley Transform MCM (DHT-MCM), DFrFT-OCDM and DFrCT- OCDM MCM systems was done by means of simulation over 100,000 multicarrier blocks for each one and showed that our proposed scenario gave the best performance.

ENGLISH

Fractional Cosine Transform (FRCT) is a generalization of the ordinary cosine transform and it has similar relationship with Fractional Fourier Transform (FRFT). In actual computations of fractional cosine transform (FCT) and fractional sine transform (FST), the basic integrations are performed with quadratures. Because the data are sampled and the duration is finite, most of the quadratures can be implemented via matrix computations. In this paper we obtained some relations between fractional cosine and sine transform with other transforms. .

EER, Robustness and Generation Condition of Fingerprint Templates Based on the Fractional Cosine and Sine Transforms

Recently, we have proposed new fingerprint templates based on the 1D discrete fractional Fourier, cosine and sine transforms (DFRT, DFCT and DFST) in order to realize the fingerprint recognition system with high recognition accuracy and high robustness against attacks. In this study, the generation condition of the fingerprint templates, i.e., a range of the transforms’ orders, pi, is investigated in more detail to realize higher recognition accuracy. In addition, the EERs and the robustness are compared between the cases that a set of the transforms’ orders is changed for each person and the set is unchanged for each person. As a result, in the case that the set is changed for each person, the most appropriate templates can be obtained as the phase distributions (PDs) of the DFCTs and the DFSTs under the condition that 0.0< pi ≤0.3 and 0.0< pi ≤0.5.The EER is an order of 10 -7 %

Joint image compression–encryption using discrete fractional transforms

Exchange of data in the form of text and image on internet is in fast progression and it is spawning new compression and encryption algorithms for bandwidth and security respectively. We have proposed a new kind of joint algorithm using discrete fractional transforms for compression–encryption of image. In this algorithm, the discrete fractional Fourier transform which is discrete version of fractional Fourier transform, is used to compress the images with variation of its parameter ‘α’ (order of transform). The compressed image is encrypted using discrete fractional cosine transform to provide security. The advantage of this method is its feasible implementation in practice, superior, robustness, security and sensitivity of keys, which has a good prospect and practicability in information security field. Results of computer simulations are presented to verify the validity of the proposed method such as mean square error (MSE) and peak signal to noise ratio between the original image and decrypted image. Sensitivity for right decryption key is proved with respect to MSE.

A new characteristic property of Mittag-Leffler functions and fractional cosine functions

A new characteristic property of the Mittag-Leffler function $E_\alpha (at^\alpha )$ with $1<\alpha <2$ is deduced. Motivated by this property, a new notion, named $\alpha $-order cosine function, is developed. It is proved that an $\alpha $-order cosine

A Discrete Fractional Cosine Transform Based Speech Enhancement System Through Adaptive Kalman Filter Combined with Perceptual Weighting Filter with Pitch Synchronous Analysis

The speech enhancement plays a vital role commonly used in noisy environment to develop the performance of speech identification in mobile phones or in car navigation system. Thus the quality of the performance of the speech recognition is becoming worse due to the presence of noises in the surrounding. The objective is to increase the evident quality of the speech and to develop the transparency. Signal representation and enhancement in cosine transformation is observed to provide significant results. As an alternative of using DCT, a combination of conventional Discrete Cosine Transform (DCT) and Discrete Fourier Transform (DFT) which forms the another transform called as the Discrete Fractional Cosine Transform (DFrCT). The DFrCT have a free parameter, its fraction. In order to deal with the issue of frame to frame deviations of the Cosine Transformations, DFrCT is integrated with Pitch Synchronous Analysis (PSA). Also Pitch Synchronous OverLap and Add (PSOLA) method are used to enhance the performance of PSA. Moreover, in order to improve the noise minimization of the system, Improved Iterative Wiener Filtering approach called Adaptive Kalman Filter Combined with Perceptual Weighting Filter is used in this approach. This filter is used to eliminate the matrix operations, reduces both the calculation time and complexity. Thus, a novel DFrCT based speech enhancement using improved iterative filtering algorithm integrated with PSA is used in this approach