Ratio Mathematica Volume 46, 2023 BIBO stability and decomposition analysis of signals and system with convolution techniques C. B. Sumathi * R. Jothilakshmi † Abstract In this paper control system’s stability is arrived based on Bounded Input Bounded Output (BIBO) when bounded input is given in the form of discrete values. The control system allows the state estima- tion constraints to reach the convergence even when fluctuations in the parameters of the input system occur. To overcome this DTFT (Discrete Time Fourier Transform) is used when the signal is com- pletely absolutely summable. Stability of the LTI (Linear time invari- ant) system is showed and is depending on the absolute summable of their impulse response. Simultaneously for continuous signal the sta- bility occurs if it is absolutely integrable. In addition to that the linear- ity and time-invariance properties are discussed. This provide a new way to decompose the periodic signals into Fourier series by convolv- ing the fundamental signals. Continuous and discrete time signals are focused in this paper to get linear time invariant system (LTI) through complex exponentials. Finally filtering techniques were used to elim- inate the noisy frequency component in a signal. Keywords: Stability, DTFT, CTFT, Dirichlet conditions. 2020 AMS subject classifications: 39A12, 39A30, 39A60. 1 *1PG and Research Department of Mathematics, Marudhar Kesari Jain College for Women,Tamil Nadu, India. c.bsumathi@yahoo.in. †P G and Research Department of Mathematics, Mazharul Uloom College, Tamil Nadu, India. jothilakshmiphd@gmail.com. 1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20, 2023. DOI: 10.23755/rm.v46i0.1086. ISSN: 1592-7415. eISSN: 2282-8214. ©C. B. Sumathi et al. This paper is published under the CC-BY licence agreement. 291 C. B. Sumathi and R. Jothilakshmi 1 Introduction Difference equations are described the evolution of applied mathematics phe- nomenon of latest technology from artificial intelligence to IoT. Higher order difference equations and time invariant systems are focusing more real world problems in the diverse fields like communication technology Srinivasan et al. [2022]. When we classify these difference equations, each category of classifi- cation are solving specific engineering and technological problems R.P.Agarwal [2000], Thamvichai and T.Bose [2002]. Various methodologies are used to re- solve error reduction and quality enhancement techniques. In such cases ana- lytical solutions are possible to address particular specified problems Kayar and KaymakçalanBistritz [2022]. To analyze these difference equations finite difference methods can be uti- lized. In particular when the errors are magnified, then the difference equations will be unstable and for smaller components these equation guarantees the stabil- ity Camouzis and Ladas [2008], Liu1 et al. [2022]. When the larger components are involved, tri-diagonal system (based on Crank-Nicolson Method) will be com- pared with other explicit and implicit methods with higher derivatives Camouziss and G.Ladas [2010], Liu [2008]. Time Delay is one of the reasons for instability which appears in dynamical systems such as biological systems, chemical systems, communication systems, nuclear systems, electrical systems, etc., and it is one of the key performances in these systems Bose [1995], Kaczorek [2011], Oppenheim et al. [2009]. As the signal is impulsive, it goes to infinity at any time and hence, the system is unstable even when an input is bounded but an output is infinite. Bounded Signal is a signal which is having a finite value at all instants of time. In general a signal is bounded if it has finite value M > 0, and the signal does not exceed M, i.e. |y(n)| ≤ M,∀n ∈ Z for discrete-time signals. This paper is structured as follows: Section II provides the fundamental con- cepts of LTI systems in time as well as frequency domain. The systems represents through linear time invariant difference equations are discussed in section III. Sec- tion IV dealt with the development of Fourier series in difference equations with time dependent variable. Section V depicts the Filtering Techniques Related to LTI systems that change the shape of the input signal. Finally section VI con- cludes the paper. 292 BIBO stability analysis of signals and system 2 LTI systems 2.1 Time domain conditions Theorem 2.1.1 (Sufficient condition) : The BIBO stability of discrete time LTI system and its impulse response is absolutely summable i.e. ∑∞ n=−∞ |h[n]| < ∞. Proof :Consider the following By convolution, y[n] = ∞∑ n=−∞ h[k]x[n−k] (1) Then |y[n]| = | ∞∑ n=−∞ h[n−k]x[k]| (2) Applying triangular inequality |y[n]| = ||x||∞ ∞∑ n=−∞ |h[k]| (3) Therefore h[n] is absolutely summable. Theorem 2.1.2 (sufficient condition of continuous time) : The BIBO stability of continuous time LTI is ∫ ∞ −∞ |h(t)|dt < ∞. Proof : Consider the output function |y(t)| = | ∫ ∞ −∞ x(t−T)h(T)dT|, (4) ≤ ∫ ∞ −∞ M|h(T)|dT = M ∫ ∞ −∞ |h(T)|dT (5) Hence the proof. 2.2 LTI systems - frequency domain conditions Discrete Time signal In general, for BIBO stability a unit circle in the Z-plane must contain all the 293 C. B. Sumathi and R. Jothilakshmi poles of a system. The condition for stability can be obtained by the above time domain condition ∞∑ n=−∞ |h[n]| = ∞∑ n=−∞ |h[n]||e−jwn| Continuous Time signal In the continuous case, Laplace transform must include the imaginary axis. For BIBO stability s-plane must contain all the poles of a system. The condition for stability can be obtained by the above condition∫ ∞ −∞ |h(t)|dt = ∫ ∞ −∞ |h(t)||e−jwt|dt (6) = ∫ ∞ −∞ |h(t)|(e−st|dt (7) where s = σ + jw and Re(s) = σ = 0 In addition to above condition, the continuous time signals is convergence if it encounter the following Dirichlet conditions • if the interval is finite then x is of bounded variation , • for all finite number of points, x is continuous, if the interval is finite. These conditions are satisfied by the periodic interval. Assume that ∞∑ m=0 am = 1 (1−a) (8) where |a| < 1. On multiplying 1−a both side we get ∞∑ m=0 am −a ∑ m = 0∞am = 1 a0 = 1 , since |a| < 1, the sums converge. As an example, if h(t) = atu(t) for every t ∈ R, where a > 0. Since the integral is infinite if a ≥ 1, it is unstable and it is finite if 0 < a < 1, and thus∫ ∞ 0 atdt = −1 ln a Therefore, if 0 < a < 1, the system becomes stable. 294 BIBO stability analysis of signals and system 3 Representation of systems via LTI difference equa- tions The output form of difference equation of discrete time system is defined like Haung and P. M.Knopf [2012] y[n] = y[n−1] + x[n] If the system h[n] is LTI system then y[n] = x[n]∗h[n] (9) In causal system, the impulse response is zero for all t < 0,n < 0 in both discrete and continuous system respectively Alzabut et al. [2021], Oppenheim et al. [2009], Thamvichai and T.Bose [2002]. For example consider the following system, y[n]− 1 2 y[n−1] = x[n] If x[n] = δ [n], then we have y[0] = 1 y[1] = 1 2 y[2] = 1 4 . . . y[n] = ( 1 2 )n Then h[n] = ( 1 2 )n u[n] 4 Development of Fourier series in different equa- tions In this section we calculate the transform signals through Fourier series Hu [2011], Bistritz [2004]. We consider some simple transformations with time vari- able. The ouput function y(t) is y(t) = est ∫ ∞ −∞ h(τ)e−sτdτ 295 C. B. Sumathi and R. Jothilakshmi The input signal x(t) = cosw0t, where w0 > 0, and then x(t) = 1 2 ejw0t − 1 2 e−jw0t Filtering is used to modify or eliminate some frequency components in the discrete signals. Consider H(ejw) = 1 (1−ae(−jw)) Now define y(t) as y(t) = e(−at) ∫ 1 0 ea τdτ y(t) = 1 a [1−e(−at)] Here x(τ) and h(t− τ) does not overlap, and hence y(t) = 0. 5 Conclusions This paper analyzed the stability of the BIBO system in time as well as fre- quency domain. Based on the state of time domain, a linear time invariant system is stable only if its impulse response is absolutely stable. In addition, it is focused the decomposition of signals into LTI system through suitable examples. The de- velopment of these techniques has been used in implementation of time-varying convolution filters. The signals are convolved to produce the linear time invariant system and non overlapping systems. References J. Alzabut, M.Bohner, and S. Grace. Oscillation of nonlinear third-order differ- ence equations with mixed neutral terms. Adv. Difference Equ, 2021:1–18, 2021. Y. Bistritz. Testing stability of 2-d discrete systems by a set of real 1-d stability tests,. IEEE Transactions on Circuits and Systems I,, 51:1312 – 1320, 2004. T. Bose. Stability of 2-d state-space system with overflow and quantization. IEEE Transactions on Circuits and Systems II,, 42:432–434, 1995. E. Camouzis and G. Ladas. Dynamics of third order rational difference equations with open problems and conjectures,. Chapman and Hall CRC, Boca Raton, FL,, 2008. 296 BIBO stability analysis of signals and system E. Camouziss and G.Ladas. Global results on rational systems in the plane. Part I, Journal of Difference and Applications,, 16(8):975–1013, 2010. Y. Haung and P. M.Knopf. Global convergence properties of first order homoge- neous system of rational difference equations,. Journal of Difference Equations and Applications,, 18:1683 – 1707, 2012. D. Hu. New stability tests of positive standard and fractional linear systems. Cir- cuits and Systems, 2(4):261–268, 2011. T. Kaczorek. New stability tests of positive standard and fractional linear systems. Circuits and Systems, 2(4):261–268, 2011. Z. Kayar and B. KaymakçalanBistritz. Applications of the novel diamond alpha hardy–copson type dynamic inequalities to half linear difference equations,. Journal of Difference Equations and Applications,, 28:457 – 484, 2022. T. Liu. Stability analysis of linear 2-d systems. Signal Processing,, 3(4):2078 – 2084, 2008. Z. Liu1, L.Jiang, and R.Qu. A machine-learning based fault diagnosis method with adaptive secondary sampling for multiphase drive systems. IEEE transac- tions on power electronics,, 2022. A. Oppenheim, R. W.Schaffer, and PHI. Oscillation of nonlinear third-order dif- ference equations with mixed neutral terms. Discrete Time Signal Processing, 2009. R.P.Agarwal. Difference Equations and Inequalities. Marcel Dekker,, New York,NY, USA, 2nd Edition,, 2000. R. Srinivasan, R.Graef, and E. Thandapani. Asymptotic behaviour of semi- canonical third-order functional difference equations. Journal of Difference Equations and Applications,, 28(4):547 –560, 2022. R. Thamvichai and T.Bose. Stability of 2-d periodically shift variant filters,. IEEE Transactions on Circuits and Systems II,, 49:61 – 64, 2002. 297