Combination Of Neural Network Research Articles

Electric field detection method of 500 kV porcelain deteriorated insulator based on Combination Neural Network

The operation state of insulators is closely related to the safety and reliability of the power network. Affected by the environment, the insulator strings will gradually age and become low-value or zero-value in insulators. Therefore, it is of great engineering significance to correctly and timely detect whether there are faults in insulators. It has a good engineering application prospect to evaluate the insulation condition of insulators by detecting the surface electric field value of insulators. In this paper, the external axial electric field value of the insulator skirt is obtained through finite element simulation, and the combined neural network model is used. Finally, the effectiveness of the model is verified by the measured data. The experimental results show that this method can effectively identify whether the insulator has deteriorated.

Noise pollution prediction and seasonal comparison in urban parks using a coupled GIS- artificial neural network model.

Noise pollution is a challenging environmental issue in densely built urban areas and requires a holistic understanding of its sources and alleviation processes. Taking Isfahan City in Iran as a typical case, this study developed a combined GIS-artificial neural network (ANN) model to predict the spatio-temporal contribution of low-width parks to poise pollution mitigation. The 30-min equivalent sound level was measured at 100 stations in six urban parks (with a total area of 55.84ha) under stable and controlled winter and summer conditions. The noise level predicting variables were hypothesized to be the area of vegetation cover; NDVI-based vegetation density and standard deviation (std); vegetation height; and road coverage measured within 100-, 200-, and 300-m radius buffer rings drown around each noise sampling station. These predictors were introduced to a multi-layer perceptron ANN model to identify and compare the most important noise alleviation variables among the selected predictors. The mean noise levels ranged from 67.23 to 70.57dB. The number of vehicles showed an insignificant temporal difference, indicating that the noise source was relatively constant between the seasons. The ANN model performed satisfactorily in both seasons with SSE values of < 0.03. The Mann-Whitney U test showed a significant difference in the predicted noise levels between summer and winter. This study highlighted the efficiency of the combined GIS-ANN model in predicting distant-dependent urban processes, especially noise pollution whose levels and variability are essential in formulating urban land-use management.

CSI-Based Joint Location and Activity Monitoring for COVID-19 Quarantine Environments

The recent SARS-COV-2 virus, also known as COVID-19, badly affected the world’s healthcare system due to limited medical resources for a large number of infected human beings. Quarantine helps in breaking the spread of the virus for such communicable diseases. This work proposes a nonwearable/contactless system for human location and activity recognition using ubiquitous wireless signals. The proposed method utilizes the channel state information (CSI) of the wireless signals recorded through a low-cost device for estimating the location and activity of the person under quarantine. We propose to utilize a Siamese architecture with combined one-dimensional convolutional neural networks (1-D-CNNs) and bi-directional long short-term memory (Bi-LSTM) networks. The proposed method provides high accuracy for the joint task and is validated on two real-world testbeds, first, using the designed low-cost CSI recording hardware, and second, on a public dataset for joint activity and location estimation. The human activity recognition (HAR) results outperform state-of-the-art machine and deep learning methods, and localization results are comparable with the existing methods.

Mapping soil available copper content in the mine tailings pond with combined simulated annealing deep neural network and UAV hyperspectral images

Improper discharge of slag from mining will pollute the surrounding soil, thereby affecting the ecology and becoming an important global problem. The available copper (ACu) content in polluted soil is an important factor affecting plant growth and development. When investigating a large area of soil with ACu, manual sampling by points and inspection are mainly used, due to the heterogeneity of soil, the efficiency and accuracy are lower. The Unmanned aerial vehicle (UAV) equipped with a hyperspectral sensor as a remote sensing technology is widely used in soil indicator monitoring because of its rapid and convenience. Meanwhile, using the relationship between soil organic matter and available copper has the potential to predict available copper. In this study, we selected the study area with tailings area in the Jianghan Plain of China and used a UAV equipped with a hyperspectral sensor to predict ACu and soil organic matter (SOM) in the soil with two datasets. Firstly, 74 soil samples were collected in the study area, and the ACu and SOM of the soil samples were determined. Second, a hyperspectral image of the study area is obtained using a UAV equipped with a hyperspectral sensor. Thirdly, we combine hyperspectral data with competitive adaptive reweighted sampling (CARS) to obtain feature bands and utilize simulated annealing deep neural network (SA-DNN) to generate estimation models. Finally, maps of the distribution of ACu and SOM in the area were generated using the model. In two datasets, the model of ACu with R2 values both are 0.89, and R2 on the model of SOM is 0.89 and 0.88. The results show that the combination of UAV hyperspectral imagery with the SA-DNN model has good performance in the prediction of organic matter and available copper, which is helpful for soil environmental monitoring.

5G Indoor Positioning for Manufacturing using Convolutional Neural Networks

Current disrupting trends in manufacturing like mobile robot-based intralogistics and digital twins require precise indoor positioning. 5G-based positioning has the potential to be more cost-efficient than other positioning systems due to shared hardware for positioning and communication. But first it has to achieve high positioning accuracy and stability. No positioning technique has yet been established as standard in the industry in combination with 5G because traditional positioning approaches can deliver unstable results when multi-path fading, reflections, or missing line-of-sight effects occur. Identifying promising techniques is still an open research topic. It is established in the literature that deep learning-based approaches can mitigate these effects better than traditional ones by learning unique fingerprints that correspond to precise positions. However, fully-connected neural networks tend to overtrain on these fingerprints which again leads to unstable positioning due to a lack of generalization. This can be explained by the inability of fully connected neural networks to learn structural interrelationships in high dimensional data. This is because their input data is flattened, which means the structure is removed. Convolutional neural networks are being widely used with image data and can learn structural interrelationships. This paper presents a technique that uses convolutional neural networks in combination with a channel-smoothing approach to position mobile robots in the research factory ARENA 2036. The results presented in this paper indicate an indoor positioning accuracy similar to other industrial positioning systems, with improved stability compared to a fully-connected approach.

Resilient Composite Learning Neuro-Adaptive Integrated Guidance and Control for reusable Launch Vehicle

An integrated guidance and control system for tracking a reference flight path trajectory of a reusable launch vehicle in the presence of model uncertainties, external disturbances, and measurement noises is proposed. To achieve this goal, a properly-trained feedforward neural network to estimate uncertainties in conjunction with a disturbance observer, which estimates external disturbances and the estimation error made by the neural network, is implemented in both the guidance and control loops. In addition, a state observer is designed whose state estimation error is included (along with the tracking error) in the learning algorithm of the neural network and the disturbance observer; such learning is called composite learning. The combined neural network and disturbance observer (employed in both the guidance and control loops) can consistently compensate for complex uncertain terms in the dynamic model of a reusable launch vehicle. More specifically, the introduced design results in an asymptotic tracking of the reference command, thereby providing a resilient flight control system. Extensive simulations are performed to show the effectiveness and robustness of the proposed integrated guidance and control system for trajectory tracking in the presence of uncertainties, aerodynamic disturbances, and measurement noises.

CNN-LSTM Model for Recognizing Video-Recorded Actions Performed in a Traditional Chinese Exercise.

Identifying human actions from video data is an important problem in the fields of intelligent rehabilitation assessment. Motion feature extraction and pattern recognition are the two key procedures to achieve such goals. Traditional action recognition models are usually based on the geometric features manually extracted from video frames, which are however difficult to adapt to complex scenarios and cannot achieve high-precision recognition and robustness. We investigate a motion recognition model and apply it to recognize the sequence of complicated actions of a traditional Chinese exercise (ie, Baduanjin). We first developed a combined convolutional neural network (CNN) and long short-term memory (LSTM) model for recognizing the sequence of actions captured in video frames, and applied it to recognize the actions of Baduanjin. Moreover, this method has been compared with the traditional action recognition model based on geometric motion features in which Openpose is used to identify the joint positions in the skeletons. Its performance of high recognition accuracy has been verified on the testing video dataset, containing the video clips from 18 different practicers. The CNN-LSTM recognition model achieved 96.43% accuracy on the testing set; while those manually extracted features in the traditional action recognition model were only able to achieve 66.07% classification accuracy on the testing video dataset. The abstract image features extracted by the CNN module are more effective on improving the classification accuracy of the LSTM model. The proposed CNN-LSTM based method can be a useful tool in recognizing the complicated actions.

SHORT-TERM WIND SPEED FORECASTING USING DEEP VARIATIONAL LSTM

Wind speed and power at wind power stations affect the efficiency of a wind farm, so accurate wind forecasting, a nonlinear signal with high fluctuations, increases security and better efficiency than wind power. We are looking for wind speed for a wind farm in Iran. In this research, a combined neural network created from variational autoencoder (VAE), long-term, short-term memory (LSTM), and multilayer perceptron (MLP) for dimension Reduction and encoding is proposed for predicting short-term wind speeds. The data used in this research is related to the statistics of 10 minutes of wind speed in 10- meter, 30-meter, and 40-meter wind turbines, the standard deviation of wind speed, air temperature, and humidity. To compare the proposed model (V- LSTM-MLP), we implemented three deep neural network models, including Stacked Auto-Encoder (SAE), recurrent neural networks (Regular LSTM), and hybrid model Encoder-Decoder recurrent network (LSTM-Encoder-MLP) presented on this dataset. According to the RMSE statistical index, the proposed model is worth 0.1127 for a short time and performs better than other types on this dataset.

Wireless Sensing Technology Combined with Facial Expression to Realize Multimodal Emotion Recognition.

Emotions significantly impact human physical and mental health, and, therefore, emotion recognition has been a popular research area in neuroscience, psychology, and medicine. In this paper, we preprocess the raw signals acquired by millimeter-wave radar to obtain high-quality heartbeat and respiration signals. Then, we propose a deep learning model incorporating a convolutional neural network and gated recurrent unit neural network in combination with human face expression images. The model achieves a recognition accuracy of 84.5% in person-dependent experiments and 74.25% in person-independent experiments. The experiments show that it outperforms a single deep learning model compared to traditional machine learning algorithms.

Objective hearing threshold identification from auditory brainstem response measurements using supervised and self-supervised approaches

Hearing loss is a major health problem and psychological burden in humans. Mouse models offer a possibility to elucidate genes involved in the underlying developmental and pathophysiological mechanisms of hearing impairment. To this end, large-scale mouse phenotyping programs include auditory phenotyping of single-gene knockout mouse lines. Using the auditory brainstem response (ABR) procedure, the German Mouse Clinic and similar facilities worldwide have produced large, uniform data sets of averaged ABR raw data of mutant and wildtype mice. In the course of standard ABR analysis, hearing thresholds are assessed visually by trained staff from series of signal curves of increasing sound pressure level. This is time-consuming and prone to be biased by the reader as well as the graphical display quality and scale.In an attempt to reduce workload and improve quality and reproducibility, we developed and compared two methods for automated hearing threshold identification from averaged ABR raw data: a supervised approach involving two combined neural networks trained on human-generated labels and a self-supervised approach, which exploits the signal power spectrum and combines random forest sound level estimation with a piece-wise curve fitting algorithm for threshold finding.We show that both models work well and are suitable for fast, reliable, and unbiased hearing threshold detection and quality control. In a high-throughput mouse phenotyping environment, both methods perform well as part of an automated end-to-end screening pipeline to detect candidate genes for hearing involvement. Code for both models as well as data used for this work are freely available.

Data-driven failure prediction and RUL estimation of mechanical components using accumulative artificial neural networks

It is necessary to develop advanced fault prognostics techniques for improving maintenance planning and scheduling to avoid unexpected shutdowns, additional expenses and decreased productivity. Such techniques include advanced failure prediction and remaining useful life (RUL) estimation of mechanical components comprising manufacturing systems with complicated structures. This study proposes a novel data-driven prognostic analysis approach for predicting the failure of a mechanical component based on its degradation path and estimating the RUL. A simulated labelled degradation dataset of a mechanical component with a predefined failure threshold was exploited. In order to increase the ability to maintain the increasing trend and the monotony of the degradation path, supervised machine learning models, including combined artificial neural network architectures and an improved version of the neuron-by-neuron training algorithm using accumulative neural networks design were applied for the prediction process.The expected degradation path was extrapolated as a testing dataset of the trained prediction model using an accumulative function. The predicted values are updated with each new data point during the training process until a failure occurs. The results show that the used approach is efficient in predicting the failure and estimating the RUL of mechanical components with high accuracy and a high prediction success rate, and it can maintain the monotonous trend of the degradation path. On the other hand, the used network architectures enable the prediction of the failures of mechanical components within a manufacturing system having a complex structure and providing a vast amount of data.

A Novel Combination Neural Network Based on ConvLSTM-Transformer for Bearing Remaining Useful Life Prediction

A sensible maintenance strategy must take into account the remaining usable life (RUL) estimation to maximize equipment utilization and avoid costly unexpected breakdowns. In view of some inherent drawbacks of traditional CNN and LSTM-based RUL prognostics models, a novel combination model of the ConvLSTM and the Transformer, which is based on the idea of “Extracting spatiotemporal features and applying them to RUL prediction”, is proposed for RUL prediction. The ConvLSTM network can directly extract low-dimensional spatiotemporal features from long-time degradation signals. The Transformer, based entirely on attention mechanisms, can deeply explore the mapping law between deep-level nonlinear spatiotemporal feature information and equipment service performance degradation. The proposed approach is validated with the whole-life degradation dataset of bearings from the PHM 2012 Challenge dataset and the XJTU-SY public dataset. The detailed comparative analysis shows that the proposed method has higher RUL prediction accuracy and outstanding comprehensive prediction performance.

Early prediction of pathological complete response to neoadjuvant chemotherapy in breast cancer MRI images using combined Pre-trained convolutional neural network and machine learning

ObjectiveTimely diagnosis of breast cancer can ameliorate the treatment plan, thus reducing the mortality rate. We propose a model integrating pre-trained Convolutional neural network (CNN) with machine learning for prognosticating pathologic complete response(PCR) using breast cancer dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) prior to commencement of neoadjuvant chemotherapy(NACT).For predicting pathologic complete response (PCR) using dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) for breast cancer prior to the start of neoadjuvant chemotherapy, we present a hybrid model integrating a pre-trained Convolutional neural network (CNN) with machine learning (NACT). Material & MethodsIn this retrospective study, 64 patients receiving NACT for invasive breast cancer are examined. Deep learning-based pre-trained CNN models ResNet-50 and ResNet-18 were used to extract features from patient visit 1 MRI images (before the initiation of NACT). Mann-Whitney U tests is used to assess features and their relevance (significance level p less than 0.05 and confidence interval is 95 %). Furthermore, features extracted and features selected were independently given as an input to different machine learning classifiers for the prediction of response of NACT. Classification performance was assessed under different data division protocols using accuracy, specificity, sensitivity, and area under the receiver operating characteristic curve (AUROC). ResultThe hybrid combination using ResNet-18 as feature extractor, fine K-nearest neighbor(KNN) as classifier and feature selected using Mann-Whitney U test outperformed the result. Accuracy of 99.8% and AUROC of 1 is obtained under hold-out validation protocol while accuracy of 99.3%, and AUROC of 0.99 is obtained under 10-fold cross-validation. ConclusionThe proposed model employing DCE-MRI images acquired before starting chemotherapy has considerable accuracy in classifying PCR and non-PCR patients. The efficacy of the prediction model can improve considerably on the back of a larger dataset.

PID-based CNN-LSTM for accuracy-boosted virtual sensor in battery thermal management system

Battery thermal management is essential to achieve good performance and a long battery system lifespan in electric vehicles and stationary applications. Such a thermal management system is dependent on temperature monitoring, which is frequently hampered by the limited sensor measurements. The virtual sensor is brought forward to overcome this physical restriction and provide broader access to the battery’s temperature distribution. Through leveraging the combined convolutional neural network (CNN) and long short-term memory (LSTM) networks to extract both spatial and temporal information from the data, this paper proposes a novel virtual sensing platform. A PID compensator is included to offer auxiliary correction to the inputs and drive the prediction error to zero over time in a feedback loop. Off-line and online modes of this CNN-LSTM virtual sensor are considered. The network, which is trained off-line, will work with the PID compensator in the online mode with real-time sensor data. With the PID-based accuracy-boosted virtual sensor, the performance of the trained CNN-LSTM prediction on real-time data inputs is improved. Besides, this PID compensator reduces the number of hyper-parameters to be tuned. Based on control theory, the design of PID and its analysis are presented as well. With generated battery thermal data, numerical simulations show that the CNN-LSTM-PID virtual sensing framework can achieve the real-time prediction error reduction rate to 35.52% on average with 18.78% less online calculation time.

Long-distance dependency combined multi-hop graph neural networks for protein–protein interactions prediction

BackgroundProtein–protein interactions are widespread in biological systems and play an important role in cell biology. Since traditional laboratory-based methods have some drawbacks, such as time-consuming, money-consuming, etc., a large number of methods based on deep learning have emerged. However, these methods do not take into account the long-distance dependency information between each two amino acids in sequence. In addition, most existing models based on graph neural networks only aggregate the first-order neighbors in protein–protein interaction (PPI) network. Although multi-order neighbor information can be aggregated by increasing the number of layers of neural network, it is easy to cause over-fitting. So, it is necessary to design a network that can capture long distance dependency information between amino acids in the sequence and can directly capture multi-order neighbor information in protein–protein interaction network.ResultsIn this study, we propose a multi-hop neural network (LDMGNN) model combining long distance dependency information to predict the multi-label protein–protein interactions. In the LDMGNN model, we design the protein amino acid sequence encoding (PAASE) module with the multi-head self-attention Transformer block to extract the features of amino acid sequences by calculating the interdependence between every two amino acids. And expand the receptive field in space by constructing a two-hop protein–protein interaction (THPPI) network. We combine PPI network and THPPI network with amino acid sequence features respectively, then input them into two identical GIN blocks at the same time to obtain two embeddings. Next, the two embeddings are fused and input to the classifier for predict multi-label protein–protein interactions. Compared with other state-of-the-art methods, LDMGNN shows the best performance on both the SHS27K and SHS148k datasets. Ablation experiments show that the PAASE module and the construction of THPPI network are feasible and effective.ConclusionsIn general terms, our proposed LDMGNN model has achieved satisfactory results in the prediction of multi-label protein–protein interactions.

A combined convolutional neural network model and support vector machine technique for fault detection and classification based on electroluminescence images of photovoltaic modules

Nowadays, photovoltaic (PV) systems are gaining increasing momentum due to their ability to generate clean and affordable electric power. However, many factors can impede the production of the PV panels totally or partially. As such, it has become imperative to develop fault detection/classification models to ensure best operating conditions at the maximum energy conversion efficiencies. To address this challenge, a new model for detecting and classifying the faults in electroluminescence images of PV panels has been proposed in this paper. The model combines two machine learning algorithms named Convolutional Neural Network (CNN) and Support Vector Machine (SVM) that are employed for features exaction and classification, respectively. The current model is trained and evaluated using two databases D1 and D2 that contains the electroluminescence images of PV cells. By comparing the proposed model with the previous similar works, this study demonstrates that the CNN combined with SVM provides a higher classification performance with an accuracy of 99.49 % and 99.46 % for databases D1 and D2, respectively.

Combined segmentation and classification of Alzheimer’s disease pathology distribution within human hippocampal subregions

AbstractBackgroundAlzheimer’s disease (AD) is a complex neurodegenerative disease characterized by progressive amyloid and tau pathology. Quantifying and classifying AD pathology across the human brain is challenging due to its large size and expertise needed for critical evaluation of the morphological variety in structure types (e.g. diffuse vs dense plaques, neurofibrillary tangles, neuropil threads, etc.). High‐throughput microscopy imaging and artificial intelligence allow for large‐scale, quantitative ‘digital pathology’ surveys. Due to their structural differences, distinct pathological aggregates in microscopy images can be identified and quantified by trained artificial neural networks (Tang et al 2019). However, current approaches do not incorporate anatomical segmentation to map the distribution of AD pathology across whole tissue sections. Understanding where and when AD pathology is distributed across different brain regions is critical to defining cell type susceptibility to neurodegeneration. This project’s goal was to create a combined neural network classification and segmentation quantification approach to generate anatomically‐accurate pathology maps of the human hippocampus and investigate the spatial distribution of distinct types of pathology.MethodWhole slide images of 4G8 and Gallyas stained coronal tissue sections from the USC ADRC Neuropathology Core were imaged using a slide scanning microscope (29 patients with low, intermediate, or high AD stages). 4G8 sections were manually annotated and segmented using the Allen Human Brain Atlas into thirteen layered hippocampal subregions and delineated using Quantitative Imaging Toolkit (QIT). The machine learning toolkit Ilastik was utilized to quantify and classify 4G8 amyloid staining (dense vs diffuse plaques) and Gallyas (neurofibrillary tangles) within each hippocampal subregion and overlaid to segmentation maps.ResultIn the hippocampus, we found diffuse amyloid pathology increased more rapidly across progressive AD stages compared to dense amyloid plaques or tau neurofibrillary tangles. The ratio of diffuse/dense labeling was highest in the presubiculum and parasubiculum in high AD patients. Patient APOE4 genotype also significantly increased the fraction of diffuse amyloid labeling in high AD patients.ConclusionIntegrating machine learning quantification and classification approaches with anatomical segmentation maps allows for rapid characterization of AD pathology across hippocampal subregions and provides new insights into AD progression across the brain.

Real-Time Monitoring of Parameters and Diagnostics of the Technical Condition of Small Unmanned Aerial Vehicle’s (UAV) Units Based on Deep BiGRU-CNN Models

The paper describes an original technique for the real-time monitoring of parameters and technical diagnostics of small unmanned aerial vehicle (UAV) units using neural network models with the proposed CompactNeuroUAV architecture. As input data, the operation parameter values for a certain period preceding the current and actual control actions on the UAV actuators are used. A reference parameter set model is trained based on historical data. CompactNeuroUAV is a combined neural network consisting of convolutional layers to compact data and recurrent layers with gated recurrent units to encode the time dependence of parameters. Processing provides the expected parameter value and estimates the deviation of the actual value of the parameter or a set of parameters from the reference model. Faults that have led to the deviation threshold crossing are then classified. A smart classifier is used here to detect the failed UAV unit and the fault or pre-failure condition cause and type. The paper also provides the results of experimental validation of the proposed approach to diagnosing faults and pre-failure conditions of fixed-wing type UAVs for the ALFA dataset. Models have been built to detect conditions such as engine thrust loss, full left or right rudder fault, elevator fault in a horizontal position, loss of control over left, right, or both ailerons in a horizontal position, loss of control over the rudder and ailerons stuck in a horizontal position. The results of estimating the developed model accuracy on a test dataset are also provided.

A Combined Neural Network Approach for the Prediction of Admission Rates Related to Respiratory Diseases

In this paper, we investigated rates of admission to hospitals (or other health facilities) due to respiratory diseases in a United States working population and their dependence on a number of demographic and health insurance-related factors. We employed neural network (NN) modelling methodology, including a combined actuarial neural network (CANN) approach, and model admission numbers by embedding Poisson and negative binomial count regression models. The aim is to explore the gains in predictive power obtained with the use of NN-based models, when compared to commonly used count regression models, in the context of a large real data set in the area of healthcare insurance. We used nagging predictors, averaging over random calibrations of the NN-based models, to provide more accurate predictions based on a single run, and also employed a k-fold validation process to obtain reliable comparisons between different models. Bias regularisation methods were also developed, aiming at addressing bias issues that are common when fitting NN models. The results demonstrate that NN-based models, with a negative binomial distributional assumption, provide improved predictive performance. This can be important in real data applications, where accurate prediction can drive both personalised and policy-level interventions.

Electromagnetic Parametric Modeling Using Combined Neural Networks and RLGC-Based Eigenfunctions for Two-Port Microstrip Structures

This article proposes a novel electromagnetic (EM) parametric modeling technique using combined neural networks and resistance, inductance, conductance, and capacitance (RLGC)-based eigenfunctions (neuro-EF) for microwave components with two-port microstrip structures. In the proposed technique, neural networks are used to learn the unknown relationship between the parameters of RLGC of the eigenfunction and geometrical parameters. The generation of training data depends on obtaining the correct eigenvalues of different modes. However, for different geometrical parameter samples, there is no uniform correspondence between the calculated eigenvalues and the modes. The incorrect correspondence may cause the two modes to be swapped, defined as the mode-swap issue. We propose a mode-matching method based on eigenvectors to solve this mode-swap issue. After the training data of eigenfunction parameters is obtained, a preliminary training of the neural networks and a two-step refinement training of the neuro-EF model are proposed to develop the overall EM parametric model. By the proposed modeling technique, the trained model can provide a fast and accurate prediction of EM responses for two-port microstrip structures as geometrical parameters change. For the parametric modeling of microwave components with microstrip structures, the proposed technique can obtain better accuracy in larger geometrical variations compared with the existing methods. Two examples of microstrip structures are used to illustrate the proposed technique.