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The short-time Fourier transform allows calculation of the amplitude and initial phase distribution of the real signal as functions of time and frequency, whereas the wavelet transform allows calculation of the amplitude and instantaneous phase distribution of the real signal as functions of time and scale. However, for a complete description of the non-stationary signal, we should obtain not only the amplitude, initial phase, and instantaneous phase distribution as functions of time and frequency simultaneously with high precision but also the amplitude distribution as a function of time and phase referred to as the time–phase amplitude spectrum. In this paper, the time–phase amplitude spectrum is presented based on the high-precision time–frequency amplitude spectrum and initial and instantaneous phase spectra that are generated simultaneously by the proposed modified short-time Fourier transform. To minimise the effect of noise on the high-precision time–frequency amplitude spectrum, initial and instantaneous phase spectra, and time–phase amplitude spectrum, the modified short-time Fourier transform is applied to the real signal reconstructed by the peak high-precision time–frequency amplitude spectrum and the high-precision time–frequency instantaneous phase spectrum at that location to obtain the stable high-precision time–frequency amplitude spectrum, initial and instantaneous phase spectra, and stable time–phase amplitude spectrum. Compared with the short-time Fourier transform and wavelet transform, the time–frequency amplitude spectrum and initial and instantaneous phase spectra obtained by the modified short-time Fourier transform have higher precision than those obtained by the short-time Fourier transform and wavelet transform. Analysis of synthetic data shows that the modified short-time Fourier transform can be used not only for the calculation of the high-precision time–frequency amplitude spectrum, initial and instantaneous phase spectra, and time–phase amplitude spectrum but also for signal reconstruction, stable high-precision time–frequency amplitude spectrum, initial and instantaneous phase spectra, and stable time–phase amplitude spectrum. Analysis of real seismic data applications demonstrates that the stable time–phase amplitude spectrum reveals seismic events with high sensitivity and is well-matched for seismic data processing and interpretation.

We propose and comprehensively analyze a computationally efficient algorithm to estimate the parameters of a real sinusoidal signal in noise. This method uses the fast Fourier Transform and is, therefore, computationally efficient. Accounting for the interference due to the negative spectral component allows the frequency to be estimated very accurately. Estimates of the amplitude and phase are derived in the process and are necessary for the suppression of the leakage. Theoretical analysis establishes that the estimator is asymptotically unbiased and achieves the Cramer–Rao lower bound. Simulation results are presented to verify the theory and demonstrate that the estimation performance is superior to other estimators in the literature.

Phase-shifting interferometry (PSI) is an effective technique in optical measurement. In this paper, a noniterative algorithm for retrieving the unknown phase shifts in three frame generalized PSI is proposed, in which the interference term of each frame is eliminated by the aid of a reference interferogram with the phase shift of π. With this algorithm, the phase shifts can be determined by solving a system of transcendental equations with trigonometric functions, in which this system consists of three different ratios between the similar matrix 1-norms of the intensity difference and the intensity sum. After the determination of phase shifts, a phase can be retrieved easily. The algorithm provides a possible method to solve the limitation of the fringe number existing in some common algorithms. The feasibility and accuracy of this algorithm were demonstrated by the simulation and experimental results.

In this letter, a hybrid algorithm based on the genetic algorithm (GA) and modified iterative Fourier transform (MIFT) technique called HGAMIFT is proposed for large thinned array synthesis. By employing a perturbation mechanism, the MIFT can achieve a better solution in terms of the peak sidelobe level when compared to the iterative Fourier transform technique. Moreover, a control factor to determine the proportion of individuals from GA and MIFT as well as the crossover and mutation rate is introduced to help the HGAMIFT maintain the diversity of population in the early phase while avoiding stagnation in the late phase. Thus, a resulting enhanced search ability and fast convergence velocity can be obtained simultaneously. Several thinned arrays that comprise different array sizes and aperture shapes have been synthesized, which validate the superior performance of the proposed algorithm.

In the context of the non-abelian Fourier transform, the natural extension of the amplitude and phase response to a convolution by a given filter mask are shown to be the polar decompositions of the Fourier transform matrices. A specific example regarding amplitude and phase response of a certain filter mask over the symmetric group S_3 is given.

A novel method composed of phase retrieval is proposed to precisely recover the phase distribution in the double random phase encoding system. The model of a point light source is employed to illuminate the system, in the output side of which several diffracted intensity patterns are recorded in an orderly way. The phase at the output plane of the system is calculated by a parallel iterative phase retrieval algorithm with the recorded images. With the help of the reconstructed phase, the two phase masks can be determined by inverse Fourier transform. As a kind of chosen-plaintext attack, the secret image with high quality can be recovered with the proposed method.

This study investigates the synthesis of multiple-pattern periodic phase-only control planar arrays. A hybrid generalised iterative Fourier technique is proposed to iteratively derive the array element excitations efficiently from the prescribed array factor, in which the two-dimensional Woodward-Lawson method is employed in order to generate good initial values. During the synthesis process, the phase-only control reconfigurable array is obtained by changing the corresponding phase for different array patterns while keeping the excitation amplitude constant. The robustness of the proposed algorithm has been testified by investigating the sensitivity of the radiation pattern affected by the amplitude and phase errors. Furthermore, three totally different types of reconfigurable numerical examples are discussed to validate the proposed method. Simulation results reveal the superior performance, the efficiency and the robustness of the proposed algorithm in dealing with multiple-pattern arrays.

Quality control of micro-nano structured and freeform surfaces is becoming increasingly important, which leads to challenging requirements in the measurement and characterization of rough and highly reflective surfaces. As an important measurement technique, white light scanning interferometry (WLSI) is a fast noncontact method to measure three-dimensional (3D) surface profiles. Nevertheless, the existing WLSI 3D surface reconstruction algorithms are prone to environmental vibrations and phase changes caused by reflections on the tested surface. A novel peak detecting algorithm that combines the white light phase-shifting interferometry (WLPSI) method and fast Fourier transform (FFT) coherence-peak-sensing technique is proposed in this paper, which can accurately determine the local fringe peak and improve the vertical resolution of the measurement. A microcomponent (10 μm standard step height) and a spherical surface were used as test specimens to evaluate the proposed method. Both simulated and experimental results show that the proposed algorithm improves the precision and anti-interference ability of the WLPSI and FFT methods, which can effectively reduce the batwing effects at the edges and solve the problem of positioning error in the maximum modulation.

Fringe analysis in two-shot phase-shifting interferometry is important but meets challenges due to a limited number of images, corrupting noise, and background modulation. Here we propose an effective algorithm for phase retrieval from two interferograms with unknown phase shifts. The algorithm first evaluates the phase shift in a local mask through phase fitting and global optimization and then computes a full-field phase map using an arctangent function. Since the phase shift evaluation is performed within a local mask, the algorithm is fast compared with conventional optimization-based algorithms and typically needs tens of seconds to complete the processing. Computer simulation and experimental results show that the proposed algorithm has excellent performance compared with state-of-the-art algorithms. A complete software package of the algorithm in MATLAB is available at http://two-shot.sourceforge.io/.

This paper presents an automatic Fourier synthesis-based input-shaping algorithm for repetitive force-curve trajectories, applied to a piezo-electric actuator. A periodic scan-signal is generated by Fourier synthesis, consisting of the fundamental frequency and its even and odd harmonics. In order to compensate for the system dynamics of the used piezo-electric actuator, amplitude and phase response at these frequencies are identified. By adjusting amplitude and phase of each individual frequency component, the tracking error is reduced significantly. The presented method also demonstrates its capability of improving the non-linear behavior of piezoelectric actuators. The technique is experimentally evaluated on a piezo-electric stack-actuator, used as fast Z-actuator within an atomic force microscope (AFM). Comparing the presented method to an uncompensated triangular signal as well as input shaping methods, presented in literature, shows a significant reduction of tracking error of 84% and 65%, respectively.

Recovering phase information with Deterministic approaches as the Transport of Intensity Equation (TIE) has recently emerged as an alternative tool to the interferometric techniques because it is experimentally easy to implement and provides fast and accurate results. Moreover, the potential of employing partially coherent illumination (PCI) in such techniques allow obtaining high quality phase reconstructions providing that the estimation of the corresponding Phase Transfer Function (PTF) is carried out correctly. Hence, accurate estimation of the PTF requires that the physical properties of the optical system are well known. Typically, these parameters are assumed constant in all the set of measurements, which might not be optimal. In this work, we proposed the use of an amplitude Spatial Light Modulator (aSLM) for tuning the degree of coherence of the optical system. The aSLM will be placed at the Fourier plane of the optical system, and then, band pass filters will be displayed. This methodology will perform amplitude modulation of the propagated field and as a result, the state of coherence of the optical system can be modified. Theoretical and experimental results that validate our proposed technique will be shown.

The relationships between the amplitude and phase of the short-time Fourier transform (STFT) are investigated. By choosing the Gaussian window for the STFT, we reveal that the group delay and instantaneous frequency of each signal segment, both of which are derived from the phase by definition, can also be explicitly linked with the amplitude. As a result, the amplitude and phase can also be linked through the group delay or instantaneous frequency without making any assumptions for the phase property of the target signals, e.g., minimum, maximum, or linear phase. The theoretical basis is also confirmed in numerical simulations.

We report the design of an all-optical fast Fourier transform (FFT) on a silicon-photonics on-chip platform, show the phase-, amplitude-, and delay sensitivity of the transfer function, and a delay performance of 20 Tbps.

This paper proposes an improved amplitude spectrum based Fourier-Mellin algorithm Fourier by studying the nature of the Fourier transform image of the amplitude spectrum and the application of phase spectrum in estimation for image motion vector according to the shortcomings of traditional Fourier-Mellin algorithm when the video image translation, rotation and scaling of the situation exist. Experimental results show that the algorithm can estimate the rotation, translation and other vector parameters of a complicated motion model .The complexity of the algorithm has greatly improved comparing to the traditional Fourier-Mellin algorithm.

This paper expounded the principle of using Fast Fourier transform measuring the phase difference and completed the simulation of measuring the phase differences of a plurality of signal. The simulation results show that the method can rapidly and accurately realize the phase difference measurement without considering the AD quantization error and noise interference. In considering the AD quantization error and random noise, Fast Fourier transform measuring the phase difference still has high measurement accuracy, and the measurement error can be controlled within 0.01°. The method can be applied in practical engineering.

In most cases, the distribution of an object can be calculated by measuring its Fourier transform distribution. When a general laser wave plane is used, the measurement of Fourier transform distribution has a large error because of the digitizing of Fourier amplitude. In this study, we suggest a method to generate a laser light distribution with pseudo random phase that can reduce the error of Fourier amplitude.

Recently, a variety of the optical comb-based interferometries has been developed for profilometry and tomography. However the interference amplitude and phase characteristics involving the center frequency and mode spacing of the optical comb have not been sufficiently studied. To investigate these multi-frequency interference characteristics, we proposed a broadband frequency variable quasi-comb generator utilizing 4-f optical system and a spatial spectral filter which can perform unrestricted scanning of the center frequency and mode spacing. By using a sinusoidal phase modulating interferometer with the quasi-comb generator, fundamental proof-of-principle experiments were successfully demonstrated. The interference phase fixation during the symmetrical varying of the mode spacing produced the interference amplitude peak envelope without fringes. On the other hand, it was confirmed that the interference phase was changed linearly without the amplitude change by the center frequency shift of the multi-frequency spectrum.

An alternative method for phase retrieval based on spatial and binary non-quadrature amplitude modulation (NQAM) is presented. This proposal is based on the superposition of a probe beam with a reference beam modulated in phase and amplitude (PAM) by NQAM, which is implemented by two neutral density filters (NDF) in a three-beam Mach–Zehnder interferometer (MZI). The principal advantage of this proposal lies in an analytical relationship between the variations of phase and visibility in an interferogram with the variations in the amplitudes of the reference beams used to implement NQAM; thus, the interferograms can be normalized and their introduced phase variations can be known from the measured intensities. Consequently it is possible to successfully retrieve the object phase. It is worthy to note that this method is capable of accepting that the phase and visibility variations in the interferograms could be spatial functions.

An algorithm for retrieval of the complex amplitude of the recorded object in blind two-step phase-shifting interferometry is proposed, in which the complex amplitude of the object wave can be found out directly from the recorded intensity patterns without the process of extracting the phase-shifting value in advance. A simple analytical formula for calculating the complex amplitude (including both the amplitude and the phase) of the object wave based on the recorded intensities is derived. Both the theoretical analysis and experimental results demonstrated the feasibility of this kind of blind phase-shifting algorithm in small phase shift situations.

A method to create homogenous polarized light based on non-quadrature amplitude modulation is proposed. This method consists of the addition of two optical fields out of phase, different from m π and in the variation of their amplitudes, only for obtaining a resulting field modulated in both phase and amplitude. This principle is used to modulate the vertical components in both phase and amplitude, while the horizontal component is varied in amplitude but with a constant phase. Thus, any amplitude relation and phase difference between components can be created, and therefore any polarization state could be obtained. A theoretical model is shown and supported with numerical simulations of several polarization examples.

International audience; In this paper the accuracy of the complex sinusoid amplitude and phase estimators provided by the Interpolated Weighted Discrete-Time Fourier Transform (IpWDTFT) algorithm is analyzed. It is shown that the use of the WDTFT spectral sample corresponding to the estimated sinusoid frequency allows to minimize the estimator asymptotic Mean Square Error (MSE) due to additive wideband noise. Also, the expression of the asymptotic expected sum-squared fitting error in the reconstruction of the complex sinusoid by means of the IpWDTFT algorithm is derived. All the derived expressions are verified through computer simulations.

We propose a multiple-image hiding scheme based on the amplitude- and phase-truncation approach, and phase retrieval iterative algorithm in the fractional Fourier domain. The proposed scheme offers multiple levels of security with asymmetric keys. Multiple input images multiplied with random phase masks are independently fractional Fourier transformed with different orders. The individual keys and common keys are generated by using phase and amplitude truncation of fractional spectrum. After using two fractional Fourier transform, the resultant encrypted image is hided in a host image with phase retrieval iterative algorithm. Using the correct universal keys, individual keys, and fractional orders, one can recover the original image successfully. Computer simulation results with four gray-scale images support the proposed method. To measure the validity of the scheme, we calculated the mean square error between the original and the decrypted images. In this scheme, the encryption process and generation of decryption keys are complicated and should be realized using computer. For decryption, an optoelectronic setup has been suggested.

A phase-shifting algorithm, called a (4,4) algorithm, which takes four phase-shifting interferograms before a specimen is deformed and four interferograms after a specimen is deformed, is presented first. This method is most widely used for phase extraction. Its drawback limited it to be used in dynamic measurements. Also shown is an algorithm called a (4,1) algorithm that takes four phase-shifting interferograms before a specimen is deformed and one interferogram after a specimen is deformed. Because a high-speed camera can be used to record the dynamic interferogram of the specimen, this algorithm has the potential to retain the phase-shifting capability for ESPI in dynamic measurements. The quality of the phase map obtained using (4,1) algorithm is quite lower compared to using (4,4) algorithm. In order to obtain high-quality phase map in dynamic measurements, a direct-correlation algorithm was integrated with the (4,1) algorithm to form DC-(4,1) algorithm which is shown to improve significantly the quality of the phase maps. The theoretical and experimental aspects of this newly developed technique, which can extend ESPI to areas such as high-speed dynamic measurements, are examined in detail.

Based on the small scale power system frequency fluctuation frequency leakage problems due to the FFT phase measurement error problem, this paper presents a high accuracy algorithm for phase difference measurement based on windowed discrete Fourier transform, the signal truncated using 4 order Blackman-Harris window, and corrects the phase correction method for discrete spectrum thus to calculate the phase difference. The algorithm for phase difference measurement theory and signal frequency independent, so there is no need to sample the signals of the whole cycle. A simple algorithm, less calculation, high precision, has strong inhibition on the harmonic and noise, simulation results verify the feasibility and effectiveness of the algorithm.

This paper advances a new method, double phase differences correction method, which aims at correcting the errors of frequency, phase and amplitude of the harmonic signal. The employment of this method involves that two phase differences of two highest spectral lines in the main lob are used to get the correcting value of frequency and then phase and amplitude can be rectified by corresponding formula. It solves the problem that the traditional formula of translation of window center is only for rectangle window. This new formula can correct for the different window by the principle of translation of window center. Simulation shows that the maximum error of frequency of the signal with white noise is less than 1% of frequency resolution. The maximum error of phase is around 1.5º, and the maximum error of amplitude is within 0.01. The average errors in frequency, phase and amplitude are approximately 0.002Hz, 1º and 0.005, respectively.

High-accuracy frequency, amplitude, and phase estimation methods for asynchronous sampling are presented. The proposed estimation methods are based on phase difference estimation, compensation of number of samples, and a modified discrete Fourier transform. This study focused on processing signals in substation automation systems that comply with IEC 61850. Some simulation tests were conducted in cases of pure and distorted sinusoids, and the frequency, amplitude, and phase errors of the fundamental and harmonics are evaluated at fundamental frequencies around 50 Hz at fixed sampling rates of 4 kHz and 12.8 kHz (i.e., 80 and 256 samples per period). Dependence of each estimation accuracy on a fraction of number of samples is discussed.

To extract the exact phase of the measured object in the environmental conditions with random noise, a novel kind of frequency domain phase extraction method for phase-shifting interferometry with random phase-shifting amount and low-sampling rate is proposed, in which the time-varied intensity signal of an arbitrary pixel of phase-shifting interferograms is cut into a lot of sub-sections intensity signals with the same length, and then the frequency domain weighted stack operation is performed for the corresponding Fourier spectrum of these sub-section intensity signals. Importantly, the phase obtained from the proposed method coincides well with the conventional four-step phase-shifting algorithm, but the requirement for the environmental conditions in the former are greatly lowered than that in the latter, moreover, while the same number of phase-shifting interferograms is employed to perform the phase extraction, the phase measuring accuracy in the proposed method is greatly higher than that in the time domain Fourier transform (TFT) method.

In order to improve the rapidity and accuracy of phase detection in power system, this paper proposes a method to calculate amplitude and phase of AC fundamental voltage based on the iterative Fourier transform. In each sample interval, it uses the new sample data of phase-A to calculate the phase angle by an add operation and a subtract operation. According to the phase relationship between the three-phase voltages, angles of phase-B and phase-C are calculated by subtracting and adding 120 degrees respectively. In the first part, this paper analyzes principle of the iterative Fourier transform and introduces steps of amplitude and phase calculating for the AC fundamental voltage. In the second part, it introduces the real-time phase angle calculation of three fundamental voltages. Finally, the proposed phase-lock technique is tested through a simulation, and the simulation results indicate that the iterative Fourier transform is more accurate and faster than the traditional phase-lock method.

The problem of estimating the frequency of a complex signal is considered. Through analyzing the performance of the Quinn algorithm and Aboutanios iterative algorithm, when the frequency of signal is located in the central region of two neighboring quantized frequency in discrete Fourier transform (DFT), Quinn algorithm's precision is very high and its root-mean-square error (RMSE) is close to Cramer-Rao bound (CRB). Meanwhile, the variance of frequency estimated by the Aboutanios iterative algorithm is big. However, when the frequency is located in the vicinity of quantitative frequency point, the Quinn algorithm's accuracy is poor and Aboutanios iterative algorithm has high precision. In this paper, through combining the advantages of two kinds of algorithms and obtaining the correct interpolation direction by spectral shift in the vicinity of quantitative frequency point, there is a new comprehensive algorithm which has better computational efficiency and its RMSE is close to the CRB in the entire frequency estimation range.

In order to calculate the lost phase from the intensity information effectively, a new method of phase retrieval which based on cosine grating modulation and transport of intensity equation is proposed. Firstly, the cosine grating is loaded on the spatial light modulator in the horizontal and vertical direction respectively, and the corresponding amplitude of the light field is modulated. Then the phase is calculated by its gradient which is extracted from different direction modulation light illumination. The capability of phase recovery of the proposed method in the presence of noise is tested by simulation experiments. And the results show that the proposed algorithm has a better resilience than the traditional Fourier transform algorithm at low frequency noise. Furthermore, the phase object of different scales can be retrieved using the proposed algorithm effectively by changing the frequency of cosine grating, which can control the imaging motion expediently.

Interferometry is an optical measuring method with the character of non-destructive, high sensitivity and high accuracy. However, its measurement range is limited by the phase ambiguity. Hence the method with two separate different wavelengths light source is introduced to enlarge the measurement range. As for the two-wavelength interferometry case, phase shifting is the traditional way to acquire the phase map, it needs to repeat the measurement twice, which means the measurement cannot be accomplished in real time. Hence to solve the problem, a temporal sequence interferometry has been used. This method can obtain the desired phase information in real time by using the Fourier transform methods of the interferogram recorded in a sequence while the object is being deformed. But, it is difficult to retrieve the phase information directly due to the multi extreme points in one period of the cosine function. In this paper, an algorithm based on the wavelet ridge analysis is adopted to retrieve the two wavelength phase fluctuation caused by the displacement simultaneously. The preliminary experiment is conducted and the results are compared with theoretical simulations to validate the proposed approach. The laser emits light with two wavelengths 532 nm and 473 nm, two separated interference patterns in time sequence are detected by the CCD camera in the same time. The overlapped interferograms of two colors are analyzed by this algorithm and the corresponding phase information are obtained. The maximum error value between the simulation and theory is 0.03 um and the relative error is 0.33%.

A moveable lens is used for determining amplitude and phase on the object plane. The extended fractional Fourier transform is introduced to address the single lens imaging. We put forward a fast algorithm for the transform by convolution. Combined with parallel iterative phase retrieval algorithm, it is applied to reconstruct the complex amplitude of the object. Compared with inline holography, the implementation of our method is simple and easy. Without the oversampling operation, the computational load is less. Also the proposed method has a superiority of accuracy over the direct focusing measurement for the imaging of small size objects.

Systematic phase errors in Fourier transform spectroscopy can severely degrade the calculated spectra. Compensation of these errors is typically accomplished using post-processing techniques, such as Fourier deconvolution, linear unmixing, or iterative solvers. This results in increased computational complexity when reconstructing and calibrating many parallel interference patterns. In this paper, we describe a new method of calibrating a Fourier transform spectrometer based on the use of artificial neural networks (ANNs). In this way, it is demonstrated that a simpler and more straightforward reconstruction process can be achieved at the cost of additional calibration equipment. To this end, we provide a theoretical model for general systematic phase errors in a polarization birefringent interferometer. This is followed by a discussion of our experimental setup and a demonstration of our technique, as applied to data with and without phase error. The technique's utility is then supported by comparison to alternative reconstruction techniques using fast Fourier transforms (FFTs) and linear unmixing.

Compared with the existing conventional frequency correlation algorithm which needs both Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT), an modified frequency discrete correlation algorithm only needing FFT is presented and deduced in new ways based on the correlation theorem. Also simulation experiments of the algorithm for periodic signals and mixed signals composed by periodic components and random components have been completed. The results of the experiments prove that the algorithm is correct and effective. The deviation caused by the algorithm is less than 10-6 in terms of taking time domain correlation calculation as criterion.

The object of this paper is to describe a phase step with axial resolution for phase shifting interferometry (PSI) based on recursive algorithm, which can eliminate the DC and autocorrelation terms of Fourier-domain optical coherence tomography (FDOCT). The phase step of this algorithm is equivalent to axial resolution of FDOCT system, which is determined by the wavelength and full width at half maximum (FWHM) of light source. Compared with other PSI algorithm methods, the significant feature of the proposal is that it deeply lowers the demand for the accuracy of reference plane movement, and only one time movement is needed for eliminating DC signals and self-interference signals inside the sample.

Spatial carrier fringe pattern analysis is an effective tool in optical measurement, e.g. in interferometry and fringe projection technique. With it, the very large phase deformations in a spatial carrier fringe pattern may increases the bandwidth of fringe component thus leading to difficulties in retrieving its phase map. In order to overcome this problem, many local-adaptive methods have been developed for processing the spatial carrier fringe pattern with large phase variations, and in fact, the local spatial frequency estimation is central to these methods. This paper introduces a simple algorithm for estimating the local frequencies of a fringe pattern with spatial carrier. First, the intensity gradients of the fringe pattern are calculated, and then the standard deviations (SDs) of the intensity gradients at each pixel are estimated from its neighborhood. Finally the local frequencies are estimated from the SDs just calculated simply using an arccosine function. This algorithm is potential in developing effective techniques for retrieving phases from a spatial carrier fringe pattern with large phase variations. For example, we can recover the phase map by directly integrating the local frequencies or by use of an adaptive spatial carrier phase shifting algorithm (SCPS) with the local frequencies being the local phase shifts. It can also be used in Fourier transform method for exactly determining the carrier frequencies, or for extrapolating aperture in order to reduce the boundary effect. Combined with time-frequency techniques such as windowed Fourier transform and wavelet transform methods, it is helpful for alleviating the computational burdens.

In this paper, it is shown that N spatially separated harmonically related carriers may be generated by a Generalised Mach-Zehnder Interferometer (GMZI) based architecture by replacing the N×1 combiner of a GMZI by an N×N optical Discrete Fourier Transform (DFT) network; thereby avoiding the requirements of an optical de-multiplexing filter to separate the subcarriers released from the single port of an optical comb generator.

A new phase shift method that requires only a single image to create the phase map is needed to expand the application of the phase shift digital shearography technique to include dynamic loading conditions. In this case, the acquisition rate will only be limited by the camera instead of by the multiplephase-stepping procedure. As only one image is needed, there is no need to make assumptions about the phase distribution between each phase-stepping image. This new phase shift method is called the spatial phase shift method. The spatial phase shift technique can be divided into two major categories: the multiple-channel spatial phase shift approach and the frequency carrier spatial phase shift method. The earlier approach to solve dynamic loading problems was to use the multichannel spatial phase shift method. Later, the frequency carrier spatial phase shift method, a more advanced technique, was introduced. This method records multiple speckle interferograms on a single image. A Fourier transform method provides a way to extract and separate the phase information from this single image. In the remainder of this chapter, the multiple-channel spatial phase shift approach will be introduced, and the frequency carrier spatial phase shift method will be discussed in detail.

Multi-beam interference exists in testing high-reflectivity surfaces with a Fizeau interferometer. In this paper, the multi-beam interference intensity was estimated as the sum of the first six order harmonics using the Fourier series expansion. Then, by adopting carrier squeezing interferometry with a π/4 phase shift, an algorithm was proposed to extract the phase from multi-beam interferograms. To ensure the separation of the lobes of phase-shift errors and the phase in the frequency domain, conditions of the necessary linear carrier in the proposed algorithm were derived. Simulation results indicated that the phase retrieving precision is better than PV 0.008λ and RMS 0.001λ, even when the reflection coefficient of the test surface is as high as 0.9 and the phase shift varies within π/4±π/20. Compared with the other algorithms, the proposed algorithm for multi-beam interference was validated by its good performance in the experiments, especially when the phase-shift error exists.

Fourier phase in Fourier-domain optical coherence tomography (FD-OCT) has been shown to estimate the sub- resolution change in the optical depth location of a strong interface in refractive index profile of a sample using spectral-domain phase microscopy (SDPM), a derivative of FD-OCT. From first principles we show that in general Fourier phase not only estimates this sub-resolution change but also the mean spatial frequency of the coherence-gated refractive index, and both SDPM and depth-resolved spatial-domain low-coherence quantitative phase microscopy (dr-SLQPM) are special cases of this general theory. We also show that for spectral source with infinite bandwidth Fourier phase is zero. We provide analytical expressions and numerical simulations to support our results.

In this paper, implementation of the complex screen approach for generating partially-coherent Schell-model sources using phase-only control, like that available using commercial-off-the-shelf liquid-crystal spatial light modulators, is presented. Previous techniques for generating Schell-model sources have used phase-only spatial light modulators in combination with amplitude filters to produce partially-coherent fields. Using amplitude filters complicates the optical set-up and significantly reduces the flexibility of the system. Here, both analytically and with simulations, it is shown that both the amplitude and phase of the source field can be manipulated using phase-only control. This contribution removes the need to use amplitude filters, thereby simplifying the optical set-up and increasing the system's flexibility.

Phase-shifting interferometry inside an optical cavity is useful for measuring the length of a cubic object and the diameter of a metal sphere. A new error-compensating phase-shifting algorithm is proposed, with which a spatially uniform phase error caused by nonlinear phase modulation can be eliminated. It is shown that the conventional algorithm design for harmonic analysis cannot compensate for this dc error unless the sampling weight definition is extended to a complex number. Specifically, a 3% quadratic nonlinearity in the phase modulation is found to yield a systematic error of 1.5 nm in the object length in many conventional algorithms. A new 13-frame algorithm is described that applies a modified discrete Fourier window and can reduce the dc error to less than 0.5 nm.

In this paper, a new frequency estimation algorithm based on the DFT (Discrete Fourier Transform) is proposed to increase the estimation accuracy. The proposed algorithm consists of three stages such as DFT, frequency estimation and error compensation, and iterative loop. First, the input voltage signal is decomposed into two components by using DFT. Secondly, the two components, the real part and the imaginary part, and their derivatives are combined to estimate the frequency. Finally, to obtain a better estimated frequency, an iterative method is applied. To evaluate the performance of the proposed algorithm, frequency change is considered when the test signals are generated. The performance of the proposed method was also evaluated for a dynamic condition in a 5-bus model system. PSCAD/EMTDC was used to generate the voltage signal for a load increase case. The performance evaluation results show that the proposed algorithm can accurately estimate the power system frequency in the presence of the amplitude variation and the frequency change.

Fourier ptychography is a recently developed computational framework for wide-field, high-resolution imaging. Here, we propose to use a fast converging and error reducing phase retrieval algorithm to retrieve the complex sample information.

For optical coherent M-ary quadrature amplitude modulation (M-QAM) system, the frequency offset can be extracted directly by applying Fast Fourier Transform (FFT) to the signal's argument, without removing the modulated data phase. By categorizing the constellation points and rotating some constellation points by π / 4 , this algorithm is robust to extract the frequency offset against the noise. Numerical simulations of 16-QAM and 256-QAM coherent systems are presented to demonstrate this algorithm.

Phase retrieval is concerned with the reconstruction of a signal or its Fourier phase from the magnitude of its Fourier transform. In this paper, we address the problem of reconstructing an unknown signal from the following information: the magnitude of the Fourier transform of an unknown signal, a known ‘reference’ signal, and the magnitude of Fourier transform of the signal that is given by the addition of the known reference signal and the unknown desired signal. After we briefly mention an iterative algorithm that is proposed to reconstruct a desired signal from the given information and its convergence properties, we modify the algorithm to improve the convergence property such that the total sum of the squared errors of all the steps composing the algorithm, except the one for imposing magnitude constraints, is non-increasing. Finally, we present some examples that show the properties of convergence against two kinds of noises.

SUMMARY This paper presents the design and analysis of an adaptive algorithm for tracking the amplitude, phase and frequency of the fundamental, harmonics and interharmonics present in time-varying power sinusoid in white noise. If frequency, amplitude and phase of the multiple sinusoids become nonstationary, they are estimated as an unconstrained optimization problem using robust and low complexity multi-objective Gauss–Newton algorithm. The presented algorithm deals with frequency drift and can accurately estimate frequency variation, amplitude and phase variation, as well as harmonic amplitude and phase variations. Further, the learning parameters in the proposed algorithm are tuned iteratively to provide faster convergence and better accuracy. The excellent tracking capability of proposed multi-objective Gauss–Newton algorithm is shown through simulation and experimental results in a nonstationary environment. Copyright © 2012 John Wiley & Sons, Ltd.

Frequency is an important parameter of a power system, and frequency measurement is a very important link in a power system measurement device. This paper compared the accuracy of the Fourier frequency measurement algorithm with different sampling period, chose the method of whole cycle to improve. Calculated based on the empirical model and used the least error squares fitting the error compensation curve to correct the result and get a new algorithm in frequency measurement which is convenient in calculation, high precision and real-time because it need only 2 sampling cycle and does not increase the complexity. The simulation results show the frequency value measured by improved algorithm is highly accurate and can be applied in a quick arc-suppressing system.

In this study, a 6N - 5 phase shifting algorithm comprising a polynomial window function and discrete Fourier transform is developed for the simultaneous measurement of the surface shape and optical thickness of a transparent plate with suppression of the coupling errors between the higher harmonics and phase shift error. The characteristics of the 6N - 5 algorithm were estimated by connection with the Fourier representation in the frequency domain. The phase error of the measurements performed using the 6N - 5 algorithm is discussed and compared with those of measurements obtained using other algorithms. Finally, the surface shape and optical thickness of a transparent plate were measured simultaneously using the 6N - 5 algorithm and a wavelength tuning interferometer.