Time-domain Envelope Research Articles

Interpreting Convolutional Layers in DNN Model Based on Time–Frequency Representation of Emotional Speech

Abstract The paper describes the relations of speech signal representation in the layers of the convolutional neural network. Using activation maps determined by the Grad-CAM algorithm, energy distribution in the time–frequency space and their relationship with prosodic properties of the considered emotional utterances have been analysed. After preliminary experiments with the expressive speech classification task, we have selected the CQT-96 time–frequency representation. Also, we have used a custom CNN architecture with three convolutional layers in the main experimental phase of the study. Based on the performed analysis, we show the relationship between activation levels and changes in the voiced parts of the fundamental frequency trajectories. As a result, the relationships between the individual activation maps, energy distribution, and fundamental frequency trajectories for six emotional states were described. The results show that the convolutional neural network in the learning process uses similar fragments from time–frequency representation, which are also related to the prosodic properties of emotional speech utterances. We also analysed the relations of the obtained activation maps with time-domain envelopes. It allowed observing the importance of the speech signals energy in classifying individual emotional states. Finally, we compared the energy distribution of the CQT representation in relation to the regions’ energy overlapping with masks of individual emotional states. In the result, we obtained information on the variability of energy distributions in the selected signal representation speech for particular emotions.

Attenuation characteristics of coda wave in southern Arabian Shield

This study aims to estimate regional attenuation of coda wave in the southernmost part of the Arabian Shield. The decay rate of the time-domain coda envelope was analyzed and used for the estimation of coda wave quality factor (Qc) following the single isotropic-scattering model. The dataset comprises of a total number of 78 earthquakes of magnitude ranges from 1.5 to 4.5 recorded by 13 seismic stations within a hypocentral distance of 11 to 205 km and a lapse time between 10 and 40 s. Seismograms from vertical and two horizontal components have been analyzed at central frequencies of3, 6, 9, 12, 18, 21, and 24 Hz, obtaining frequency-dependent attenuation relationships in the form of Qc=(36 ± 10)f(0.98±0.1), Qc=(53 ± 13)f(0.89±0.1), Qc=(63 ± 12)f(0.86±0.1), and Qc=(75 ± 13)f(0.81±0.14)at lapse time 10, 20, 30, and 40 s respectively. The regional frequency-dependent relationship is approximated by the power law of Qc=57±11f0.88±0.1 for the study area. For 10 sec lapse time, the average Qc ranges between 124 ± 41 at 3 Hz and 890 ± 193 at 24 Hz, while for 40 sec lapse time it varies from 200 ± 51 at 3 Hz to 1077 ± 130 at 24 Hz, implying a continuous increase in Qc with lapse time as a consequence of subsurface homogeneity with depth. The relationship between Qc and frequency shows that the coda quality factor is frequency dependent. The low attenuation characteristics can be attributed to the geodynamic process that causes a thin lithosphere beneath the study area due to active rifting in the Red Sea combined with magma flow and migration into the Arabian Shield.

Hilbert-envelope features for cardiac disease classification from noisy phonocardiograms

Cardiovascular disease (CVD) is considered a significant public health concern around the world. Automated early diagnostic tools for CVDs can provide substantial benefits, especially in low-resource countries. This study proposes a time-domain Hilbert envelope feature (HEF) extraction scheme that can effectively distinguish among different cardiac anomalies from heart sounds even in highly noisy recordings. The method is motivated by how a cardiologist listens to the heart murmur configurations, e.g., the intensity of the heart sound envelope over a cardiac cycle. The proposed feature is invariant to the heart rate, the position of the first and second heart sounds, and robust in extracting the murmur configuration pattern in the presence of respiratory noise. Experimental evaluations are performed compared to two different state-of-the-art methods in the presence of respiratory noise with signal-to-noise ratio (SNR) values ranging from 0–15 dB. The proposed HEF, fused with standard acoustic and Resnet features, yields an average accuracy, sensitivity, specificity, and F1-score of 94.78% (±2.63), 87.48 %(±6.07), 96.87% (±1.51) and 87.47% (±5.94), respectively, while using a random forest (RF) classifier and applied on a mixture of clean and noise mixed recordings of an open-access dataset. Compared to the best-performing baseline model, this feature-fusion scheme provides a significant performance improvement (p<0.05), notably achieving an absolute improvement of 6.16% in averaged sensitivity. When applied to noisy heart sound recordings collected from a local hospital, this method significantly outperforms the existing systems by achieving an average accuracy of 80.54% (±2.65) and sensitivity of 68.52% (±4.54). The achieved sensitivity yields an absolute improvement of 12.65% compared to the best-performing baseline model in this real-world dataset.

Seismology on Mars: An analysis of direct, reflected, and converted seismic body waves with implications for interior structure

The InSight (Interior Exploration using Seismic Investigations, Geodesy and Heat Transport) mission has been collecting high-quality seismic data on Mars since early 2019 that provide the first direct observations of its interior structure. Here we report on a complete analysis of the part of the marsquakes known as the low-frequency seismic events (main energy below 1 Hz) that are sensitive to the deep interior. To identify body-wave arrivals in the highly-scattered martian seismograms, we employ four complementary approaches: 1) time-domain envelopes; 2) polarised waveforms and their time-domain envelopes; 3) polarisation analysis; and 4) waveform matching. Through careful application of this processing scheme to each marsquake, we are able to significantly increase the number of phase picks relative to earlier analyses (from 41 to 76), including body-wave arrivals from direct (P and S), reflected (PP, SS, PPP, SSS, and ScS), and converted (Ps and Sp) phases. To constrain the depth of the marsquakes, we also identify depth phases (pP and sS). Following this, we invert an initial set of phase picks for models of interior structure, event distance, and depth, while predicting travel times for seismic phases not identified at the outset. Based on the predictions, we repick (every pick is subject to our processing scheme), thereby enlarging our differential travel-time data set (all picks are relative to the main P-wave arrival), and subsequently re-invert for an updated set of interior structure models, distances, and depths. Proceeding thus, we present updated radial seismic velocity models of the crust, mantle, and core. We observe crustal interfaces at average depths of 10, 25, and 45 km, respectively, of which the former two are interpreted as intra-crustal interfaces and the latter as the crust-mantle boundary. We find an upper mantle structure consistent with a low-velocity zone associated with a thermal lithosphere and a thermal gradient in the range 2.4–2.9 K/km that extends to a depth of ~450 km. The thermal structure of the Martian mantle indicates potential and core-mantle-boundary temperatures in the ranges 1650–1750 K and 1900–2100 K, respectively, implying an entirely liquid core at present. Based on the identification of ScS phases, we obtain an improved estimate of the Martian core radius (1820–1870 km) and mean core density (6–6.2 g/cm3).

Identification of MOSFET Working State Based on the Stress Wave and Deep Learning

Nondestructive testing methods are gradually used to monitor the state of power devices, including acoustic emission (AE) testing. However, the application of AE in power devices is still in the research stage and relies on manual analysis. In this article, the stress wave signal of the device working under different electrical parameters is combined with deep learning (DL) for the first time to judge the working state of the device. Time-domain signals, frequency-domain amplitudes, and envelope of the stress wave and their corresponding label are used for the training of the artificial neural network (ANN), 1-D convolutional neural network (1DCNN), long short-term memory (LSTM), and recurrent neural network (RNN). The training effect of 1DCNN combined with time-domain series is the best, and the accuracy rate can reach up to 100%. Compared with other classic 1DCNN models, the proposed 1DCNN method does not require complex feature extraction and model calculations, which accurately identifies the working state of the device and also improves its speed.

Research on the Advantages of Digital Sensor Equipment in Language Audio‐Visual and Oral Teaching

Digital sensors use biotechnology and information processing technology to strengthen the processing of relevant visual and auditory information, which is helpful to ensure that the receiver can obtain more accurate information, so as to improve the learning effect and reduce the impact on the environment. This paper designs an experiment to explore the role of digital sensors in language audio‐visual teaching, which provides a reference for the application of digital sensors in the future. The impulse response function in sensor technology is introduced. The speech time domain envelope and time‐varying mouth area of the sensor device are calculated. The auditory attention transfer detection based on line of sight rotation estimation is carried out through the auditory attention decoding fusion technology and the sensor auditory attention conversion detection method. At the same time, the characteristic of sensor heog signal is analyzed. The results show that the algorithm proposed in this paper has good results.

Design of an Acoustic-Based Nondestructive Test (NDT) Instrument to Predict the Modulus of Elasticity of Wood

Modulus of elasticity (E) can predict the mechanical characteristics, as well as grade the quality of wood. The Destructive Test (DT) method is the more commonly used, where the wood sample is split up when being tested. This research used the NDT method based on the longitudinal stress wave method (LSWM) whilst utilizing handheld instruments. The calculated E-dynamic (Ed) from the NDT method was compared with the E-static (Es) from the DT method to validate the technique. Six different wood types were tested with ten samples of each kind. An average R-value of 0.898 was obtained, indicating a high correlation between the Ed values and Es values. The LSWM method requires reliable hardware and software to record the impulse response. It starts by hitting the wood with a hammer to create an impulse, finding the resonance frequency (f0), and later calculating the Ed. A more practical and easy-to-use handheld mobile instrument was developed using a Raspberry Pi-2 microcomputer as the signal processor, an LCD touchscreen, a USB soundcard, and a dynamic microphone that covers 0.1-5kHz and -64 ± 3dB. An internal telecommunication system is provided to support measurements conducted at lumber mills. The software includes band-pass filtering of the recorded spectrum where the f0 is depicted. A time-domain envelope fitting is then applied to the filtered spectrum to obtain the R2envelope. A low R2envelope value indicates an inadequate impulse response, and therefore, the test should be redone.

Artificial bandwidth extension of speech signals

Artificial Bandwidth Extension (ABWE) algorithms try to reconstruct frequency components of speech signals which have been lost by coding/transmission. From a system theoretical point of view, this seems to be questionable. However, by exploiting redundancy of speech production as well as psychoacoustic properties of auditory perception, missing parts can subjectively be recovered or at least mimicked to a certain extent. The motivation for ABWE is to bridge the quality gap between the traditional narrowband telephony (200–3400 Hz) and the recent wideband services (HD voice, 50–7000 Hz). An HD terminal which is connected to a traditional phone, may activate ABWE for improving the audio quality and possibly the intelligibility of the received signal. In this contribution, state-of-the-art ABWE algorithms and coding standards including ABWE techniques are reviewed. ABWE algorithms are based on a source-filter model. The three main sub-tasks are to estimate the excitation signal, the spectral envelope and the time domain envelope, given either just the narrowband signal or in addition a few bits of side information. Furthermore, insights with respect to the relative importance of these three components are presented.

An Experimental Approach for Detection of the Acoustic Radiation Induced Static Component in SolidsSupported by the National Natural Science Foundation of China (Grant No. 11834008).

We propose an experimental approach to directly detect the acoustic radiation induced static component (SC) of primary longitudinal (L) wave propagation in solids using an ultrasonic pitch-catch technique, where a low-frequency ultrasonic transducer is used to detect the SC generated by the co-propagating primary L-wave tone burst that is excited by a high-frequency ultrasonic transducer. Essentially, the experimental approach proposed uses a dynamic method to detect the SC generated. The basic requirement is that the central frequency of the low-frequency ultrasonic transducer needs to be near the center of the main lobe frequency range of the time-domain envelope of the primary L-wave tone burst. Under this condition, the main lobe of the frequency spectrum of the SC pulse generated adequately overlaps with that of the low-frequency ultrasonic transducer. This will enable the generated SC pulse to be directly detected by the low-frequency ultrasonic transducer. The performed experimental examination validates the feasibility and effectiveness of the proposed approach for direct detection of the acoustic radiation induced SC generated by L-wave propagation in solids.

Underwater Target Noise Recognition and Classification Technology based on Multi-Classes Feature Fusion

As one of the main signal sources of underwater acoustic target recognition, the target noise signal is difficult to characterize the characteristics of the target from clearly comparing with the multi-sensor detection technology, which may lead to lower recognition rate and higher false alarm rate and seriously restricts the function of underwater acoustic detection system. In order to solve this problem, a multi-layers LSTM underwater acoustic target noise feature extraction model is established by using the long short term memory network. The information features such as time-domain envelope of target noise, Demon line spectrum and Mel frequency cepstrum coefficient are extracted, and a subset of multi-classes features is constructed. On this basis, the feature level fusion recognition and classification model based on the multi-classes features subset and the decision level fusion recognition and classification model based on the D-S evidence theory are established, and the above-mentioned models are tested by using the sample database. The difference of classification result between the multi-classes feature fusion and the single class feature recognition classification is compared, and the above model is tested and verified by using the relevant test data of port basin verification experiment. The correlation results show that the present intelligent recognition and classification method of underwater target noise based on the multi-classes feature fusion is more robust, and the recognition rate and false alarm rate of underwater target are better than those of single category feature discrimination method.

High performance distributed acoustic sensor based on digital LFM pulse coherent-optical time domain reflectometer for intrapulse event

A distributed fiber-optic acoustic sensor based on digital linear frequency modulated (LFM) pulse and coherent detection is proposed and its proof-of-concept experiment is demonstrated. High-performance measurement has been achieved for the intrapulse acoustic event, which is much shorter than the pulse spatial length. The actual acoustic sensing is achieved using the polystyrene membrane adhered to an only 9 cm long fiber at frequency of 1.5 kHz and 3 kHz. Compared with the direct detection configuration, the time domain envelope correlation coefficient of the coherent detection is twice that of the direct detection, and the standard deviation is less than one third.

Adaptive Vector Method for Motion Compensation in Ultra-Wideband Coherent Doppler Tomography

Ultra-wideband coherent Doppler tomography (UWB-CDT) is a method whereby complex frequency data are evaluated and back-projected at uniform angles around a subject under test (SUT). In the presence of phase shifts brought upon by the unwanted motion in the test equipment, generating a focused image becomes complex due to the nature of the back-projection integration process. This article will demonstrate a vector-based correction algorithm, which can be implemented during the nondestructive evaluation (NDE) of a target to compensate for down-range phase errors (brought upon by the factors external to the ideal measurement setup). This new method will be known as point scatterer vector tracking (PSVT). A coherent image can be constructed by using the returned scattering information contained within a complex time-domain envelope. For the proposed method, multiple vectors based on the estimated peak locations are used to track, correct, and reconstruct the path of an ideal point scatter. The demonstration of the algorithm with measured tomographic data combined with a simulated additive noise component is presented, along with a step-by-step approach to reconstruct the desired data set. It will be shown that by using this proposed method, a coherent CDT image with an average correlation coefficient of 0.71 can be recovered from the data that have been corrupted by the unwanted phase shifts.

A continuous vocoder for statistical parametric speech synthesis and its evaluation using an audio-visual phonetically annotated Arabic corpus

In this paper, we present an extension of a novel continuous residual-based vocoder for statistical parametric speech synthesis by addressing two objectives. First, because the noise component is often not accurately modelled in modern vocoders (e.g. STRAIGHT), a new technique for modelling unvoiced sounds is proposed by adding time domain envelope to the unvoiced segments to avoid any residual buzziness. Four time-domain envelopes (Amplitude, Hilbert, Triangular and True) are investigated, enhanced, and then applied to the noise component of the excitation in our continuous vocoder, i.e. of which all parameters are continuous. With the future aim of producing high-quality Arabic speech synthesis, we secondly apply this vocoder on a modern standard Arabic audio-visual corpus which is annotated both phonetically and visually, and dedicated to emotional speech processing studies.In an objective experiment, we investigated the Phase Distortion Deviation, whereas a MUSHRA type subjective listening test was conducted comparing natural and vocoded speech samples. As a result, both experiments based on the proposed noise modelling have shown satisfactory results in terms of naturalness and intelligibility, while outperforming STRAIGHT and other earlier residual-based approaches.

A Novel Scheme for Single-Channel Speech Dereverberation

This paper presents a novel scheme for speech dereverberation. The core of our method is a two-stage single-channel speech enhancement scheme. Degraded speech obtains a sparser representation of the linear prediction residual in the first stage of our proposed scheme by applying orthogonal matching pursuit on overcomplete bases, trained by the K-SVD algorithm. Our method includes an estimation of reverberation and mixing time from a recorded hand clap or a simulated room impulse response, which are used to create a time-domain envelope. Late reverberation is suppressed at the second stage by estimating its energy from the previous envelope and removed with spectral subtraction. Further speech enhancement is applied on minimizing the background noise, based on optimal smoothing and minimum statistics. Experimental results indicate favorable quality, compared to two state-of-the-art methods, especially in real reverberant environments with increased reverberation and background noise.

Utilization of wave attenuation in the time-lapse sonic logging at the Nagaoka site for a near well monitoring of CO2 migration

This paper discusses the seismic wave attenuation observed by time-lapse sonic logging at the Nagaoka CO2 injection site, in consideration of the many geological on-site experimental results asserting the occurrence of amplitude attenuation, as well as reduction of P-wave velocity, by CO2 invasion. The amplitude attenuation of seismic wave can be caused by a variety of physical phenomena and that the seismic data is obtained in a noisy environment, where acquisition of reliable amplitude data is technically challenging. In this paper we analyzed time-lapse sonic logging data at the Nagaoka site for evaluating the amplitude attenuation by CO2 invasion. We re-examined the semblance analysis conducted in the site using more robust methods and evaluated data quality by utilizing the obtained coherency as the quality of the waveform data. Then, we examined the P-wave amplitude from the envelope in time and spectrum domain. Although the amplitude at each logging was erroneous, averaging several logging runs revealed the amplitude attenuation. Moreover from the spectrum of P-wave, the peak frequency changed depending on the degree of CO2 saturation. We also examined the relationships that exist among the CO2 saturation, velocity reduction, and amplitude attenuation at the Nagaoka site. These relationships exhibited a dependency pattern similar to the model expected from a squirt flow. As a whole, we confirm the amplitude attenuation in the time-lapse sonic logging data, indicating that P-wave attenuation is applicable for a reliable evaluation of CO2 saturation, and for the early detection of CO2 leakage along the well.

A waveform-based convolutional neural net for source range and ocean environment classification

Neural networks learn features that are useful for classification directly from a source, such as a recorded signal, which removes the need for feature extraction or domain transformations necessary in other machine learning algorithms. To take advantage of these benefits and have a finer temporal resolution than a spectrogram, we built a one-dimensional convolutional neural network to classify source range and ocean environment from a received signal. The neural network was trained on simulated signals generated in different environments (sandy, muddy, or mixed-layer sediment layers) for several range classes. We found significant potential in a neural network of this type, given a large amount of varied training samples for the network to learn important features to make range and environment predictions. This type of network provides an alternative for frequency-domain learning and is potentially useful for impulsive sources. Benefits of using a time-domain envelope are also explored. Success in the time domain also reduces the computational requirements of conversion to frequency domain and increases the temporal resolution, which might be beneficial for real-time applications. [Work supported by the Office of Naval Research.] Neural networks learn features that are useful for classification directly from a source, such as a recorded signal, which removes the need for feature extraction or domain transformations necessary in other machine learning algorithms. To take advantage of these benefits and have a finer temporal resolution than a spectrogram, we built a one-dimensional convolutional neural network to classify source range and ocean environment from a received signal. The neural network was trained on simulated signals generated in different environments (sandy, muddy, or mixed-layer sediment layers) for several range classes. We found significant potential in a neural network of this type, given a large amount of varied training samples for the network to learn important features to make range and environment predictions. This type of network provides an alternative for frequency-domain learning and is potentially useful for impulsive sources. Benefits of using a time-domain envelope are also explored. Success in the ti...

Sparse Envelope Spectra for Feature Extraction of Bearing Faults Based on NMF

Periodic impulses and the oscillation response signal are the vital feature indicators of rolling bearing faults. However, finding the suitable feature frequency band is usually difficult due to the interferences of other components and multiple resonance regions. According to the characteristics of non-negative matrix factorization (NMF) on a spectrogram, the feature extraction method from a sparse envelope spectrum for rolling bearing faults is proposed in this paper. On the basis of the time–frequency distribution (TFD) of the periodic transient oscillations, the basic matrix can be interpreted as the spectral bases, and the time weight matrix corresponding to spectral bases can be extracted by NMF. Because the bases and the weights have a one-to-one correspondence, the frequency band filtering with the basic component and the time domain envelope of the weight vector are calculated respectively. Then, the sparse envelope spectrum can be derived by the inner product of the above results. The effectiveness of the proposed method is verified by simulations and experiments. Compared with band-pass filtering and spectral kurtosis methods, and considering the time weights and corresponding the spectral bases for the periodic transient oscillations, the weak fault-rated feature can be enhanced in the sparse spectrum, while other components and noise are weakened. Therefore, the proposed method can reduce the requirement of selecting frequency band filtering.

Principal component analysis-based features generation combined with ellipse models-based classification criterion for a ventricular septal defect diagnosis system.

In this study, a simple and efficient diagnostic system, which adopts a novel methodology consisting of principal component analysis (PCA)-based feature generation and ellipse models-based classification criterion, is proposed for the diagnosis of a ventricular septal defect (VSD). The three stages corresponding to the diagnostic system implementation are summarized as follows. In stage 1, the heart sound is collected by 3M-3200 electronic stethoscope and is preprocessed using the wavelet decomposition. In stage 2, the PCA-based diagnostic features, [[Formula: see text]], are generated from time-frequency feature matrix ([Formula: see text]). In the matrix TFFM, the time domain features [Formula: see text] are firstly extracted from the time domain envelope [Formula: see text] for the filtered heart sound signal [Formula: see text], and frequency domain features, [Formula: see text], are subsequently extracted from a frequency domain envelope ([Formula: see text]) for each heart sound cycle automatically segmented via the short time modified Hilbert transform(STMHT). In stage 3, support vector machines-based classification boundary curves for the dataset [Formula: see text] are first generated, and least-squares-based ellipse models are subsequently built for the classification boundary curve. Finally, based on the ellipse models, the classification criterion is defined for the diagnosis of VSD sounds. The proposed diagnostic system is validated by sounds from the internet and by sounds from clinical heart diseases. Moreover, comparative analysis to validate the usefulness of the proposed diagnostic system, mitral regurgitation and aortic stenosis sounds are used as examples for detection. As a result, the higher classification accuracy, which is achieved by this study compared to the other methods, is [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for diagnosing small VSD, moderate VSD, large VSD and normal sounds, respectively.

Time domain characterization of high power solid state amplifiers for the next generation linear accelerators

AbstractThis paper presents the time‐domain characterization of high power pulsed solid state amplifiers to be used for linear accelerator applications. The study comprises nonlinear circuit envelope simulations and time domain envelope measurements. Measurements and simulations are performed under the pulsed conditions (3.5 ms pulse width, 5% duty cycle) specific to the European Spallation Source (ESS) high intensity proton accelerator. We measure the characteristics of pulsed LDMOS based power amplifiers such as: pulse droop along the pulse, efficiency, average envelope pulse amplitude and phase, pulse drain current waveform, pulse drain voltage waveform, etc. A comparison between the measured results and the simulated results is also presented. In addition to the pulse profile characterization, the pulse to pulse (P2P) stability of the presented solid state power amplifier (SSPA) is investigated as variations of amplitude and phase. The P2P stability simulations are introduced as a combination of the Monte‐Carlo simulations and the nonlinear circuit envelope simulations. The simulated results are used for fitting the P2P measurements to give an early insight of causes of instabilities of the nonlinear LDMOS models.

Fast computation of far-field pulse-echo PSF of arbitrary arrays for large sparse 2-D ultrasound array design

Large wideband two-dimensional (2-D) arrays are essential for high-resolution three-dimensional (3-D) ultrasound imaging. Since the tremendous element number of a full sampled large 2-D array is not affordable in any practical 3-D ultrasound imaging system, it is necessary to reduce the element number through sparse 2-D array design. Sparse array design requires that both the positions and weights of the array elements should be arbitrarily alterable. Hence a proper evaluation tool that can deal with arbitrary array is integral to optimizing the array structure and apodization function. It is known that pulse-echo point spread function (PSF) has been a common tool used to evaluate the performance of wideband arrays in ultrasound imaging all along, which also plays an important role in wideband ultrasound simulations. However, so far the conventional ultrasound simulation tools can only calculate pulse-echo PSF of arbitrary wideband arrays in the time domain because of the existence of nonuniform nodes in the spatial impulse response expressions, which obstructs their application of FFT to do fast computation of the time-domain convolutions. As a result, ultra-high time consumption of pulse-echo PSF computation of a large arbitrary wideband array hampers it to be taken as the evaluation tool by any stochastic optimization methods which need massive iterations in designing large sparse 2-D arrays. This paper aims to make available the pulse-echo PSF tool in designing large sparse 2-D arrays by proposing a fast computation method of far-field pulse-echo PSFs of arbitrary wideband arrays. In the paper, fast computation of wideband spatial impulse responses of a 2-D array is first realized in frequency domain by employing the nonuniform fast Fourier transform (NUFFT), under the point source assumption in far-field. On the basis of that, fast computation of time-domain convolutions is made possible by using FFT. In addition, a short inverse FFT (IFFT) is applied in recovering the time-domain envelopes rather than the detailed waveforms of beam pulses to extract the pulse-echo PSF, which further accelerates the computation. Compared with the computation speed of the time domain method, i.e. Field II, the proposed method achieves an improvement of three orders of magnitude with comparable accuracy for a 100×100 wideband 2-D array. The proposed method makes it possible for applying stochastic optimization methods to design arbitrary large wideband 2-D arrays using pulse-echo PSF as the evaluation tool.