generalized sampling expansions with multiple sampling rates for lowpass and bandpass signals in the fractional fourier transform domain the objective of generalized sampling expansion gse is the reconstruction of an unknown continuously defined function f left t right from samples of the responses from m linear time invariant lti systems that are each sampled using the 1 m th nyquist rate in this paper we investigate the gse for lowpass and bandpass signals with multiple sampling rates in the fractional fourier transform frft domain first we propose an improvement of papoulis gse which has multiple sampling rates in the frft domain based on the proposed gse we derive the periodic nonuniform sampling scheme and the derivative interpolation method by designing different fractional filters and selecting specific sampling rates in addition the papoulis gse and the previous gse associated with frft are shown to be special instances of our results second we address the problem of the gse of fractional bandpass signals a new gse for fractional bandpass signals with equal sampling rates is derived we show that the restriction of an even number of channels in the gse for fractional bandpass signals is unnecessary and perfect signal reconstruction is possible for any arbitrary number of channels further we develop the gse for a fractional bandpass signal with multiple sampling rates lastly we discuss the application of the proposed method in the context of single image super resolution reconstruction based on gse illustrations and simulations are presented to verify the validity and effectiveness of the proposed results

a multicarrier architecture based upon the affine fourier transform recently innovative multicarrier schemes have been proposed that exploit the transmission of chirp shaped waves e sup j2 spl pi ct2 by optimally choosing the chirp parameter c on the basis of the channel characteristics and are more robust to time varying channels than ordinary ofdm schemes this concept was applied to continuous time and to discrete time systems in the present paper we aim at developing those ideas using the affine fourier transform aft which is a very general formulation of chirp transforms we present a multicarrier modulation based upon the discrete form of the aft that is therefore inherently discrete and strictly invertible moreover it allows to define a circular prefix concept that is coherent with the chirp nature of the transmission the system can be efficiently implemented by adding a simple phase correction block to standard ofdm modulators demodulators and can effectively combat interchannel interference when the propagation channel is made of few multipath components affected by independent frequency offsets our discrete time multicarrier scheme is an improved version of martone s approach as we also show by simulation results and exhibits analogous characteristics to barbarossa s continuous time system

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

fast algorithms for polynomial time frequency transforms of real valued sequences this paper presents fast algorithms for computing the polynomial time frequency transform that deals with a real valued sequence of length apb where a b and p are positive integers in particular it shows that the polynomial time frequency transform has a conjugate symmetric property similar to that of the discrete fourier transform if the input sequence is real valued the computational complexities needed by these proposed algorithms are analyzed in terms of the numbers of real additions and real multiplications when a 3 4 and 8 comparisons show that the computational complexities required by the proposed algorithms are less than 60 of those needed by the fast algorithms for complex valued sequences

signal to noise ratio and frequency analysis of continuous loop averaging deconvolution clad of overlapping evoked potentials in this study a frequency domain formulation of continuous loop averaging deconvolution clad of overlapping evoked potentials is developed and applied for the extraction of transient responses from recordings obtained at high stimulation rates this formulation allows for a faster execution of clad by using fast fourier transform algorithms the frequency characteristics of the deconvolution filter depends exclusively on the stimulus sequence and determines whether the noncoherent noise is amplified or attenuated in different frequencies a formula for calculating the signal to noise ratio snr achieved by the deconvolution process is developed the newly developed theory and the methodology is applied to the extraction of the auditory brainstem and middle latency responses using various sequences the effects of the sequence used and the number of sweeps averaged in ongoing acquisition on snr are examined by using single sweep recordings the results verify the deconvolution theory and the methodology and show its limitations depending on the frequency characteristics of the sequence the deconvolution process can amplify or attenuate the eeg noise proper selection of the stimulus sequence can increase the snr enhancement obtained with conventional averaging

generalized fast algorithms for the polynomial time frequency transform this paper presents a general class of fast algorithms for computing the polynomial time frequency transform ptft of length apb where a b and p are positive integers the process of derivation shows some interesting properties that are effectively used for minimization of the computational complexity by assigning values of a b and p various algorithms for example radix o and split radix 2 2a can be easily obtained to provide the flexibility supporting polynomial time frequency transforms of various sequence lengths the detailed analysis on the computational complexities needed by these algorithms is also presented in terms of the numbers of additions and multiplications it is shown that the proposed algorithms significantly reduce the computational complexity for applications that deal with polynomial phase signals

baseline correction of overlapping event related responses using a linear deconvolution technique the time it takes an event related response err to subside is often longer than the interval between successive events so that the response to a new event interferes with a baseline formed by responses to preceding events thus without proper baseline correction the interpretation of an event triggered average eta of recorded data can be problematic as the spectral compositions of err and baseline typically overlap filtering the eta is not always an adequate solution the approach introduced here exploits that eta and err are linearly related unless the series of events is exactly periodic the err can be derived from the eta by linear deconvolution the performance of the method is illustrated with simulated examples as well as data from an auditory evoked field aef study it is also outlined how to handle experiments with two or more different events intriguing applications beyond the scope of baseline correction arise from the fact that the err is invariably estimated for a time window longer than the mean interval between successive events the method may help for example to better understand the relationship between transient and steady state responses or to delineate the component structure of a specific err

complex scaled tangent rotations cstar for fast space time adaptive equalization of wireless tdma a new update algorithm for space time equalization of wireless time division multiple access signals is presented the method is based on a modified qr factorization that reduces the computational complexity of the traditional qr decomposition based recursive least squares method and maintains numerical stability square roots operations are avoided due to the use of an approximately orthogonal transformation defined complex scaled tangent rotation

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

a comparison of maximum length and legendre sequences for the derivation of brain stem auditory evoked responses at rapid rates of stimulation experiments were performed in which brain stem auditory evoked responses baers were elicited by two types of pseudorandom pulse trains maximum length sequences mls and legendre sequences lgs in experiment 1 each pulse sequence was presented at 50 db nhl with minimum pulse intervals varying from 1 to 10 ms wave v latency increased and wave v amplitude decreased with decreasing minimum pulse intervals with no significant effect of the type of sequence mls vs lgs and no significant interaction between sequence and minimum pulse interval in terms of wave v amplitude or latency in a second experiment the minimum pulse interval was held constant at 4 ms while mls and lgs levels were varied from 20 to 60 db nhl with increasing click intensity there is a decrease in wave v latency and an increase in wave v amplitude there was no significant effect of type of sequence lgs vs mls or interaction between type of sequence and stimulus intensity for wave v amplitude or latency despite the obvious

generalized set valued variational inclusions and resolvent equations in this paper we establish the equivalence between the generalized set valued variational inclusions the resolvent equations and the fixed point problem using the resolvent operator technique this equivalence is used to suggest and analyze some iterative algorithms for solving the generalized set valued variational inclusions and related optimization problems

a sampling theorem for fractional wavelet transform with error estimates as a generalization of the ordinary wavelet transform the fractional wavelet transform frwt is a very promising tool for signal analysis and processing many of its fundamental properties are already known however little attention has been paid to its sampling theory in this paper we first introduce the concept of multiresolution analysis associated with the frwt and then propose a sampling theorem for signals in frwt based multiresolution subspaces the necessary and sufficient condition for the sampling theorem is derived moreover sampling errors due to truncation and aliasing are discussed the validity of the theoretical derivations is demonstrated via simulations

a multicarrier system based on the fractional fourier transform for time frequency selective channels traditional multicarrier techniques perform a frequency domain decomposition of a channel characterized by frequency selective distortion in a plurality of subchannels that are affected by frequency flat distortion the distortion in each independent subchannel can then be easily compensated by simple gain and phase adjustments typically digital fourier transform schemes make the implementation of the multicarrier system feasible and attractive with respect to single carrier systems however when the channel is time frequency selective as it usually happens in the rapidly fading wireless channel this traditional methodology fails since the channel frequency response is rapidly time varying the optimal transmission reception methodology should be able to process nonstationary signals in other words the subchannel carrier frequencies should be time varying and ideally decompose the frequency distortion of the channel perfectly at any instant in time however this ideally optimal approach presents significant challenges both in terms of conceptual and computational complexity the idea disclosed in this work is that a nonstationary approach can be approximated using signal bases that are especially suited for the analysis synthesis of nonstationary signals we propose in fact the use of a multicarrier system that employs orthogonal signal bases of the chirp type that in practice correspond to the fractional fourier transform signal basis the significance of the methodology relies on the important practical consideration that analysis synthesis methods of the fractional fourier type can be implemented with a complexity that is equivalent to the traditional fast fourier transform

transform order division multiplexing multiplexing in the fractional fourier domain frfd only transmits the signals in a single direction in the time frequency plane and the transform order of the fractional fourier transform frft is not used sufficiently in this paper we apply the perfect reconstruction transmultiplexer in the frfd to develop the transform order division multiplexing todm which transmits the original information in the multidirections in the time frequency plane simultaneously the proposed todm systems can be divided into two types one needs the multitransmission of the original information while the other needs the modifications of the original information by the interpolation and the inverse discrete frft the performance analysis of the todm systems is also presented including the system capacity the reconstruction accuracy the security and the limitation compared with the multiplexing in the frfd the todm is more flexible and the security of the system is highly strengthened finally simulation results confirm the obtained analytical results

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

a new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings in this paper we introduce and study a new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings an existence theorem of solutions is proved without compactness assumptions a new iterative algorithm is suggested and analysed the convergence of iterative sequences generated by the new algorithm are also given as special cases some known results are also discussed

the feasibility of maximum length sequences to reduce acquisition time of the middle latency response maximum length sequences mls have been used to improve the signal to noise ratio snr of otoacoustic emissions thornton j acoust soc am 94 132 136 1993 and the auditory brainstem response thornton and slaven br j audiol 27 205 210 1993 by implication a shorter recording time would be required to give equal signal to noise ratio snr this study aimed to establish whether it is also possible to improve the snr of the auditory evoked potential termed the middle latency response mlr using maximum length sequences mls recordings of 180 s each were made using a conventional recording rate and mls rates of 42 89 and 185 clicks s three different stimulus intensities were used in the range 30 to 70 db nhl the rate of 89 clicks s was found to produce most improvement in snr for both the na pa region of the mlr and the na pb region this improvement in snr using mls implies that an mls rate of 89 clicks s would produce a fourfold reduction in recording time for equal snr over conventional recording for the pa nb region of the mlr at a stimulus intensity of 70 db nhl the latency of the nb wave was found to reduce significantly using mls an mlr could not be recorded from every subject in this study but more subjects had an identifiable response for mls than for conventional recordings use of mls to record the mlr appears to offer the potential for reduction in test time and better wave identification

iterative methods for generalized nonlinear variational inequalities abstract in this paper we establish the equivalence between the generalized nonlinear variational inequalities and the generalized wiener hopf equations this equivalence is used to suggest and analyze a number of iterative algorithms for solving generalized variational inequalities we also discuss the convergence analysis of the proposed algorithms as a special case we obtain various known results from our results

signal to noise ratio analysis of maximum length sequence deconvolution of overlapping evoked potentials in this study a general formula for the signal to noise ratio snr of the maximum length sequence mls deconvolution averaging is developed using the frequency domain framework of the generalized continuous loop averaging deconvolution procedure ozdamar and bohorquez j acoust soc am 119 429 438 2006 this formulation takes advantage of the well known equivalency of energies in the time and frequency domains parseval s theorem to show that in mls deconvolution snr increases with the square root of half of the number of stimuli in the sweep this increase is less than that of conventional averaging which is the square root of the number of sweeps averaged unlike arbitrary stimulus sequences that can attenuate or amplify phase unlocked noise depending on the frequency characteristics the mls deconvolution attenuates noise in all frequencies consistently furthermore mls and its zero padded variations present optimal attenuation of noise at all frequencies yet they present a highly jittered stimulus sequence in real recordings of evoked potentials the time advantage gained by noise attenuation could be lost by the signal amplitude attenuation due to neural adaptation at high stimulus rates

split radix algorithms for arbitrary order of polynomial time frequency transforms the polynomial time frequency transform is one of important tools for estimating the coefficients of the polynomial phase signals ppss with the maximum likelihood method the transform converts a one dimensional 1 d data sequence into a multidimensional output array from which the phase coefficients of the data sequence are estimated a prohibitive computational load is generally needed for high order polynomial phase signals although the 1 d fast fourier transform fft algorithm can be used based on the split radix concept this paper derives a fast algorithm for arbitrary order of polynomial time frequency transforms to significantly reduce the computational complexity comparisons on the computational complexity needed by various algorithms are also made to show the merits of the proposed algorithm

deconvolution of evoked responses obtained at high stimulus rates continuous loop averaging deconvolution clad is a new general mathematical theory and method developed to deconvolve overlapping auditory evoked responses obtained at high stimulation rates using clad arbitrary stimulus sequences are generated and averaged responses deconvolved until now only a few special stimulus series such as maximum length sequences mls and legendre sequences lgs were capable of performing this task a clad computer algorithm is developed and implemented in an evoked potential averaging system computer simulations are used to verify the theory and methodology auditory brainstem responses abr and middle latency responses mlr are acquired from subjects with normal hearing at high stimulation rates to validate and show the feasibility of the clad technique

fractional fourier series expansion for finite signals and dual extension to discrete time fractional fourier transform conventional fourier analysis has many schemes for different types of signals they are fourier transform ft fourier series fs discrete time fourier transform dtft and discrete fourier transform dft the goal of this article is to develop two absent schemes of fractional fourier analysis methods the proposed methods are fractional fourier series frfs and discrete time fractional fourier transform dtfrft and they are the generalizations of fourier series fs and discrete time fourier transform dtft respectively

auditory brainstem and middle latency responses recorded at fast rates with randomized stimulation randomized stimulation and averaging rsa allows auditory evoked potentials aeps to be recorded at high stimulation rates this method does not perform deconvolution and must therefore deal with interference derived from overlapping transient evoked responses this paper analyzes the effects of this interference on auditory brainstem responses abrs and middle latency responses mlrs recorded at rates of up to 300 and 125 hz respectively with randomized stimulation sequences of a jitter both greater and shorter than the dominant period of the abr mlr components additionally this paper presents an advanced approach for rsa iterative randomized stimulation and averaging i rsa which includes the removal of the interference associated with overlapping responses through an iterative process in the time domain experimental results show that a rsa can be efficiently used in the recording of aeps when the jitter of the stimulation sequence is greater than the dominant period of the aep components and b i rsa maintains all the advantages of rsa and is not constrained by the restriction of a minimum jitter the significance of the results of this study is discussed

a framework for assessing the relative efficiency of stimulus sequences in evoked response measurements a general matrix based framework for characterizing the performance of cross correlation techniques for recovering the response to an arbitrary stimulus sequence is presented in general an infinite number of recovery sequences can be identified for any given stimulus sequence the recovery sequence that minimizes the noise in the recovered response is derived and the effect of the recovery operation on the noise is analyzed this general framework is employed to develop analytic expressions for the relative efficiencies of conventional signal averaging and binary mls based methods for the most commonly used recovery operation and the one that optimizes snr assuming the background noise is uncorrelated the results depend on the ratio of response length to mls minimum pulse interval and the amplitude loss as a function of stimulus rate examples based on measured amplitude loss functions for abr recordings are employed to evaluate the relative efficiencies of mls techniques when the noise is uncorrelated conventional signal averaging is from two to five times as efficient as mls techniques the relative advantage of averaging is shown to decrease when the noise is dominated by low frequency components

existence and algorithm of solutions for generalized mixed implicit quasi variational inequalities1this project was supported by the national science foundation of people s republic china 1 in this paper by applying the auxiliary variational principle technique some existence theorems of solutions for a class of generalized mixed implicit quasi variational inequalities are proved in hilbert spaces a novel and innovative iterative algorithm to compute approximate solutions is suggested and analyzed the convergence criteria are also given as special cases of these results the open problem put forward by noor is answered positively and some results of existence and algorithm of solutions for generalized implicit quasi complementarity problems are also obtained our results are new and generalize a number of known results of mixed variational inequalities mixed quasi variational inequalities and quasi complementary problems involving single valued and set valued mappings in recent literatures

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