Computers and Intelligent Systems - Jyväskylän yliopistousers.jyu.fi/~timoh/isast2012.pdf ·...

ISAST Transactions on No. 1, Vol. 4, 2012 (ISSN 1798-2448)

Computers and Intelligent Systems

Florian Zitzelsberger, Michael Shafae:

Markerless Performance Capture of Hand Gestures …………………………………………………1

Jonathan Blackledge, Eugene Coyle, Niall McCoy,

Derek Kearney, Keith Sunderland, and Thomas Woolmington:

Analysis of Wind Velocity and the Quantification of Wind Turbulence in Rural and Urban

Environments using the L´evy Index and Fractal Dimension ………………………………………...7

Vu Van Yem, Nguyen Xuan Quyen, and Kyandoghere Kyamakya:

A Chaotic Modulation Method based on Combination of

CPWPM and BPSK for Digital Communication …………………………………………………….18

Jonathan M. Blackledge, Matthew D. Blackledge, and Jane. N. Courtney:

Non-Gaussian Anisotropic Diffusion for Medical Image Processing using the OsiriX DICOM…….24

Ashish Patel, Michael Shafae:

A Cake Cutting Approach to Rezoning School Districts……………………………………………..33

Jonathan Blackledge, Dmitri Dubovitskiy, and Fiona Lyng:

Targeting Cell Nuclei for the Automation of Raman Spectroscopy in Cytology…………………….42

Santhosh Kumar Kadari, Bala Soujanya Bhupathi Raju, and Nguyen Xuan Quyen:

Digital Image Encryption based on Chaotic Behavior of A Modified Tent Map…………………….52

Jarmo Siltanen, Jyrki Joutsensalo, Timo Hämäläinen, and Kari Luostarinen:

Least Squares Bandwidth Allocation Scheme for 4G Heterogeneous Networks…………………….58

http://www.isastorganization.org/ISAST_CIS_no1_vol1_09.pdfhttp://www.isastorganization.org/ES_no2_vol3.pdf

Markerless Performance Capture of HandGestures

Florian Zitzelsberger, Michael ShafaeCalifornia State University, Fullerton

F

Abstract—Performance capture is where computer vision techniquesare applied to extract information regarding a performer’s body move-ments and gestures which are then mapped to a synthetic actor. In thispaper, a system for controlling a simple virtual puppet using consumer-grade webcams to capture the performance of an untrained puppeteer’shand. The system operates in real-time by first identifying the user’shand based on a skin color mask, tracking the hand’s position usingflocks of KLT features, and identifying gestures through a machinelearning process. The system does not require any markers on theuser’s hand.

Index Terms—Computer vision, image processing, performance cap-ture, gesture recognition, hand tracking, hand detection, skin toneclassifier

1 INTRODUCTIONCommunicating with digital computers has often meantthat the human adapted to the limited input and outputcapabilities of the machine. Slowly but suredly, technol-ogy has advanced to give more natural and intuitivemeans for us to express our wishes to our computers.Especially within the realm of entertainment softwaressuch as games, joysticks and buttons have been thepinnacle of the human-computer interface.

There are entirely new input control methods andtechnologies, such as the Nintendo WiiMote, SonyPlayStation Move and Microsoft Kinect. The electronicentertainment industry is rethinking the traditional, but-ton and controller based approach to human-machineinteraction in video games and entertainment applica-tions. Similarly, more natural user interfaces are heavilyresearched in the medical and military fields. However,across the board, these new input technologies requirevery specific and potentially expensive hardware inorder to reliably interpret human gestures. Specifically,these devices include sensors like depth-perceptive cam-eras, gyroscopes, accelerometers, or visual trackers.

This is understandable, considering the difficultiesinvolved with recognizing human gestures, posture and

M. Shafae is with the Department of Computer Science, California StateUniversity, Fullerton, CA, 92834 USA e-mail: [email protected]. phone:657-278-3291 fax: 657-278-1341F. Zitzelsberger is now with Pixar Animation Studios, Emeryville, CA 94608USA e-mail: [email protected] received September 7, 2012.

appearance as they are inherently ambiguous and differbetween ages and genders. Additionally ethnic and cul-tural influences add meaning and context to our bodylanguage. Moreover, human gestures are not generallyassociated with the same meaning across all cultures andsocial settings. Nodding one’s head, for example, is inter-preted as a form of acknowledgement by most, but notall cultures. Direct eye contact may be interpreted as aform of attentive listening, but may easily be understoodas a provocative gesture.

In this study, we explore the possibilities of usinginexpensive and readily available hardware, such as awebcam and a desktop or laptop computer, as a meansof detecting and interpreting natural hand gestures tocontrol a virtual puppet. The objective is to identify apotential hand object in a scene, track the position andmotion of the hand object, extract gesture informationand apply the gesture’s meaning to manipulate a virtualpuppet. Most importantly, the entire process is to bedone without markers to aid in recognizing a handobject.

This is rather difficult, considering the limitations ofcomputer vision. As visual creatures, we perceive imagesfundamentally differently from the way a computerperceives an image. ”The human brain divides the visionsignal into many channels that stream different kinds ofinformation into your brain. Your brain has an attentionsystem that identifies [...] important parts of an image toexamine while suppressing examination of other areas.There is massive feedback in the visual stream that is,as yet, little understood.” [1] A computer, on the otherhand, sees an image as merely a set of discrete samples,potentially falsified by a noise component. Moreover,where as human eyes feed our brains a wealth of infor-mation instantaneously, consumer-quality webcams arefar inferior. Webcams do not feature automatic controlof focus and aperture and colors are severely distortedunder varying lighting conditions. Most importantly,there is no automatic pattern recognition or a database ofyears of experience in identifying objects and extractingimportant information from this stream of images.

ISAST Transactions on Computers and Intelligent Systems, No. 1, Vol. 4, 2012 (ISSN 1798-2448)

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2 PREVIOUS WORKWith respect to computer vision, the primary focus isnot a novel computer vision algorithm, but rather tocombine a set of existing algorithms to identify a handobject, track the hand object and to extract gestureseffectively in real-time. Furthermore, the identification,tracking and gesture extraction is to be done withoutthe use of artificial markers. The problem domain canbe divided into three subproblems:

1) Hand Detection2) Hand Tracking3) Gesture Recognition

2.1 Hand DetectionOne of the most common approaches to solve the handdetection problem consist of building a skin color maskfrom the input image. In order to build this mask,it is often desirable to convert the input image to alighting condition agnostic color space. This is a colorspace which would allow us to separate the lighting andcolor information from one another. The hue, saturation,and value (HSV) or hue, saturation, and intensity (HSI)color spaces, both a derivation of the RGB color modelsatisfy this requirement. With HSV, variations in light-ing are largely reflected in the saturation component.Sometimes, the normalized RGB or RG chromaticitycolor spaces may be used to achieve similar results. Thenormalized components Rn, Gn, and Bn are computedas follows:

Base = R+G+B

Rn =R

Base

Gn =G

Base

Bn =B

Base= 1−Rn +Gn

Note, that only Base, Rn and Gn must be stored andthat Bn may be derived. If Base is omitted, the con-version becomes lossy, as it is usually the case with RGchromaticity, a variation of the normalized RGB model.

Once the saturation component has been separatedor removed from the input image, a skin color maskmay be created by using histogram back-projection orthresholding. With histogram back-projection, a previ-ously created histogram of skin tones is back-projectedonto the input image, resulting in a distribution prob-ability image. A distribution probability image’s pixel’scolor value reflects the probability of the correspondingpixel in the input image being within the range of thehistogram. This approach can be further improved byusing an adaptive model, such that the color histogramis created and updated on-the-fly accounting for possiblechanges in the illumination conditions of the scene. Onesuch approach is detailed in [2].

Another successful approach to creating a skin colormask consists of filtering the input image by a range

threshold. This approach results in a binary skin colormask. The sensitivity of this classifier may be adjustedby changing the minimum and maximum values ofthe threshold applied. These minimum and maximumvalues are often empirically determined and may beadapted over time. This idea of an adaptive skin colorclassifier using a threshold is explained in [3].

An entirely different approach to object recognitioninvolves using a Haar cascading classifier. This approachis now widely used in face detection and was firstintroduced by Viola and Jones [4]. The idea is based onHaar wavelets and uses black and white rectangles asillustrated in Figure 1. Haar features are calculated bysubtracting the sum of pixels within the white regionsfrom the sum of pixels in the black region of the blackand white rectangle.

Fig. 1. Some examples of Haar feature rectangles.

This computation can be done in constant time, byfirst generating the integral image from the input image.Essentially, the value at any given pixel ii(x, y) on theintegral image indicates the sum of all the pixel valuespreceding ii(x, y):

ii(x, y) =∑x′≤x

∑y′≤y

i(x′, y′)

Once the integral image has been generated, no morethan four array lookups are required to compute the sumof any given rectangle on the input image. As describedin [4], the Viola-Jones object detection framework movesa window of a specified size across the input image,calculating the Haar features for every subregion withinthat window. It then compares the value of the featureagainst a database of learned objects and non-objects. If agiven region falls below the threshold of being classifiedas an object, early rejection of the window results in adramatic performance gain. Only if subregions of thewindow are classified as objects, further processing isdone on the window. It must be noted that in order forthe Viola-Jones approach to work reliably, a considerablylarge, trained database of objects and non-objects mustbe available.

2.2 Hand TrackingOnce the hand detection phase concludes, the user’shand is tracked in order to map it’s movement to the vir-tual puppet’s screen-space position. Several algorithmsexist for this purpose, most notably Mean-Shift andOptical Flow.

The Mean-Shift algorithm is based on the idea offinding the maxima of a density function, represented


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by a number of discrete samples. The algorithm movesa convolution kernel over a predefined search window,detecting maxima within the image region covered bythe kernel, and ultimately the search window. The algo-rithm returns the mode(s) of the density map.

Continuously Adaptive Mean-Shift (CamShift) is anadaptation of the Mean-Shift algorithm, most com-monly used in face, head and object tracking. Insteadof using a static, precomputed probability distribution,CamShift employes an adaptive approach, by recom-puting the probability distribution after every frame.With CamShift, the probability distribution image iscommonly generated by creating a skin probability maskvia histogram back-project, as explained above. For amore detailed explanation of Mean-Shift and CamShiftin particular, see [5].

Optical Flow is not an algorithm per se. The termgenerally describes a pattern of motion in a streamof images. However, a number of algorithms exist tomeasure optical flow. These algorithms are based onphase correlation, which can be measured efficiently us-ing a fast Fourier transform. However, some of the mostpopular algorithms to measure optical flow in computervision are based on differential methods. One examplefor such a method is Lucas-Kanade Optical Flow. Lucas-Kanade Optical Flow is an iterative approach, whichuses spatial intensity gradient information. This ideais explained in [6] and a Kanade-Lucas-Tomasi (KLT)feature tracker is introduced in [7].

2.3 Gesture Recognition

A simple sock puppet can be created by placing one’shand in a sock. By pinching one’s fingers together thesock puppet’s mouth closes. By opening one’s hand, thesock puppet’s mouth closes. Intuitively, a hand gesturemanipulates the sock puppet to take on different formswhich in turn are interpreted as such ideas as talking,nodding, smiling, grimacing, etc. In order to achieve thesame ends with a virtual puppet, the user’s gesture’smust be first detected and then mapped onto an actionthe virtual puppet must take.

Doing hand gesture recognition often involves a cre-ative approach to pattern recognition or template match-ing. A number of algorithms exist in this field, ofteninvolving a machine learning process for training adatabase with recognizable gestures. A comparison ofhand gesture recognition techniques is given in [8].

One of the most straight-forward ways of performingtemplate matching is to use the subtraction method.This method involves converting an image of the handobject into a binary mask (for example from the skincolor mask), then comparing the hand mask against aset of gesture masks saved in a trained database. Inorder to perform this comparison, the pixel values on thehand mask are subtracted from the corresponding pixelvalues on the stored gesture masks. Then, the euclideandistance d is computed. These steps are repeated for

every gesture stored in the database and the mask withthe smallest euclidean distance is picked as the closestmatch.

d =

√ ∑x≤width

∑y≤height

(imask(x, y)− igesture(x, y))2

Another approach has its roots in Principle Com-ponent Analysis (PCA). Besides being widely used inimage compression, one of the most popular implemen-tations of PCA is Eigenfaces. Although generally used asa face recognition technique, Eigenfaces can be adaptedto recognize other features, such as hand gestures. Thegeneral idea is to look at all the gesture masks storedin the training set, then generate a mean image as wellas and alternative representations as Eigenvectors fromthe data set. The Eigenvectors will serve as the prin-ciple components in this implementation. Every maskin the training set will then be expressed as a vectorv of dimension n, where n be the number of principlecomponents and the value of the vector component xi,0 < i ≤ n be an index for similarity between themask and the respective principle component. The handmask extracted from the current video frame is then alsoconverted into such a vector and compared against thevectors in the training set. The stored gesture mask withsimilarity indices closest in value to the current handmask will then be identified as the recognized gesture.

An alternative method of gesture recognition involvesfinding the contours of the detected hand object. Thepositions of all visible fingertips may then be extractedby converting the contours to a convex hull and lookingfor start and end points of convexity defects. Based onhow many fingertips have been found and their distanceto each other, some specific hand gestures may thenbe detected without prior training of a database. Thisapproach is also robust in that it is largely agnostic tothe orientation of the hand.

3 IMPLEMENTATIONFor our particular implementation we needed to selectan appropriate algorithm for each of the three stepsinvolved in extracting the necessary information fromthe input image stream. Our general requirements wereas follows:• Performance: the image analysis is required to con-

clude within less than 30 milliseconds. This leavesus enough frame time to further draw the virtualpuppet, while maintaining a minimum frame rateof 15 frames per second.

• Flexibility: the implementation needs to be flexibleenough to work under somewhat unfavorable light-ing conditions, with less than perfect backgroundcolors, and with a wide variety of skin tones. More-over, the system must function well even with a low-quality webcam.


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3.1 Hand Detection

In order for our hand detection algorithm to providethe necessary flexibility as proposed in Section 3, wewould like to employ and adaptive approach. Due toits ease of implementation and overwhelmingly greatresults in a practical environment, we chose to imple-ment the Adaptive Skin Color Classifier as proposed in[3], operating on the normalized RGB color space. Inorder to initialize the Skin Color Classifier, we use a HaarCascading Classifier as proposed by Viola and Jones [4]to first create a bounding box containing a face foundin the scene. Second, a number of discrete skin colorsamples are taken from the face using an empiricallydetermined mask. Figure 2 illustrates this idea.

Fig. 2. Sampling skin colors using a mask. The discretesampling points are indicated by white dots.

We repeat these two steps for a number of frames inan effort to collect a large enough number of samples.Once enough samples have been collected, our imple-mentation computes the mean and standard deviation ofthe data set of samples and calculates lower and upperbounds of the thresholding range as proposed in [3].

Reliably detecting skin color in the input image istantamount to identify potential areas where hand ob-jects may be located. In order to do so, we try to firsteliminate some of the image noise then use a largelyillumination condition and ethnicity agnostic skin colorclassifier. We will also try to eliminate a number of falsepositives returned by the skin color classifier, by usinga simple background segmentation filter. We assumethat only one human hand object will be present inthe scene at any given time, hence after detecting skincolor in the image, we can assume that the largest areasof skin colored pixels will be part of the user’s faceor hand(s). In order to minimize the potential risk offalsely identifying the person’s face as a hand, we canuse a Haar cascading classifier to calculate a bounding

box from the user’s facial features, then subtract thisimage area from our skin color mask. In order to trackthe hand movement, we will set up a search rectanglein the image, containing the detected skin color pixelsand a buffer zone around the bounding box borders. Wefurther use a contour detection algorithm and compute abounding box of the skin color mask, in order to extracta scale value and interpret potential hand gestures. Afterall this information has been extracted from the originalinput image, we can compute a 3D transform and applyit to a model of a virtual puppet.

The image processing core of our application is orga-nized as a deterministic finite state machine. Once thelower and upper bounds for our skin color mask havebeen determined we transition to the hand tracking stateso to not waste any more processing resources on theskin color threshold computation. However, in order tocompensate for drastically changing lighting conditionswe monitor the average white coverage of the binaryskin color mask. If the mask contains an unreasonableamount of skin color, such as too much or too little whitecoverage, our image processing core transitions back intothe hand detection state, resampling the image for skincolors.

3.2 Hand TrackingBesides the requirements mentioned in Section 3, ourhand tracking algorithm needs to be able to cope withmoderately fast movements and the motion blur intro-duced by such movements. Unfortunately, CamShift isknown to be somewhat susceptible to this particularproblem. However, we have found the algorithm ex-plained in [9] to perform very well in our case. The ideais based on using flocks of KLT features, collected aroundthe center of the current hand bounding box. Opticalflow is measured by a differential algorithm in order todetermine the positions of these KLT features in the nextframe. If a feature ceased to exist or is now outside thearea of the bounding box, a new KLT feature close thethe center is picked. Figure 3 illustrates this approach.

3.3 Gesture RecognitionSince our application has fairly specific requirementsin terms of gesture recognition, we opted for a morespecialized, but easy to implement approach. For thisresearch, we require to primarily detect mouth openingand mouth closing gestures in order to control the virtualpuppet. To do so we find the contours of the handon the binary skin mask, using the tracking boundingbox. Figure 4 illustrates this idea. The hand contours areshown as red lines. Once the contours have been found,we convert them into a convex hull. Using the derivedconvex hull and the original contours, we then search forconvexity defects along the contours. As a result, we canderive information about the spaces between the fingers.The depth of the convexity defect gives us an idea of howopen or closed the hand is. Moreover, we can estimate


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Fig. 3. Hand tracking by means of flocks of KLT features:Small dots indicate KLT features. The circle shows thecenter of the bounding box used.

how many fingers are shown, by counting the number ofdetected convexity defects, which are the spaces betweenthe fingers.

From the information we have about the convexitydefects, we can derive a mouth closing and openinggesture: If one or less fingers are detected, we considerthe mouth closed. If three or more fingers are shown,the mouth is considered open. We can also derive anopenness factor from computing the mean of convexitydefect depths.

Since based on image noise and quality of the skincolor mask, recognition errors may be made, we can usea simple pole-filter, such as a low-pass filter, in order toderive a more stable value for the openness factor.

4 FUTURE WORKOur application suffers from a handful of issues, whichmay be solved in a future revision. For example, one ofthe problems with our current implementation consist ofthe user’s face being part of the skin color mask. This isespecially problematic if the tracked hand moves closerto the user’s face. In that case, parts of the face mighterroneously be considered part of the user’s hand. KLTfeatures are then also searched for on the face, whichmakes the tracking become somewhat unreliable.

Moreover, parts of the background may falsely beidentified as skin colored pixels, if some backgroundfeatures are similar in color to human skin. This is es-pecially problematic under imperfect lighting conditionsor with poor quality webcams. An adaptive background

Fig. 4. Hand gesture recognition using contours: Thetracking position and bounding box are tinted blue. Thegreen circle indicate detected finger tips. The red linesillustrate the hand contours.

segmentation algorithm could potentially alleviate thisproblem. This algorithm would keep a history of videoframes and consider rarely changing pixels as being partof the background.

Much better image segmentation results may addition-ally be achieved by using a time-of-flight (TOF) depth-perceptive camera, such as the Microsoft Kinect.

REFERENCES[1] G. Bradski and A. Kaehler, Learning OpenCV: Computer Vision with

the OpenCV Library, 1st ed. O’Reilly Media, October 2008.[2] M. Soriano, B. Martinkauppi, S. Huovinen, and M. Laaksonen,

“Skin detection in video under changing illumination conditions,”in Proceedings. 15th International Conference on Pattern Recognition,2000, pp. 839 –842.

[3] M. Wimmer and B. Radig, “Adaptive skin color classificator,” inProc. of the first ICGST International Conference on Graphics, Visionand Image Processing GVIP 05, A. A. et al., Ed., vol. 5, InternationalCongress for Global Science and Technology. Cairo, Egypt:International Congress for Global Science and Technology, 2005,pp. 324–327.

[4] P. Viola and M. Jones, “Rapid object detection using a boostedcascade of simple features,” in Proceedings of the 2001 IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition, 2001.

[5] J. G. Allen, R. Y. D. Xu, and J. S. Jin, “Object tracking using camshiftalgorithm and multiple quantized feature spaces,” in Proceedingsof the Pan-Sydney area workshop on Visual information processing, ser.VIP ’05. Darlinghurst, Australia, Australia: Australian ComputerSociety, Inc., 2005, pp. 3–7.

[6] B. D. Lucas and T. Kanade, “An iterative image registrationtechnique with an application to stereo vision,” in Proceedings ofImageing Understandings Workshop, 1981, pp. 121–130.

[7] J. Shi and C. Tomasi, “Good features to track,” in Proceedings ofIEEE Computer Society Conference on Computer Vision and PatternRecognition, June 1994, pp. 593–600.

[8] P. Chakraborty, P. Sarawgi, A. Mehrotra, G. Agarwal, and R. Prad-han, “Hand gesture recognition: A comparative study,” in Proceed-ings of the International MultiConference of Engineers and ComputerScientists, vol. 1, March 2008.


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[9] M. Kölsch and M. Turk, “Fast 2d hand tracking with flocks offeatures and multi-cue integration,” in In IEEE Workshop on Real-Time Vision for Human-Computer Interaction (at CVPR), 2004, p. 158.


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Analysis of Wind Velocity and the Quantification ofWind Turbulence in Rural and Urban Environments

using the Lévy Index and Fractal DimensionJonathan Blackledge, Eugene Coyle, Niall McCoy, Derek Kearney, Keith Sunderland and Thomas Woolmington

Abstract— This paper is concerned with a quantitative andcomparative analysis of wind velocities in urban and rural envi-ronments. It is undertaken to provide a route to the classificationof wind energy in a rural and urban setting. This is a commonproblem and the basis of a significant focus of research into windenergy. In this paper, we use a non-Gaussian statistical model toundertake this task, and, through a further modification of thedata analysis algorithms used, extend the model to study theeffect of wind turbulence, thereby introducing a new metric forthis effect that is arguably superior to a more commonly used andqualitatively derived measure known as the Turbulence Intensity.

Starting from Einstein’s evolution equation for an elasticscattering process, we consider a stochastic model for the windvelocity that is based on the Generalised Kolmogorov FellerEquation. For a specific ‘memory function’ - the Mittag-Lefflerfunction - it is shown that, under specified conditions, this modelis compatible with a non-Gaussian processes characterised by aLévy distribution that, although previously used in wind velocityanalysis, has been introduced phenomenologically. By computingthe Lévy index for a range of wind velocities in both rural andurban environments using industry standard cup anemometers,wind vanes and compatible data collection conditions (in terms ofheight and sampling rates), we show that the intuitive notion thatthe ‘quality’ of wind velocity in an urban environment is poorcompared to a rural environment is entirely quantifiable. Thisquantifies the notion that a rural wind resource is, on average, ofhigher yield when compared to that of the urban environment inthe context of the model used. In this respect, results are providedthat are based on five rural and urban locations in Ireland andthe UK and illustrate the potential value of the model in theconsideration of locating suitable sites for the development ofwind farms (irrespective of the demarcation between an urbanand rural environment). On this basis, the paper explores anapproach whereby the same model is used for evaluating windturbulence based on the Fractal Dimension using the ‘polar windspeed’ obtained from three-dimensional data sets collected inurban environments.

Index Terms— Wind velocity, wind turbines, non-Gaussianstatistics, Lévy index, rural and urban analysis, wind turbulence,Fractal Dimension.

Manuscript completed in December, 2012.Jonathan Blackledge ([email protected]) is the Science Foun-

dation Ireland Stokes Professor and Professor Eugene Coyle ([email protected]) is Head of Research Innovation and Partnerships at DublinInstitute of Technology. Niall McCoy ([email protected]) worksfor the Wind Prospect Group and Derek Kearney ([email protected]),Keith Sunderland ([email protected]) and Thomas Woolmington([email protected]) are staff members of the Electrical ServicesEngineering Department at Dublin Institute of Technology where they areregistered for a doctoral research degree programme.

I. INTRODUCTION

A primary factor in the development of a wind farm is anunderstanding of the potential wind energy associated withthe site, i.e. the geographical location of the farm, [1], [2].This is the key to the economic viability of any wind energyproject which must focus on the development of wind farmsin effective and efficient regions, subject to the structural andenvironmental conditions that provide an optimum solutionwithin the engineering and commercial constraints imposed,[3], [4]. Understanding the relationship between on-shore ruraland urban environments (and off-shore wind energy schemes),has been, and remains fundamental in the development ofthe wind industry throughout the world, [5]. For some timenow, it has been ‘understood’ by industry experts that thewind velocity in a rural environment is of a higher ‘quality’and energy yield when compared to the wind velocity inan urban environment, [6]. The term ‘understood’ is oftentaken for granted, rather than taking all of the facts intoconsideration and fully justifying the actual results, [7], [8].In this context, the purpose of this paper is to look at usingrecently developed stochastic models (originally developed foralgorithmic financial trading and used to launch and develophttp://tradersnow.com, for example) to investigatea possible correlation between the wind velocity in bothrural and urban environments based on a statistical parametercalled the Lévy index. This represents a significant departurefrom conventional statistical analysis of wind velocity datawhich is typically based on Gaussian-type models where thewind velocity is taken to be a Rayleigh-type distributed. Thestatistical model considered in this paper is non-Gaussian andis used to provide two distinctive and original contributions:• quantification of the intuitive understanding that potential

wind energy is less in urban regions based on computingthe Lévy index;

• quantification of wind turbulence in terms of a new metricthat is arguably superior to the conventional TurbulenceIntensity based on computing the Fractal Dimension(which is simply related to the Lévy Index).

II. FUNDAMENTALS OF WIND ENERGY

The power generated by a wind turbine is based on a rangeof design factors but they all relate indirectly to Betz law,which states that the power P in Watts is given by, [9]

P =1

2αρAv3 (1)


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where v is the upwind speed (i.e. the wind velocity that isincident on the turbine) in metres per second (ms−1), A is thearea mapped out by the turbine blades in m2, ρ is the density ofair in kgm−3 and α ' 0.593 is the coefficient of performance.This result is derived by considering the energy generated bya change in the upwind and downwind velocities together withthe change in the mass of air that occurs as it ‘travels’ throughthe turbine. A derivation of equation (1) is given in [10],for example, which includes the idealised conditions uponwhich this equation is based. This law is the ‘design guide’associated with the development of wind turbines world-wide.There are two important factors associated with optimising theoutput power: (i) the diameter of the turbine blades; (ii) thewind velocity which scales of the velocity cubed. The cubicvelocity scaling law cannot be maintained in practice overall ranges of wind velocity, and, depending upon the designcharacteristics of the turbine, there is a natural threshold forthe wind velocity beyond which the output power does notincrease. This is due to a range of influencing factors includingthe turbulence phenomena that occur at high wind velocitiesthrough interaction with the turbine blades when Betz lawbreaks down. However, within the framework of Betz law, andgiven a turbine with a fixed blade diameter, the velocity cubedscaling law is of fundamental importance in determining theoutput power. Clearly, the wind velocity is time dependentand this dependence cannot, in general, be classified in adeterministic sense. Stochastic models that lead to the designof statistical data analysis algorithms are therefore requiredthat ideally provide a statistical parameter or parameter setthat can quantify the wind resource subject to a range ofinfluencing factors.

III. INFLUENCING FACTORS

There are a range of influencing factors that can affect theperformance of a wind turbine and a wind farm. The currentindustry knowledge is based on the ‘roughness principle’.Commonly found in the rural environment, there tends to belittle in the way of large obstacles to cause sufficient turbulencewhich affect the wind quality, and, in turn, the energy yield ofwind turbines. This is due to the relatively laminar flow thatis a characteristic of a ‘good site’ as illustrated in Figure 1.

By contrast, in the urban environment, there is generally anabundance of built obstacles representing adverse roughness ofthe ‘ground truth’ generating turbulence and thereby curtailingthe potential energy yield and the output power of windturbines located in such an environment as illustrated inFigure 2.

Understanding the ‘quality of the wind velocity’ is ofparticularly interest to those in the wind energy industry, as itallows the developer to identify specific sites to develop, con-centrating on which sites produces the greatest energy yields.Within the urban environment, there are numerous factorsinfluencing the wind velocity. The overriding factor of the builtenvironment in the urban setting is that of roughness. Thereare also the numerous properties of the urban environmentand atmospheric influences to be accounted for resulting in anextremely complex environment to accurately model.

Fig. 1. Illustration of the laminar flow that is typical of a rural site in whichthere can be an increase or ‘speed up effect’ of the upwind velocity overa smooth hill thereby providing a greater power output of a wind turbinethrough the ‘v3 scaling law’ (top illustration). In contrast, ‘bad sites’ (bottomillustration) such as cliff tops dissipate wind energy through the turbulencegenerate by discontinuities that ‘break-up’ laminar flows (Source: Greenspec2011).

Fig. 2. Illustration of the effect of obstacles such as buildings which generatea zone of maximum turbulence downwind but with decreasing height and ofthe same order of magnitude as the height of the building H (top). Turbineslocated downwind are placed away from the turbulence zone at a distance ofthe order of 10H and the turbine blades placed at a height greater than H .Typical wind speeds are shown in the bottom left-hand diagram and the basicissues associated with roof mounted wind turbines illustrated in the bottomright-hand side diagram (Source: Greenspec, 2011).

Referring to the idealised model illustrated in Figure 3, astandard scaling law for the effect of roughness on the windvelocity at a height z is given by, [11]

v(z) =vfκ

log

(z − dz0

)(2)

where v(z) denotes the wind velocity at a height z, vf isthe friction velocity (which is dependent on the roughnessof the ground), κ is the (dimensionless) von Karman constant(typically of the order of 0.41), z is the height above the earth’ssurface, d is the displacement height and z0 is the height abovethe earth’s surface roughness (where the wind velocity appearsto approach zero). The friction velocity depends upon the shearstress T at the boundary of the flow and is given by, [12]vf =

√T/ρ, where ρ is the density of air. This log-based


8

scaling law describes the velocity profile of a turbulent fluidflow near a boundary with a non-slip condition and must befully and accurately introduced when taking into account windgusts and strong winds (especially in urban environments).Equation (2) is a semi-empirical relationship used to describethe vertical distribution of horizontal mean wind speeds withinthe lowest portion of the boundary layer.

As the wind speed decreases to zero, closer to ground level,this results in an atmospheric boundary layer. Thus, equation(2) can be accurate up to 200 m. In rural environments,however, the impact of such ‘roughness factors’ are lesscommon and mostly attributed to forestry etc. (as illustratedin Figure 3) which can be removed as required and with therequired permission of the appropriate environmental agencies.

Fig. 3. Rural (top) and Urban (bottom) roughness models associated withequation (2) (Source: Mertens 2006).

It is intuitively obvious that, whatever the cause, turbulencereduces the energy output from a wind turbine since turbulencedissipates energy over a larger volume (at least for an adiabaticsystem). It is also clear that turbulence is extremely difficultto model in a fully deterministic sense, based on the principlesof fluid dynamics. Thus, in the following section we developa stochastic model from first principles.

IV. STOCHASTIC MODEL FOR THE WIND VELOCITY

We consider the temporal behaviour of the wind velocity interms of the space-time evolution of a field v(x, t) working inone-dimension. The temporal behaviour of the wind velocityis then taken to be the time dependent behaviour of this fieldat a point in space x. In [10] and [13], a stochastic modelfor the wind velocity is developed using a fractional partialdifferential equation of the type

∂γ

∂xγv(x, t)− ∂

∂tv(x, t) = −δ(x)s(t), γ ∈ (0, 2]

where γ is the Lévy index and s(t) is a ‘white noise’ stochastic‘source function’ with a uniformly distributed Power SpectralDensity Function (PSDF) and arbitrary Probability Density

Function (PDF). Ignoring scaling constants, it is shown thatthe Green’s function solution to this equation is

v(t) =1

t1−1/γ⊗t s(t) (3)

where ⊗t denotes the convolution integral over t and v(t) ≡v(0, t). This solution has the self-affine scaling relationship

Pr[v(at)] = a1/γPr[v(t)]

where Pr denotes the PDF and a PSDF given by (for scalingconstant c)

| V (ω) |2= c| ω |2/γ

where V (ω) =

∞∫−∞

v(t) exp(−iωt)dt

Following [14] and [15], we now consider an extension andgeneralisation to this model which is based on developinga solution to the Generalised Kolmogorov-Feller Equation(GKFE) which is derived in the following section. The aimis to show that the solution to the GKFE considered providesa model for the PSDF that is effectively the same as thatconsidered in [10] and [13] and that the wind velocity fieldcan be considered to by a random scaling fractal signalcharacterised by a Lévy index. In turn, this index is related tothe fractal dimension of the signal, and, as discussed later onin this paper, this dimension can be used to characterise theturbulent behaviour of the wind velocity providing a index thatis arguably superior to the conventional Turbulence Intensity.

A. Derivation of the Generalised Kolmogorov-Feller Equation

For an arbitrary PDF p(x), Einstein’s evolution equation is,[16]

u(x, t+ τ) = u(x, t)⊗x p(x)

where u(x, t) is a ‘density function’ representing the concen-tration of a canonical ensemble of particles undergoing elasticcollisions. This function is interpreted as a field representingthe distribution of physical properties such as the mass,velocity, temperature and pressure, for example.

Consider a Taylor series for the function u(x, t+ τ), i.e.

u(x, t+ τ) = u(x, t) + τ∂

∂tu(x, t) +

τ2

2!

∂2

∂t2u(x, t) + ...

For τ

than by taking an increasingly larger number of terms in theTaylor expansion of u(x, t+ τ) which is not of (closed-form)analytical value.

For arbitrary values of τ ,

τ∂

∂tu(x, t) +

τ2

2!

∂2

∂t2u(x, t) + ... = −u(x, t) +u(x, t)⊗x p(x)

We model the effect on a solution for u(x, t) of the serieson the left hand side of this equation in terms of a ‘memoryfunction’ m(t) and write

τm(t)⊗t∂

∂tu(x, t) = −u(x, t) + u(x, t)⊗x p(x) (5)

where ⊗t is taken to denote the causal convolution integralover t. This is the Generalised KFE (GKFE) which reducesto the Classical KFE when

m(t) = δ(t)

Note that for any memory function for which there exists afunction or class of functions of the type n(t), say, such that

n(t)⊗t m(t) = δ(t)

then we can write equation (5) in the form

τ∂

∂tu(x, t) = −n(t)⊗t u(x, t) +n(t)⊗t u(x, t)⊗x p(x) (6)

where the Classical KFE is recovered when n(t) = δ(t).Any solution obtained to the GKFE will be dependent upon

the choice of memory function m(t) used. There are a numberof choices that can be considered, each or which is taken tobe a ‘best characteristic’ of the stochastic system in terms ofthe influence of its time history. However, it may be expectedthat the time history of physically significant random systemsis relatively localised in time. This includes memory functionssuch as the Mittag-Leffler function [19]

m(t) =1

Γ(1− β)tβ, 0 < β < 1

wheren(t) =

1

Γ(β − 1)t2−β

given that∞∫0

exp(−st)Γ(β)t1−β

dt =1

sβand

∞∫0

δ(t) exp(−st)dt = 1

B. Green’s Function Solution to the GKFE

Equation (6) can be written in the form

τ∂

∂tu(x, t) + u(x, t) = u(x, t)− n(t)⊗t u(x, t)

+n(t)⊗t u(x, t)⊗x p(x)

so that the Green’s function solution is given by

u(x, t) = g(t)⊗t u(x, t)− g(t)⊗t n(t)⊗t u(x, t)

+g(t)⊗t n(t)⊗t u(x, t)⊗x p(x) (7)

where the Green’s function is given by

g(t) =1

τexp(−t/τ), t > 0

which is the solution to

τ∂

∂tg(t− t0) + g(t− t0) = δ(t− t0)

and we assume the initial conditions u(x, t = 0) = 0 andg(t = t0) = 0. We can now analyse this solution in Fourier-Laplace space by taking the Fourier transform and the Laplacetransform of equation (7) and using the convolution theoremsfor the Fourier and Laplace transform, respectively, to obtain

¯̃u(k, s) = ḡ(s)¯̃u(k, s)−ḡ(s)n̄(s)¯̃u(k, s)+ḡ(s)n̄(s)¯̃u(k, s)p̃(k)(8)

where

¯̃u(k, s) =

∞∫0

∞∫−∞

u(x, t) exp(−ikx)dx exp(−st)dt

ḡ(s) =

∞∫0

g(t) exp(−st)dt, n̄(s) =∞∫0

n(t) exp(−st)dt

and

p̃(k) =

∞∫−∞

p(x) exp(−ikx)dx

From equation (8) we can write

¯̃u(k, s) = − ḡ(s)1− ḡ(s)

n̄(s)¯̃u(k, s) +ḡ(s)

1− ḡ(s)n̄(s)¯̃u(k, s)p̃(k)

= − n̄(s)τs

¯̃u(x, t) +n̄(s)

τs¯̃u(k, s)p̃(k)

given that ḡ(s) = (1 + τs)−1 and thus obtain the equation

¯̃u(k, s) = h̄(s)¯̃u(k, s)p̃(k)

whereh̄(s) =

n̄(s)

τs+ n̄(s)

or, upon inverse transformations

u(x, t) = h(t)⊗t u(x, t)⊗x p(x) (9)

withh(t)↔ n̄(s)

τs+ n̄(s)

where↔ denotes the Laplce transformation, i.e. mutual trans-formation from t-space to s-space.

Consider the iteration of equation (9) defined by

un+1(x, t) = h(t)⊗t un(x, t)⊗x p(x) (10)

for an initial solution u0(x, t) where n = 1, 2, ..., N Theequivalent iteration in Fourier-Laplace space is

¯̃un+1(k, s) = h̄(s)¯̃un(k, s)p̃(k)

with initial solution ¯̃u0(k, s) so that, after N iterations,

¯̃uN (k, s) = [h̄(s)]N [p̃(k)]N ¯̃u0(k, s)


10

and upon inverse Fourier-Laplace transformation, has theiterative form

uN (x, t) =N∏j=1

⊗ p(x)N∏k=1

⊗ h(t)⊗x ⊗tu0(x, t) (11)

whereN∏j=1

⊗ f(t) ≡ f(t)⊗t f(t)⊗t f(t)⊗t ...

denoting the N th convolution of f(t).The criterion for convergence of this (iterative) solution can

be considered by introduction of the error function �n(x, t) atany iteration n so that un(x, t) = u(x, t) + �n(x, t). Fromequation (10) we can then write (transforming to Fourier-Laplace space)

¯̃�n+1(k, s) = h̄(s)p̃(k)¯̃�n(k, s)

so that¯̃�n(k, s) = [h̄(s)p̃(k)]

n¯̃�0(k, s)

and it is clear that, since we require ¯̃�n → 0 and n →∞, [h̄(s)p̃(k)] < 1 ∀(k, s). The condition for convergencetherefore becomes

‖h̄(s)p̃(k)‖ ≤ ‖h̄(s)‖ × ‖p̃(k)‖ < 1

or, for Euclidian norms, and, using Rayleigh’s theorem,

‖h(t)‖2 × ‖p(x)‖2 <1√2π

C. Impulse Response for the Mittag-Leffler Memory Function

Form equation (11), if the initial solution is an impulse(i.e. u0(x, t) = δ(x)δ(t)) then the Impulse Response Function(IRF), denoted by r(x, t), is given by

r(x, t) =N∏j=1

⊗ p(x)N∏k=1

⊗ h(t)

with ‘transfer function’

¯̃r(k, s) = [h̄(s)p̃(k)]N

For a memory function m(t) modelled by the Mittag-Lefflerfunction (for 0 < β < 1)

m(t)↔ 1s1−β

and h̄(s) =1

1 + τsβ∼ 1τsβ

so thath(t) ∼ 1

τΓ(β)t1−β

Similarly, if we consider a Mittag-Leffler PDF of the form

p(x) =1

Γ(1− γ) | x |γ, 0 < γ < 1

then the IRF becomes

r(x, t) ∼N∏j=1

⊗1

Γ(1− γ) | x |γN∏k=1

⊗1

τΓ(β)t1−β

D. Temporal IRF for Early Evolutionary Behaviour

The function r(x, t) is a space-time IRF. A temporal IRFcan be considered by integrating over x. Physically, theresulting IRF can be taken to be a characteristic of a timeseries recorded at an arbitrary point in space. Further, if weconsider the early evolutionary behaviour of uN (x, t) (i.e. thecase when N = 1), we obtain the simplified expression forthe field u(t) given by

u(t) ≡∞∫−∞

u1(x, t)dx =1

τΓ(β)t1−β⊗t s(t) (12)

where

s(t) =

∞∫−∞

p(x)⊗x u0(x, t)dx

This result demonstrates that the model developed in [10] and[13], where the wind velocity is given by equation (3), is aspecial case of the solution to the GKFE considered here (i.e.equation (12) where, ignoring scaling constants, β = γ−1 andthe field u is taken to be the wind velocity) and describes theearly evolution of a time series governed by the GKFE. In turn,the GKFE is an expression of Einstein’s evolution equationsubject to a specialised Memory Function - the Mittag-Lefflerfunction - which yields the fractional diffusion equation. Thisrelationship is compounded further in the following analysisfor the case when τ

the rate at which mass enters a system is equal to the rate atwhich mass leaves the system and is given by (for a three-dimensional space vector r)

∂

∂tρ+∇ · (ρv)

where ρ(r, t) is the fluid density and v(r, t) is the flow velocityvector field. Thus, for a one-dimensional system characterisedby a constant velocity field v (which is constant over x andt) and a density field ρ(x, t) we obtain

∂

∂xρ+ v

∂

∂tρ(x, t) = 0

In this sense, the field u(x, t) for γ = 1 may be taken todescribe the flow of mass subject to a constant fluid velocityv = τ/a. The case of γ = 1 is therefore representative of asteady state process. Moreover, the PDF associated with thisprocess is a Chauchy function since

1

2π

∞∫−∞

exp(−a | k |) exp(ikx)dk = 1π

a

a2 + x2

Similarly, given that

1

2π

∞∫−∞

exp(−ak2) exp(ikx)dk = 12π

√π

aexp

(−x

2

4a

)it is clear that the case when γ = 2 describes a Gaussiansystem, the field u(x, t) being the solution to the ClassicalDiffusion Equation

∂2

∂x2u(x, t)− σ ∂

∂tu(x, t) = 0

We note that, in general [20],

1

2π

∞∫−∞

exp(−a | k |γ) exp(ikx)dk ∼ 1x1+γ

Thus, in terms of using the field u to model a single or com-bined velocity field (such as the polar wind speed discussed inSection VII), on the basis of the physical systems describedby the cases when γ = 1 and γ = 2 given above, we canexpect that for γ ∈ [1, 2], larger values of γ correspond tomore urbanised environments where wind turbulence (whichtends towards fully diffusive behaviours but is still fractionallydiffusive according to our model) is greater. This idea appearsto be validated in the data analysis associated with the casestudy discussed in the following section.

V. DATA ANALYSISOn the basis of the stochastic model discussed in the

previous section, it is possible to estimate the Lévy index,relatively simply. This is achieved using the PSDF methoddiscussed in [21], for example. It is based on exploiting thebasic relationship (which ignores scaling factors) [21]

1

t1−1/γ↔ 1| ω |1/γ

where ↔ denotes transformation from real to Fourier space(i.e. t- to ω space). Using the convolution theorem, equations

(3) and (11) with β = γ−1, and ignoring scaling by [τΓ(β)]−1,transform to

ũ(ω) =s̃(ω)

| ω |1/γ

Thus, assuming s̃(ω) is a white noise spectrum that can betaken to be a ‘phase only’ function (with unit amplitude),

| ũ(ω) |2= 1| ω |2/γ

This idealised model for the power spectrum is used toestimate the Lévy index based on standard linear regressionmethods. For the work reported in this paper, and using aMATLAB7 programming environment, the Orthogonal LinearRegression Method based on the m-code available at [22] isused. We note that the power spectrum of a random scalingfractal signal scales as [21] | ω |−(5−2D)/2) where D is thefractal dimension. Thus, the relationship between the LévyIndex and the Fractal Dimension is

1

γ=

5− 2D4

(13)

To accurately model both urban and rural environments,historical data from five rural and five urban wind measure-ment sites were used. All measurement devices were located at50m above local ground level to allow an accurate comparisonand the locations spread across Ireland and the UK1. Themeasurement devices were all located on lattice towers of thetype shown in Figure 4 with industry standard data loggers tostore the data. The raw data sets were taken from their raw10 minute average from industry standard cup anemometersand wind vanes. All data sets were calibrated by industryprofessionals and the author’s are in receipt of all relevanttest certificates to verify the credibility of the calibrations.

Fig. 4. A typical Met Mast used to record wind velocity data at 50m heightwith a 10 minute average using industry standard cup anemometers and windvanes. (Source: Wind Prospect Group, 2012.)

The key factor in determining a possible correlation betweenrural and urban wind velocity is the use of stochastic mod-elling. The modelling is often based on a statistical analysisof the available wind velocity data which is used to assessoptimum regions for the construction of wind farms. In this

1Much of the specific data is confidential and the exact location of the datasources cannot be mentioned in the paper. However, for creditability reasons,the locations are available on receipt of a non-disclosure agreement betweenthe authors and the reader


12

TABLE IMEAN VALUES OF THE LÉVY INDEX γ̄ FOR FIVE rural sites.

Site 1 2 3 4 5Location Newport Waterford Limerick Cavan Dundalk

Longitude 51.5853 52.2527 52.6438 53.8995 53.9979Latitude -2.9796 -7.1256 -8.6903 -6.8051 -6.4059γ̄ 1.3407 1.3672 1.3477 1.3708 1.4174

TABLE IIMEAN VALUES OF THE LÉVY INDEX γ̄ FOR FIVE urban sites.

Site 1 2 3 4 5Location Mayo Wexford Louth Tyrone Limerick

Longitude 53.9050 52.6401 53.7965 54.6029 52.3459Latitude -9.2756 -6.6045 -6.5348 -7.0941 -8.9736γ̄ 1.3692 1.3586 1.3128 1.3481 1.3397

paper, data from five rural and five urban sites were analysedthrough determination of the Lévy index over a period of 12months. The results are tabulated in Tables I and II and showthat, bar one anomaly, the trend is that the mean values ofthe Lévy index γ̄ for the urban sites is consistently higherin comparison to the mean values of the same index for therural sites. The average value of this index for the urbansites considered is 1.3688 which should be compared to theequivalent average value for the rural sites of 1.3457. Thus, inthis case study, the Lévy index Rural-to-Urban ratio is 0.9832.

Some of the data used to generate the results in Table I arerecorded at urban locations that are not ‘deeply embedded’ inan urban environment. For example, Sites 1 in Newport, SouthWales, is in a costal location on an urban boundary whichmay explain why the mean is relatively low compared withthe other values. Site 3 in Limerick is a similar location closeto the coast and on the boundary of the urban environment.However, both sites are impacted by the urban setting witha high density of ‘urban features’ in close proximity. Thus,the urban locations chosen are not optimal in terms of all thesites being fully embedded in an urban environment, but werechosen to provide data consistency given the limited data setscurrently available.

VI. TURBULENCE INTENSITY

Urban wind regimes are characterised as having low windspeeds with more turbulent flows which result in limitedenergy realisation. Research has shown that the lower meanspeeds are linked to the higher surface roughness lengths z0prevalent in urban environments, [23] and [24]. The manifesta-tion of turbulence, however, is less well understood. Turbulentflows can be described as those in which the fluid velocityvaries significantly and irregularly in both position and time[25]. While turbulently fluctuating flow impacts directly thedesign of wind turbines, they also influence the productivityof turbines particularly in areas of complex morphologies.

The Turbulence Intensity (TI) is the most common metricused to quantify the effect of wind turbulence as it is generallymore useful to develop descriptions of turbulent in terms ofstatistical properties [26]. TI is defined in [27] as ‘the ratioof wind speed standard deviation to the mean wind speed,

determined from the same set of measured data samples ofwind speed, and taken over a specified time’ and should beconsidered as the standard deviation of the longitudinal windspeed σv normalised with the mean wind speed v̄, i.e.

TI =σvv̄

The complex morphology experienced in an urban environ-ment results in a modified flow and turbulence structure inthe urban atmosphere in contrast to the flow over ‘ideal orhomogenous’ surfaces [28]. Thus, in [27], for example, it isproposed that the TI can be ‘linked’ to the surface roughnessparameter via the following equation

TI =1

log∣∣∣ z−dz0 ∣∣∣

where d is the displacement height, which is taken to beequal to 0.66 of the average building height (denoted byzH ) and z0 is the surface roughness length. This equationis predicated on z (the observation height) being in excess ofthe wake diffusion height - z∗, which is taken to be abovethe surface roughness sub-layer and into the inertial sub-layeras illustrated in Figure 5. This result suggests that there is anincreasing level of turbulence with increasing roughness anddecreasing height relative to the earth’s surface.

Fig. 5. Wind Speed in the urban context with respect to the boundary layertransitions.

With respect to the impact on the power output of windturbines subjected to turbulence, the majority of the availableresearch considers utility scale systems with capacities in theMW ranges [29]. For example, [30] considers empiricallylinking surface roughness and the power law wind shear coef-ficient to turbulence manifestation and presents a descriptionof TI within the lower portion of atmospheric boundary layer,again, based on surface roughness, and concluding that the(kinetic) energy available at the turbine hub height can varyby as much as 20% depending on the level of TI present ata site. The effect of turbulence intensity on the wind turbinepower curve is summarised in Figure 6 [31], [32] and [33].High TI contributes to increased output power from a turbineat moderate wind speeds (cut-in), whereas low TI results inreduced output power at rated wind speed.

The evaluation of TI relies on the standard deviation.Therefore, an asymptotic characteristic is derived at relatively


13

Fig. 6. Typical Effects of Turbulence on Power Curves (Source:[31])

low wind speeds (< 3.5m/s). Micro/small wind turbines aredesigned to commence generating at such wind speeds and inurban environments, mean wind speeds are characteristicallylow. Thus there is a lack of confidence in the quantification ofTI in these environments. Wind speeds below the cut-in speedof a turbine are normally regarded as being non productive;however, this is not the case. In order to have an average windspeed that equals the cut-in speed of say 3.5m/s some valuesmust be above and below 3.5m/s over a 10 minute windowso that the mean is 3.5m/s. The question is how erratic is thisdeviation from the mean and can it be power productive?

Another issue concerning the evaluation of the TI is thequalitative nature of its definition. Given the theoretical modelpresented in this paper, in the following section we pro-pose a method for evaluating the turbulence intensity basedcomputing the Fractal Dimension of a time series of two-dimensional velocity data. This approach implies that tur-bulence (as measured by a statistic computed from a windvelocity field) is a self-affine phenomenon and we refer tothis metric as the Fractal Turbulence Intensity. In turn, thismetric is related to the Lévy Index used to characterise ruraland urban environments via equation (13) which provides a‘link’ between the approach discussed in Section V and thatof the following section given the stochastic model developedin Section IV.

VII. FRACTAL TURBULENCE INTENSITY

Observations are made at two urban locations in Dublin,Ireland. St. Pius X National (Girls) School (Site 1), locatedin Terenure, Dublin 6W (53o20’15.96”N, 6o18’19.02”W) andDublin City Council Buildings (Site 2), in Marrowbone Lane,located in Dublin 8 (53o20’15.96”N, 6o17’10.27”W) as shownin Figure 7. Site 2 is located closer to the city centre than Site1 and is therefore more urbanised with a higher associatedroughness length. This Site is also characterised by a higherbuilding density in comparison to Site 1 which has a muchlower concentration of buildings. As site 2 is closer to thecity centre, the buildings consist mostly of office blocks andhigh-rise residential building. Buildings in the area often reachheights of 20 m and beyond, with some reaching 25 m withtopographical complexities located at all angles relative to theanemometer used to record the wind velocity data. Site 1 hasa more consistent building morphology and the anemometer

is surrounded by a relatively lower average building heightthat consists mostly of two-storey residential buildings andvegetation which is also at similar heights - see Figure 7.

Fig. 7. Satellite image of Dublin city showing the relative positions of Sites1 and 2.

Fig. 8. The high-resolution observation site located at Site 1.

At both sites, high-resolution wind speed measurements aretaken with a Campbell Scientific CSAT3 three-dimensionalsonic anemometer [34]. The observations are at 10Hz at anassociated resolution-between 0.5 and 1.0 mm/s, with datathat includes date and time-stamp, wind-speed, wind-directionand standard deviation. The CSAT3 measures wind speedemploying a right handed orthogonal coordinate system Threeorthogonal wind components, which relate to the three di-mensions in space, are each measured. Wind entering straightinto the anemometer is from the x-direction giving windvelocity component vx; wind approaching from the left ofthe anemometer is from the y-direction giving wind velocitycomponent vy; and, wind advancing upwards from the ground


14

is from the z-direction generating wind velocity componentvz . Thus, effectively, the Easterly, Northerly and verticalcomponents of the wind velocity are vx, vy and vz , respec-tively, giving a wind velocity vector field v = (vx, vy, vz).Measurements of this field are taken to an accuracy of 0.01m/sat a frequency of 10 Hz over a 40 day period from 4/4/2012to 15/5/2012. Although, on a theoretical basis the FractalDimension of any signal is scale invariant so that the samplerate should not matter, in practice, because the computationof the Fractal Dimension uses a Power Spectral DensityFunction (as discussed in Section V), high data rates in agiven sample subset are required to obtain reasonable accuracy.Since turbulence models in general are based on a 10 minutesampling period bench mark, this period is used to compute theFractal Dimension on a moving window basis, each windowconsisting of 6000 samples (10 minutes at 10Hz).

The field used to compute the Fractal Dimension from thethree-dimensional data available is given by the followingmodel:

u(t) =√vx(t) + vy(t)

This provides a measure of the ‘polar wind speed’ in thehorizontal plane which is taken to be the mid (x, y)-planeof the three dimensional data field. Application of a com-bined wind speed model of this type is significant in thesense that, from a physical view point, a turbulence effectis not compounded in a single wind speed direction, anymeasure of turbulence ultimately having to rely on some multi-dimensional mapping of a fully three-dimensional physicaleffect. Computation of the longitudinal TI at low wind speedscan have excessive values. This is due to the asymptotic natureof the formula which makes the TI measurement in urban areasparticularly problematic with the standard turbulence model.Firstly it is generally accepted that the standard deviation ofwind speeds in an urban area is large due to a increasedturbulence. Secondly the average wind speed is considerablylower than that of laminar air flows due to the increased surfaceroughness. The net result is that the TI becomes asymptoticallylarge as the mean wind speed approaches zero. To compensatefor this effect is is possible to filter the data by truncating allvalues of the TI that exceed 1. Using this approach to filterthe TI and the data processing method discussed in Section Vto compute the Fractal Dimension (and as detailed further in[10]), Figure 9 compares the TI with the Fractal Dimension,normalisation of the data with respect to null entries resultingin the use of 4502 samples.

These results clearly shows that there is correlation betweenthe TI the Fractal Dimension of the horizontal polar windspeed, although it is noted that the Fractal Dimension which,for a Random Scaling Fractal Signal, is a value D ∈ [1, 2],exceeds the upper bound in an analogues way to when TI >1. Figure 10 shows a scatter-plot of the filtered TI denotedby TIf and the Fractal Dimension and application of linearregression clearly shows that these metrics are correlated, acorrelation that, for this the data considered, is compoundedin the equation

TIf = 0.1928D + 0.1385

Fig. 9. The (filtered) Longitudinal Turbulence Intensity (Red) calculated inaccordance with IEC 61400-2 and the Fractal Dimension of the horizontalpolar wind speed (Blue).

Fig. 10. Scatter-plot (Blue) of the (filtered) Turbulence Intensity (verticalaxis) and the Fractal Dimension (horizontal axis) together with a best fit linearregressed estimate (Red) showing a linear correlation between the two metrics.

VIII. CONCLUSION

It is well known that the differences in wind resourcein rural and urban environments are curtailed due to theinfluencing factors such a surface roughness. The aim ofthis paper has been twofold: (i) to quantify the differencesthrough determination of the Lévy index; (ii) to investigateuse of the Fractal Dimension as a measure of the TurbulenceIntensity. In the first case, a direct comparison is consideredbetween the urban and rural wind resources at selected locationacross Ireland and the UK using similar reference heights andfully calibrated equipment so that there is data consistencywithin the bounds of the practical constraints associated withthe technology used to measure the wind velocities. Theresults confirm that a rural resource is generally of a higherenergy yield when compared to the urban resource at leastin terms of the Lévy index as computed from the PowerSpectrum. This is compounded in lower values of the Lévyindex and, as a first study, paves the way for using this non-Gaussian statistical index to evaluate wind resource in general.With regard to the second principal contribution, the factthat conventional turbulence models cannot cater for erraticlow mean wind speeds associated with an urban environmentrequires quantification of alternatives to be considered as givenin Section VI.

The approach reported in Section VI is an alternative wayof computing a Turbulence Intensity that has two advan-tages. First, it is based on a more fundamental concept ofturbulence in terms of the model provided in Section IVand a fractal geometric interpretation thereby providing a


15

greater conceptual understanding of turbulence compared theheuristic conventional definition of the TI; second, the problemassociated with asymptotic behaviour, which is characteristicof the conventional definition of the TI, and occurs at relativelylow wind speeds, is eradicated. Moreover, it is noted that theFractal Dimension of the polar wind speed and the filteredTI are correlated thereby providing evidence that the conven-tional qualitative and quantitative measure of wind turbulenceproposed have an underlying connectivity.

The model, methodology and results reported in this papernow require a quantification procedure to be developed inorder to assess and predict the power performance of windturbines in rural environments and the degradation of thisperformance in urban environments. This is predicated onthe basis, that, with respect to turbulence assessment, thesignificant reduction in processing overheads associated withcomputing the Fractal Dimension implies a more efficientmeans of quantification as well as a conceptual qualificationof the model used (at least in terms of the Fractal Geometryof Nature [35] and methods of processing two- and three-dimensional data under a fractal based model [21]). For ex-ample, in [10] and [13], the following scaling law is proposedfor the mean turbine power output 〈logP 〉τ (over a period oftime τ ):

〈logP 〉τ ∝1

γ

where γ is computed from the wind velocity over the sametime period. Quantification of this scaling relationship is nowrequired based on known turbine output power and wind veloc-ity measurements. Finally with regard to urban environmentsin particular, it may be possible to find a correlation betweenthe Fractal Dimension of the polar wind velocity and theroughness of the local area from a high resolution satelliteimage of the type given in Figure 8. By computing fractalparameters such as the Image Dimension (Fractal Dimensionof a surface), the Information Dimension, Lacunarity and otherMulti-Fractal parameters [21], for example, it may be possibleto generate a single or combined image roughness measure.A correlation of this measure with the Fractal TurbulenceIntensity reported could provide a way of estimating the windturbulence and hence, subject to quantifying the inverse scalingrelationship given above, predict the power performance ofwind turbines in a rural environment from an satellite imagealone! Such a solution would provide a simple and effectiveway of prospecting for wind resources in urban environmentsusing on-line facilities such as Google Earth, for example.

ACKNOWLEDGMENTS

The research reported in this paper is supported by theScience Foundation Ireland Stokes Professorship Programme.The authors also acknowledge the support of Dublin Instituteof Technology and the Dublin Energy Lab. The wind velocitydata provided for the study given in Section V is providedby Wind Prospect Group. The urban data used in the windturbulence analysis (Section VII) has been provided by Dr.Rowan Fealy, National University of Ireland, Maynooth andDr. Gerald Mills, University College Dublin, Ireland.

REFERENCES

[1] Danish Wind Industry Association. Guided Tour, 2003. http://www.windpower.org/en/pictures/lacour.htm

[2] World Wind Energy Association, Wind Energy: Technology and Plan-ning, 2006. http://www.world-wind-energy.info

[3] Irish Wind Energy Association, Frequently Asked Questions, 2008.http://www.iwea.com/index.cfm/page/faqs?rfaqId=2

[4] Sustainable Energy Ireland, Wind Farms & the Environment, 2008.http://www.sei.ie/index.asp?locID=271&docID=-1

[5] Global Wind Energy Council, Wind: A Global Power Source, 2005.http://www.gwec.net/index.php?id=13

[6] S. Mertens, Wind Energy in the Built Environment, Brentwood, Multi-Science, 2006.

[7] Met Éireann, Climate of Ireland, 2008. http://www.met.ie/climate/climate-of-ireland.asp

[8] M. R. Patel, Wind and Solar Power Systems, Taylor & Francis Group,LLC, 2006.

[9] A. V. D. Rosa, Fundamentals of Renewable Energy Processes, Aca-demic Press, 2005.

[10] J. M. Blackledge, D. Kearney and E. Coyle, Non-Gaussian Analysis ofWind Velocity Data for the Determination of Power Quality Control,ISAST Trans. On Computing and Intelligent Systems, Vol. 3, No. 1,56-73, 2011.

[11] T. R. Oke, Boundary Layer Climates, Methuen, 1987.[12] G. B. Bonan, Land Surface Model (LSM 1.0) for Ecological, Hy-

drological, Atmospheric Studies. Model product, Available on-line athttp://daac.ornl.gov/ from Oak Ridge National LaboratoryDistributed Active Archive Centre, Oak Ridge, Tennessee, USA, 2005.

[13] J. M. Blackledge, D. Kearney and E. Coyle, Wind Turbine PowerQuality Estimation using a Levy Index Analysis of Wind Velocity Data,International Conference on Environment and Electrical Engineering,Rome 8-11 May, EEEIC 2011 IEEE Xplore Digital Object Identifier:10.1109/EEEIC.2011.5874695, 2011.

[14] J. M. Blackledge, M. Lamphiere, K. Murphy, S. Overton and A. Panahi,Stochastic Volatility Analysis using the Generalised Kolmogorov-FellerEquation, International Conference of Financial Engineering, WorldCongress on Engineering (WCE2012), London 4-6 July, IAENG, 453-458, 2012.

[15] J. M. Blackledge, M. Lamphiere, K. Murphy, S. Overton, Fi-nancial Forecasting using the Kolmogorov-Feller Equation, IAENGTransactions on Engineering Technologies - Special Issue of theWorld Congress on Engineering, Springer, 2013. (To be Published.Pre-publication available at http://eleceng.dit.ie/papers/244.pdf).

[16] A. Einstein, On the Motion of Small Particles Suspended in Liquids atRest Required by the Molecular-Kinetic Theory of Heat, Annalen derPhysik, Vol. 17, 549-560, 1905.

[17] A. N. Kolmogorov, On Analytic Methods in Probability Theory, Se-lected Works of A. N. Kolmogorov, Volume II: Probability Theoryand Mathematical Statistics (Ed. A. N. Shiryaev), Kluwer, Dordrecht,1992, pp. 6208, 1931. (Original: Uber die analytischen Methoden inder Wahrscheinlichkeitsrechnung, Math. Ann. 104, 41558, 1931).

[18] W. Feller, On Boundaries and Lateral Conditions for the KolmogorovDifferential Equations, The Annals of Mathematics, Second Series, Vol.65, No. 3, pp. 527-570, 1957.

[19] F. W. Olver and L. C. Maximon, Mittag-Leffler function, Handbookof Mathematical Functions in Olver, (Eds. W. J. Frank et al.), NIST,Cambridge University Press, 2010.

[20] J. M. Blackledge, The Fractal Market Hypothesis: Applications toFinancial Forecasting, Lecture Notes 4, Centre for Advanced Studies,Warsaw University of Technology, ISBN: 978-83-61993-02-5, 2010(Appednix B).

[21] M. Turner, J. M. Blackledge and P. Andrews, Fractal Geometry inDigital Imaging, Academic Press, ISBN: 0-12-703970- 8, 1998.

[22] m-Code for Othognal Linear Regression, http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=6716&objectType=File

[23] L. Landberg et al., Wind Resource Estimation - An Overview, WindEnergy, Vol. 6, 261-271, 2003.

[24] S. L. Walker, Building Mounted Wind Turbines and Their Suitabilityfor the Urban Scale: A Review of Methods of Estimating Urban WindResource,”Energy and Buildings, Vol. 43, 1852-1862, 2011.

[25] S. B. Pope, Turbulent Flows, Cambridge University Press, 2000.[26] T. Burton et al., Wind Energy Handbook, Wiley, 2001.


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[27] S. Mertens, Wind Energy in the Built Environment: ConcentratorEffects of Buildings, Technische Universiteit Delft, PhD Thesis, 2006.

[28] M. W. Rotach, Profiles of Turbulence Statistics in and Above an UrbanStreet Canyon, Atmospheric Environment, Vol. 29, 14731486, 1995.

[29] A. Rossen and Y. Sheinman, The Average Output Power of a WindTurbine in a Turbulent Wind, Wind Engineering & Industrial Aerody-namics, Vol. 51, 287-302, 1994.

[30] B. Cochran, The Influence of Atmospheric Turbulence on the KineticEnergy Available During Small Wind Turbine Power PerformanceTesting, IEA Expert Meeting on Power Performance of Small WindTurbines Not Connected to the Grid, 2002.

[31] W. Langreder et al., Turbulence Correction for Power Curves, Pre-sented at the EWEC, London, 2004.

[32] A. Tindal et al., Site-specific Adjustments to Wind Turbine PowerCurves, Presented at the AWEA Wind Power Conference, Houston,2008.

[33] R. Wagner et al., Simulation of Shear and Turbulence Impact onWind Turbine Power Performance, Riso DTU (National Laboratory forSustainable Energy), 2010.

[34] Campbell Scientific, 2011, CSAT3, 3D Sonic Anemometer. http://www.campbellsci.com/csat3.

[35] B. B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman,1982.

Jonathan Blackledge holds a PhD in Theoretical Physicsfrom London University and a PhD in Mathematical Infor-mation Technology from the University of Jyvaskyla. Hehas published over 200 scientific and engineering researchpapers including 14 books, has filed 15 patents and has

been supervisor to over 200 MSc/MPhil and PhD research graduates. He is theScience Foundation Ireland Stokes Professor at Dublin Institute of Technologywhere he is also an Honorary Professor and Distinguished Professor at WarsawUniversity of Technology. He holds Fellowships with leading Institutes andSocieties in the UK and Ireland including the Institute of Physics, the Instituteof Mathematics and its Applications, the Institution of Engineering andTechnology, the British Computer Society, the Royal Statistical Society andEngineers Ireland.

Eugene Coyle is Head of Research Innovation and Part-nerships at the Dublin Institute of Technology. His researchspans the fields of control systems and electrical engineer-ing, renewable energy, digital signal processing and ICT,and engineering education and has published in excessof 120 peer reviewed conference and journal papers in

addition to a number of book chapters. He is a Fellow of the Institution ofEngineering and Technology, Engineers Ireland, the Energy Institute and theChartered Institute of Building Services Engineers, was nominated to chair theInstitution of Engineering and Technology (IET) Irish branch committee for2009/10 and a member, by invitation, of the Engineering Advisory Committeeto the Frontiers Engineering and Science Directorate of Science FoundationIreland, SFI.

Niall McCoy graduated with a Diploma in Electrical Engi-neering in 2008 and Honours Degree in Electrical Services& Energy Management in 2010 from Dublin Instituteof Technology. He is a Member of Engineers Ireland, aChartered Engineer with the Energy Institute, an Accredited

Professional Engineer with the South Africa Engineering Institute and joinedthe renewable energies company Wind Prospect Group in 2007 where heis currently responsible for wind farm developments world-wide such as in

Canada, US, South Africa, UK & Ireland, having successfully project managedover 1.4GW of wind farm installations to date. Following a gap-analysis inthe wind industry with regard to the development of more quantitative windenergy forecasting models in September 2011, Niall registered for a PhDdegree at Dublin Institute of Technology with the provisional title of WindPower Analysis using Non-standard Statistical Models.

Derek Kearney graduated in 2007 from Ulster Universitywith a First Class Masters Degree in Renewable Energy,and is currently working on a PhD in the area of windenergy. He is a lecturer in the College of Engineering andBuilt Environment at Dublin Institute of Technology and is

programme manager for the MSc courses in Energy Management, SustainableElectrical Energy Systems and Sustainability, Technology and Innovation. Heis also programme manager for the BSc in Electrical Services and EnergyManagement. His principal area of specialty for both teaching and researchis in the area of renewable energy and energy conversion systems. Recentachievements include winner of the 2006 President’s Award for TeachingExcellence. He was chairman of the Honours Degree in Electrical Servicesand Energy Management for three years and previously worked on mainlandEurope for a number of years.

Keith Sunderland is an electrical engineering graduateof Dublin Institute of Technology (DIT) with a first classhonours degree in Electrical/Electronic Engineering. He isa member of the (DIT) Power Research Group as well asthe Dublin Energy Lab and his research focuses on theapplications of micro wind generation. More specifically,

his interests are with respect to urban wind profiling and (distribution) networktolerance to increased technology proliferation and his PhD research is incollaboration with the School of Geography, Planning & Environmental Policyat University College Dublin (UCD). He is the DIT member of the DublinUrban Boundary Layer Experiment (DUBLex), which is a cross institutioncollaboration (DIT, National University of Ireland, Maynooth and UCD)investigating urban climatology with respect to energy budgets, CO2 fluxesand energy applications. He is currently the Assistant Head of Department,Electrical Services Engineering within the School of Electrical EngineeringSystems (at DIT).

Thomas Woolmington joined the lecturing staff of DublinInstitute of Technology of the Electrical Services Depart-ment in September 2006 and has qualifications in theelectrical services and education areas including a nationalcraft certificate (Electrician), a BSc in Electrical ServicesEngineering and Energy Management and a Postgraduate

Certificate in Third Level Teaching and Learning. In April 2009 he co-authored a paper with other members of the Solar Energy Society of Ireland(SESIE) on The Uptake of Micro-gen Training on Third Level CoursesBetween Ireland and Wales presented at the PV-SAT5 Conference heldin Glyndwr University, Wales. In October 2009 he secured funding fromEnterprise Ireland to undertake a feasibility study into photometric testing ofLED luminaries using proprietary instruments and in April 2011 he registeredon a PhD research programme at Dublin Institute of Technology which isconcerned with the statistically self-affine modelling of turbulence on thepower performance of wind turbines.


17

Abstract—In this paper, we present and investigate a chaotic

modulation method based on combination of chaotic pulse width-

position modulation (CPWPM) and binary phase-shift keying

(BPSK) for digital communication. Firstly, binary information is

modulated onto chaotically-varied intervals of position and width

of pulses by CPWPM. Output pulses from CPWPM are then

conveyed by a sinusoidal carrier by means of BPSK modulation.

Schemes for modulator and demodulator are proposed and their

operation is described in detail. In addition, chaotic behavior and

its effect on average parameters of the method are investigated.

Theoretical calculation and numerical simulation of a CPWPM

system with specific parameters are carried out. Simulation

results on time and frequency domains as well as bit error rate

(BER) performance in additive white Gaussian noise (AWGN)

channel are shown in order to verify the operation and

performance of the proposed method.

Index Terms—Nonlinear dynamics, chaos, chaotic modulation,

CPPM, CPWPM, BPSK, chaos-based digital communications

I. INTRODUCTION

VER THE past decade, increasing efforts have been

devoted to study the possibility of using chaotic behavior

to improve the features of communication systems [1], [2].

Many chaotic modulation methods have been proposed [3]–[5]

and most of them exploited chaotic signals generated by

dynamical systems to convey information. A robust

modulation method named chaotic pulse-position modulation

(CPPM) was introduced [6], [7] to reduce the impact of noise

and distortion from communication channel on chaotic

synchronization. CPPM signal is in the pulse train format, in

which the binary information is modulated onto chaotically-

varied intervals of inter-pulses. The principal advantage of

CPPM is the automatic synchronization without the need of

specific hand-shaking protocols [8]. In [9], [10], a chaotic

modulation method which is considered as a development of

CPPM, named chaotic pulse width-position modulation

(CPWPM) was proposed, where the binary information is

modulated onto both chaotically-varied intervals of position

and width of output pulses. It means that two bits are encoded

on a single pulse. There, the position and width of the present

pulse are determined by time intervals from its rising edge to

the rising edge of the previous pulse and to its falling edge,

respectively. Since CPWPM signal is also in the pulse train

format as the CPPM signal, the demodulation process can

establish and maintain the synchronization state automatically.

The CPWPM signal can be considered as a time-modulated

baseband binary signal which is able to become the input

signal of a binary sinusoidal carrier modulation method such

as on-off keying (OOK), binary frequency-shift keying

(BFSK) or binary phase-shift keying (BPSK) [11], etc. In

practice, the BPSK technique provides a good bit error rate

(BER) performance in noise-affected environments [11] with a

simple structure and thus it is selected to investigate in the

combination with CPWPM in this research. This combination

method is called CPWPM+BPSK modulation.

The rest of this paper is organized as follows: In Section II,

schemes for CPWPM+BPSK modulator and demodulator are

presented and described in detail. Section III investigates the

chaotic behavior with tent map and from that average

parameters of the system are determined. A CPWPM+BPSK

system with specific parameters is calculated and simulated,

and their results are shown in Section IV. Finally, our

conclusion is given in Section V.

II. CPWPM+BPSK MODULATOR AND DEMODULATOR

Schemes for CPWPM+BPSK modulator and demodulator are

presented as in Fig. 1b and Fig. 1c, respectively, where each

scheme consists of CPWPM and BPSK parts.

A. Chaotic-Position Pulse Generator (CPPG)

Basically, the CPWPM parts in the modulator and

demodulator are built around chaotic-position pulse generators

(CPPG). Configuration of the CPPG is shown in Fig. 1a. A

counter operates in free running mode to produce a linearly

increasing signal, ( ) , where is the time duration from the reset instance and is count-step (the slope of the signal). This linearly increasing signal is reset to zero by each

input pulse. Before the reset time , the output value of the counter,

, is stored in the sample-and-hold circuit (S&H) whose output is fed to the nonlinear converter ( ). An amplifier with a gain-factor, , is used to produce another linearly increasing signal, ( ) , which has a higher slope compared with that of the input signal. When the

magnitudes of the output signals of the amplifier and that of

counter reach the same value ( ) at the output of the ( ),

two narrow pulses at Outputs 2 and 1 are generated at the

times, (

) and (

) , respectively. It is easy to find that the time

is earlier than and these times

can be controlled by the values of the gain-factor and count-step . With a proper choice of parameters, when Output 1 is connected back to the input to form a closed loop, CPPG will

Vu Van Yem1, Nguyen Xuan Quyen

1 and Kyandoghere Kyamakya

2

1School of Electronics and Telecommunications, Hanoi University of Science and Technology, Hanoi, Vietnam 2Institute for Smart-system Technologies, Alpen-Adria Klagenfurt University, Klagenfurt, Austria

A Chaotic Modulation Method based on

Combination of CPWPM and BPSK for Digital

Communication

O


18

1 0

Data_in 1

CPPG

Data_in 2

1 0

UCPWPM(t)

1

2

1

2

O1(t)

O2(t)

M1(t)

M2(t)

PTEG

BPSK part

1

UCPWPM+BPSK(t)

LO

M BPF

-1

CPWPM part

Data_out 1

CPPG

Data_out 2

10

1 0

1

2

1

2ETPG

BPSK part

1

-1

M TDLPF 1

-1

CPWPM part(a)

(b) (c)

S&H F(.)

Input

Output 1

Reset

K Output 2

Counter

Amplifier

C(t)

A(t)

1

nX1

nF(X )

CR

Delay

modulator 1

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