solution of multi delay systems using hybrid of block pulse functions and taylor series a method for finding the solution of linear time varying multi delay systems using a hybrid function is proposed the properties of the hybrid functions which consist of block pulse functions plus taylor series are presented the method is based upon expanding various time functions in the system as their truncated hybrid functions operational matrices of integration delay and product are presented and are utilized to reduce the solution of multi delay systems to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

the discrete rotational fourier transform we define a discrete version of the angular fourier transform and present the properties of the transform that show it to be a rotation in time frequency space this new transform is a generalization of the dft efficient algorithms for its computation can then be based on the fft and the eigenstructure of the dft

short time fractional fourier transform and its applications the fractional fourier transform frft is a potent tool to analyze the chirp signal however it fails in locating the fractional fourier domain frfd frequency contents which is required in some applications the short time fractional fourier transform stfrft is proposed to solve this problem it displays the time and frfd frequency information jointly in the short time fractional fourier domain stfrfd two aspects of its performance are considered the 2 d resolution and the stfrfd support the time frfd bandwidth product tfbp is defined to measure the resolvable area and the stfrfd support the optimal stfrft is obtained with the criteria that maximize the 2 d resolution and minimize the stfrfd support its inverse transform properties and computational complexity are presented two applications are discussed the estimations of the time of arrival toa and pulsewidth pw of chirp signals and the stfrfd filtering simulations verify the validity of the proposed algorithms

closed form discrete fractional and affine fourier transforms the discrete fractional fourier transform dfrft is the generalization of discrete fourier transform many types of dfrft have been derived and are useful for signal processing applications we introduce a new type of dfrft which are unitary reversible and flexible in addition the closed form analytic expression can be obtained it works in performance similar to the continuous fractional fourier transform frft and can be efficiently calculated by the fft since the continuous frft can be generalized into the continuous affine fourier transform aft the so called canonical transform we also extend the dfrft into the discrete affine fourier transform daft we derive two types of the dfrft and daft type 1 is similar to the continuous frft and aft and can be used for computing the continuous frft and aft type 2 is the improved form of type 1 and can be used for other applications of digital signal processing meanwhile many important properties continuous frft and aft are kept in the closed form dfrft and daft and some applications such as filter design and pattern recognition are also discussed the closed form dfrft we introduce has the lowest complexity among all current dfrfts that is still similar to the continuous frft

maxwellian circuits of conducting circular loops circular loops are useful fundamental devices subjects of many investigations before the era of the method of moments mom it was difficult to analyze a circular loop because it was hard to find the current distribution on the loop the classical investigations on circular loops involve fairly complicated mathematical analyses in this paper based on the theory of maxwellian circuits mc it is discovered that the current of a circular loop is the solution of a classical wave equation with complex constant parameters once the parameters are determined the current may be solved analytically or by simple numerical methods very broad band computations of circular loops up to 40 ghz have been performed it shows that the maxwellian circuits formulations can indeed simplify the analyses of conducting circular loops and provide valuable insights to rf engineers

parameter estimation of chirp signals the problem of the parameter estimation of chirp signals is addressed several closely related estimators are proposed whose main characteristics are simplicity accuracy and ease of online or offline implementation for moderately high signal to noise ratios they are unbiased and attain the cramer rao bound monte carlo simulations verify the expected performance of the estimators it should be easy to extend this approach to signals having polynomials of any degree in the exponent all the derivations will be done under the assumption that the signal to noise ratio is sufficiently high

the fractional fourier transform and time frequency representations the functional fourier transform frft which is a generalization of the classical fourier transform was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community to which it may however be potentially useful the frft depends on a parameter spl alpha and can be interpreted as a rotation by an angle spl alpha in the time frequency plane an frft with spl alpha spl pi 2 corresponds to the classical fourier transform and an frft with spl alpha 0 corresponds to the identity operator on the other hand the angles of successively performed frfts simply add up as do the angles of successive rotations the frft of a signal can also be interpreted as a decomposition of the signal in terms of chirps the authors briefly introduce the frft and a number of its properties and then present some new results the interpretation as a rotation in the time frequency plane and the frft s relationships with time frequency representations such as the wigner distribution the ambiguity function the short time fourier transform and the spectrogram these relationships have a very simple and natural form and support the frft s interpretation as a rotation operator examples of frfts of some simple signals are given an example of the application of the frft is also given

discrete fractional fourier transforms based on closed form hermite gaussian like dft eigenvectors in this paper we construct discrete fractional fourier transforms dfrft using recently introduced closed form hermite gaussian like hgl eigenvectors with respect to such eigenvectors we discuss the convergence of their components to samples of the corresponding continuous hermite gaussian functions and propose solutions to deal with some restrictions related to their construction this allows us to give new procedures for obtaining orthonormal bases of hgl eigenvectors which are used to fractionalize the discrete fourier transform we illustrate the application of the resulting dfrft in the scenario of filtering in the fractional domain and compare the results with existing dfrft approaches

spectral analysis and reconstruction for periodic nonuniformly sampled signals in fractional fourier domain nonuniform sampling occurs in many applications due to imperfect timebase or random events periodic nonuniform sampling is a special type of nonuniform sampling the properties and applications of periodic nonuniform sampling signals in traditional fourier domain have been extensively studied but so far none of the research papers covering the spectral analysis and reconstruction of periodic nonuniformly sampled signals in fractional fourier domain have been published this correspondence is to explore the spectral properties of periodic nonuniformly sampled signals associated with the fractional fourier transform first based on the uniform fractional fourier transform kernel the general spectral representation of periodic nonuniformly sampled signals has been derived second detailed analysis of periodic nonuniformly sampled chirp signals in the fractional fourier domain have been performed the results can be used to estimate the chirp rate and the sampling offsets finally a more simple relationship between the discrete fractional spectrum of periodic nonuniformly sampled signals and the continuous spectrum has been obtained based on this relationship the original continuous spectrum can be reconstructed from periodic nonuniformly sampled signals in fractional fourier domain in addition the simulations are carried out to verify the correctness of the results

fractional power spectrum in this paper by investigating the definitions of the fractional power spectrum and the fractional correlation for the deterministic process we consider the case associated with the random process in an explicit manner the fractional power spectral relations for the fractional fourier domain filter are derived and the expression for the fractional power spectrum in terms of the fractional correlation is obtained in addition the definitions and the properties of the fractional white noise and the chirp stationary process are presented simulation results verify the theoretical derivations and demonstrate the potential applications such as detection and parameter estimation of chirp signals fractional power spectral estimation and system identification in the fractional fourier domain

multiple parameter discrete fractional transform and its applications in recent years several special multiple parameter discrete fractional transforms mpdfrts have been proposed and their advantages have been demonstrated in the fields of communication systems and information security however the general theoretical framework of mpdfrts has not yet been established in this paper we propose two separate theoretical frameworks called the type i and ii mpdfrt that can include existing multiple parameter transforms as special cases the properties of the type i and ii mpdfrt have been analyzed in detail and their high dimensional operators have been defined under the theoretical frameworks we can construct new types of transforms that may be useful in signal processing and information security finally we perform two applications about image encryption and image feature extraction in the type i and ii mpdfrt domain the simulation results demonstrate that the typical transforms constructed under the proposed theoretical frameworks yield promising results in these applications

eigenvectors of ordinary generalized centered and offset discrete fourier transforms based on lookup table methods efficiency and approximation uses in this paper we first propose a methodology to construct an eigenvector of the n n discrete fourier transform dft matrix from any eigenvector of the k 2 n k 2 n dft matrix where k is a positive integer specifically using the procedure to down sample by k any k 2 n point dft eigenvector with zero as the initial index and then fold modulus n and add the resultant down sampled kn point vector we obtain an n point dft eigenvector computing an n point dft eigenvector using the proposed method is very efficient which requires only n k 1 additions and no multiplications with a pre stored lookup table of k 2 n point dft eigenvectors similar efficient methods are also developed to compute n point eigenvectors of other dft related transforms including the generalized dft gdft centered dft cdft and the offset dft odft from k 2 n point eigenvectors of their corresponding transforms or the dft as application examples we apply the proposed methods to compute n point dft gdft cdft and odft eigenvectors which are closer to samples of the continuous hermite gaussian function hgf than existing n point eigenvectors from their corresponding k 2 n point eigenvectors these improved n point hermite gaussian like eigenvectors can be used to define fractional versions of the various n n dft transforms whose outputs are closer to samples of the continuous fractional fourier transform than outputs of the corresponding existing fractional transforms finally we perform computer experiments to verify the effectiveness and demonstrate applications of our proposed methods

optimal filtering in fractional fourier domains for time invariant degradation models and stationary signals and noise the classical fourier domain wiener filter which can be implemented in o n log n time gives the minimum mean square error estimate of the original undistorted signal for time varying degradations and nonstationary processes however the optimal linear estimate requires o n sup 2 time for implementation we consider filtering in fractional fourier domains which enables significant reduction of the error compared with ordinary fourier domain filtering for certain types of degradation and noise especially of chirped nature while requiring only o n log n implementation time thus improved performance is achieved at no additional cost expressions for the optimal filter functions in fractional domains are derived and several illustrative examples are given in which significant reduction of the error by a factor of 50 is obtained

discrete chirp fourier transform and its application to chirp rate estimation the discrete fourier transform dft has found tremendous applications in almost all fields mainly because it can be used to match the multiple frequencies of a stationary signal with multiple harmonics in many applications wideband and nonstationary signals however often occur one of the typical examples of such signals is chirp type signals that are usually encountered in radar signal processing such as synthetic aperture radar sar and inverse sar imaging due to the motion of a target the radar return signals are usually chirps and their chirp rates include the information about the target such as the location and the velocity in this paper we study discrete chirp fourier transform dcft which is analogous to the dft besides the multiple frequency matching similar to the dft the dcft can be used to match the multiple chirp rates in a chirp type signal with multiple chirp components we show that when the signal length n is prime the magnitudes of all the sidelobes of the dcft of a quadratic chirp signal are 1 whereas the magnitude of the mainlobe of the dcft is spl radic n with this result an upper bound for the number of the detectable chirp components using the dcft is provided in terms of signal length and signal and noise powers we also show that the n point dcft performs optimally when n is a prime

applications of the maxwellian circuits to linear wire antennas and scatterers the recently proposed theory of maxwellian circuits is demonstrated for applications to linear wire scatterers as well as to linear antennas it is shown that for each integral equation of thin wire type there exist coupled linear ordinary differential equations of currents and voltages the solutions of which are identical to the integral equation if the same boundary conditions of the integral equation are applied the subsequence is that the coupled differential equations can be interpreted as equivalent circuit of new type named maxwellian circuit the equivalent circuit can provide physical insights to design engineers and computational advantages for broadband calculations the highlight of this paper is to show both theoretically and numerically that the maxwellian circuit components depend only on the geometry of the problem not on the excitation or boundary conditions at the terminals

fractional spectrum analysis for nonuniform sampling in the presence of clock jitter and timing offset this paper focuses on analyzing the fractional spectrum of nonuniform sampling which is affected by clock jitter and timing offset both cases of single channel and recurrent samplings with limited bandwidth in the fractional fourier domain are considered for which we study two types of discrete time fractional fourier transforms dtfrfts for the first type dtfrfts we perform their statistical analyses and meanwhile establish relationships between the fractional spectra of the input and output signals of non ideal single multi channel analog to digital converter such relationships indicate that the statistical mean of each fractional spectrum contains periodic replicas of the spectrum for the input signal and these replicas are particularly modulated by the characteristic function of the perturbations introduced by the clock jitter moreover we interpret the first type dtfrfts from the perspective of linear frequency modulation signal decomposition for nonuniform sampling the second type dtfrfts serve as approximate dtfrfts and they especially deal with the general situation when the nonuniform sampling instants are unknown to this end we study modified forms of the first type dtfrfts with known inputs in kernels based on which the statistical mean of the approximate fractional spectra for both cases are derived then we compensate the fractional spectrum bias with high accuracy by developing optimal filters that minimize the mean square error between the original and the compensated fractional spectra for both cases different from single channel sampling the compensation for recurrent sampling involves an additional step before optimal filtering which gives better accuracy simulation results show that the proposed methods outperform existing methods

the discrete polynomial phase transform the discrete polynomial phase transform dpt is a new tool for analyzing constant amplitude polynomial phase signals the main properties of the dpt are its ability to identify the degree of the phase polynomial and to estimate its coefficients the transform is robust to deviations from the ideal signal model such as slowly varying amplitude additive noise and nonpolynomial phase the authors define the dpt derive its basic properties and use it to develop computationally efficient estimation and detection algorithms a statistical accuracy analysis of the estimated parameters is also presented

spectral analysis of randomly sampled signals suppression of aliasing and sampler jitter nonuniform sampling can facilitate digital alias free signal processing dasp i e digital signal processing that is not affected by aliasing this paper presents two dasp approaches for spectrum estimation of continuous time signals the proposed algorithms named the weighted sample ws and weighted probability wp density functions respectively utilize random sampling to suppress aliasing both methods produce unbiased estimators of the signal spectrum to achieve this effect the computational procedure for each method has been suitably matched with the probability density function characterising the pseudorandom generators of the sampling instants both proposed methods are analyzed and the qualities of the estimators they produce have been compared with each other although none of the proposed spectrum estimators is universally better than the other one it has been shown that in practical cases the wp estimator produces generally smaller errors than those obtained from ws estimation a practical limitation of the approaches caused by the sampling instant jitter is also studied it has been proven that in the presence of jitter the theoretically infinite bandwidths of ws and wp signal analyses are limited the maximum frequency up to which these analyses can be performed is inversely proportional to the size of the jitter

filterbank reconstruction of bandlimited signals from nonuniform and generalized samples this paper introduces a filterbank interpretation of various sampling strategies which leads to efficient interpolation and reconstruction methods an identity which is referred to as the interpolation identity is developed and is used to obtain particularly efficient discrete time systems for interpolation of generalized samples as well as a class of nonuniform samples to uniform nyquist samples either for further processing in that form or for conversion to continuous time the interpolation identity also leads to new sampling strategies including an extension of papoulis 1977 generalized sampling expansion

optimal control method with time delay in control abstract optimal control method for active vibration control of linear time delay systems is investigated in this paper in terms of two cases that time delay is integer and non integer times of sampling period motion equation with time delay is transformed as standard discrete forms which contain no time delay by using zero order holder respectively discrete quadratic function is used as objective function in design of controller to guarantee good control efficiency on sampling points in every step of computation of the deduced controller it contains not only current step of state feedback but also linear combination of some former steps of control because the controller is deduced directly from time delay differential equation system stability can be guaranteed easily thus this method is generally applicable to ordinary control systems the performance of the control method proposed and system stability when using this method are all demonstrated by numerical simulation results simulation results demonstrate that the presented method is a viable and attractive control strategy for applications to active vibration control instability in responses occurs possibly if the systems with time delay are controlled using controller designed in case of no time delay

fast simple and accurate computation of the currents on an arbitrarily large circular loop antenna this paper is presenting a very efficient method for computing the currents on a circular loop antenna with arbitrarily large size pocklington s integral equation is formulated for the circular loop geometry and the method of moments with point matching is applied to cast the equation s discrete counterpart the basis functions are chosen in such a way that the relevant square matrix is circulant and therefore amenable to exact eigenvalue analysis subsequently the matrix is diagonalized and inverted analytically yielding simple analytical expressions for the weights of the basis functions and hence the current and the input admittance of the antenna the algorithm utilizes mainly elementary mathematical functions as opposed to already known expressions in the literature and yields results in the form of a rapidly convergent single series applicable to extremely large loops

digital computation of the fractional fourier transform 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

unified fractional fourier transform and sampling theorem the fractional fourier transform frt is an extension of the ordinary fourier transform ft applying the language of the unified ft we develop frt expressions for discrete and continuous signals introducing a particular form of periodicity chirp periodicity the frt sampling theorem is derived as an extension of its ordinary counterpart

robust saturation controller for linear time invariant system with structured real parameter uncertainties abstract this paper is focused on a robust saturation controller for the linear time invariant lti system involving both actuator s saturation and structured real parameter uncertainties the controller suggested in this paper can analytically prescribe the upper and lower bounds of parameter uncertainties and guarantee the closed loop robust stability of the system in the presence of actuator s saturation the suboptimal bang bang control method is extended to lti system with parameter uncertainties based on affine quadratic stability and multi convexity concept the robust optimal bang bang controller is newly derived by minimizing the time derivative of affine lyapunov function subjected to the limit of control force since this controller is a gain scheduled type it requires the exact knowledge of uncertain parameters another robust saturation controller with a fixed gain is proposed and the linear matrix inequality lmi based sufficient existence conditions for a fixed gain controller are derived the effectiveness and the availability of the proposed controller are investigated by a practical numerical example through numerical simulations it is confirmed that the proposed robust saturation controller is robustly stable with respect to parameter uncertainties over the prescribed range defined by the upper and lower bounds

robust h control for a class of uncertain mechanical systems in this article the problem of h control is investigated for a class of mechanical systems with input delay and parameter uncertainties which appear in all the mass damping and stiffness matrices two approaches norm bounded and linear fractional transformation lft uncertainty formulations are considered by using a new lyapunov krasovskii functional approach combined with the advanced techniques for achieving delay dependence improved robust h state feedback controller design methods are developed the existence condition for admissible controllers is formulated in the form of linear matrix inequalities lmis and the controller design is cast into a convex optimisation problem subject to lmi constraints if the optimisation problem is solvable a desired controller can be readily constructed the result for the norm bounded uncertainty case improves the existing ones in terms of design conservatism and that for the lft uncertainty case represents the first attempt in this direction an ill

computing the ambiguity surface the computation of the discrete ambiguity function is considered two straightforward methods are developed depending upon whether we write the discrete ambiguity as a filter or as a discrete fourier transform a modification of the transform method produces an approximation to the discrete ambiguity function but has increased computational efficiency this method is based on a recent work 1 in most major applications we need to compute limited portions of the dft description of the discrete ambiguity function to do so we first pass a long sequence of data through a decimated fir filter and then use the fft algorithm on the results the remez algorithm is employed to control the resulting aliasing errors in the final section approximation theory is applied directly to the problem of computing the discrete ambiguity function the algorithm required to carry out the necessary computations turns out to be the same as on the prevoius approximation method based on a decimating fir filter

sparse discrete fractional fourier transform and its applications 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

discrete fractional fourier transform based on orthogonal projections the continuous fractional fourier transform frft performs a spectrum rotation of signal in the time frequency plane and it becomes an important tool for time varying signal analysis a discrete fractional fourier transform has been developed by santhanam and mcclellan see ibid vol 42 p 994 98 1996 but its results do not match those of the corresponding continuous fractional fourier transforms we propose a new discrete fractional fourier transform dfrft the new dfrft has dft hermite eigenvectors and retains the eigenvalue eigenfunction relation as a continous frft to obtain dft hermite eigenvectors two orthogonal projection methods are introduced thus the new dfrft will provide similar transform and rotational properties as those of continuous fractional fourier transforms moreover the relationship between frft and the proposed dfrft has been established in the same way as the conventional dft to continuous fourier transform

the discrete fractional fourier transform we propose and consolidate a definition of the discrete fractional fourier transform that generalizes the discrete fourier transform dft in the same sense that the continuous fractional fourier transform generalizes the continuous ordinary fourier transform this definition is based on a particular set of eigenvectors of the dft matrix which constitutes the discrete counterpart of the set of hermite gaussian functions the definition is exactly unitary index additive and reduces to the dft for unit order the fact that this definition satisfies all the desirable properties expected of the discrete fractional fourier transform supports our confidence that it will be accepted as the definitive definition of this transform

time delay estimation of chirp signals in the fractional fourier domain a new time delay estimator is presented in this paper it is evaluated based on the delay property of the fractional fourier transform with less computation and is suitable for chirp signals the statistical analysis in terms of signal to noise ratio snr and estimation accuracy for this estimation is also studied the proposed method for the time delay yields a variance which is theoretically equal to the cramer rao lower bound the validity of this estimation method is demonstrated via simulations

on sampling of band limited signals associated with the linear canonical transform sampling is one of the fundamental topics in the signal processing community theorems proposed under this topic form the bridge between the continuous time signals and discrete time signals several sampling theorems which aid in the reconstruction of signals in the linear canonical transform lct domain have been proposed in the literature however two main practical issues associated with the sampling of the lct still remain unresolved the first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically focusing on these issues this paper seeks to address the above from the lct point of view first we extend several previously developed theorems for signals band limited in the fourier domain to signals band limited in the lct domain followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the lct simulation results and the potential applications of the theorem are also proposed

sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain the fractional fourier transform frft has become a very active area in signal processing community in recent years with many applications in radar communication information security etc this study carefully investigates the sampling of a continuous time band limited signal to obtain its discrete time version as well as sampling rate conversion for the frft firstly based on product theorem for the frft the sampling theorems and reconstruction formulas are derived which explain how to sample a continuous time signal to obtain its discrete time version for band limited signals in the fractional fourier domain secondly the formulas and significance of decimation and interpolation are studied in the fractional fourier domain using the results the sampling rate conversion theory for the frft with a rational fraction as conversion factor is deduced which illustrates how to sample the discrete time version without aliasing the theorems proposed in this study are the generalizations of the conventional versions for the fourier transform finally the theory introduced in this paper is validated by simulations

sampling and reconstruction of signals in function spaces associated with the linear canonical transform the linear canonical transform lct has been shown to be useful and powerful in signal processing optics etc many results of this transform are already known including sampling theory most existing sampling theories of the lct consider the class of bandlimited signals however in the real world many analog signals arising in engineering applications are non bandlimited in this correspondence we propose a sampling and reconstruction strategy for a class of function spaces associated with the lct which can provide a suitable and realistic model for real applications first we introduce definitions of semi and fully discrete convolutions for the lct then we derive necessary and sufficient conditions pertaining to the lct under which integer shifts of a chirp modulated function generate a riesz basis for the function spaces by applying the results we present a more comprehensive sampling theory for the lct in the function spaces and further a sampling theorem which recovers a signal from its own samples in the function spaces is established moreover some sampling theorems for shift invariant spaces and some existing sampling theories for bandlimited signals associated with the fourier transform ft the fractional ft or the lct are noted as special cases of the derived results finally some potential applications of the derived theory are presented