On the Usage of Dirac Delta Functions withNonlinear Argument in High Order I/O Integrals

Huber Nieto-ChaupisUniversidad Nacional de Ingenierıa - Instituto de Investigacion - FIEE - Lima

Universidad Nacional Mayor de San Marcos - Unidad de Postgrado - FIEE - Limahubernieto@uni.edu.pe

Abstract—The effect of using a Dirac Delta function withpolynomial argument in mathematical methodologies based onnonlinear I/O integrals, is studied. In concrete, we present thecase where a Dirac Delta function plays the role of input ortransfer function which would turn out to be of importance tomodel nonlinear systems containing distortion or degradationin their output signals. In effect, is seen the presence of sub-harmonics as consequence of quadratic argument into the DiracDelta function when it enters into an I/O convolution integral.This mathematical methodology presents clearly some virtueswhich might be employed to the case where a fully disturbedsimulated signal can be modeled with 6 parameters identificationcurve based on the Dirac Delta functions with polynomialargument. The discrepancy between simulated data and modelyields an error of order of up to 8%. In other case, is required upto 9 parameters when the formalism is adjusted to model path-loss signal in wireless communications, with an error of orderof 5%, thereby indicating that the modeling with Dirac Deltafunctions might be advantageous to model nonlinear systemsvariables plagued of stochastic disturbs.

I. INTRODUCTION

Inside the impulse response theory, the so-called Dirac Deltaor simply Delta functions are introduced as examples of inputfunctions representing very instantaneous impulses but witha well-defined output signal which can be well measured inthe frequency domain [1]. Some interesting properties of theDelta function might be of importance in the extension ofmathematical methodologies aimed to extract information ofsystems whose intensity of inputs signals is limited in time.Normally, Delta functions contain linear argument and theirinclusion in convolution integrals makes the result trivial asseen in abundant examples in literature [2]. Indeed the Deltafunctions are useful in many applications of the Physics andMathematics because it provides simplification in calculatingoperations of the type I/O. On the other hand, it serves asa medium for calculating integrals with a dependence multi-variable [3].In this note, the central point is the usage of Delta functionscontaining a polynomial function in their arguments and theirinclusion in nonlinear integrals to model nonlinear systems.Our view might be of importance in those cases where amodeling of distorted output signal created by instantaneousinput functions is needed. By means of simple calculationsit is shown that the polynomial argument makes the outputmorphology more complex as viewed in the apparition ofsecondary harmonics. For this exercise, the integer order

Bessel functions are used. Numerical simulations have yieldedthat shift of harmonic peak in output curves, occurs as well.All these interesting features are taken into account for systemidentification of simulated scenarios (through the Monte Carlomethod). Thus, simulated data which is plagued of randomdisturbs was prepared. Two scenarios were selected: Firstly,one consisting of plant data in which the system is attackedby stochastic disturbs. Secondly, the case of path-loss signalin wireless communication is considered. Here, a middle-sizeurban zone is computationally drawn in order to test the wire-less signal amplitude degradation through power emission-reception phenomenological models.In this manner, nonlinear system identification parameter ex-traction error around 8% and 5% in cases of nonlinear plantdata and path-loss signal was respectively estimated. Thisnote is structured as follows: second section explains all thatrespect to the Dirac Delta functions. In third section someillustrations for the simplest cases is given. Fourth sectioncovers the applicability of the Dirac Delta theory to model”plant” data generated from the Monte Carlo algorithm. Fifthsection uses again the Dirac Delta methodology to measurethe path-loss signal in a simulated urban zone, and thereforecompared to the well-known Okumura-Hata model, amongothers. Finally in last section, some conclusions regarding thisstudy are sketched.

II. THE ”SPIKE” PROPERTY OF THE DELTA FUNCTIONS

A. The ’Spike’ Property

One of the most notable applications among the diverseproperties of the well-known Delta function is that of thetrivial convolution integral written as f(t) =

∫f(τ)δ(t−τ)dt

where the Delta function argument depends linearly on s−u,the integration variable. An interesting extension of the Deltafunction considers modifying its argument from linear tononlinear function and which is known as the ’spike’ propertyof the Dirac Delta function,

δ [f(τ)] = δ

[N∏i=1

(τ − ai)

]=

N∑i=1

δ(τ − ai)df(τ)dτ |τ=ai

(1)

where f(τ) = (τ − a1)(τ − a2)...(τ − aN−1)(τ −aN ) is a continue function whose zeros are denoted bya1, a2, ..., aN−1, aN . For instance f(t) =

∫f(τ)δ(t2 − τ2)dt

≈ f(t)/2t that makes us to argument in the case that

978-1-4673-6386-0/14/$31.00 ©2014 IEEE

δ(tm−τn) the resulting convolution gets the form f(t)/tn−1,thereby demonstrating that the resulting integration dependson 1/tn−1. It is also possible express f(τ) as f(t, τ) therebysuggesting to rewrite (1) in a most general manner

δ [f(t, τ)] =

δ[∏M

i=1(t− τ − ai)]

uniform

δ[∏N

j=1(γjt− ρjτ − aj)]

composed

(2)where has been coined the terminology ’uniform’ and ’com-posed’. Clearly the ’uniform’ case obeys to a definition of theDirac Delta function argument where the variables are freeof parameters or constants but a shift or delay is introducedby means the ai parameter. The ’composed’ case engages γjand ρj parameters which are adjudicated as coefficients of thevariables t and τ .

B. Dirac Delta Functions in I/O Integrals

In praxis, exists there models which have considered delaysin input functions such as the Volterra series as used for param-eters extraction at nonlinear system identification. For instanceconsider truncated second order Volterra series. Under a fullylinear scenario it is traduced by y(t) =

∫h(τ)x(t − τ)dτ +∫

h(τ, β)x(t−τ)x(t−β)dτdβ where h(τ) and h(τ, β) denotethe kernels. Normally one can to reconstruct the kernels frominformation of input and output data, but it is a complicatedtask involving unnecessary time consumption. On the otherhand, kernel expansion onto orthogonal polynomials is high-lighted, and appears as an interesting strategy to solve theVolterra kernels. This methodology consist in essence as h(τ)=∑r CrPr(τ) and h(τ, β) =

∑r,p Cr,pPr(τ)Pp(β) where

Cr,Cr,p the system parameters. Studies performed in controlsystems using the Volterra formalism have unveiled interestingadvantages as to model MIMO systems with reduced kernelsnumber [5]. This fact would demonstrate that in certaincases, linear systems might be adjusted by a reduced set ofparameters, even though this parametrization can be extendedto the nonlinear ones.It is noteworthy that these concepts can be translated to anintegral formulation of Volterra-like formalism, but with DiracDelta functions playing the role of kernel. As seen in (2)the ’composed’ case where the Dirac Delta function argumentcontains a set of parameters which can be interpreted as thesystem parameters depending upon the grade of approxima-tion. So that, the changes h→ δ and x→ N would constitutealternatively the following approach for the Q order,

yT (t) =

Q∑`=1

[∫ ∏i=1

δ [f(t, τi)]N`(τi − α`i)τi

], (3)

where it is assumed that the integration variable is delayedby τ → τi−α`i in according to the Volterra formulation thatforces us to delay the input function by a quantity namely α`i.The input function N (τ) ⊂ R, with τ ⊂ R. Consider the casewhere Q = 1, which means that the entire formulation opts

Fig. 1. Resulting spectra obtained with Eq. (6). Top: Resulting output withρj <1. A noticeable dip is seen near to 7. Middle: spectra with ρ1 and ρ2= 3.0 and 5.0, respectively and ρ3=0.1. Several dips are observed, being themost pronounced on 2.5, in comparison with the minor ones. Bottom: variousdips appear as consequence of high values of ρ1 and ρ2 = 6.0 and 11.0,respectively and ρ3 = 0.5. The former indicates that the signal amplitude isreduced by a 1/100 of its initial value, roughly.

the first order of approximation. Then one gets

yT (t) =

∫ ∞−∞

δ [f(t, τ)]N (τ − α11)dτ. (4)

The next step in to introduce one of the definitions given in (2)to resemble the kernel concept under the input/output view. Inother words, the formulation should kept the Volterra approachstructure, then the choice of the ’composed’ case turns outto be the most appropriate since the Dirac Delta argumentexhibits the incorporation of the model parameters. To notethat these parameters are real numbers. Thus, one can express(4) in a manner most detailed,

yT (t) =

∫ ∞−∞

δ

N∏j=1

(γjt− ρjτ − aj)

N (τ − α11)dτ (5)

where N runs over the maximum value of polynomial grade.For model test, N = 3 is taken. In conjunction for threemodel parameters is to possible to find three real roots, whichmakes the result consistent to some extent. Then a dedicateprocess of parameter identification is necessary in order todiscard complex roots and avoid misinterpretation of DiracDelta polynomial argument.

C. Numerical Examples

The presence of parameters γj and ρj deserves a careas to determine their values appropriately. For instance, weattempt to model the case when the system is order N=3and resulting in the curves plotted in Fig. 1(a-c). The set ofparameters are presented for three different cases, α11 = 0,γ1 = γ2 = γ3 = 1, ρ1=[0.3,3.0,6.0], ρ2=[0.9,5.0,11.0],

ρ3=[0.6,0.1,0.5], a1=10.5, a2=12.0, and ρ3=1.1. Thus, thecorresponding formulation of output function is given by

yT (r) =

∫ ∞−∞

J22 (τ)δ[(r − 6τ − 10.5)(r − 11τ − 12)×

×(r − 0.5τ − 1.1)]dτ (6)

with N (τ) = J22 (τ), with J2(τ) the second order Bessel

function. For this exercise no any delay is considered. Theresulting spectra are plotted in Fig. 1. One can note that inFig.1(a) the curve is falling up to one third of its initial valuereaching on a well formed dip near to 7. One can argue thatthis shape comes from the fact that the ρj runs over valuesless than 1. In Fig.1(b) various dips are seen since ρ1=3.0and ρ2=5.0 whereas ρ3=0.1. A consequence the curve fallsmore rapidly as to observed in the amplitude degradation ofup to 1/10 of its initial value. The most pronounced dips isobserved in 2.5. However, one interesting morphology of curveis viewed in Fig.1(c) where ρ1=6.0 and ρ2=11.0 and ρ3=0.5,and dips are repeatedly created and decreasing dramaticallyits amplitude.

III. SIMPLE CASES I/O CONVOLUTION INTEGRALS WITHBESSEL Y DIRAC DELTA FUNCTIONS

A. Triviality in the Linear Case

Triviality is defined in the sense that any convolutionoperation by taking into account the Delta function (withlinear argument) is performed in a straightforward manner.For instance, it can be applied in those system identificationschemes which are defined at least with diagonal terms in sometruncated order as the well-known Volterra truncated series [5]

y(t) =

∫h(τ)x(t− τ)dτ +

∫ ∫h(τ)h(β)x(t− τ)x(t− β).

(7)When Delta functions replace the input functions given by xthe resulting integration turns out to be a triviality. In otherwords the output depends on the morphology of the kernelsh,

y(t) =

∫h(τ)δ(t− τ)dτ +

∫ ∫h(τ)h(β)δ(t− τ)δ(t− β)

(8)y(t) = h(t) + h2(t) (9)

One can see that the usage of kernels like the so-called or-thogonal polynomial would give a defined shape to the outputy(t). So that depending on the N -order of approximation theny(t)=

∑Nk h

k.

B. The Nonlinear Case

The nonlinear case is then defined as the convolution inte-gral containing a Delta function with a polynomial argumentand it reads

y(t) = y1(t) + y2(t) =

∫dτ1h(τ1)δ(G1(τ1, t)) +∫

dτ1dτ2h(τ1)h(τ2)δ(G1(τ1, t))δ(G2(τ2, t)) (10)

Fig. 2. Top: output y(t) obtained with Eq. (8) and kernel as Bessel(1,t).Bottom: The y(t) as given but Eq. (15). Note the distortion of Bessel functionas well as the reduction of its amplitude. For large values of the independentvariable, it is observed that the output becomes negligible.

where now the Delta function argument express the fact thatit depends on a generalized G1(τ1, t) function. Furthermore,it can contain free parameters which in somewhat should beabsorbed by the kernel through the convolution operation. Ina first instance one can assume

G1(τ1, t) = (τ1 − α1t− α2t2)(τ1 − β1t− β2t2). (11)

One can see in Eq. (11) the dependency on the variable tis given in a power expansion and accompanied of the freeparameters α1,2 and β1,2. For the sake of the simplicity weshall use the truncated case of up to second order. When Eq.(11) is inserted in Eq. (10) then one gets

y1(t) =

∫dτ1

h(τ1)

(α1 − β1)t+ (α2 − β2)t2×

×[δ(τ1 − α1t− α2t

2)− δ(τ1 − β1t− β2t2)]

(12)

after a short algebra. This expression can be written in anotherway,

y1(t) =1

(α1 − β1)t+ (α2 − β2)t2

∫dτh(τ1)×

×[δ(τ1 − α1t− α2t

2)− δ(τ1 − β1t− β2t2)]

(13)

reflecting that the output is already decreasing in large valuesof t. The evaluation of the integral is then

y1(t) =h(α1t+ α2t

2)− h(β1t+ β2t2)

(α1 − β1)t+ (α2 − β2)t2(14)

Fig. 3. Top left (right) panel: it is shown the output y(t) with Delta functionsworking as Eq. (8) and kernel as Bessel2 (Bessel3). Bottom left (right) panel:the output y(t) but working with structure of Eq. (15) and kernel Bessel2(Bessel3).

and the second order turns out to be,

y2(t) =h(α1t+ α2t

2)− h(β1t+ β2t2)

(α1 − β1)t+ (α2 − β2)t2×

×h(α11t+ α22t2)− h(β11t+ β22t

2)

(α11 − β11)t+ (α22 − β22)t2. (15)

with α1,11 and β2,22 the free parameters for the second orderapproximation. So that, (14) plus (15) are tested when aninteger order Bessel function replaces the kernel. In the next,the sum y(t)=y1(t) + y2(t) is considered. The plotting ofspectra with and without polynomial argument is given withthe help of the software Mathematica, as shown in Fig 2. It isquite interesting to note the presence of additional harmonicsover the true ones (i.e., to the case of the top panel of Fig.2where no polynomial argument is introduced in the Deltafunction). Integer order Bessel functions were employed. InFig. 3 is shown the cases where the order of the Besselfunctions takes a different order (2 and 3). Although it doesnot changes dramatically the spectra one sees that the principalas well as the secondary harmonics are shifted. Besides, theamplitude is degraded by a substantial fraction by makingnegligible the value of the ”wave” in large values of t. In otherwords, the inclusion of quadratic terms of t has the direct effectin decreasing the wave over the horizon until its disappearance.A point that should be stressed is the value of this shift onthe wave as observed in Fig. 4. In fact, the plot does not onlyshow such shift but also occurs degradation by a quarter of itsamplitude by action of the denominator as seen in (13).

Fig. 4. A shift of the first harmonic of the Bessel3 polynomial by effect ofincluding a Delta function with a quadratic harmonic. It is also observed theapparition of secondaries sub harmonics and the reduction of the amplitudeby a 1/4 of the value on the first harmonic compared to the case when theDelta function is not used.

IV. APPLICABILITY IN NONLINEAR SYSTEMSIDENTIFICATION

A possible application of those schemes would be in thecase of nonlinear response containing a high degree of distor-tions as consequence of the instability of system, as commonlyoccurs in MIMO systems. For instance, consider a systemwhose physical observable (e.g. temperature, current, voltage,engine velocity, etc) makes a rapid evolution per unit of time.However, it is observed the presence of episodes of disturbswhich are trying down the systems stability.We have simulatedsuch system through the Monte Carlo method in which eachindividual register is accepted only if the probability of havinga stochastic fluctuation is greater than the probability of systemstability. We call to this set of point plant data. Our subsequentstep is the searching of the identification curve by means themethodology of Delta functions. In other words, we try tomodel the plant by means the Delta and Bessel functions. Inthis way, below is written the proposal of identification curve,replacing the kernels by the integer order Bessel functions,

y(t) =

∫P1(x)δ(x− α1t− α2t

2)dx+∫P1(x)P1(y)δ(x− β1t− β2t2)δ(y − β3t− β4t2), (16)

where the αi and βi denote the system parameters.

A. Monte Carlo Data Simulation Procedure

In order to test the validity of model, a set of data is needed.To this end, we have simulated data through the Monte Carlotechnique up to 100 points between 0 and 3s. These data hastaken as template the curve given by fig.5 (top panel). TheMonte Carlo procedure is as follows: (i) 100 data points andtheir associated errors are thrown, (ii) the first one is takennamely tk+1 and the resulting curve is evaluated g(tk+1), (iii)gk+1 is compared to a random number rk+1, if gk+1 is greaterthan rk+1 the abscissa number is computed, (iv) 99 data pointsare again thrown and repeat previous procedure for k + 2.Thus, up to 100 data points were simulated, each one withina small statistic error and therefore, statistic fluctuations wereassigned.

Fig. 5. (Top panel) Example of a representation of model from Eq.16. Forthis exercise α1=0.52, α2=0.045, β1=0.15, β2=6.6, β3=6.0, and β4=27.0.(Bottom panel) Identification curve or model (continuous line) superimposedto plant data (dots) during the first 3s. To note that the model adjusts wellto simulated data between 1 to 1.4s. The extracted parameters are α1=0.50,α2=0.07, β1=0.1, β2=6.7, β3=0.1, and β4=3.07. The most representativefitting result is that of the α1 which is less than 10%. Although themodel differs notably of data, it is observed that the Delta scheme followsapproximately the data, within an error of order of 8%.

B. Procedure for System Parameter Extraction

We use the model given by Eq.(16) by replacing P → Jn,with Jn the integer-order Bessel function. However, a casemost realistic is through the Monte Carlo method in whicheach individual register is accepted only if the probability ofhaving a stochastic fluctuation is greater than the probabilityof system stability. From 100 simulated points, only 53 pointspassed the selection. All of them are plotted as dots in fig.5(bottom). One can note the presence of multiple peaks asconsequence of the random nature of data. One can see thatthe simulated data starts its evolution at the first second.Then the system gets fully disturbed between 1s and 3s. Itis clear that the system is plagued by stochastic fluctuationsas consequence of the Monte Carlo step. We call to this setof point ”plant data”. Our subsequent step is the searching ofthe identification curve by means the methodology of Deltafunctions. In other words, we try to model the ”plant” bymeans the Delta and Bessel functions. We have performed a 6-parameters system identification and the extracted parametersyields the following values: α1=0.5, α2=0.07, β1=0.1, β2=6.7,β3=0.1, and β4=3.07. All these parameters were the best valueswith errors ranging between 1 to 8%. The average parametererror is of order of 5%. It should be noted that the identificationprocedure was performed between 1s and 3s. A point whichshould be stressed is that of the αi and βi parameters byplaying the role of adjusting the peaks of the y(t) to data.

TABLE IRELEVANT MODELS AND THE DIRAC DELTA APPROACH BY INDICATING

THE NOMINAL PATH LOSS (DB) DETERMINATION FOR DIFFERENTSCENARIOS. SIGNAL OF 1.5 GHZ IS CONSIDERED.

DiracRun Free Okumura Hata Cost-123 Delta

Space Hata Composed6 90 103 110 105 108

11 91 102 115 109 11021 91 111 111 110 11134 91 110 117 107 10250 90 104 109 106 11369 91 116 110 111 109

According to our scheme the fast change of the βi parametersis reflected on the presence of peaks on the y(t) curve. In orderto be most precise, y(t) is well adjusted to data between 1s and1.4s. It can be traduced that the system identification preservesits memory until a time given by 1.4s. which is largely affectedby the disturbs.

V. ATTEMPTS TO MODEL PATH-LOSS SIGNAL WITHDIRAC-DELTA FORMALISM IN SIMULATED URBAN ZONE

A. Estimation of Model Parameters

Another possible example to illustrate the application ofthe Dirac Delta methodology to measure system parametersis that of path-loss signal measurement [6] [7]. To this end,we simulate a middle-size urban zone of 0.25 Km2 with theassistance of Google-Sketchup software. In figure 6 is depictedthe optimal height of 28 meters of a source of signal located ina base station (cylinder) radiating onto a suburbia composedby middle buildings of height between 5 and 40 m. In addition,up to 75 runs for different scenarios of urban zones weregenerated with Google-Sketchup software.In the mathematical side this, the output nonlinear integralcontaining a Delta function with a polynomial argumentwould resemble the full mechanism among the transmitter,communication medium and receiver (it is claimed that thewell-known delay spread should occurs and its measurementshould be registered) [8].Therefore, it enters in a full formulation of path loss in signalstrength at receivers

PL = ALog

{ Q∑`=1

[∫ ∏i=1

δ [f(t, τi)]N`(τi − α`i)dτi

]}(17)

where A=45, being a normalization quantity.It was assumed up to 42 obstacles in average with statistics

error of ± 4. These obstacles lie inside the region painted byblue en Fig. 7. For computational ends, the full suburb lengthis of 0.5 Km. On the other hand, the extraction of the modelparameters the calculation have yielded Q=3, γ1=22.4, γ2=-6.3, γ3=31.9, ρ1=-0.4, ρ2=37.3, ρ3=23.5, a1=-0.5, a2=25.0,a3=32.7. Again, the integer order Bessel functions were usedbecause their smoothness character and capability in to pro-duce successive dips overall integration. It should be noted thatthe all values presented above denote the average for all runs.

Fig. 6. Location of cross plane over suburb which indicates that the upperarea is of optimal signal reception. It is given by the run 21. Below this planereceivers undergo of path loss and signal degradation. Cross plane is on 28m., from the ground level.

In addition, a statistics error of 7% had have to be attainedfor all values. One can see that the model parameters are oforder of 20 and 37 which quite consistent with the number ofobstacles already defined. The simulation have revealed thatthe apparition of successive dips might be absorbed by thecontiguous dip and subtract one of the net count. It is actuallytroublesome because the model depends on the number ofobstacles and the miscount of them give rise to systematicerrors of model. The main source of error came form therandom generation of blocks distribution with a specific valueof wireless signal. The model parameters extraction havebeen achieved with the support of Mathematica software.The encountered values of path loss through the Dirac Deltafunction model is listed in Table I, together to the nominalvalues with same inputs but with different models. Values areexpressed in dB.

B. Comparison with Known Models

To assess the capability of model, in table I encloses thevalues of path loss given also for known models as Free Space,Okumura-Hata, Hata, and Cost-123. According to the simula-tion, base station height of 28 m. ± 2 m., is considered. Thecoberture distance is of 0.5 Km. However, we assume that thenominal suburb lenght is near to 1 Km. This threshold valueis inside the valid range of Hata, Okumura-Hata and Cost-123models. It is interesting that the proposed model fits well themean values of models as for example occurs in the run 21. Ineffect, on this run, a height of 30 m, coberture distance of 0.5Km, height receiver of 28 m, and received signal of -65 dBwas obtained by the simulation. For other runs, errors beyond5% appears to be consistent with the known models. It isexemplified in the run 50 whose differences with the Okumura-Hata model is of order 10%. It is because the receiver gotup to -70 dB because the height antenna of 25 m. Apartof that, it should be mentioned that the terrain morphologyis nearly to be flat, without any pronounced deformation. Itmakes us to state that the model might be disadvantageousin other terrain situations and different strategies for suburbsimulation should be taken [9][10][11]. In addition, the DiracDelta model appears to be congruent with well-used models,but a discrepancy of at least 5% is encountered.

VI. CONCLUSIONS

In this note, a possible application of the I/O integralcontaining the so-called Dirac Delta functions with polynomialargument was studied. Interestingly, the case of data plagued of

Fig. 7. Possible geographical locations with highest probability of signalreception at -65 dBm (blue) given by the run 21. It follows from the resultingsimulations where optimal height of base station is 28 m.

stochastic fluctuations and random behavior is of importancein order to face the problem of control MIMO systems witha minimum of error for small lapses of time, and sensitivesto disturbs. An I/O second order integral is proposed havingbeen applied to simulated data by means of Monte Carlomethod. As consequence an identification error of order of8% is found. On the other side, the Dirac Delta formalism wasused to model the path-loss of degraded signal when it is sentfrom base stations to end users. It was found a discrepancy oforder of 5% with known models like the Okumura-Hata. Here,the existence of dips on the resulting output functions canbe interpreted as the possible location of obstacles. From allthese quantitative results, is suggested that a dedicate analysisfor high order I/O integrals is needed, in order to assessthe validity of the Dirac Delta formalism into control systemtheory and wireless communications.

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