Proposed Concept Of Signals For Unit Step Functions
Proceedings of the World Congress on Engineering and Computer Science 2010 Vol IWCECS 2010, October 20-22, 2010, San Francisco, USAProposed Concept of Signals forUnit Step FunctionsSatyapal Singh Abstract— There are several elementary signals which playvital role in the study of signals. These elementary signals serveas basic building blocks for the construction of more complexsignals. In fact, these elementary signals may be used to model alarge number of signals which occur in nature. One of theseelementary signals on which the article is based is Unit stepfunction. This paper explains a new approach to explain UNITSTEP FUNCTION hence it is named as PROPOSEDCONCEPT OF SIGNALS, which, if recognized may be knownas ‘SP’s ANGLES BASED UNIT STEP FUNCTION’.Index Terms — SP’s – Satyapal’s, Angle – The angles atwhich shape of the elementary signal changes, Clockwise – Thedirection of watch, Anticlockwise – the opposite direction ofwatch.I. INTRODUCTIONWhen we are asked to construct a shape from a givenequation, then normally we are provided with an equationthat usually contains basic or elementary signals. Most of thestudents and engineers may be unaware to what to do for agiven equation even after learning the existing theory. Forthis I have tried to develop “Concept of Angles” theory thatmay be helpful in constructing the shapes from the givenequation and in understanding the basic signals. Let us takethe unit step signal function to explore the concept of angles.II. SP’S CONCEPT OF ANGLESSignals can be represented by using angles also. Thisrepresentation gives more clarity to understand the signals.Generally, signals are represented in equation form [1], [2],[3]. For example –Satyapal Singh, Al-Falah School of Engineering and Technology, Dhauj,Faridabad, Haryana, India where the author is pursuing M.Tech. (Electronics& Communication) IVth semester (final) and Priyadarshini College ofComputer Sciences, Greater Noida, UP, India where the author has gainedteaching experience, e-mail: satyapalsingh67@yahoo.co.in; RegistrationNo.: 1278146363; Paper No.: ICCSA 43; Contact No.: 09968554717,09868347416; Qualification: M.Tech. (Computer Science & Engineering),Master of Computer Management, Master of Management Science(Marketing), B.Tech. (Electronics & Telecommunication), B.Sc. (PCM),Diploma in Electronics & Communication, Diploma in MaterialsManagement, Diploma in Business Management; Address: H.No. 460, NearDeepak Public School, Sec 9, Vijay Nagar, Shivpuri, District – Ghaziabad,State – UP, Country – India Pin(Zip) : 201009.ISBN: 978-988-17012-0-6ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)a. u(t) 1, when t 00, otherwise (that is for t 0)b. r(t) t, if t 00, otherwise (that is for t 0)When these elementary signals are put in equation form,then this form of equation representation may be difficult tounderstand by a student and it may become more tedious taskwhen the student is asked to draw the shape. For betterunderstanding, the concept of angles is tried to develop[4].The concept of angles says that these signals can berepresented by using angles too. In this method, the signal isbroken into different angles as per the given signal. Thisconcept does not change the original shape of the signal but itsimplifies the process. With the help of concept of angles,complex signal equations can be broken into simple steps andcan be plotted on the paper. This concept explores step bystep procedure to how to draw the elementary signals.III. UNIT STEP SIGNALUnit step signal u(t) states that the signal will start fromtime zero and instantly will take unit height (amplitude) anddepending upon given time characteristics (i.e. eitherpositive or negative, here positive) the signal will follow thestraight path either towards right or left, here towards right.Thus, the unit step function is a type of elementary functionu(t) which exists only for positive side and is zero fornegative. Also, the unit step function is discontinuous at t 0.The continuous time unit step function is denoted by u(t) andmay be represented in equation form as shown below. Thisequation is pictorially depicted as in figure 1[1], [2], [3]. Inother words, the unit step function is a type of elementaryfunction which exists only positive side and is zero fornegative side. Also the unit step function is discontinuous at t 03.The continuous-time unit-step function is denoted by u(t)and it is mathematically expressed as –u(t) 1, when t 00, otherwise (that is for t 0)To understand this, let us understand the example of u(t). Itcan be depicted as –WCECS 2010
Proceedings of the World Congress on Engineering and Computer Science 2010 Vol IWCECS 2010, October 20-22, 2010, San Francisco, USAlet x(t) u(-t)u(t)x(t) u(t)1- 00ttFigure 2. Sketching function u(t) using angle theory, step 1.Figure 1. Unit step function u(t) based on existing theory.2. As soon as the signal u(t) comes/appears then the signaltakes a 90 degree shift in anticlockwise direction and takesone straight line in u(t) direction i.e. in positive y-axisdirection line. This condition is shown in figure 3.u(t)Now if someone asks to depict the signal of u(t) – u(t-2) u(t-3) – 2u(t-5) then it creates ambiguities that is whencomplex equations are given to draw then it becomescomplex to draw.Now if someone asks to depict the signal u(t) – 2u(t-2) 4u(t-3) – 2u(t-5) then it creates ambiguities that is whencomplex equations are given to draw then it becomescomplex to draw.Here, I will try to present the logic regarding each andevery elementary signal. My theory[4] says if we areprovided with a set of unit step signals in the form ofequations and are asked to depict on paper then it will be veryeasy to depict the diagram if we use the concept of angles.Here, for unit step signals remember to have 90 degree angleshift concept. How? Solution – first we learn how to drawu(t) for which shape is given in figure 1 and then we willlearn how to draw u(-t), -u(t) and lastly –u(-t).IV. ANGLES IN UNIT STEP FUNCTION[4]Unit step signal uses 900 Concept. How is it used, we willsee in the coming paragraphs.Drawing of Different Unit Step Functions UsingAngles[4].19000tFigure 3. Sketching function u(t) using angle theory, step 2.3. After attaining a unit height for this case again the shapetakes a shift of 90 degree in clockwise direction and finallyextends to infinity towards right. This is shown in figure 4.u(t)90010tA. Drawing of u(t)Figure 4. Sketching function u(t) using angle theory, step 3.Figure 5 shows that there are actually two 90 degrees shiftsin unit step function, it is explained with the help of first bytaking a unit step function u(t) –For u(t) case, u(t) can be represented in terms ofangles as shown below –u(t) 4. Thus u(t) signal takes two 90 degrees shifts – first shiftin anticlockwise direction and second shift in clockwisedirection with a unit height (amplitude) and finally extendstowards right side (positive) infinity. Hence, plotting of u(t)by using angles can be shown as in figure 5.0, when t 01, 900 anticlockwise at t 01, 900 clockwise for t 0u(t)90011. First assume that, in idle case when no signal is there,then signal u(t) is assumed to come on x-axis from negativeinfinity to origin. This condition is shown in figure 2.9000tFigure 5. Sketching function u(t) using angle theory, step 4.ISBN: 978-988-17012-0-6ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCECS 2010
Proceedings of the World Congress on Engineering and Computer Science 2010 Vol IWCECS 2010, October 20-22, 2010, San Francisco, USAB. Drawing of u(-t)[4]u(-t)Mathematically u(-t) is represented[1], [2], [3] as –1u(-t) 1, when t 00, otherwise (that is for t 0)To understand this, let us understand the example of u(-t).It can be depicted as –let x(t) u(-t)x(t) u(-t)1-t900-tFigure 8. Sketching function u(-t) using angle theory, step 2.3. After attaining a unit height again the shape takes a shiftof 90 degree in anticlockwise direction and finally extends toinfinity towards left. This is shown in figure 9.u(-t)9000Figure 6. Function u(-t) based on existing theory.01-t0Figure 6 shows that there are actually two 90 degreesshifts, it is explained with the help of first by taking a unit Figure 9. Sketching function u(-t) using angle theory, step 3.step function in negative direction i.e. u(-t) – For u(-t) case,4. Thus u(-t) signal takes two 90 degrees shifts – first shiftu(-t) can be represented in terms of angles as shown below – in clockwise direction and second shift in anticlockwisedirection with a unit height (amplitude) and finally extends0, when t 0towards left side (negative) infinity. The final figure is shownu(-t) 1, 900 clockwise at t 0in figure 10.u(-t)1, 900 anticlockwise for t 0900For u(-t) case –17. First assume that, in idle case when no signal is therethen the depiction line comes from positive infinityto origin. This situation is shown in figure 7.u(-t)900-t01Figure 10. Sketching function u(-t) using angle theory, step 4.-t0Figure 7. Sketching function u(-t) using angle theory, step 1.C. Drawing of -u(t)[4]Mathematically -u(t) is re presented as –-u(t) -1, when t 00, otherwise (that is for t 0)2. As soon as the signal u(-t) comes/appears then the signaltakes a 90 degree shift in clockwise direction and takes onestraight line in signal u(-t) direction i.e. in positive y-axisdirection line. This condition is shown in figure 8.ISBN: 978-988-17012-0-6ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCECS 2010
Proceedings of the World Congress on Engineering and Computer Science 2010 Vol IWCECS 2010, October 20-22, 2010, San Francisco, USATo understand this, let us understand the example of -u(t).It can be depicted as –if x(t) -u(t)3. And after attaining a negative unit height again theshape takes a shift of 90 degree in anticlockwise directionand finally extends to infinity towards right. This is shown infigure 14.t00-1-1-u(t)Figure 11. Function -u(t) based on existing theory.Figure 11 shows that there are actually two 90 degreesshifts, it is explained with the help of first by taking aunitstep function in positive direction i.e. -u(t) –For -u(t) case, -u(t) can be represented in terms of anglesas shown below –-u(t) t0, when t 0-1, 900 clockwise at t 01, 900 anticlockwise for t 0900-u(t)Figure 14. Function -u(t) using angle theory, step 3.4. Thus -u(t) signal takes two 90 degrees shifts – first shiftin clockwise direction and second shift in anticlockwisedirection with a unit height (amplitude) in –y axis and finallyextends towards right side (positive) infinity. The final figurecan be shown as –-u(t)tFor -u(t) case 1. First assume that, in idle case when no signal is therethen the depiction line comes from negative infinity to origin.It is shown in figure 12.9000-1900Figure 15. Function -u(t) using angle theory, step 4.tLikewise, one can plot the signals –u(-t) unit step function.After learning this, it is the time to move to have somepractical examples.0-1-u(t)Figure 12. Function -u(t) using angle theory, step 1.2. As soon as the signal –u(t) comes/appears then thesignal takes a 90 degree shift in clockwise direction and takesone straight line in -u(t) direction i.e. in negative y-axisdirection line. This situation is shown in figure 13.t0V. EXAMPLES BASED ON THEORY DEVELOPEDExample 1. – Drawing of u(t-2)Solution : u(t-2) function can be shown as –u(t-2) 1, when t 20, otherwiseIt can be depicted as –u(t-2)9001-1-u(t)Figure 13. Function -u(t) using angle theory, step 2.012tFigure 16. Function u(t-2) based on existing theory.ISBN: 978-988-17012-0-6ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCECS 2010
Proceedings of the World Congress on Engineering and Computer Science 2010 Vol IWCECS 2010, October 20-22, 2010, San Francisco, USALet us see the case step by step, which is more convincingExample 2. – Drawing of -u(t-2)Solution : -u(t-2) function can be shown as ––1. First assume that, in idle case when no signal is therethen the depiction line comes from negative infinity to t 2as shown in figure 17.u(t-2)1012-u(t-2) -1, when t 20, otherwiseIt can be depicted as –- u(t-2)t0Figure 17. Function u(t-2) using angle theory, step 1.2. As soon as the signal u(t-2) comes/appears then thesignal takes a 90 degree shift at t 2 in anticlockwisedirection and takes one straight line parallel to x(t) i.e.parallel to y-axis line. This situation is shown in figure 18.12t-1Figure 21. Function –u(t-2) based on existing theory.Let us see the case step by step, which is more convincingx(t) u(t-2)–1. First assume that, in idle case when no signal is therethen the depiction line comes from negative infinity to t 2as shown in figure 22.- u(t-2)1900012012ttFigure 18. Function u(t-2) using angle theory, step 2.3. After attaining a unit height again the shape takes a shiftof 90 degree in clockwise direction and finally extends toinfinity towards right. This is shown in figure 19.u(t-2)900-1Figure 22. Function -u(t-2) using angle theory, step 1.2. As soon as the signal -u(t-2) comes/appears then thesignal takes a 90 degree shift at t 2 in clockwise directionand takes one straight line parallel to x(t) i.e. parallel toy-axis line. This situation is shown in figure 23.x(t) -u(t-2)1001212 t900tFigure 19. Function u(t-2) using angle theory, step 3.4. The whole diagram can be shown as below for two 900shifts.u(t-2)9001-1Figure 23. Function -u(t-2) using angle theory, step 2.3. After attaining a unit height in negative direction, againthe shape takes a shift of 90 degree in anticlockwise directionand finally extends to infinity towards left. This is shown infigure 24.900012tFigure 20. Function u(t-2) using angle theory, step 4.ISBN: 978-988-17012-0-6ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCECS 2010
Proceedings of the World Congress on Engineering and Computer Science 2010 Vol IWCECS 2010, October 20-22, 2010, San Francisco, USALikewise some other complex examples can be consideredfor better understanding of the theory developed.This is what I call concept of 90 degree related to unit stepfunction u(t). This concept plays a vital role while solving therelated numerical.x(t) -u(t-2)012tVI. CONCLUSION-1900Figure 24. Function -u(t-2) using angle theory, step 3.4. The whole diagram can be shown as below for two 900shifts.x(t) -u(t-2)012t900-1900When I applied this theory to the B.Tech. (Subject :Signals and Systems) students, then I found that students notonly grasped this theory but also solved a number ofproblems based on this.This paper is an outcome from the teaching experiencewhere the students faced a lot of problems to understand theramp function numerical problems. This work is an attemptto teach the students step by step construction procedure oframp signal functions and this work has an attempt to explorethe new and easy theory specially written for ramp related tothe basic signal functions. No doubt the future studies willfurther explore my work in deep.On the basis of this theory, some other signals could bedeveloped that will go long to the scientists and students. Asno matter is available on the internet, hence I claim that thistheory is purely based on my research work/affords and has abright chance to explore new theory and ideas on this. .ACKNOWLEDGMENTFigure 25. Function -u(t-2) using angle theory, step 4.Let us take one complex example without explaining steps.Example 3. Sketch the shape of the given equation –x(t) u(t-2)-2u(t-5) 3u(t-7)-2u(t-9)Solution : The solution is given below by keeping thegiven theory in the previous section.I would like to thank to my B.Tech. pursuing students whoposed a lot of questions in the form of doubts and inclined meto think more and more to clarify their complex doubts. WhatI feel in this context that it is only the students for a teacherwho can make a teacher gold from silver. Hence, Iacknowledge my students and again I pay special thanks tomy students who made me to reach at this stage where I couldproduce this paper.REFERENCES[1] A.V. Oppenheim, A.S. Willsky with S. Hamid Nawab,Signals and Systems – by – Pearson Education, Secondedition, 2002.3900 3u(t-7)2900 u(t-2)9001900900-2u(t-9)012345-16 78 9900[2] Simon Haykin, Barry Van Veen, Signals and systems –by – John Wiley & Sons (Asia) Pte. Ltd., Second Edition,2004 .[3] P. Ramakrishna Rao, Signals and Systems – by – TataMcGraw Hill, First edition, 2008.[4] Satyapal Singh own concept, based on teachingexperience.900900 -2u(t-5)-2tFigure 26 Function -u(t-2) using angle theory.Now it might be cleared that each and every unit functionis having two 900 angles hence the theory developed fits bestin the numerical examples.ISBN: 978-988-17012-0-6ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCECS 2010
Abstract— There are several elementary signals which play vital role in the study of signals. These elementary signals serve as basic building blocks for the construction of more complex signals.
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