An Explicit Form of Signum Function

Abstract

In this paper, the author derives an analytical exact form of signum function, which evidently constitutes a fundamental concept of Communication Systems and Control Theory along with digital control systems and is also involved in many other fields of applied mathematics and engineering practices. In particular, this significant function is performed in a simple manner as a finite combination of purely algebraic representations. The novelty of this work when compared to other analytical expressions of this nonlinear function is that the proposed explicit representation is not performed in terms of miscellaneous special functions, such as Bessel functions, error function, and beta function, and also is neither the limit of a function nor the limit of a sequence of functions with a point-wise or uniform convergence.

1. Introduction

In mathematics, the signum function (also known as sign function) is a piecewise single-valued function that returns the sign of a real number [1]. The notation of the signum function is often represented as $sgn$.

Signum function constitutes a fundamental concept of Communication Systems and Control Theory. Indeed, the versatility of the signum function in the design and analysis of communication systems can be highlighted by elucidating the following key applications [2,3]:

(a)

Signal Detection: The signum function is used to determine the presence of a signal. By the application of the signum function to a received signal, one may identify the sign changes which are crucial for detecting binary signals in digital communication systems

(b)

Quantization: In digital communication, signals are often quantized to reduce the number of bits required for transmission. Signum function contributes to the quantization process by simplifying the representation of signals, especially in the case of binary quantization

(c)

Phase Modulation: The signum function is used in phase modulation techniques. It helps in determining the phase shifts of a carrier signal, which is essential for encoding information in phase-modulated signals.

(d)

Error Detection and Correction: In error detection and correction algorithms, the signum function may be used to identify errors in transmitted signals. By comparing the sign of the received signal with the expected sign, errors can be detected and corrected.

(e)

Adaptive Filtering: The signum function is used in adaptive filtering algorithms to update filter coefficients. This is particularly useful in echo cancelation and noise reduction in communication systems.

On the other hand, the ability of the signum function to switch control inputs based on the sign of the error renders it a powerful tool in control systems, specifically for applications requiring high robustness and fast response times [4]. Actually, in control systems, the signum function plays a crucial role, particularly in the design of robust controllers like sliding mode controllers (SMCs). Indeed, it contributes to the robustness of the control law to disturbances and uncertainties. In this context, by switching the control input based on the sign of the sliding surface, the system can quickly correct deviations and maintain desired performance. Moreover, in digital control systems, the signum function is frequently used in control algorithms in order to implement bang–bang control or relay control. This type of control switches the control input between two extreme values depending on the sign of the error signal. Actually, it is useful in systems where accurate control is needed without intermediate values [5].

Meanwhile, the signum function is able to be expressed in terms of the well-known Heaviside step function (or unit step function) (which is a step function named after Oliver Heaviside, whose value is zero for negative arguments and one for positive arguments and its notation is $H$) as indicated below [6]

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

(1)

or

$$sgn\left(x\right)=H\left(x\right)+H(-x)$$

(2)

where $x\in R$.

In addition, the signum function can be directly obtained from the following formula [7,8]

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

(3)

The general form of the above representation is the following

$$sgn\left(x\right)={\displaystyle \frac{2}{\pi }}\cdot \left(arctan\left(x^{2n-1}\right)+arctan\left({\displaystyle \frac{1}{x^{2n-1}}}\right)\right)$$

(4)

Nonetheless, a shortcoming of the above formulae,, i.e., Equations (3) and (4), is that they cannot be defined at zero, i.e., when $x$ = 0.

Moreover, another serious shortcoming of all mathematical formulae consisting of inverse trigonometric functions is that such functions do not have unique definitions [9,10]. Moreover, one may point out that the singularity structure is left ambiguous [10].

Further, another interesting topic that deserves to be mentioned is the relationship between the signum function and Bessel functions. In fact, although the signum function and Bessel functions are not linked with each other in a direct manner, they can both appear in the context of solving differential equations or in signal processing. For instance, the signum function may be used to define boundary conditions or initial conditions in a boundary value problem where Bessel functions are the solutions [11]. A remarkable case where both the signum function and Bessel functions occur is in the analysis of Fourier transforms of certain types of functions. Indeed, the Fourier transform of the signum function involves the signum function itself and leads to an expression involving Bessel functions [12].

Now, as it was stated beforehand, there are many applications of the signum function as can be seen in the literature. In [13], a linear model reduction along with a solution of the algebraic Riccati equation was carried out by the use of the signum function whereas for remarkable studies on an integral representation and direct integration of the signum function, one may refer to [14,15].

In [16], the signum function was taken into consideration during the structure of a flexible independent component analysis algorithm in the framework of the natural Riemannian gradient, where self-adaptive nonlinear functions were used, whilst in [17], a uniform approximation of the signum function by the use of polynomials and entire functions was carried out. In [18], a valuable investigation on the spectral properties of singular Sturm—Liouville operators with indefinite weight was performed. Here, the signum function played the role of the indefinite weight. In [19], a sinusoidally driven system with a simple signum nonlinearity term was investigated through an analytical analysis as well as dynamic simulation. To obtain the correct Lyapunov exponents, the signum function was replaced by a sharply varying continuous hyperbolic tangent function. In [20], a novel optimal sliding mode control strategy was proposed on the basis of an optimal Linear Quadratic Regulator (LQR) approach. In this investigation, the signum function was taken into account during the calculation process of the eigenvectors which are used to design the stable sliding surface. In [21], an analytical exact form of Heaviside step function was proposed as the summation of two inverse tangent functions. This representation in association with Equations (1) and/or (2) may result in a closed form expression for the signum function. Nonetheless, one may point out that the singularity structure is left ambiguous. In [22], a rigorous modified form of the signum function was exhibited. In this framework, nonlinear approximations of this form were proposed by means of trigonometric generating functions. Next, by the aid of Gaussian type quadratures, the convergence and error terms were obtained. In [23], a signum function rigorous approach to solve algebraically an interval system of linear equations for nonnegative solutions was presented, whereas in [24], the effect of severe clipping of leak noise signals on time delay estimation was simulated with the aid of the signum function. In [25], explicit forms of Heaviside step function, ramp function, and signum function were performed, whilst in [26], a novel four-dimensional chaotic system with only one equilibrium with non-hyperbolic feature was presented. In this context, some initial values in the system were obtained by means of sine and/or the signum function. In [27], another analytical representation of Heaviside step function was proposed. This expression in combination with Equations (1) and/or (2) may yield a closed form expression for the signum function. In [28], a simple non-equilibrium chaotic system with only one signum function for generating multidirectional variable hidden attractors and its hardware implementation was introduced, whilst in [29,30], analytical expressions of Heaviside step function, ramp function, and signum function were presented.

Next, in [31], a novel continuous-time approach to convex optimization with linear equality constraints regarding sign projected gradient flow was performed. In this framework, a sign projected gradient flow, which attempted to combine nonsmooth control and projected gradient methods. In order to ensure the finite-time convergence property, a nonsmooth signum function was used. In [32], a valuable novel investigation concerning a new multi-scroll megastable oscillator on the basis of the signum function was carried out. In [33], a considerable study on time-synchronized stability and control was performed. In particular, the authors revisited two distinct types of multi-variable signum functions—the classical sign function and the norm-normalized sign function, as well as their influence on the control effect. This latter function was also described and called the unit vector. Essentially, the classical signum function maps all vectors from the same orthant into one vector excluding vectors on the axes, while the norm-normalized signum function maps each vector into the direction of itself. In this context, new and deeper insights were provided into the established finite-time stability, by defining a new fixed-time synchronized control problem with fixed-time-synchronized stability, and formulating new Lyapunov stability results.

In [34], a remarkable mathematical analysis of hidden attractors of non-equilibrium fractal–fractional chaotic systems was carried out by the use of a unique signum function. In particular, the Caputo fractal–fractional operator was taken into consideration in order to explore a chaotic system which contains only one signum function. The existence theory was developed by using the fixed-point result of Leray–Schauder to prove that the considered chaotic system possesses at least one solution. The proposed chaotic system has a unique solution, according to Banach’s fixed-point theorem. The authors demonstrated that the suggested chaotic system is Ulam–Hyres (UH) stable under the novel operator of the power law kernel by employing nonlinear functional analysis. In [35], a static security assessment of an electric–gas coupling system was carried out by the use of an equivalent signum function, whereas in [36], an analogous law of the iterated logarithm (LIL) was derived for the sums of signum functions.

Next, in [37], a valuable approximate method for the computation of step functions in homomorphic encryption was proposed. In particular, a novel linear relationship between Heaviside step function and signum function was presented. On the basis of this connection, any kind of step function is able to be homomorphically evaluated by the use of the proposed approximations of the signum function. In this framework, two methods were developed. The first method leverages the fact that any step function can be expressed as a linear combination of shifted sign functions. This connection enables the homomorphic evaluation of any step function using known polynomial approximations of the sign function. The second method boosts computational efficiency by employing a composite polynomial approximation strategy.

Further, in [38,39,40], analytical representations of Heaviside step function and ramp function were performed. These formulae were derived in a rigorous manner, without the involvement of Integral Calculus quantities (gamma function, error function, complementary error function, etc.).

In [41], a novel Krylov–Bogolyubov averaging method-based analytical solution of an unforced nonlinear coulomb damped oscillator was performed. The signum function was approximated using harmonic balance and it was established by direct comparison with numerical results that only the first harmonic is sufficient to obtain an approximate solution. Indeed, the ‘signum’ function in spite of its discontinuous nature can provide a continuous analytical solution and the characteristic linear decay of amplitude. This statement can be established by a combination of the harmonic balance method and the solution to ordinary differential equations using the inverse differential operator method.

In [42], a duffing-like oscillator with signum nonlinearity was proposed, whilst in [43], according to the principle of PWL (piecewise linear function) interpolation, a PWL interpolation approximation was carried out for the third-degree terms in a JCS system (caching system written in Java). On this basis, the signum function and the absolute value function (ABS) were further used to approximate effectively the PWL interpolation function.

Finally, in [44], a rigorous exact form of signum function was performed as a summation of five inverse tangent functions. However, as it was clarified beforehand, a serious disadvantage of all mathematical formulae consisting of inverse trigonometric functions is that such functions do not have unique definitions. In addition, the singularity structure is left ambiguous.

The present paper aims at performing the signum function in a simple manner as a finite combination of purely algebraic representations. In particular, this nonlinear and discontinuous single-valued function is presented as the product of an exponential function the basis of which is equal to $-1$, whereas its exponent is the integer part of an irrational quantity with a fraction where both numerator and denominator consist of trigonometric functions. Concurrently, the integer part of an exponential term is subtracted from this product. The novelty of this formula when compared to other analytical formulae approaching the signum function is that the proposed explicit representation is not performed in terms of miscellaneous special functions, such as Bessel functions, error and/or complementary error function, and is also neither the limit of a function nor the limit of a sequence of functions with a point-wise or uniform convergence.

2. Towards an Analytic Form of the Signum Function

Let us introduce the following single-valued parametric function

$$f:R-\left\{\kappa \pi \pm {\displaystyle \frac{\pi }{2}}\right\}\to Z,\forall \kappa \in Z$$

such that

$$f(a,x)={(-1)}^{\left[{\left(2+\sqrt{3}\right)}^{\left[x\right]}\right]}\cdot {\displaystyle \frac{tan\left(2x-\frac{\kappa \pi }{5}\right)+tan\left(\frac{2\kappa \pi }{5}-x\right)+tan\left(\frac{4\kappa \pi }{5}-x\right)}{tan\left(\frac{\kappa \pi }{5}-2x\right)\cdot tan\left(\frac{2\kappa \pi }{5}-x\right)\cdot tan\left(\frac{4\kappa \pi }{5}-x\right)}}-\left[a^{-\left|x\right|}\right]$$

(5)

where $\left[x\right]$ denotes the integer part of the real variable $x$ whereas $\left[{\left(2+\sqrt{3}\right)}^{\left[x\right]}\right]$ and $\left[a^{-\left|x\right|}\right]$ are the integer parts of the irrational quantity ${\left(2+\sqrt{3}\right)}^{\left[x\right]}$ and the exponential quantity $a^{-\left|x\right|}$, respectively. Here, we elucidate that the parameter $a$ denotes an arbitrary real number such that it is strictly greater than 1.

3. Claim

Claim 1.

The function f coincides with the signum function over its domain of definition.

4. Proof

Proof. 

We shall prove that the values of the function $f$ are equal to $-$1 for strictly negative arguments whereas they are equal to +1 for strictly positive arguments, as well as at zero. To this end, let us initiate our analysis by focusing on the product $tan\left({\displaystyle \frac{\kappa \pi }{5}}-2x\right)\cdot tan\left({\displaystyle \frac{2\kappa \pi }{5}}-x\right)\cdot tan\left({\displaystyle \frac{4\kappa \pi }{5}}-x\right)$ which occurs on the denominator of the fraction appearing on the right hand side of Equation (5). Since tan is an odd function, this product is obviously equal to $-tan\left(2x-{\displaystyle \frac{\kappa \pi }{5}}\right)\cdot tan\left({\displaystyle \frac{2\kappa \pi }{5}}-x\right)\cdot tan\left({\displaystyle \frac{4\kappa \pi }{5}}-x\right)$.

Next, for facility reasons, let us introduce the following auxiliary real variables:

$$A=2x-{\displaystyle \frac{\kappa \pi }{5}}$$

(6)

$$B={\displaystyle \frac{2\kappa \pi }{5}}-x$$

(7)

$$C={\displaystyle \frac{4\kappa \pi }{5}}-x$$

(8)

Here, one may observe that

$$A+B+C=\kappa \pi$$

(9)

On the other hand, Euler’s Identity [45] asserts that

$$e^{i\pi }=-1$$

(10)

and therefore

$$e^{i\kappa \pi }={\left(-1\right)}^{\kappa }$$

(11)

Since $\kappa \in {\rm Z},$it implies that ${\left(-1\right)}^{\kappa }\in R$. Thus, the imaginary part of this quality vanishes.

Hence, one infers

$$Im\left(e^{i\kappa \pi }\right)=0$$

(12)

Equation (12) can be combined with Equation (9) to yield

$$\begin{gathered} Im{e}^{i(A+B+C)}=0\Rightarrow \\ Im\left(e^{iA}{e}^{iB}{e}^{iC}\right)=0\Rightarrow \\ Im\left((cosA+isinA)(cosB+isinB)(cosC+isinC)\right)=0 \end{gathered}$$

(13)

and therefore

$$\begin{gathered} sinAcosBcosC+cosAsinBcosC+cosAcosBsinC-sinAsinBsinC=0\Rightarrow \\ {\displaystyle \frac{1}{cosAcosBcosC}}\cdot \left(sinAcosBcosC+cosAsinBcosC+cosAcosBsinC\right)={\displaystyle \frac{sinAsinBsinC}{cosAcosBcosC}}\Rightarrow \\ {\displaystyle \frac{sinA}{cosA}}+{\displaystyle \frac{sinB}{cosB}}+{\displaystyle \frac{sinC}{cosC}}={\displaystyle \frac{sinAsinBsinC}{cosAcosBcosC}}\Rightarrow \\ tanA+tanB+tanC=tanA·tanB·tanC \end{gathered}$$

(14)

Equation (5) can be combined with Equation (14) to yield

$$\begin{gathered} f(a,x)={(-1)}^{\left[{\left(2+\sqrt{3}\right)}^{\left[x\right]}\right]}\cdot {\displaystyle \frac{tanA+tanB+tanC}{tan(-A)\cdot tanB\cdot tanC}}-\left[a^{-\left|x\right|}\right]\Rightarrow \\ f(a,x)={(-1)}^{\left[{\left(2+\sqrt{3}\right)}^{\left[x\right]}\right]}\cdot \left(-{\displaystyle \frac{tanA+tanB+tanC}{tanA\cdot tanB\cdot tanC}}\right)-\left[a^{-\left|x\right|}\right]\Rightarrow \\ f(a,x)={-(-1)}^{\left[{\left(2+\sqrt{3}\right)}^{\left[x\right]}\right]}-\left[a^{-\left|x\right|}\right] \end{gathered}$$

(15)

Next, let us focus on the irrational quantity ${\left(2+\sqrt{3}\right)}^{\left[x\right]}$ where $x$ lies over the set $(-\infty ,0)\cup [0,+\infty )-\left\{\kappa \pi \pm {\displaystyle \frac{\pi }{2}}\right\}$.

Here, one may distinguish three cases concerning the real variable $x:$

(a)

$x$ is strictly positive. Then, $\forall x\in (0,+\infty )-\left\{\kappa \pi \pm {\displaystyle \frac{\pi }{2}}\right\}\exists (m,q)\in N\times N$ such that

$${\left(2+\sqrt{3}\right)}^{\left[x\right]}=m+q\sqrt{3}$$

(16)

and

$${\left(2-\sqrt{3}\right)}^{\left[x\right]}=m-q\sqrt{3}$$

(17)

and therefore

$$\begin{gathered} {\left(2+\sqrt{3}\right)}^{\left[x\right]}+{\left(2-\sqrt{3}\right)}^{\left[x\right]}=2m\Rightarrow \\ {\left(2+\sqrt{3}\right)}^{\left[x\right]}=2m-{\left(2-\sqrt{3}\right)}^{\left[x\right]} \end{gathered}$$

(18)

Then, by applying the integer part operation in Equation (18), one finds

$$\left[{\left(2+\sqrt{3}\right)}^{\left[x\right]}\right]=\left[2m-{\left(2-\sqrt{3}\right)}^{\left[x\right]}\right]$$

(19)

Evidently, the quantity $2m$ constitutes a natural number. Hence, one infers

$$\left[{\left(2+\sqrt{3}\right)}^{\left[x\right]}\right]=2m+\left[-{\left(2-\sqrt{3}\right)}^{\left[x\right]}\right]$$

(20)

On the other hand, the following inequality holds

$$\begin{gathered} 0<2-\sqrt{3}<1\Rightarrow \\ 0<{\left(2-\sqrt{3}\right)}^{\left[x\right]}<1\Rightarrow \\ -1<-{\left(2-\sqrt{3}\right)}^{\left[x\right]}<0\Rightarrow \\ \left[-{\left(2-\sqrt{3}\right)}^{\left[x\right]}\right]=-1 \end{gathered}$$

(21)

Next, Equation (20) can be combined with Equation (21) to yield

$$\left[{\left(2+\sqrt{3}\right)}^{\left[x\right]}\right]=2m-1$$

(22)

Here, one may pinpoint that Equation (22) asserts that the term $\left[{\left(2+\sqrt{3}\right)}^{\left[x\right]}\right]$ constitutes an odd natural number when $x\in (0,+\infty )-\left\{\kappa \pi \pm {\displaystyle \frac{\pi }{2}}\right\}$.

In this context Equation (15), which is equivalent to Equation (5), can be combined with Equation (22) to yield

$$f(a,x)={-(-1)}^{2m-1}-\left[{\displaystyle \frac{1}{a^{\left|x\right|}}}\right]$$

(23)

Now, one may observe that $\alpha >1$ and, moreover, $-\left|x\right|<0$ since $x\in (0,+\infty )$.

Thus, one infers

$$\begin{gathered} {\log }_{\mathsf{\alpha }}\left(\mathsf{\alpha }\right)\cdot \left(-\left|x\right|\right)<0\Rightarrow \\ 0<a^{-\left|x\right|}<1 \end{gathered}$$

(24)

Hence, the fraction ${\displaystyle \frac{1}{a^{\left|x\right|}}}$ is strictly positive and simultaneously strictly less than unity, and therefore its ingerer part vanishes.

Finally, Equation (23) yields

$$f(a,x)=-(-1)=1$$

(25)

(b)

x is strictly negative.

Then, the following inequality is evident

$${0<\left(2+\sqrt{3}\right)}^{\left[x\right]}\le {\displaystyle \frac{1}{2+\sqrt{3}}}$$

(26)

Hence, one obtains

$$\left[{\left(2+\sqrt{3}\right)}^{\left[x\right]}\right]=0$$

(27)

Equation (15) can be combined with Equation (27) to yield

$$f(a,x)={-(-1)}^0-\left[{\displaystyle \frac{1}{a^{\left|x\right|}}}\right]$$

(28)

Again, one may observe that $\alpha >1$and, moreover, $\left(-\left|x\right|\right)<0$since $x\in (-\infty ,0)$. Thus, inequality (24) holds when $x$ lies over the set $(-\infty ,0)$ too. This implies that the term $\left[{\displaystyle \frac{1}{a^{\left|x\right|}}}\right]$ vanishes. In this framework, Equation (28) yields

$$f(a,x)=-1$$

(29)

(c)

x equals zero.

Then, one may deduce that

$$f(a,x)={-(-1)}^{\left[{\left(2+\sqrt{3}\right)}^0\right]}-\left[{\displaystyle \frac{1}{a^0}}\right]$$

(30)

and therefore

$$f(a,x)={-(-1)}^1-1=0$$

(31)

Summarizing, one may conclude that when the real variable $x$ lies over the set $(-\infty ,0)-\left\{\kappa \pi \pm {\displaystyle \frac{\pi }{2}}\right\}$, it implies that $f(a,x)=-1$. Moreover, when $x$ lies over the set $(0,+\infty )-\left\{\kappa \pi \pm {\displaystyle \frac{\pi }{2}}\right\}$, it follows that $f(a,x)=1$. Finally, when the variable $x$ vanishes, $f(a,x)$ equals zero.

After all, the single-valued parametric function $f$introduced by Equation (5) was definitely proved to be synonymous to the signum function, over the set $(-\infty ,0)\cup [0,+\infty )-\left\{\kappa \pi \pm {\displaystyle \frac{\pi }{2}}\right\}$. □

5. Discussion

In Section 3, we introduced a discontinuous single-valued parametric function. Then, we claimed that this function constitutes an explicit form of the signum function. In particular, this nonlinear and discontinuous single-valued function introduced by Equation (5) consisted of a finite combination of irrational, trigonometric, and exponential functions. To be more specific, it was formulated as the product of an exponential function the basis of which is equal to $-1$, whereas its exponent is the integer part of an irrational quantity and a fraction where both numerator and denominator consist of trigonometric functions. Concurrently, the integer part of an exponential quantity, whose basis is a parameter named α such that it is strictly greater than 1, is subtracted from this product. The domain of definition of this function is the set $(-\infty ,0)\cup [0,+\infty )-\left\{\kappa \pi \pm {\displaystyle \frac{\pi }{2}}\right\}$. Next, in Section 4, we gave a rigorous proof that the function introduced in Section 3 is indeed equivalent to the signum function over the set $(-\infty ,0)\cup [0,+\infty )-\left\{\kappa \pi \pm {\displaystyle \frac{\pi }{2}}\right\}$. Here, one may pinpoint that an advantage of the single-valued function $f$ introduced by Equation (5) is that it can indeed be defined at$x$ = 0. Nevertheless, a shortcoming of the aforementioned proposed formula is that it cannot be defined when $x=\kappa \pi \pm {\displaystyle \frac{\pi }{2}}$ with $\kappa \in Z$. Here, one may point out that the parametric single-valued function $f$ introduced in Equation (5) depends on the parameter α in a rather qualitative manner. Indeed, although the output of the function $f$ is not influenced by the values that the parameter $a$ may take over the set $(1,+\infty )$, this parameter is quite necessary in order for the exponential term $-\left[a^{-\left|x\right|}\right]$ to yield $-1$ at $x=0$ and also to vanish over the set $(-\infty ,0)\cup (0,+\infty )$. Therefore, it can be said that the existence of this parameter guarantees the fact that the introduced function is identical to the signum function.

In general, it is known that a disadvantage of all mathematical formulae consisting of inverse trigonometric functions is that these functions do not have unique definitions. Hence, we avoided these functions during the process of structuring the function proposed by Equation (5) which was proved to be synonymous to the signum function.

Moreover, one should elucidate that mathematical representations containing the integer parts of real quantities are not appropriate for differentiating them with respect to variable $x$ [46]. However, the formula introduced by Equation (5) is mathematically rigorous and seems to be flexible and practical. Thus, it may have good prospects towards the computational procedures that concern the applications of Heaviside step function in Operational Calculus, as well as in other engineering practices.

On the other hand, let us remark [7,8] that to obtain the decimal part of a real number, the fractional part function is used. This single-valued function is known with the notation $frac:R\to \left[\mathrm{0,1}\right)$. The fractional part function and the signum function are linked with each other by means of the following relationship:

$$frac\left(x\right)=x-\left[x\right]\cdot sgn\left(x\right)$$

(32)

In this framework, Equation (32) can be combined with Equation (5) to yield the decimal part of any real number.

6. Conclusions

The objective of this theoretical investigation was to introduce an analytical form of the signum function.

The proposed formula constitutes a product of an exponential function the basis of which is equal to $-1$, whereas its exponent is the integer part of an irrational quantity and a fraction where both numerator and denominator consist of trigonometric functions.

Coincidently, the integer part of an exponential quantity, whose basis is a parameter named α such that it is strictly greater than 1, is subtracted from this product. Hence, this analytical representation does not contain either generalized integrals or any other infinitesimal quantities. In addition, no other special functions are involved (e.g., gamma function, beta function, or error function). The novelty of this work when compared with other analytical representations is evident according to the author’s opinion. In fact, the proposed formula seems to be more flexible and practical and thus may have good prospects towards the computational procedures that concern the applications of the signum function in Communication Theory, Operational Calculus, and other engineering practices. Nevertheless, one should pinpoint that a shortcoming of the proposed closed-form expression is that it cannot be defined when $x=\kappa \pi \pm {\displaystyle \frac{\pi }{2}}$ with $\kappa \in Z$. In closing, as a future work, one may propose an analytical representation to fractional part function, which directly yields the decimal part of any real number, by taking into consideration the proposed exact analytical approach for the signum function in association with Equation (32) which was referred to in the last paragraph of the previous section.