Accurate Analytical Forms of Heaviside and Ramp Function

Abstract

In this paper, explicit exact representations of the Unit Step Function and Ramp Function are obtained. These important functions constitute fundamental concepts of operational calculus together with digital signal processing theory and are also involved in many other areas of applied sciences and engineering practices. In particular, according to a rigorous process from the viewpoint of Mathematical Analysis, the Unit Step Function and the Ramp Function are equivalently performed as bi-parametric single-valued functions with only one constraint imposed on each parameter. The novelty of this work, when compared with other investigations concerning accurate and/or approximate forms of Unit Step Function and/or Ramp Function, is that the proposed exact formulae are not exhibited in terms of miscellaneous special functions, e.g., Gamma Function, Biexponential Function, or any other special functions, such as Error Function, Complementary Error Function, Hyperbolic Function, or Orthogonal Polynomials. In this framework, one may deduce that these formulae may be much more practical, flexible, and useful in the computational procedures that are inserted into operational calculus and digital signal processing techniques as well as other engineering practices.

1. Introduction

The Ramp Function, the notation of which is R($x$), where $x$ denotes the argument, is a discontinuous single-valued function of a real variable with a point discontinuity located at zero. For negative arguments, the output of this function is zero, while for positive arguments, R($x$) is simply $x$ [1,2].

In addition, its first derivative with respect to $x$ is the Heaviside Step Function, also known as the Unit Step Function, the notation of which is $H\left(x\right)$. The Heaviside Function is commonly used in the mathematics of control theory as well as in signal processing in order to represent a signal that switches on at a specified time and stays switched on indefinitely [2,3]. The Heaviside Step Function represents a sudden change in value at a specific point. Moreover, it is commonly used in engineering and applied physics to simulate discontinuous events. Indeed, it has many applications in signal processing, control systems, and circuit analysis. For instance, it is used to model events such as turning on or off a switch, sudden changes in voltage or current, and the activation of a system based on a threshold value. Another example of the utility of the Unit Step Function concerns electrical circuits, where this function may be used to predict the behavior of a switch. Moreover, in control systems, it can represent the activation of a system when a certain condition is met. It can also be used in signal processing to model the sharp rise or fall of a signal. Additionally, the second derivative of the Ramp Function is the Dirac delta distribution (or δ distribution), also known as the unit impulse [4,5]. Step, Ramp and parabolic functions are called singularity functions [4,5]. Nonetheless, regardless of its name, the Dirac delta function does not truly constitute a function, and indeed, the Heaviside Step Function is not stringently differentiable. A rigorous treatment of the Dirac delta distribution necessitates the concepts of Measure Theory [5]. Indeed, the Dirac delta function, in the traditional sense, is not assumed to be a function. It is considered as a generalized function or a measure in most recent mathematical literature. The Dirac delta function has a domain over the set of real numbers, and it vanishes everywhere except at zero argument, where it takes an indeterminate value. Actually, both the Heaviside Step Function and the Ramp Function have many applications in applied sciences and engineering and are mainly involved in digital signal processing and electrical engineering. In fact, the Heaviside Step Function may serve as a test input signal for characterizing the response of linear systems throughout linear system theory and feedback control theory. In addition, it may serve as a multiplicative weighting function to create causal (one-sided) functions [4,5]. For instance, a causal exponential time function may be expressed as follows:

$$f\left(t\right)=H\left(t\right)\cdot EXP(-t)$$

(1)

Here, one may elucidate that, in general, exact representations of the Heaviside Function, which evidently constitutes a step function, are often less practical in computations (when compared with approximate expressions) due to the discontinuity of this function [4,5]. On the other hand, its approximations allow for smooth, differentiable functions that are easier to work with in computational methods and numerical simulations [6]. In fact, approximations may involve continuous functions such as logistic, Cauchy, or normal distributions, therefore allowing for easier manipulation and computation [7]. Moreover, such approximations are practical when dealing with numerical simulations where sharp discontinuities may lead to phenomena like Gibbs oscillations. Evidently, the majority of numerical methods, such as finite difference and finite element analysis, require smooth and differentiable functions. Hence, the discontinuity of the Heaviside Step Function offers challenges for the implementations of these methods. Indeed, approximations such as Sigmoid Functions provide smooth transitions that can be readily used. Further, one may point out that exact forms of the Heaviside, Ramp, and Signum Functions may lead to singularities in calculations, particularly when their derivatives are also involved. Actually, approximations may help to avoid these singularities by maintaining numerical stability. Additionally, in many physical systems, signals do not switch instantaneously. In this context, approximations like Sigmoid Functions may represent in a more rigorous manner the gradual transition observed in realistic situations, e.g., the charging or discharging of a capacitor.

Nonetheless, the exact representations of the Heaviside Function play a crucial role in order to preserve the integrity of theoretical results. For instance, in operational calculus, the Heaviside Function is defined rigorously, and its properties are used in solving differential equations.

Apart from the previously mentioned implementations, a rigorous analytical technique to estimate the deflection of a linearly elastic simply supported beam under off-axis pointwise loading and for high rates of its curvature was performed in Ref. [8]. A rectangular two-dimensional Cartesian frame of reference was considered such that the origin of which coincides with the left end of the beam and the primary horizontal axis of abscissas coincides with the centroid axis of the beam. The novelty of this work was a presentation of an analytical method to find the deflection throughout the simply supported beam, in a unified manner, without its separation in distinguished domains (sub-domains). This was guaranteed by the introduction of the Unit Step Function. Nonetheless, to obtain the final closed-form solution, the authors proposed an exact form of the Heaviside Step Function, in particular as the summation of two inverse tangent functions. However, according to this approach, the singularity structure was left ambiguous.

In addition, one may consider the bending of a straight prismatic Timoshenko beam [9] of length L resting on simple supports as it was presented in Refs. [9,10,11,12]. In this framework, a rectangular Cartesian coordinate system is adopted such that the x- and z-axes are to be located along the neutral surface (axis) and perpendicular to the neutral surface (axis), respectively. The transverse deflection slope of the deformed beam is expressed in terms of the independent variable x via an Ordinary Differential Equation (ODE) in which the Heaviside Step Function occurs as well. The Euler–Bernoulli hypothesis [9,10] asserts that the cross section and the neutral surface remain perpendicular during deformation, and therefore, the rotation of the cross section of the beam at an arbitrary position is equal to the transverse defection slope. In some cases, such as centrally loaded force and symmetrically loaded forces, the abovementioned ODE reduces to a separable differential equation, which can be solved by integrating directly on both sides. In this context, to evaluate the transverse deflection of the deformed elastic beam along with the rotation of the cross section at an arbitrary position x, it is necessary to calculate integrals of the forms $\int H\left(x\right)dx$ and $\int x\cdot H\left(x\right)dx$.

Hence, one may deduce that analytical representations of the Heaviside Step Function are indeed very helpful.

In addition to the above, one may elucidate that the Ramp Function constitutes a signal whose amplitude varies linearly with respect to time and can be expressed by several definitions [4,5]. In digital signal processing, the unit Ramp Function is a discrete time signal that starts from zero and increases linearly.

Here, one may remark that a signal is defined as a physical phenomenon that carries some information or data [13]. The signals are usually functions of the independent variable time. Nonetheless, there are some cases where the signals are not functions of time. For instance, the electrical charge distributed in a body constitutes a signal that is a function of space and not time.

The signal that is specified for every value of time t is called continuous-time signal, while the signal that is specified at a discrete value of time is called discrete-time signal.

Basic signals play a very important role in signals and systems analysis. The basic continuous time signals are as follows [14,15]:

• Unit impulse function;
• Unit step function;
• Unit ramp function;
• Unit parabolic function;
• Unit rectangular pulse function;
• Unit area triangular function;
• Unit signum function;
• Unit sinc function;
• Sinusoidal signal;
• Real exponential signal;
• Complex exponential signal.

It is worth noting that the basic continuous-time (CT) and discrete-time (DT) signals include impulse, step, ramp, parabolic, rectangular pulse, triangular pulse, Signum Function, sinc function, sinusoid, and finally real along with complex exponentials [13,14,15]. The Ramp Function states that the signal will start from time zero and will instantly take a slant shape, and depending upon given time characteristics (i.e., either positive or negative, here positive), the signal will follow the straight slant path either towards the right or left, here towards the right [9,10,11]. In this context, the Ramp Function constitutes a type of elementary function that exists only for positive values of the argument and is zero for negative values [15,16,17]. Moreover, the impulse function is obtained by differentiating the Ramp Function twice [9,10]. On the other hand, it is well known that an electrical network consists of passive elements like resistors, capacitors, and inductors. They are connected in series, parallel, and series parallel combinations [17,18,19]. The currents through and voltages across these elements are obtained by solving integro-differential equations. Alternatively, the elements in the network are transformed from the time domain, and an algebraic equation is obtained that is expressed in terms of input and output [17,19,20]. The commonly used inputs are impulse, step, ramp, sinusoids, exponentials, etc.

In fact, there are many applications of the aforementioned function to similar problems, along with other engineering practices. Specifically, in Ref. [21], an in-memory analog-to-digital conversion (ADC) technique was introduced. This conversion is significant since most modern electronic devices operate on digital signals, which are easier to store, manipulate, and transmit. The novelty of this work is that, instead of generating a linear ramp voltage as in conventional ADCs, a nonlinear ramp voltage was generated by different properly selected values of memristive conductances. In this framework, the Ramp Function was used as a discrete approximation of a time-varying ramp signal.

In Ref. [22], the modeling and fault diagnosis of a fiber-optic gyroscope were presented. Here, the ramp noise is simulated by means of a random Ramp Function.

In Ref. [23], an experimental investigation of a PID (Proportional–Integral–Derivative) control strategy for a three-phase induction motor speed control system was carried out with implementation on a scaled railway vehicle roller test rig drive, which simulates wheel–rail interaction by the use of rotating rollers instead of a track. The closed-loop system is also subjected to a Ramp Function in accordance with the requirements and objective of the operating condition. Actually, PID control is a widely used feedback control loop mechanism that adjusts input to preserve a desired output value by calculating an error signal (the difference between the desired and actual output) and then applying corrective actions on the basis of integral and derivative terms.

In Ref. [24], a survey of inverter-based resource (IBR)-penetrated power system stability analysis was performed. Here, it is signified that the ramp + chirp perturbation combines the smoothness of a Ramp Function with the broad frequency coverage of a chirp signal.

Meanwhile, in addition to what was previously mentioned regarding the applications of the Heaviside Step Function, one may also focus on applications of this function to cutting-edge issues, as demonstrated in the following recent publications:

In Ref. [25], a rolling bearing fault diagnosis technique based on a multidomain feature and whale optimization algorithm–support vector machine was proposed. The role of the Heaviside Function is important since a Recurrence Plot (RP) is the basis of recurrence quantification analysis (RQA). The point R(i,j) of RP is a value that is either 0 or 1.

In Ref. [26], the Parametric level-sets method (PaLS) was redefined aiming at a rigorous mathematical treatment of inverse imaging problems involving piecewise constant contrasts. To facilitate the analysis, the exact Heaviside Step Function was replaced by a smooth differentiable approximation.

In Ref. [27], novel techniques for computing multi-domain features to detect cognitive load in two datasets—Multi-Aspect Trajectory (MAT) and Simultaneous Task Electroencephalography Workload (STEW)—were introduced. In this framework, two scenarios for analysis were employed: lead-wise and overall. Here, the role of the Heaviside Function is significant in the feature extraction process, since this function is characterized by its “leaping” feature, which refers to its sudden jump from 0 to 1 at a specific point (usually at zero).

In Ref. [28], a novel and interpretable data-driven learning strategy based on explainable artificial intelligence (AI) and rapid ultrasonic detection is used to effectively estimate the battery state of health (SoH) in real time. Here, since the differentiable oblivious decision tree (ODT) is parameterized by response vector threshold values, scale factors, and the feature selection matrix, the Heaviside Function was replaced by its continuous counterparts.

In addition to the above, one may observe that, in the past years, there have been many research works carried out aiming at the analytical approximation of the Unit Step Function along with the Ramp Function and Dirac delta distribution. Specifically in Ref. [29], a rigorous representation of the Heaviside Step Function was accomplished by a linear combination of exponential functions, whereas for a rigorous study on the numerical approximation of the Heaviside Step Function by means of a finite difference method, one may refer to Ref. [30]. On the other hand, the limiting form of many sigmoid-type functions centered on $x=0$ may also serve as an approximation to the Heaviside Step Function. For instance, one may remark the following expressions [31]:

$$H(x)=\underset{k\to \infty }{lim}{\displaystyle \frac{1}{2}}\left(1+tanh\left(kx\right)\right)$$

(2)

$$H(x)=\underset{k\to \infty }{lim}\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{\pi }}arctan(kx)\right)$$

(3)

$$H(x)=\underset{k\to \infty }{lim}{\displaystyle \frac{1}{2}}\left(1+erf\left(kx\right)\right)$$

(4)

where the notation $erf$ denotes the well-known Error Function.

Evidently, by multiplying both sides of the above three equations with the argument $x$, one shall obtain closed-form expressions of the Ramp Function. Additionally, there are many explicit forms of the Heaviside Step Function and Ramp Function that can be found in the literature.

Specifically, in Ref. [4], an elegant explicit representation of this function was proposed as the summation of two inverse trigonometric functions by means of the following relationship:

$$H(x)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{\pi }}\cdot \left(arctan(x)+arctan({\displaystyle \frac{1}{x}})\right)$$

(5)

In this framework, by multiplying both sides of the above equation with the argument $x,$one shall derive the Ramp Function as

$$R(x)={\displaystyle \frac{x}{2}}+{\displaystyle \frac{x}{\pi }}\cdot (arctan(x)+arctan({\displaystyle \frac{1}{x}}))$$

(6)

In Ref. [32], another analytic exact form of the Unit Step Function, as a summation of two inverse trigonometric functions, was presented, as follows:

$$H(x)={\displaystyle \frac{3}{4}}+{\displaystyle \frac{1}{\pi }}\cdot \left(arctan(x-1)+arctan({\displaystyle \frac{x-2}{x}})\right)$$

(7)

and therefore,

$$R(x)={\displaystyle \frac{3x}{4}}+{\displaystyle \frac{x}{\pi }}\cdot \left(arctan(x-1)+arctan({\displaystyle \frac{x-2}{x}})\right)$$

(8)

In Ref. [33], the following elegant analytical form of the Heaviside Step Function was performed:

$$H(x)={\displaystyle \frac{1}{2}}+i {\displaystyle \frac{ ln \left(x\right) -ln (-x)}{2\pi }}$$

(9)

where $i$ denotes the imaginary unit.

In this context, one may also obtain the Ramp Function as

$$R(x)={\displaystyle \frac{x}{2}}+i {\displaystyle \frac{ xln \left(x\right)-xln (-x)}{2\pi }}$$

(10)

In addition, in Ref. [34], a stringent approximation of the Unit Step Function by the use of cumulative distribution functions was carried out, while in Refs. [35,36], the Heaviside Step Function was approached via Raised-Cosine and Laplace Cumulative Distribution Functions and Sigmoid Functions. Additionally, in Ref. [37], an analytical exact form of the Unit Step Function was exhibited by the use of the floor function, which evidently yields the greatest integer less than or equal to its argument. Moreover, in Ref. [38], a rigorous approximation of the Unit Step Function was exhibited as the pointwise limit of a sequence of functions, whereas an analogous investigation towards an analytical approach of the Ramp Function was performed in Ref. [39].

Further, in Ref. [40], the Unit Step Function was represented as the summation of six inverse trigonometric functions. In particular, the novel element of this analytical formula is that it contains two arbitrary single-valued functions that satisfy only one single condition and does not contain any other special functions such as Gamma Function or generalize integrals compared with other representations of the Unit Function. On the other hand, there are many applications of the Heaviside and Ramp Function in signal processing, as it can be seen in the literature. In Ref. [41], a rigorous study on the relations between control signal properties and robustness measurements was carried out. In particular, the Heaviside Step Function was used to express the specified output at a load disturbance. In Ref. [42], a single-image super-resolution by means of approximated forms of the Heaviside Step Function was performed. In addition, a valuable investigation concerning the measurement of regularity in discrete time signals is presented in Ref. [43]. The Heaviside Step Function was used in the evaluation procedure of the entropy of a regular system with orderly behavior. In Ref. [44], a remarkable study on the use of Generalized Functions in the Study of Signals and Systems was carried out. Specifically, a physical system was simulated as a linear time-invariant (LTI) system. Evidently, such a system is represented mathematically by an Ordinary Differential Equation (ODE) or by a set of coupled ODEs. In this context, the Heaviside Step Function was taken into account towards the analytical treatment of this original problem. In Ref. [45], a Green’s function analysis of nonlinear thermoacoustic effects under the influence of noise in a combustion chamber was carried out, whereas in Ref. [46], an integrated vibration energy harvesting–storage–injection device based on a piezoelectric bistable device was performed. Finally, in Ref. [47], a rigorous nonlinear vibration analysis and defect characterization using an entropy measure-based learning algorithm in defective rolling element bearings was presented, while enhancing convolutional neural network robustness against image noise via an artificial visual system was carried out in Ref. [48]. In the current theoretical investigation, the Heaviside Step Function and its antiderivative, i.e., the Ramp Function, are performed as bi-parametric single-valued functions with only one single restriction imposed on each parameter. The novelty of this work when compared with other theoretical investigations concerning analytical exact forms of these seminal functions is that the proposed explicit formulae are not exhibited in terms of miscellaneous special functions, e.g., Gamma Function, Biexponential Function, or any other special functions, such as Error Function, Hyperbolic Function, or Orthogonal Polynomials.

2. Towards Explicit Forms of Unit Step and Ramp Function

Let us introduce the following two bi-parametric single-valued functions:

$$f: R\to { R}^{+}\mathrm{such} \mathrm{that}$$

$$f(a,b,x)=\left[{ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}\right]$$

(11)

and

$$g:R\to { R}^{+}\mathrm{such} \mathrm{that}$$

$$g(a,b,x)=x\cdot \left[{ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}\right]$$

(12)

where $a$ and $b$ are two arbitrarily selected real numbers such that $a>1$ and $b<0$.

In addition, the term $\left[{ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}\right]$ denotes the integer part of the exponential quantity ${ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}$.

Here, it can be observed that only one constraint is imposed on each of the two parameters $a$ and $b$.

3. Claim

The functions $f$ and $g$ coincide with Heaviside Step Function and Ramp Function respectively over the set $(-\infty ,0)\cup [0,+\infty )$.

4. Proof

First, we shall prove that, for strictly positive arguments, every output of the bi-parametric function $f$ equals unity, whereas the outputs of the bi-parametric function $g$ coincide with those of the argument itself, i.e., the real variable $x$. Next, we shall prove that, for strictly negative arguments, the outputs of both bi-parametric functions $f$ and $g$ vanish. Finally, at $x=0$, we shall prove that the output of the function $f$ yields unity, while the output of the function $g$ yields zero.

To this end, let us initiate our mathematical analysis by distinguishing the following three cases concerning the domain that the independent variable $x$ belongs.

(i)

$x\in (0,+\infty )$

In this context, let us concentrate on the exponential term ${ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}$, which appears in both Equations (1) and (2), and then focus on its exponent, i.e., the quantity $b\left|x\right|+x\left|b\right|-bx-\left|bx\right|$. Since the parameter $b$ has been considered beforehand to be a strictly negative number, and also the variable $x$ has been currently assumed to lie over the set $(0,+\infty )$, one may infer that the term $\left|bx\right|$ is a strictly positive quantity. Hence, the following relationship holds:

$$\begin{array}{cc}\hfill b\left|x\right|+x\left|b\right|-bx-\left|bx\right|& =\left|bx\right|\cdot {\displaystyle \frac{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}{\left|bx\right|}}\iff \hfill \\ \hfill b\left|x\right|+x\left|b\right|-bx-\left|bx\right|& =\left|bx\right|\cdot \left({\displaystyle \frac{b\left|x\right|}{\left|b\right|\cdot \left|x\right|}}+{\displaystyle \frac{x\left|b\right|}{\left|b\right|\cdot \left|x\right|}}-{\displaystyle \frac{bx}{\left|bx\right|}}-{\displaystyle \frac{\left|bx\right|}{\left|bx\right|}}\right)\iff \hfill \\ \hfill b\left|x\right|+x\left|b\right|-bx-\left|bx\right|& =\left|bx\right|\cdot \left({\displaystyle \frac{x}{\left|x\right|}}+{\displaystyle \frac{b}{\left|b\right|}}-{\displaystyle \frac{bx}{\left|bx\right|}}-1\right)\iff \hfill \\ \hfill b\left|x\right|+x\left|b\right|-bx-\left|bx\right|& =\left|bx\right|\cdot \left(sgn\left(x\right)+sgn\left(b\right)-sgn\left(bx\right)-1\right)\hfill \end{array}$$

(13)

Evidently, the notation $sgn$, which appears in Equation (13), denotes the well-known Signum Function.

Since Signum Function yields the sign of any real number, it implies that

$$sgn\left(x\right)=1 ; sgn\left(b\right)=-1; sgn\left(bx\right)=-1$$

Thus, Equation (13) can be equivalently written out as

$$\begin{gathered} b\left|x\right|+x\left|b\right|-bx-\left|bx\right|=\left|bx\right|\cdot \left(1-1-\left(-1\right)-1\right)\iff \\ b\left|x\right|+x\left|b\right|-bx-\left|bx\right|=0 \end{gathered}$$

(14)

In the sequel, Equation (11) and Equation (12), respectively, can be combined with Equation (14) to yield

$$\begin{gathered} f\left(a,b,x\right)=\left[{ a}^0\right]\iff \\ f(a,b,x)=1 \end{gathered}$$

(15)

and

$$\begin{gathered} g\left(a,b,x\right)=x\cdot \left[{ a}^0\right]\iff \\ g(a,b,x)=x \end{gathered}$$

(16)

(ii)

$x\in (-\infty , 0)$

According to the same reasoning as before, let us center on the exponential term ${ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}$ and examine the sign of its exponent, i.e., the quantity $b\left|x\right|+x\left|b\right|-bx-\left|bx\right|$. Since the parameter $b$ has been primarily considered to be a strictly negative number and the independent variable $x$ is now supposed to lie over the set $(-\infty ,0)$, one may deduce that the term $\left|bx\right|$ is again a strictly positive number. Thus, in the same framework as in the previous case, one may conclude that Equation (13) still holds. Hence, one may deduce that $sgn \left(x\right)=-1; sgn\left(b\right)=-1; sgn\left(bx\right)=1$.

Thus, one obtains

$$\begin{gathered} b\left|x\right|+x\left|b\right|-bx-\left|bx\right|=\left|bx\right|\cdot \left(-1-1-1-1\right)\iff \\ b\left|x\right|+x\left|b\right|-bx-\left|bx\right|=\left(-4\right)\cdot \left|bx\right| \end{gathered}$$

(17)

In this context, the following inequality is evident:

$$b\left|x\right|+x\left|b\right|-bx-\left|bx\right|<0$$

(18)

Then, Equations (11) and (12), respectively, can be combined with Equation (17) to yield

$$f(a,b,x)=\left[{ a}^{\left(-4\right)\cdot \left|bx\right|}\right]\iff f(a,b,x)=\left[ {\displaystyle \frac{1}{{ a}^{4\left|bx\right|}}} \right]$$

(19)

and

$$g(a,b,x)=x\cdot \left[{ a}^{\left(-4\right)\cdot \left|bx\right|}\right]\iff g(a,b,x)=x\cdot \left[ {\displaystyle \frac{1}{{ a}^{4\left|bx\right|}}} \right]$$

(20)

Now, since the base of the exponential term is ${ a}^{4\left|bx\right|}$, which appears in Equation (20)—i.e., the parameter $a$ has been considered beforehand to be strictly greater than 1—one may easily conjecture that the following double inequality holds:

$$0<{\displaystyle \frac{1}{{ a}^{4\left|bx\right|}}}<1$$

(21)

According to inequality (21), it implies that the values of the strictly positive fraction ${\displaystyle \frac{1}{{ a}^{4\left|bx\right|}}}$ belong to the interval $\left(\mathrm{0,1}\right) \forall x\in (-\infty ,0)$. In this framework, one may infer that, for strictly negative arguments, the integer part of this aforementioned fraction vanishes, i.e.,

$$\left[ {\displaystyle \frac{1}{{ a}^{4\left|bx\right|}}} \right]=0$$

(22)

or equivalently

$$\left[{ a}^{\left(-4\right)\cdot \left|bx\right|}\right]=0$$

(23)

Equations (19) and (20), respectively, can be associated with Equation (22) to yield

$$f(a,b,x)=0$$

(24)

and

$$g(a,b,x)=0$$

(25)

(iii)

$x=0$

Then one obtains

$$f(a,b,0)=\left[{ a}^0\right]\iff f(a,b,0)=1$$

(26)

and

$$g(a,b,0)=0\cdot \left[{ a}^0\right]\iff g(a,b,0)=0$$

(27)

5. Discussion

In Section 2, two bi-parametric single-valued functions were proposed with the claim that they are synonymous with the Unit Step Function and the Ramp Function over the set of real numbers. Next, in Section 4, rigorous proofs were given to justify this claim. However, one may elucidate that a shortcoming of the bi-parametric single-valued function defined by Equation (11) is that it cannot be differentiated with respect to argument $x$ to yield Dirac delta distribution. Indeed, functions of this type (due to the existence of an integer part operator, also known as floor division) cannot be measured, neither in the Lebesgue sense nor in the Riemann sense.

In the sequel, according to the same reasoning, one may carry out an analogous procedure in order to derive a proportional bi- parametric representation of the Signum Function. Actually, it is well known that both the Signum Function and the Heaviside Step Function are linked with each other by means of the following linear relationship [49]:

$$sgn\left(x\right)=2H\left(x\right)-1$$

(28)

In this context, Equation (28) can be combined with Equation (11) to yield

$$sgn\left(x\right)=2\cdot \left[{ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}\right]-1$$

(29)

where $a>1$ and $b<0$.

In this framework, one may point out that the Signum Function was also performed as a bi-parametric single-valued function, i.e.,

$$sgn\left(x\right)\equiv h(a,b,x)$$

(30)

where $h:R\to \left\{-\mathrm{1,0},+1\right\}$ is a bi-parametric single-valued function.

Moreover, given an arbitrary real constant named $C$, one may remark from calculus [50] that, regarding its integer part, notated by $\left[C\right]$, the following relationships hold:

$$2\left[C\right]=\left[2C\right]$$

(31)

when $C-\left[C\right]<{\displaystyle \frac{1}{2}}$

And

$$2\left[C\right]=\left[2C\right]-1$$

(32)

when $C-\left[C\right]\ge {\displaystyle \frac{1}{2}}$

Hence, by focusing on the exponential term that appears in Equation (29), i.e., ${ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}$, one may deduce that, for non-negative values of the variable $x$, the following relationship holds:

$$\begin{gathered} { a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}-\left[{ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}\right]=0\Rightarrow \\ { a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}-\left[{ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}\right]<{\displaystyle \frac{1}{2}} \end{gathered}$$

(33)

Thus, Equation (29) can be combined with (31) to yield

$$sgn\left(x\right)=\left[{ 2a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}\right]-1$$

(34)

However, for strictly negative arguments, it was previously proved that the quantity $\left[{ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}\right]$ vanishes.

Thus it implies that

$${ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}-\left[{ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}\right]={ a}^{b\left|x\right|+x\left|b\right|-bx-\left|bx\right|}$$

(35)

In addition, according to inequality (21), one infers

$$-{\displaystyle \frac{1}{2}}<{ a}^{\left(-4\right)\cdot \left|bx\right|}-{\displaystyle \frac{1}{2}}<{\displaystyle \frac{1}{2}}$$

(36)

Here, one may observe that the quantity ${ a}^{\left(-4\right)\cdot \left|bx\right|}-{\displaystyle \frac{1}{2}}$ does not keep the same sign for strictly negative values of the variable $x$, and therefore, Equation (29) cannot be combined with Equation (32).

6. Conclusions

The objective of this analytical investigation was to introduce closed-form representations of the Heaviside Step Function and the Ramp Function. The novelty of this work, when compared with other analytical treatments to these important functions, is that the proposed mathematical formulae are not exhibited in terms of miscellaneous special functions or any other special functions such as Error Function, Supplementary Error Function, Hyperbolic Function, or Orthogonal Polynomials.

In closing, in future work, one may also propose, taking into consideration this theoretical approach, an analytical representation to the Ceiling Function (also known as the least integer function), which is defined as the smallest integer that is not smaller than the independent real variable $x$; i.e., it returns the smallest integer that is not smaller than the input decimal.