Showing papers in "IEEE Transactions on Intelligent Transportation Systems in 2004"

Travel-time prediction with support vector regression

Abstract: Travel time is a fundamental measure in transportation. Accurate travel-time prediction also is crucial to the development of intelligent transportation systems and advanced traveler information systems. We apply support vector regression (SVR) for travel-time prediction and compare its results to other baseline travel-time prediction methods using real highway traffic data. Since support vector machines have greater generalization ability and guarantee global minima for given training data, it is believed that SVR will perform well for time series analysis. Compared to other baseline predictors, our results show that the SVR predictor can significantly reduce both relative mean errors and root-mean-squared errors of predicted travel times. We demonstrate the feasibility of applying SVR in travel-time prediction and prove that SVR is applicable and performs well for traffic data analysis.

Automatic license plate recognition

Abstract: Automatic license plate recognition (LPR) plays an important role in numerous applications and a number of techniques have been proposed. However, most of them worked under restricted conditions, such as fixed illumination, limited vehicle speed, designated routes, and stationary backgrounds. In this study, as few constraints as possible on the working environment are considered. The proposed LPR technique consists of two main modules: a license plate locating module and a license number identification module. The former characterized by fuzzy disciplines attempts to extract license plates from an input image, while the latter conceptualized in terms of neural subjects aims to identify the number present in a license plate. Experiments have been conducted for the respective modules. In the experiment on locating license plates, 1088 images taken from various scenes and under different conditions were employed. Of which, 23 images have been failed to locate the license plates present in the images; the license plate location rate of success is 97.9%. In the experiment on identifying license number, 1065 images, from which license plates have been successfully located, were used. Of which, 47 images have been failed to identify the numbers of the license plates located in the images; the identification rate of success is 95.6%. Combining the above two rates, the overall rate of success for our LPR algorithm is 93.7%.

Challenges of intervehicle ad hoc networks

Abstract: Intervehicle communication (IVC) networks, a subclass of mobile ad hoc networks (MANETs), have no fixed infrastructure and instead rely on the nodes themselves to provide network functionality. However, due to mobility constraints, driver behavior, and high mobility, IVC networks exhibit characteristics that are dramatically different from many generic MANETs. This paper elicits these differences through simulations and mathematical models and then explores the impact of the differences on the IVC communication architecture, including important security implications.

A simple and effective method for predicting travel times on freeways

Abstract: We present a method to predict the time that will be needed to traverse a given section of a freeway when the departure is at a given time in the future. The prediction is done on the basis of the current traffic situation in combination with historical data. We argue that, for our purposes, the current traffic situation of a section of a freeway is well summarized by the current status travel time. This is the travel time that would result if one were to depart immediately and no significant changes in traffic would occur. This current status travel time can be estimated from single- or double-loop detectors, video data, probe vehicles, or any other means. Our prediction method arises from the empirical observation that there exists a linear relationship between any future travel time and the current status travel time. The slope and intercept of this relationship may change subject to the time of day and the time until departure, but linearity persists. This observation leads to a prediction scheme by means of linear regression with time-varying coefficients.

Color-based road detection in urban traffic scenes

Abstract: Road detection is a key issue for autonomous driving in urban traffic. In this paper, after a brief overview of existing methods, we present a road-area detection algorithm based on color images. This algorithm is composed of two modules: boundaries are first estimated based on the intensity image and road areas are subsequently detected based on the full color image. In the first module, an edge image of the scene is analyzed to obtain the candidates for the left and right road borders and to delimit the area that will subsequently be used to compute the mean and variance of the Gaussian distribution, assumed to be obeyed by the color components of road surfaces. The second module effectively extracts the road area and reinforces boundaries that most appropriately fit the road-extraction result. The combination of these modules can overcome basic problems due to inaccuracies in edge detection based on the intensity image alone and due to the computational complexity of segmentation algorithms based on color images. Experimental results on real road scenes have substantiated the effectiveness of the proposed method.

Visual sign information extraction and identification by deformable models for intelligent vehicles

Abstract: This paper deals with the extraction of part of the visual information presented in streets, roads, and motorways. This information, provided by either traffic or road signs and route-guidance signs, is extremely important for safe and successful driving. An automatic system that is capable of extracting and identifying these signs automatically would help human drivers enormously; navigation would be easier and would allow him or her to concentrate on driving the vehicle. The system would indicate to the driver the presence of a sign in advance, so that some incorrect human decisions could be avoided. A deformable model scheme allows us to include the knowledge used while designing the signs in the algorithm and is used for their detection and identification. Two techniques to find the minimum in the energy function are shown: simulated annealing and genetic algorithms. Some problems are addressed, such as uncontrolled lighting conditions; occlusions; and variations in shape, size, and color.

Springrobot: a prototype autonomous vehicle and its algorithms for lane detection

Abstract: This work presents the current status of the Springrobot autonomous vehicle project, whose main objective is to develop a safety-warning and driver-assistance system and an automatic pilot for rural and urban traffic environments. This system uses a high precise digital map and a combination of various sensors. The architecture and strategy for the system are briefly described and the details of lane-marking detection algorithms are presented. The R and G channels of the color image are used to form graylevel images. The size of the resulting gray image is reduced and the Sobel operator with a very low threshold is used to get a grayscale edge image. In the adaptive randomized Hough transform, pixels of the gray-edge image are sampled randomly according to their weights corresponding to their gradient magnitudes. The three-dimensional (3-D) parametric space of the curve is reduced to the two-dimensional (2-D) and the one-dimensional (1-D) space. The paired parameters in two dimensions are estimated by gradient directions and the last parameter in one dimension is used to verify the estimated parameters by histogram. The parameters are determined coarsely and quantization accuracy is increased relatively by a multiresolution strategy. Experimental results in different road scene and a comparison with other methods have proven the validity of the proposed method.

Urban traffic control structure based on hybrid Petri nets

Abstract: An urban network of signalized intersections can be suitably modeled as a hybrid system, in which the vehicle flow behavior is described by means of a time-driven model and the traffic light dynamics are represented by a discrete event model. In this paper, a model of such a network via hybrid Petri nets is used to state and solve the problem of coordinating several traffic lights with the aim of improving the performance of some classes of special vehicles, i.e., public and emergency vehicles. The proposed model has been validated using real traffic data relevant to the city of Torino, Italy. Some relevant experimental results are reported and discussed.

An enhanced 0-1 mixed-integer LP formulation for traffic signal control

Abstract: An enhanced 0-1 mixed-integer linear programming formulation based on the cell-transmission model is proposed for the traffic signal optimization problem. This formulation has several features that are currently unavailable in other existing models developed with a similar approach, including the components for handling the number of stops, fixed or dynamic cycle length and splits, and lost time. The problem of unintended vehicle holding, which is common in analytical models, is explicitly treated. The formulation can be utilized in developing strategies for adaptive traffic-control systems. It can also be used as a benchmark for examining the convergence behavior of heuristic algorithms based on the genetic algorithm, fuzzy logic, neural networks, or other approaches that are commonly used in this field. The discussion of extending the proposed model to capture traffic signal preemption in the presence of emergency vehicles is given. In terms of computational efficiency, the proposed formulation has the least number of binary integers as compared with other existing formulations that were developed with the same approach.

Modeling traffic signal control using Petri nets

Abstract: This paper focuses on the use of Petri nets (PN) to model the control of signalized intersections. The application of PN to an eight-phase traffic signal controller is illustrated. Structural analysis of the control PN model is performed to demonstrate how the model enforces the traffic operation safety rules. This is followed by a discussion of why this modeling tool has future value as the use of more advanced control strategies continue to expand.

VIRTUOUS: vision-based road transportation for unmanned operation on urban-like scenarios

Abstract: This work presents an intelligent transportation system (ITS) that was implemented on an autonomous vehicle designed to perform global navigation missions on a network of unmarked roads. This is the first step toward the complete implementation of ITS in urban environments, which is the long-term goal of this work. Using a global positioning system, global navigation is achieved by means of a global planner and a task manager that recurrently coordinate the execution of vision-based perception tasks for the road tracking of nonstructured roads and the navigation of intersections. In addition, a vision-based vehicle-detection task has been developed, which endows the global navigation system with a reactive capacity. The complete system has been tested on the BABIECA prototype vehicle, which was autonomously driven for hundreds of kilometers around a private circuit, designed to emulate an urban quarter, at speeds of up to 50 km/h, successfully carrying out different navigation missions. During the tests, the vehicle drove itself across crossroads and performed the appropriate turning maneuvers at intersections. It also demonstrated its robustness with regard to shadows, road texture, weather conditions, and changing illumination.

Modeling of traffic flow of automated vehicles

Abstract: With the development of near term automatic vehicles following concepts such as intelligent cruise control (ICC) and cooperative driving, vehicles will be able to automatically follow each other in the longitudinal direction. The modeling of traffic flow consisting of such vehicles is important for analyzing the effects of vehicle automation on the characteristics of traffic flow and for suggesting macroscopic control strategies to improve efficiency. Such analysis may also suggest ways for modifying the vehicle control characteristics in order to improve the macroscopic behavior of traffic. In this paper, we developed a mesoscopic and macroscopic model that describes the automated traffic-flow dynamics in a single highway lane. The mesoscopic model describes the speed and density continuously in time and space and at the same time retains the microscopic characteristics of traffic flow. The macroscopic model describes the average speed and density at each section of the lane and at each point in time. Even though the macroscopic model does not retain the microscopic characteristics of the vehicular traffic, computationally it is much simpler than the mesoscopic one. Simulations are used to demonstrate the effectiveness of these models in describing traffic-flow characteristics. The developed models indicate some similarities, but also some fundamental differences with existing traffic-flow models for manually driven vehicles.

Intelligence and security informatics for homeland security: information, communication, and transportation

Abstract: Intelligence and security informatics (ISI) is an emerging field of study aimed at developing advanced information technologies, systems, algorithms, and databases for national- and homeland-security-related applications, through an integrated technological, organizational, and policy-based approach. This paper summarizes the broad application and policy context for this emerging field. Three detailed case studies are presented to illustrate several key ISI research areas, including cross-jurisdiction information sharing; terrorism information collection, analysis, and visualization; and "smart-border" and bioterrorism applications. A specific emphasis of this paper is to note various homeland-security-related applications that have direct relevance to transportation researchers and to advocate security informatics studies that tightly integrate transportation research and information technologies.

A car test for the estimation of GPS/INS alignment errors

Abstract: Misalignment can be an important error source in the integration of the global positioning system (GPS) and inertial navigation systems. This paper presents car test results on the estimation of alignment errors in the integration of a low-grade inertial measurement unit (IMU) with accurate GPS measurement systems. The car test was conducted with a low-cost solid-state IMU and carrier-phase differential GPS measurement systems. Test results showed that changes in the angular velocity improve the estimation of the lever arm between the GPS antenna and IMU. They also showed that changes in acceleration improve the estimation of the relative attitude between the GPS antenna array and IMU. The lever arm was estimated with a 10-cm error. The relative attitude was estimated with a half-degree error. An iterative scheme was used to improve the estimation of the alignment errors during postprocessing. The scheme was shown to be useful when the test car could not have sufficient changes in motion due to limitations in its path. With the given set of test data, the estimation error decreased as the number of iterations increased.

A novel method for resolving vehicle occlusion in a monocular traffic-image sequence

Abstract: This paper presents a novel method for resolving the occlusion of vehicles seen in a sequence of traffic images taken from a single roadside mounted camera. Its concept is built upon a previously proposed vehicle-segmentation method, which is able to extract the vehicle shape out of the background accurately without the effect of shadows and other visual artifacts. Based on the segmented shape and that the shape can be represented by a simple cubical model, we propose a two-step method: first, detect the curvature of the shape contour to generate a data set of the vehicles occluded and, second, decompose it into individual vehicle models using a vanishing point in three dimensions and the set of curvature points of the composite model. The proposed method has been tested on a number of monocular traffic-image sequences and found that it detects the presence of occlusion correctly and resolves most of the occlusion cases involving two vehicles. It only fails when the occlusion was very severe. Further analysis of vehicle dimension also shows that the average estimation accuracy for vehicle width, length, and height are 94.78%, 94.09%, and 95.44%, respectively.

Evaluation of adaptive neural network models for freeway incident detection

Abstract: Automated incident detection is an essential component of a modern freeway traffic monitoring system. A number of neural network (NN)-based incident detection models have been tested independently over the past decade. This paper evaluates the adaptability of three promising NN models for this problem: a multilayer feed-forward NN (MLFNN), a basic probabilistic NN (BPNN) and a constructive probabilistic NN (CPNN). These three models have been developed on an original freeway site in Singapore and then adapted to a new freeway site in California. In addition to their incident detection performance, their ability to adapt to new freeway sites, and network sizes have also been compared. A novel updating scheme has been used for adjustment of smoothing parameter of the BPNN. Results of this study show that the MLFNN model has the best incident detection performance at the development site while CPNN model has the best performance after model adaptation at the new site. In addition, the adaptation method for CPNN model is less laborious. The efficient network pruning procedure for the CPNN network resulted in a smaller network size, making it easier to implement it for real-time application. The results suggest that CPNN model has good potential for application in an operational automatic incident detection system for freeways.

Simultaneous registration and fusion of multiple dissimilar sensors for cooperative driving

Abstract: The fusion of multiple sensory information plays a key role in cooperative driving for flexible platooning of automated vehicles over a couple of lanes within a short intervehicle distance. In this paper, the problem of online sensor fusion with spatially and temporally misaligned dissimilar sensors is considered. A spatial-temporal registration model for the popular intelligent vehicular sensors including radar, global positioning system, inertial navigation system, and camera is first developed for sensor alignment. An unscented Kalman filter (UKF) is proposed here to fuse and register these sensors that are installed on a platoon of vehicles simultaneously. When the road geometry information is available from a digital map database, a constrained UKF is further developed to improve the fusion accuracy. The effect of spatial-temporal sensor misalignment upon the vehicle-state estimation is also analyzed theoretically. Simulations show that the proposed UKF method not only can align the dissimilar vehicular sensors properly with both spatial and temporal biases, but can also obtain accurate fused tracks of vehicles in a platoon.

A new traffic-signal control for modern roundabouts: method and application

Abstract: When the circulatory roadway of a roundabout has more than two lanes, the vehicles' weaving and merging cause large traffic and safety problems. In this paper, a new method of traffic-signal control for modern roundabouts is proposed to solve problems by eliminating the conflict points and weaving sections at a roundabout with different traffic-flow rates on each approach, which normally appear in the real world. A second stop line is set exclusively for the left-turn traffic on the circulatory roadway. It is beside the first stop line on the approach. Traffic signals are installed before each stop line to eliminate the conflicts between the traffic flows on the approaches and the left-turn traffic flows on the circulatory roadway. Left-turn vehicles on the circulatory roadway will stop before red signals to avoid weaving. Equations are derived to compute the signal cycle length and the green time for each traffic flow, considering the limited queue on the circulatory roadway. Capacity and delay are also formulated to evaluate the roundabout's performance. This traffic-signal control has a successful application of a roundabout in Xiamen, China, to solve the very serious traffic-congestion problem. The signal-timing scheme was computed with the proposed equations, as well as the capacity and delay. Pictures taken before and after the improvement show the operation. After the improvement, the roundabout capacity increases 72.1% and the average delay of each vehicle decreases by 20 s.

Vehicle Reidentification using multidetector fusion

Abstract: Vehicle reidentification is the process of matching vehicles from one point on the roadway (one field of view) to the next. By performing vehicle reidentification, important traffic parameters including travel time, travel time variability, section density, and partial dynamic origin/destination demands can be obtained. Field traffic data were collected in Alton Parkway in Southern California for training and testing of the multidetector vehicle reidentification algorithm. These data consisted of inductive loop signatures of vehicles that traversed two detector stations spanning a section of an arterial and the corresponding video of these signatures. Even though the video collected was not optimized for pattern-recognition purposes, an investigation into the feasibility of fusing inductive vehicle signatures with video for anonymous vehicle reidentification was conducted. The resulting reidentification rate of over 90% shows that this approach merits further investigation. The results also show that the use of detector fusion provides system redundancy and yields slightly better results than the use of a single detector.

An HMM/MRF-based stochastic framework for robust vehicle tracking

Abstract: Shadows of moving objects often obstruct robust visual tracking. In this paper, we present a car tracker based on a hidden Markov model/Markov random field (HMM/MRF)-based segmentation method that is capable of classifying each small region of an image into three different categories: vehicles, shadows of vehicles, and background from a traffic-monitoring movie. The temporal continuity of the different categories for one small region location is modeled as a single HMM along the time axis, independently of the neighboring regions. In order to incorporate spatial-dependent information among neighboring regions into the tracking process, at the state-estimation stage, the output from the HMMs is regarded as an MRF and the maximum a posteriori criterion is employed in conjunction with the MRF for optimization. At each time step, the state estimation for the image is equivalent to the optimal configuration of the MRF generated through a stochastic relaxation process. Experimental results show that, using this method, foreground (vehicles) and nonforeground regions including the shadows of moving vehicles can be discriminated with high accuracy.

In-vehicle WLAN radio-frequency communication characterization

Abstract: Recently, new standards have emerged in the telecommunication industry, which provide an open global specification that enables mobile devices to access and interact with information and services instantly. These mobile devices are, for example, laptops and personal digital assistants. These emerging standards, generically called wireless "x" area networks (WxAN), will in the future be frequently operated from inside vehicles as part of the deployment of mobile offices and to support advanced intelligent transportation system services. These wireless networks currently operate between 1-6 GHz, although 60 GHz could be used when the technology will be economically viable. These mobile devices are frequently put to use in vehicles. To be ensure that portable equipment left in a jacket pocket somewhere in the trunk can interact with other car equipment or portable objects elsewhere in the car needs some investigation. Thus, this paper characterizes radio-frequency propagation inside vehicle bodies, with passengers, in order to evaluate the effectiveness of such a WxAN in vehicle environments.

Spatial-temporal traffic data analysis based on global data management using mas / he-sheng zhang... [et al.].

Abstract: The spatial-temporal traffic data analysis based on global data management is a newly developed and crucial approach to help traffic managers having the global view of urban traffic status in the level of road network, which is very clearly useful in traffic control and route guidance. The multiagent systems are used in traffic data management with full consideration of the characteristics of traffic data and the cooperation and workflow among them. In software implementation of data management, the agent-based common object request broker architecture is adopted taking the distributed urban traffic data in the large area under network environments into account. Based on the global traffic data, the approach of visualized spatial-temporal analysis is then induced. The similarity of traffic data is analyzed first for each link and its profile is achieved to undertake the primary processing of urban traffic data. Furthermore, analysis results are shown on the basis of the geographic information systems for transportation. The two types of visualization, pseudocolor and contour maps, are adopted in the demonstration to display the traffic status graphically and its changing frames. Among the applications in some big cities in China, the case of urban traffic analysis for Beijing is studied to demonstrate the implementation of the approach.

Spatial-temporal traffic data analysis based on global data management using MAS

Abstract: The spatial-temporal traffic data analysis based on global data management is a newly developed and crucial approach to help traffic managers having the global view of urban traffic status in the level of road network, which is very clearly useful in traffic control and route guidance. The multiagent systems are used in traffic data management with full consideration of the characteristics of traffic data and the cooperation and workflow among them. In software implementation of data management, the agent-based common object request broker architecture is adopted taking the distributed urban traffic data in the large area under network environments into account. Based on the global traffic data, the approach of visualized spatial-temporal analysis is then induced. The similarity of traffic data is analyzed first for each link and its profile is achieved to undertake the primary processing of urban traffic data. Furthermore, analysis results are shown on the basis of the geographic information systems for transportation. The two types of visualization, pseudocolor and contour maps, are adopted in the demonstration to display the traffic status graphically and its changing frames. Among the applications in some big cities in China, the case of urban traffic analysis for Beijing is studied to demonstrate the implementation of the approach.

Minimum sensor second-order sliding mode longitudinal control of passenger vehicles

Abstract: In this paper, a minimum sensor variable structure control strategy for cruise and tracking longitudinal control of vehicles has been proposed. It relies on the generation of "second-order" sliding regimes, i.e., sliding modes characterized by an identically null derivative of the sliding variable. Because of the lack of measurements, the use of suitably designed observers is exploited in the paper. On the whole, the proposed strategy is designed so as to guarantee a bounded jerk and to avoid too frequent changes between the use of the accelerator and the brake. The control strategy is robust with respect to matched bounded parameters variations, and uncertainties.

Stochastic car tracking with line- and color-based features

Abstract: Color- and edge-based trackers can often be "distracted", causing them to track the wrong object. Many researchers have dealt with this problem by using multiple features, as it is unlikely that all will be distracted at the same time. It is also important for the tracker to maintain multiple hypotheses for the state; sequential Monte Carlo filters have been shown to be a convenient and straightforward means of maintaining multiple hypotheses. In this paper, we improve the accuracy and robustness of real-time tracking by combining a color histogram feature with an edge-gradient-based shape feature under a sequential Monte Carlo framework.

A study on cyclist behavior at signalized intersections

Abstract: A study on the cyclist behavior at a signal-controlled intersection was conducted to determine the behavioral characteristics representative. The study focused on the cyclists' behavior at signalized intersections, including the crossing speeds, crossing gap/lag acceptance behavior, and group-riding behavior. Traffic data were collected by using video cameras from a wide and complex signalized intersection. The statistical analysis of data was conducted to determine the characteristics of bicycle traffic crossing speeds, gap/lag acceptance, and group riding. Interesting group behavior of cyclists at a signal-controlled intersection was discovered and analyzed. The results are useful for understanding the performance of mixed traffic at signalized intersections and building microscopic simulation models.

Freeway traffic stream modeling based on principal curves and its analysis

Abstract: We have proposed to use the method of principal curves to describe and analyze the interaction among freeway traffic-stream variables and their joint behaviors without utilizing conventional assumptions made on the functional forms of interactions, as in previous studies. As a nonparameter modeling approach, the performance of the proposed method depends only on the data used and involves no assumed knowledge regarding the relationship among the traffic-stream variables. First, we discuss the basic algorithm for data analysis using principal curves and the corresponding data filter algorithm for determining principal curves for application in traffic-steam analysis. Second, a case study is used to compare the performance of the proposed method to that of the classical model proposed by Greenshields; results indicate that the proposed model is better than the classical one in both data accuracy and curve shape. Finally, the traffic-stream models generated with principal curves at different locations and lanes are compared with each others and the three-dimensional traffic-stream models developed from principal curves are discussed. Clearly, our results have demonstrated the feasibility and advantages of applying principal curves in freeway traffic-stream modeling and analysis.

Infrared transmission for intervehicle ranging and vehicle-to-roadside communication systems using spread-spectrum technique

Abstract: In this paper, an infrared intervehicle ranging and vehicle-to-roadside communication systems are studied. A direct-sequence spread-spectrum (DS-SS) technology is employed to obtain the robustness against multiuser interference and ambient light noise. We compare the correlation properties for various optical spreading codes such as an optical orthogonal code (OOC), a prime code, an extended prime code, and a modified m-sequence. The performance of the infrared DS-SS ranging and communication system is evaluated by computer simulation over a channel in consideration of multipath dispersion, multiuser interference, and a background light noise. The infrared DS-SS intervehicle ranging system using an OOC has lower ranging error rate (RER) than ranging systems using a prime code, an extended prime code, and modified m-sequences even if there is the interference from other users and lightwave dispersion. In the infrared DS-SS vehicle-to-roadside communication system, L-ary pulse position modulation (L-PPM) is used as a modulation scheme due to high average power efficiency. It is shown that the proposed system achieves smaller BER performance as the modulation order L increases and the proposed system with a (361,6,1,1) OOC has a smaller BER than that with a (181,6,1,1) OOC.

Improving travel-time reliability by the use of trip booking

Abstract: The congestion of our infrastructure, particularly (urban) motorways, continues to increase. Efficient planning, for instance in freight transport, is hindered by the resulting unreliability of travel times. Another effect of this congestion is a reduced utilization rate of the road. This work presents trip booking, a method aimed at improvement of the reliability of travel times as well as an increase in the effective use of road capacity. Increased reliability facilitates better logistic planning. Furthermore, it allows the sharing of infrastructure between different modalities, with each modality having its own operational time window. The system aims at open dedicated infrastructure, such as bus lanes and dedicated freight lanes, and preserves the autonomy of both the provider and user of the infrastructure. The advantage claims are supported by simulation results for basic network configurations.

Dynamical analysis of freeway traffic

Abstract: This work presents the simulator of intelligent transportation systems (SITS). The SITS is based on a microscopic simulation approach to reproduce real traffic conditions in an urban or nonurban network and considers different types of vehicles, drivers, and roads. A dynamical analysis of several traffic phenomena is then addressed. The results of using classical system theory tools notes that it is possible to study traffic systems, taking advantage of the knowledge gathered with automatic control algorithms. In this line of thought, a new modeling formalism based on the embedding of statistics and Fourier transform is also presented.