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Mathematics for business, science, and technology [[electronic resource] ] : with MATLAB and spreadsheet applications / / Steven T. Karris
Mathematics for business, science, and technology [[electronic resource] ] : with MATLAB and spreadsheet applications / / Steven T. Karris
Autore Karris Steven T
Edizione [3rd ed.]
Pubbl/distr/stampa Fremont, CA, : Orchard Publications, c2007
Descrizione fisica 1 online resource (629 p.)
Disciplina 510
Soggetto topico Mathematics
Mathematics - Study and teaching (Secondary)
Mathematics - Study and teaching (Higher)
Business mathematics
ISBN 1-280-75114-2
9786610751143
1-934404-02-0
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Math book 3rd Edition Front cover.pdf; Mathematics THIRD Edition Front matter.pdf; Math Book Preface.pdf; Math book TOC All Chapters.pdf; Mathematics THIRD Edition Chapter 01.pdf; Mathematics THIRD Edition Chapter 02.pdf; Mathematics THIRD Edition Chapter 03.pdf; Mathematics THIRD Edition Chapter 04.pdf; Mathematics THIRD Edition Chapter 05.pdf; Mathematics THIRD Edition Chapter 06.pdf; Mathematics THIRD Edition Chapter 07.pdf; Mathematics THIRD Edition Chapter 08.pdf; Mathematics THIRD Edition Chapter 09.pdf; Mathematics THIRD Edition Chapter 10.pdf; Mathematics THIRD Edition Chapter 11.pdf
Mathematics THIRD Edition Chapter 12.pdfMathematics THIRD Edition Appendix A.pdf; Mathematics THIRD Edition Appendix B.pdf; Mathematics THIRD Edition Appendix C.pdf; Mathematics THIRD Edition Appendix D.pdf; Mathematics THIRD Edition Appendix E.pdf; Mathematics THIRD Edition Appendix F.pdf; Mathematics THIRD Edition Appendix G.pdf; Math book Bibliography.pdf; Math Book, 3rd Edition Index All Chapters.pdf;
Record Nr. UNINA-9910816272603321
Karris Steven T  
Fremont, CA, : Orchard Publications, c2007
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Numerical analysis using MATLAB and excel [[electronic resource] /] / Steven T. Karris
Numerical analysis using MATLAB and excel [[electronic resource] /] / Steven T. Karris
Autore Karris Steven T
Edizione [3rd ed.]
Pubbl/distr/stampa Fremont, CA, : Orchard Publications, c2007
Descrizione fisica 1 online resource (627 p.)
Disciplina 519.4028551
Soggetto topico Mathematical analysis
Mathematics
Soggetto genere / forma Electronic books.
ISBN 1-280-75115-0
9786610751150
1-934404-04-7
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Numerical Analysis Front Cover THIRD Edition.pdf; Numerical Analysis THIRD Edition Front Matter.pdf; Numerical Analysis THIRD Edition Preface.pdf; Numerical Analysis THIRD Edition TOC All Chapters.pdf; Numerical Analysis THIRD Edition Chapter 01.pdf; Numerical Analysis THIRD Edition Chapter 02.pdf; Numerical Analysis THIRD Edition Chapter 03.pdf; Numerical Analysis THIRD Edition Chapter 04.pdf; Numerical Analysis THIRD Edition Chapter 05.pdf; Numerical Analysis THIRD Edition Chapter 06.pdf; Numerical Analysis THIRD Edition Chapter 07.pdf; Numerical Analysis THIRD Edition Chapter 08.pdf
Numerical Analysis THIRD Edition Chapter 09.pdfNumerical Analysis THIRD Edition Chapter 10.pdf; Numerical Analysis THIRD Edition Chapter 11.pdf; Numerical Analysis THIRD Edition Chapter 12.pdf; Numerical Analysis THIRD Edition Chapter 13.pdf; Numerical Analysis THIRD Edition Chapter 14.pdf; Numerical Analysis THIRD Edition Chapter 15.pdf; Numerical Analysis THIRD Edition Chapter 16.pdf; Numerical Analysis THIRD Edition
Record Nr. UNINA-9910451446403321
Karris Steven T  
Fremont, CA, : Orchard Publications, c2007
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Numerical analysis using MATLAB and excel [[electronic resource] /] / Steven T. Karris
Numerical analysis using MATLAB and excel [[electronic resource] /] / Steven T. Karris
Autore Karris Steven T
Edizione [3rd ed.]
Pubbl/distr/stampa Fremont, CA, : Orchard Publications, c2007
Descrizione fisica 1 online resource (627 p.)
Disciplina 519.4028551
Soggetto topico Mathematical analysis
Mathematics
ISBN 1-280-75115-0
9786610751150
1-934404-04-7
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Numerical Analysis Front Cover THIRD Edition.pdf; Numerical Analysis THIRD Edition Front Matter.pdf; Numerical Analysis THIRD Edition Preface.pdf; Numerical Analysis THIRD Edition TOC All Chapters.pdf; Numerical Analysis THIRD Edition Chapter 01.pdf; Numerical Analysis THIRD Edition Chapter 02.pdf; Numerical Analysis THIRD Edition Chapter 03.pdf; Numerical Analysis THIRD Edition Chapter 04.pdf; Numerical Analysis THIRD Edition Chapter 05.pdf; Numerical Analysis THIRD Edition Chapter 06.pdf; Numerical Analysis THIRD Edition Chapter 07.pdf; Numerical Analysis THIRD Edition Chapter 08.pdf
Numerical Analysis THIRD Edition Chapter 09.pdfNumerical Analysis THIRD Edition Chapter 10.pdf; Numerical Analysis THIRD Edition Chapter 11.pdf; Numerical Analysis THIRD Edition Chapter 12.pdf; Numerical Analysis THIRD Edition Chapter 13.pdf; Numerical Analysis THIRD Edition Chapter 14.pdf; Numerical Analysis THIRD Edition Chapter 15.pdf; Numerical Analysis THIRD Edition Chapter 16.pdf; Numerical Analysis THIRD Edition
Record Nr. UNINA-9910784294403321
Karris Steven T  
Fremont, CA, : Orchard Publications, c2007
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Numerical analysis using MATLAB and excel [[electronic resource] /] / Steven T. Karris
Numerical analysis using MATLAB and excel [[electronic resource] /] / Steven T. Karris
Autore Karris Steven T
Edizione [3rd ed.]
Pubbl/distr/stampa Fremont, CA, : Orchard Publications, c2007
Descrizione fisica 1 online resource (627 p.)
Disciplina 519.4028551
Soggetto topico Mathematical analysis
Mathematics
ISBN 1-280-75115-0
9786610751150
1-934404-04-7
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Numerical Analysis Front Cover THIRD Edition.pdf; Numerical Analysis THIRD Edition Front Matter.pdf; Numerical Analysis THIRD Edition Preface.pdf; Numerical Analysis THIRD Edition TOC All Chapters.pdf; Numerical Analysis THIRD Edition Chapter 01.pdf; Numerical Analysis THIRD Edition Chapter 02.pdf; Numerical Analysis THIRD Edition Chapter 03.pdf; Numerical Analysis THIRD Edition Chapter 04.pdf; Numerical Analysis THIRD Edition Chapter 05.pdf; Numerical Analysis THIRD Edition Chapter 06.pdf; Numerical Analysis THIRD Edition Chapter 07.pdf; Numerical Analysis THIRD Edition Chapter 08.pdf
Numerical Analysis THIRD Edition Chapter 09.pdfNumerical Analysis THIRD Edition Chapter 10.pdf; Numerical Analysis THIRD Edition Chapter 11.pdf; Numerical Analysis THIRD Edition Chapter 12.pdf; Numerical Analysis THIRD Edition Chapter 13.pdf; Numerical Analysis THIRD Edition Chapter 14.pdf; Numerical Analysis THIRD Edition Chapter 15.pdf; Numerical Analysis THIRD Edition Chapter 16.pdf; Numerical Analysis THIRD Edition
Record Nr. UNINA-9910825802103321
Karris Steven T  
Fremont, CA, : Orchard Publications, c2007
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Signals and systems [[electronic resource] ] : with MATLAB computing and Simulink modeling / / Steven T Karris
Signals and systems [[electronic resource] ] : with MATLAB computing and Simulink modeling / / Steven T Karris
Autore Karris Steven T
Edizione [3rd ed.]
Pubbl/distr/stampa Fremont, CA, : Orchard Publications, 2012
Descrizione fisica 1 v. (various pagings) : ill
Disciplina 621.382/2
Soggetto topico Signal processing - Mathematics
System analysis
Soggetto genere / forma Electronic books.
ISBN 1-280-12982-4
9786613533647
1-934404-24-1
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Signals and Systems -- with MATLABÒ Computing -- and SimulinkÒ Modeling -- Fifth Edition -- Steven T. Karris -- Preface Signals and Systems Fifth -- Preface -- TOC Signals and Systems Fifth -- Chapter 01 Signals and Systems Fifth -- Chapter 02 Signals and Systems Fifth -- Chapter 2 -- The Laplace Transformation -- his chapter begins with an introduction to the Laplace transformation, definitions, and properties of the Laplace transformation. The initial value and final value theorems are also discussed and proved. It continues with the derivation of the Laplac... -- 2.1 Definition of the Laplace Transformation -- The two-sided or bilateral Laplace Transform pair is defined as -- (2.1) -- (2.2) -- where denotes the Laplace transform of the time function , denotes the Inverse Laplace transform, and is a complex variable whose real part is , and imaginary part , that is, . -- In most problems, we are concerned with values of time greater than some reference time, say , and since the initial conditions are generally known, the two-sided Laplace transform pair of (2.1) and (2.2) simplifies to the unilateral or one-sided Lap... -- (2.3) -- (2.4) -- The Laplace Transform of (2.3) has meaning only if the integral converges (reaches a limit), that is, if -- (2.5) -- To determine the conditions that will ensure us that the integral of (2.3) converges, we rewrite (2.5) as -- (2.6) -- The term in the integral of (2.6) has magnitude of unity, i.e., , and thus the condition for convergence becomes -- (2.7) -- Fortunately, in most engineering applications the functions are of exponential order. Then, we can express (2.7) as, -- (2.8) -- and we see that the integral on the right side of the inequality sign in (2.8), converges if . Therefore, we conclude that if is of exponential order, exists if -- (2.9) -- where denotes the real part of the complex variable .
Evaluation of the integral of (2.4) involves contour integration in the complex plane, and thus, it will not be attempted in this chapter. We will see in the next chapter that many Laplace transforms can be inverted with the use of a few standard pai... -- In our subsequent discussion, we will denote transformation from the time domain to the complex frequency domain, and vice versa, as -- (2.10) -- 2.2 Properties and Theorems of the Laplace Transform -- The most common properties and theorems of the Laplace transform are presented in Subsections 2.2.1 through 2.2.13 below. -- 2.2.1 Linearity Property -- The linearity property states that if -- have Laplace transforms -- respectively, and -- are arbitrary constants, then, -- (2.11) -- Proof: -- Note 1: -- It is desirable to multiply by the unit step function to eliminate any unwanted non-zero values of for . -- 2.2.2 Time Shifting Property -- The time shifting property states that a right shift in the time domain by units, corresponds to multiplication by in the complex frequency domain. Thus, -- (2.12) -- Proof: -- (2.13) -- Now, we let -- then, and . With these substitutions and with , the second integral on the right side of (2.13) is expressed as -- 2.2.3 Frequency Shifting Property -- The frequency shifting property states that if we multiply a time domain function by an exponential function where is an arbitrary positive constant, this multiplication will produce a shift of the s variable in the complex frequency domain by units... -- (2.14) -- Proof: -- Note 2: -- A change of scale is represented by multiplication of the time variable by a positive scaling factor . Thus, the function after scaling the time axis, becomes . -- 2.2.4 Scaling Property -- Let be an arbitrary positive constant -- then, the scaling property states that -- (2.15) -- Proof: -- and letting , we obtain -- Note 3:.
Generally, the initial value of is taken at to include any discontinuity that may be present at . If it is known that no such discontinuity exists at , we simply interpret as . -- 2.2.5 Differentiation in Time Domain Property -- The differentiation in time domain property states that differentiation in the time domain corresponds to multiplication by in the complex frequency domain, minus the initial value of at . Thus, -- (2.16) -- Proof: -- Using integration by parts where -- (2.17) -- we let and . Then, , , and thus -- The time differentiation property can be extended to show that -- (2.18) -- (2.19) -- and in general -- (2.20) -- To prove (2.18), we let -- and as we found above, -- Then, -- Relations (2.19) and (2.20) can be proved by similar procedures. -- We must remember that the terms , and so on, represent the initial conditions. Therefore, when all initial conditions are zero, and we differentiate a time function times, this corresponds to multiplied by to the power. -- 2.2.6 Differentiation in Complex Frequency Domain Property -- This property states that differentiation in complex frequency domain and multiplication by minus one, corresponds to multiplication of by in the time domain. In other words, -- (2.21) -- Proof: -- Differentiating with respect to and applying Leibnitz's rule for differentiation under the integral, we obtain -- In general, -- (2.22) -- The proof for follows by taking the second and higher-order derivatives of with respect to . -- 2.2.7 Integration in Time Domain Property -- This property states that integration in time domain corresponds to divided by plus the initial value of at , also divided by . That is, -- (2.23) -- Proof: -- We begin by expressing the integral on the left side of (2.23) as two integrals, that is, -- (2.24).
The first integral on the right side of (2.24), represents a constant value since neither the upper, nor the lower limits of integration are functions of time, and this constant is an initial condition denoted as . We will find the Laplace transform ... -- (2.25) -- This is the value of the first integral in (2.24). Next, we will show that -- We let -- then, -- and -- Now, -- (2.26) -- and the proof of (2.23) follows from (2.25) and (2.26). -- 2.2.8 Integration in Complex Frequency Domain Property -- This property states that integration in complex frequency domain with respect to corresponds to division of a time function by the variable , provided that the limit exists. Thus, -- (2.27) -- Proof: -- Integrating both sides from to , we obtain -- Next, we interchange the order of integration, i.e., -- and performing the inner integration on the right side integral with respect to , we obtain -- 2.2.9 Time Periodicity Property -- The time periodicity property states that a periodic function of time with period corresponds to the integral divided by in the complex frequency domain. Thus, if we let be a periodic function with period , that is, , for we obtain the transform pair -- (2.28) -- Proof: -- The Laplace transform of a periodic function can be expressed as -- In the first integral of the right side, we let , in the second , in the third , and so on. The areas under each period of are equal, and thus the upper and lower limits of integration are the same for each integral. Then, -- (2.29) -- Since the function is periodic, i.e., , we can write (2.29) as -- (2.30) -- By application of the binomial theorem, that is, -- (2.31) -- we find that expression (2.30) reduces to -- 2.2.10 Initial Value Theorem.
The initial value theorem states that the initial value of the time function can be found from its Laplace transform multiplied by and letting .That is, -- (2.32) -- Proof: -- From the time domain differentiation property, -- or -- Taking the limit of both sides by letting , we obtain -- Interchanging the limiting process, we obtain -- and since -- the above expression reduces to -- or -- 2.2.11 Final Value Theorem -- The final value theorem states that the final value of the time function can be found from its Laplace transform multiplied by , then, letting . That is, -- (2.33) -- Proof: -- From the time domain differentiation property, -- or -- Taking the limit of both sides by letting , we obtain -- and by interchanging the limiting process, the expression above is written as -- Also, since -- it reduces to -- Therefore, -- 2.2.12 Convolution in Time Domain Property -- Convolution in the time domain corresponds to multiplication in the complex frequency domain, that is, -- (2.34) -- Proof: -- (2.35) -- We let -- then, , and . Then, by substitution into (2.35), -- 2.2.13 Convolution in Complex Frequency Domain Property -- Convolution in the complex frequency domain divided by , corresponds to multiplication in the time domain. That is, -- (2.36) -- Proof: -- (2.37) -- and recalling that the Inverse Laplace transform from (2.2) is -- by substitution into (2.37), we obtain -- We observe that the bracketed integral is -- therefore, -- For easy reference, the Laplace transform pairs and theorems are summarized in Table 2.1. -- 2.3 Laplace Transforms of Common Functions of Time -- In this section, we will derive the Laplace transform of common functions of time. They are presented in Subsections 2.3.1 through 2.3.11 below. -- 2.3.1 Laplace Transform of the Unit Step Function -- We begin with the definition of the Laplace transform, that is, -- or.
Thus, we have obtained the transform pair.
Record Nr. UNINA-9910462145803321
Karris Steven T  
Fremont, CA, : Orchard Publications, 2012
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Signals and systems [[electronic resource] ] : with MATLAB computing and Simulink modeling / / Steven T Karris
Signals and systems [[electronic resource] ] : with MATLAB computing and Simulink modeling / / Steven T Karris
Autore Karris Steven T
Edizione [3rd ed.]
Pubbl/distr/stampa Fremont, CA, : Orchard Publications, 2012
Descrizione fisica 1 v. (various pagings) : ill
Disciplina 621.382/2
Soggetto topico Signal processing - Mathematics
System analysis
ISBN 1-280-12982-4
9786613533647
1-934404-24-1
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Signals and Systems -- with MATLABÒ Computing -- and SimulinkÒ Modeling -- Fifth Edition -- Steven T. Karris -- Preface Signals and Systems Fifth -- Preface -- TOC Signals and Systems Fifth -- Chapter 01 Signals and Systems Fifth -- Chapter 02 Signals and Systems Fifth -- Chapter 2 -- The Laplace Transformation -- his chapter begins with an introduction to the Laplace transformation, definitions, and properties of the Laplace transformation. The initial value and final value theorems are also discussed and proved. It continues with the derivation of the Laplac... -- 2.1 Definition of the Laplace Transformation -- The two-sided or bilateral Laplace Transform pair is defined as -- (2.1) -- (2.2) -- where denotes the Laplace transform of the time function , denotes the Inverse Laplace transform, and is a complex variable whose real part is , and imaginary part , that is, . -- In most problems, we are concerned with values of time greater than some reference time, say , and since the initial conditions are generally known, the two-sided Laplace transform pair of (2.1) and (2.2) simplifies to the unilateral or one-sided Lap... -- (2.3) -- (2.4) -- The Laplace Transform of (2.3) has meaning only if the integral converges (reaches a limit), that is, if -- (2.5) -- To determine the conditions that will ensure us that the integral of (2.3) converges, we rewrite (2.5) as -- (2.6) -- The term in the integral of (2.6) has magnitude of unity, i.e., , and thus the condition for convergence becomes -- (2.7) -- Fortunately, in most engineering applications the functions are of exponential order. Then, we can express (2.7) as, -- (2.8) -- and we see that the integral on the right side of the inequality sign in (2.8), converges if . Therefore, we conclude that if is of exponential order, exists if -- (2.9) -- where denotes the real part of the complex variable .
Evaluation of the integral of (2.4) involves contour integration in the complex plane, and thus, it will not be attempted in this chapter. We will see in the next chapter that many Laplace transforms can be inverted with the use of a few standard pai... -- In our subsequent discussion, we will denote transformation from the time domain to the complex frequency domain, and vice versa, as -- (2.10) -- 2.2 Properties and Theorems of the Laplace Transform -- The most common properties and theorems of the Laplace transform are presented in Subsections 2.2.1 through 2.2.13 below. -- 2.2.1 Linearity Property -- The linearity property states that if -- have Laplace transforms -- respectively, and -- are arbitrary constants, then, -- (2.11) -- Proof: -- Note 1: -- It is desirable to multiply by the unit step function to eliminate any unwanted non-zero values of for . -- 2.2.2 Time Shifting Property -- The time shifting property states that a right shift in the time domain by units, corresponds to multiplication by in the complex frequency domain. Thus, -- (2.12) -- Proof: -- (2.13) -- Now, we let -- then, and . With these substitutions and with , the second integral on the right side of (2.13) is expressed as -- 2.2.3 Frequency Shifting Property -- The frequency shifting property states that if we multiply a time domain function by an exponential function where is an arbitrary positive constant, this multiplication will produce a shift of the s variable in the complex frequency domain by units... -- (2.14) -- Proof: -- Note 2: -- A change of scale is represented by multiplication of the time variable by a positive scaling factor . Thus, the function after scaling the time axis, becomes . -- 2.2.4 Scaling Property -- Let be an arbitrary positive constant -- then, the scaling property states that -- (2.15) -- Proof: -- and letting , we obtain -- Note 3:.
Generally, the initial value of is taken at to include any discontinuity that may be present at . If it is known that no such discontinuity exists at , we simply interpret as . -- 2.2.5 Differentiation in Time Domain Property -- The differentiation in time domain property states that differentiation in the time domain corresponds to multiplication by in the complex frequency domain, minus the initial value of at . Thus, -- (2.16) -- Proof: -- Using integration by parts where -- (2.17) -- we let and . Then, , , and thus -- The time differentiation property can be extended to show that -- (2.18) -- (2.19) -- and in general -- (2.20) -- To prove (2.18), we let -- and as we found above, -- Then, -- Relations (2.19) and (2.20) can be proved by similar procedures. -- We must remember that the terms , and so on, represent the initial conditions. Therefore, when all initial conditions are zero, and we differentiate a time function times, this corresponds to multiplied by to the power. -- 2.2.6 Differentiation in Complex Frequency Domain Property -- This property states that differentiation in complex frequency domain and multiplication by minus one, corresponds to multiplication of by in the time domain. In other words, -- (2.21) -- Proof: -- Differentiating with respect to and applying Leibnitz's rule for differentiation under the integral, we obtain -- In general, -- (2.22) -- The proof for follows by taking the second and higher-order derivatives of with respect to . -- 2.2.7 Integration in Time Domain Property -- This property states that integration in time domain corresponds to divided by plus the initial value of at , also divided by . That is, -- (2.23) -- Proof: -- We begin by expressing the integral on the left side of (2.23) as two integrals, that is, -- (2.24).
The first integral on the right side of (2.24), represents a constant value since neither the upper, nor the lower limits of integration are functions of time, and this constant is an initial condition denoted as . We will find the Laplace transform ... -- (2.25) -- This is the value of the first integral in (2.24). Next, we will show that -- We let -- then, -- and -- Now, -- (2.26) -- and the proof of (2.23) follows from (2.25) and (2.26). -- 2.2.8 Integration in Complex Frequency Domain Property -- This property states that integration in complex frequency domain with respect to corresponds to division of a time function by the variable , provided that the limit exists. Thus, -- (2.27) -- Proof: -- Integrating both sides from to , we obtain -- Next, we interchange the order of integration, i.e., -- and performing the inner integration on the right side integral with respect to , we obtain -- 2.2.9 Time Periodicity Property -- The time periodicity property states that a periodic function of time with period corresponds to the integral divided by in the complex frequency domain. Thus, if we let be a periodic function with period , that is, , for we obtain the transform pair -- (2.28) -- Proof: -- The Laplace transform of a periodic function can be expressed as -- In the first integral of the right side, we let , in the second , in the third , and so on. The areas under each period of are equal, and thus the upper and lower limits of integration are the same for each integral. Then, -- (2.29) -- Since the function is periodic, i.e., , we can write (2.29) as -- (2.30) -- By application of the binomial theorem, that is, -- (2.31) -- we find that expression (2.30) reduces to -- 2.2.10 Initial Value Theorem.
The initial value theorem states that the initial value of the time function can be found from its Laplace transform multiplied by and letting .That is, -- (2.32) -- Proof: -- From the time domain differentiation property, -- or -- Taking the limit of both sides by letting , we obtain -- Interchanging the limiting process, we obtain -- and since -- the above expression reduces to -- or -- 2.2.11 Final Value Theorem -- The final value theorem states that the final value of the time function can be found from its Laplace transform multiplied by , then, letting . That is, -- (2.33) -- Proof: -- From the time domain differentiation property, -- or -- Taking the limit of both sides by letting , we obtain -- and by interchanging the limiting process, the expression above is written as -- Also, since -- it reduces to -- Therefore, -- 2.2.12 Convolution in Time Domain Property -- Convolution in the time domain corresponds to multiplication in the complex frequency domain, that is, -- (2.34) -- Proof: -- (2.35) -- We let -- then, , and . Then, by substitution into (2.35), -- 2.2.13 Convolution in Complex Frequency Domain Property -- Convolution in the complex frequency domain divided by , corresponds to multiplication in the time domain. That is, -- (2.36) -- Proof: -- (2.37) -- and recalling that the Inverse Laplace transform from (2.2) is -- by substitution into (2.37), we obtain -- We observe that the bracketed integral is -- therefore, -- For easy reference, the Laplace transform pairs and theorems are summarized in Table 2.1. -- 2.3 Laplace Transforms of Common Functions of Time -- In this section, we will derive the Laplace transform of common functions of time. They are presented in Subsections 2.3.1 through 2.3.11 below. -- 2.3.1 Laplace Transform of the Unit Step Function -- We begin with the definition of the Laplace transform, that is, -- or.
Thus, we have obtained the transform pair.
Record Nr. UNINA-9910790170103321
Karris Steven T  
Fremont, CA, : Orchard Publications, 2012
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Signals and systems [[electronic resource] ] : with MATLAB computing and Simulink modeling / / Steven T Karris
Signals and systems [[electronic resource] ] : with MATLAB computing and Simulink modeling / / Steven T Karris
Autore Karris Steven T
Edizione [3rd ed.]
Pubbl/distr/stampa Fremont, CA, : Orchard Publications, 2012
Descrizione fisica 1 v. (various pagings) : ill
Disciplina 621.382/2
Soggetto topico Signal processing - Mathematics
System analysis
ISBN 1-280-12982-4
9786613533647
1-934404-24-1
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Signals and Systems -- with MATLABÒ Computing -- and SimulinkÒ Modeling -- Fifth Edition -- Steven T. Karris -- Preface Signals and Systems Fifth -- Preface -- TOC Signals and Systems Fifth -- Chapter 01 Signals and Systems Fifth -- Chapter 02 Signals and Systems Fifth -- Chapter 2 -- The Laplace Transformation -- his chapter begins with an introduction to the Laplace transformation, definitions, and properties of the Laplace transformation. The initial value and final value theorems are also discussed and proved. It continues with the derivation of the Laplac... -- 2.1 Definition of the Laplace Transformation -- The two-sided or bilateral Laplace Transform pair is defined as -- (2.1) -- (2.2) -- where denotes the Laplace transform of the time function , denotes the Inverse Laplace transform, and is a complex variable whose real part is , and imaginary part , that is, . -- In most problems, we are concerned with values of time greater than some reference time, say , and since the initial conditions are generally known, the two-sided Laplace transform pair of (2.1) and (2.2) simplifies to the unilateral or one-sided Lap... -- (2.3) -- (2.4) -- The Laplace Transform of (2.3) has meaning only if the integral converges (reaches a limit), that is, if -- (2.5) -- To determine the conditions that will ensure us that the integral of (2.3) converges, we rewrite (2.5) as -- (2.6) -- The term in the integral of (2.6) has magnitude of unity, i.e., , and thus the condition for convergence becomes -- (2.7) -- Fortunately, in most engineering applications the functions are of exponential order. Then, we can express (2.7) as, -- (2.8) -- and we see that the integral on the right side of the inequality sign in (2.8), converges if . Therefore, we conclude that if is of exponential order, exists if -- (2.9) -- where denotes the real part of the complex variable .
Evaluation of the integral of (2.4) involves contour integration in the complex plane, and thus, it will not be attempted in this chapter. We will see in the next chapter that many Laplace transforms can be inverted with the use of a few standard pai... -- In our subsequent discussion, we will denote transformation from the time domain to the complex frequency domain, and vice versa, as -- (2.10) -- 2.2 Properties and Theorems of the Laplace Transform -- The most common properties and theorems of the Laplace transform are presented in Subsections 2.2.1 through 2.2.13 below. -- 2.2.1 Linearity Property -- The linearity property states that if -- have Laplace transforms -- respectively, and -- are arbitrary constants, then, -- (2.11) -- Proof: -- Note 1: -- It is desirable to multiply by the unit step function to eliminate any unwanted non-zero values of for . -- 2.2.2 Time Shifting Property -- The time shifting property states that a right shift in the time domain by units, corresponds to multiplication by in the complex frequency domain. Thus, -- (2.12) -- Proof: -- (2.13) -- Now, we let -- then, and . With these substitutions and with , the second integral on the right side of (2.13) is expressed as -- 2.2.3 Frequency Shifting Property -- The frequency shifting property states that if we multiply a time domain function by an exponential function where is an arbitrary positive constant, this multiplication will produce a shift of the s variable in the complex frequency domain by units... -- (2.14) -- Proof: -- Note 2: -- A change of scale is represented by multiplication of the time variable by a positive scaling factor . Thus, the function after scaling the time axis, becomes . -- 2.2.4 Scaling Property -- Let be an arbitrary positive constant -- then, the scaling property states that -- (2.15) -- Proof: -- and letting , we obtain -- Note 3:.
Generally, the initial value of is taken at to include any discontinuity that may be present at . If it is known that no such discontinuity exists at , we simply interpret as . -- 2.2.5 Differentiation in Time Domain Property -- The differentiation in time domain property states that differentiation in the time domain corresponds to multiplication by in the complex frequency domain, minus the initial value of at . Thus, -- (2.16) -- Proof: -- Using integration by parts where -- (2.17) -- we let and . Then, , , and thus -- The time differentiation property can be extended to show that -- (2.18) -- (2.19) -- and in general -- (2.20) -- To prove (2.18), we let -- and as we found above, -- Then, -- Relations (2.19) and (2.20) can be proved by similar procedures. -- We must remember that the terms , and so on, represent the initial conditions. Therefore, when all initial conditions are zero, and we differentiate a time function times, this corresponds to multiplied by to the power. -- 2.2.6 Differentiation in Complex Frequency Domain Property -- This property states that differentiation in complex frequency domain and multiplication by minus one, corresponds to multiplication of by in the time domain. In other words, -- (2.21) -- Proof: -- Differentiating with respect to and applying Leibnitz's rule for differentiation under the integral, we obtain -- In general, -- (2.22) -- The proof for follows by taking the second and higher-order derivatives of with respect to . -- 2.2.7 Integration in Time Domain Property -- This property states that integration in time domain corresponds to divided by plus the initial value of at , also divided by . That is, -- (2.23) -- Proof: -- We begin by expressing the integral on the left side of (2.23) as two integrals, that is, -- (2.24).
The first integral on the right side of (2.24), represents a constant value since neither the upper, nor the lower limits of integration are functions of time, and this constant is an initial condition denoted as . We will find the Laplace transform ... -- (2.25) -- This is the value of the first integral in (2.24). Next, we will show that -- We let -- then, -- and -- Now, -- (2.26) -- and the proof of (2.23) follows from (2.25) and (2.26). -- 2.2.8 Integration in Complex Frequency Domain Property -- This property states that integration in complex frequency domain with respect to corresponds to division of a time function by the variable , provided that the limit exists. Thus, -- (2.27) -- Proof: -- Integrating both sides from to , we obtain -- Next, we interchange the order of integration, i.e., -- and performing the inner integration on the right side integral with respect to , we obtain -- 2.2.9 Time Periodicity Property -- The time periodicity property states that a periodic function of time with period corresponds to the integral divided by in the complex frequency domain. Thus, if we let be a periodic function with period , that is, , for we obtain the transform pair -- (2.28) -- Proof: -- The Laplace transform of a periodic function can be expressed as -- In the first integral of the right side, we let , in the second , in the third , and so on. The areas under each period of are equal, and thus the upper and lower limits of integration are the same for each integral. Then, -- (2.29) -- Since the function is periodic, i.e., , we can write (2.29) as -- (2.30) -- By application of the binomial theorem, that is, -- (2.31) -- we find that expression (2.30) reduces to -- 2.2.10 Initial Value Theorem.
The initial value theorem states that the initial value of the time function can be found from its Laplace transform multiplied by and letting .That is, -- (2.32) -- Proof: -- From the time domain differentiation property, -- or -- Taking the limit of both sides by letting , we obtain -- Interchanging the limiting process, we obtain -- and since -- the above expression reduces to -- or -- 2.2.11 Final Value Theorem -- The final value theorem states that the final value of the time function can be found from its Laplace transform multiplied by , then, letting . That is, -- (2.33) -- Proof: -- From the time domain differentiation property, -- or -- Taking the limit of both sides by letting , we obtain -- and by interchanging the limiting process, the expression above is written as -- Also, since -- it reduces to -- Therefore, -- 2.2.12 Convolution in Time Domain Property -- Convolution in the time domain corresponds to multiplication in the complex frequency domain, that is, -- (2.34) -- Proof: -- (2.35) -- We let -- then, , and . Then, by substitution into (2.35), -- 2.2.13 Convolution in Complex Frequency Domain Property -- Convolution in the complex frequency domain divided by , corresponds to multiplication in the time domain. That is, -- (2.36) -- Proof: -- (2.37) -- and recalling that the Inverse Laplace transform from (2.2) is -- by substitution into (2.37), we obtain -- We observe that the bracketed integral is -- therefore, -- For easy reference, the Laplace transform pairs and theorems are summarized in Table 2.1. -- 2.3 Laplace Transforms of Common Functions of Time -- In this section, we will derive the Laplace transform of common functions of time. They are presented in Subsections 2.3.1 through 2.3.11 below. -- 2.3.1 Laplace Transform of the Unit Step Function -- We begin with the definition of the Laplace transform, that is, -- or.
Thus, we have obtained the transform pair.
Record Nr. UNINA-9910811064303321
Karris Steven T  
Fremont, CA, : Orchard Publications, 2012
Materiale a stampa
Lo trovi qui: Univ. Federico II
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Signals and systems [[electronic resource] ] : with MATLAB applications / / Steven T. Karris
Signals and systems [[electronic resource] ] : with MATLAB applications / / Steven T. Karris
Autore Karris Steven T
Edizione [3rd ed.]
Pubbl/distr/stampa Fremont, CA, : Orchard Publications, c2007
Descrizione fisica x, (650) p. : ill
Disciplina 621.382/2
Soggetto topico Signal processing - Mathematics
System analysis
Soggetto genere / forma Electronic books.
ISBN 1-280-75117-7
9786610751174
1-934404-00-4
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Title -- Preface -- Contents -- Chapter 01 -- Chapter 1 -- Elementary Signals -- his chapter begins with a discussion of elementary signals that may be applied to electric networks. The unit step, unit ramp, and delta functions are then introduced. The sampling and sifting properties of the delta function are defined and ... -- 1.1 Signals Described in Math Form -- Consider the network of Figure 1.1 where the switch is closed at time . -- Figure 1.1. A switched network with open terminals -- We wish to describe in a math form for the time interval . To do this, it is conve nient to divide the time interval into two parts, , and . -- For the time interval , the switch is open and therefore, the output voltage is zero. In other words, -- (1.1) -- For the time interval , the switch is closed. Then, the input voltage appears at the output, i.e., -- (1.2) -- Combining (1.1) and (1.2) into a single relationship, we obtain -- (1.3) -- We can express (1.3) by the waveform shown in Figure 1.2. -- Figure 1.2. Waveform for as defined in relation (1.3) -- The waveform of Figure 1.2 is an example of a discontinuous function. A function is said to be dis continuous if it exhibits points of discontinuity, that is, the function jumps from one value to another without taking on any intermediate values. -- 1.2 The Unit Step Function -- A well known discontinuous function is the unit step function which is defined as -- (1.4) -- It is also represented by the waveform of Figure 1.3. -- Figure 1.3. Waveform for -- In the waveform of Figure 1.3, the unit step function changes abruptly from to at . But if it changes at instead, it is denoted as . In this case, its waveform and definition are as shown in Figure 1.4 and relation (1.5) respectively. -- Figure 1.4. Waveform for -- (1.5).
If the unit step function changes abruptly from to at , it is denoted as . In this case, its waveform and definition are as shown in Figure 1.5 and relation (1.6) respectively. -- Figure 1.5. Waveform for -- (1.6) -- Example 1.1 -- Consider the network of Figure 1.6, where the switch is closed at time . -- Figure 1.6. Network for Example 1.1 -- Express the output voltage as a function of the unit step function, and sketch the appropriate waveform. -- Solution: -- For this example, the output voltage for , and for . Therefore, -- (1.7) -- and the waveform is shown in Figure 1.7. -- Figure 1.7. Waveform for Example 1.1 -- Other forms of the unit step function are shown in Figure 1.8. -- Figure 1.8. Other forms of the unit step function -- Unit step functions can be used to represent other time-varying functions such as the rectangular pulse shown in Figure 1.9. -- Figure 1.9. A rectangular pulse expressed as the sum of two unit step functions -- Thus, the pulse of Figure 1.9(a) is the sum of the unit step functions of Figures 1.9(b) and 1.9(c) and it is represented as . -- The unit step function offers a convenient method of describing the sudden application of a volt age or current source. For example, a constant voltage source of applied at , can be denoted as . Likewise, a sinusoidal voltage source that is a... -- Example 1.2 -- Express the square waveform of Figure 1.10 as a sum of unit step functions. The vertical dotted lines indicate the discontinuities at , and so on. -- Figure 1.10. Square waveform for Example 1.2 -- Solution: -- Line segment has height , starts at , and terminates at . Then, as in Example 1.1, this segment is expressed as -- (1.8) -- Line segment has height , starts at and terminates at . This segment is expressed as -- (1.9) -- Line segment has height , starts at and terminates at . This segment is expressed as -- (1.10).
Line segment has height , starts at , and terminates at . It is expressed as -- (1.11) -- Thus, the square waveform of Figure 1.10 can be expressed as the summation of (1.8) through (1.11), that is, -- (1.12) -- Combining like terms, we obtain -- (1.13) -- Example 1.3 -- Express the symmetric rectangular pulse of Figure 1.11 as a sum of unit step functions. -- Figure 1.11. Symmetric rectangular pulse for Example 1.3 -- Solution: -- This pulse has height , starts at , and terminates at . Therefore, with refer ence to Figures 1.5 and 1.8 (b), we obtain -- (1.14) -- Example 1.4 -- Express the symmetric triangular waveform of Figure 1.12 as a sum of unit step functions. -- Figure 1.12. Symmetric triangular waveform for Example 1.4 -- Solution: -- We first derive the equations for the linear segments and shown in Figure 1.13. -- Figure 1.13. Equations for the linear segments of Figure 1.12 -- For line segment , -- (1.15) -- and for line segment , -- (1.16) -- Combining (1.15) and (1.16), we obtain -- (1.17) -- Example 1.5 -- Express the waveform of Figure 1.14 as a sum of unit step functions. -- Figure 1.14. Waveform for Example 1.5 -- Solution: -- As in the previous example, we first find the equations of the linear segments linear segments and shown in Figure 1.15. -- Figure 1.15. Equations for the linear segments of Figure 1.14 -- Following the same procedure as in the previous examples, we obtain -- Multiplying the values in parentheses by the values in the brackets, we obtain -- and combining terms inside the brackets, we obtain -- (1.18) -- Two other functions of interest are the unit ramp function, and the unit impulse or delta function. We will introduce them with the examples that follow. -- Example 1.6.
In the network of Figure 1.16 is a constant current source and the switch is closed at time . Express the capacitor voltage as a function of the unit step. -- Figure 1.16. Network for Example 1.6 -- Solution: -- The current through the capacitor is , and the capacitor voltage is -- (1.19) -- where is a dummy variable. -- Since the switch closes at , we can express the current as -- (1.20) -- and assuming that for , we can write (1.19) as -- (1.21) -- or -- (1.22) -- Therefore, we see that when a capacitor is charged with a constant current, the voltage across it is a linear function and forms a ramp with slope as shown in Figure 1.17. -- Figure 1.17. Voltage across a capacitor when charged with a constant current source -- 1.3 The Unit Ramp Function -- The unit ramp function, denoted as , is defined as -- (1.23) -- where is a dummy variable. -- We can evaluate the integral of (1.23) by considering the area under the unit step function from as shown in Figure 1.18. -- Figure 1.18. Area under the unit step function from -- Therefore, we define as -- (1.24) -- Since is the integral of , then must be the derivative of , i.e., -- (1.25) -- Higher order functions of can be generated by repeated integration of the unit step function. For example, integrating twice and multiplying by , we define as -- (1.26) -- Similarly, -- (1.27) -- and in general, -- (1.28) -- Also, -- (1.29) -- Example 1.7 -- In the network of Figure 1.19, the switch is closed at time and for . Express the inductor voltage in terms of the unit step function. -- Figure 1.19. Network for Example 1.7 -- Solution: -- The voltage across the inductor is -- (1.30) -- and since the switch closes at , -- (1.31) -- Therefore, we can write (1.30) as -- (1.32).
But, as we know, is constant ( or ) for all time except at where it is discontinuous. Since the derivative of any constant is zero, the derivative of the unit step has a non-zero value only at . The derivative of the unit step function is def... -- 1.4 The Delta Function -- The unit impulse or delta function, denoted as , is the derivative of the unit step . It is also defined as -- (1.33) -- and -- (1.34) -- To better understand the delta function , let us represent the unit step as shown in Fig ure 1.20 (a). -- Figure 1.20. Representation of the unit step as a limit -- The function of Figure 1.20 (a) becomes the unit step as . Figure 1.20 (b) is the derivative of Figure 1.20 (a), where we see that as , becomes unbounded, but the area of the rect angle remains . Therefore, in the limit, we can think of as a... -- Two useful properties of the delta function are the sampling property and the sifting property. -- 1.4.1 The Sampling Property of the Delta Function -- The sampling property of the delta function states that -- (1.35) -- or, when , -- (1.36) -- that is, multiplication of any function by the delta function results in sampling the func tion at the time instants where the delta function is not zero. The study of discrete-time systems is based on this property. -- Proof: -- Since then, -- (1.37) -- We rewrite as -- (1.38) -- Integrating (1.37) over the interval and using (1.38), we obtain -- (1.39) -- The first integral on the right side of (1.39) contains the constant term -- this can be written outside the integral, that is, -- (1.40) -- The second integral of the right side of (1.39) is always zero because -- and -- Therefore, (1.39) reduces to -- (1.41) -- Differentiating both sides of (1.41), and replacing with , we obtain -- (1.42) -- 1.4.2 The Sifting Property of the Delta Function.
The sifting property of the delta function states that.
Record Nr. UNINA-9910451019703321
Karris Steven T  
Fremont, CA, : Orchard Publications, c2007
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Signals and systems [[electronic resource] ] : with MATLAB applications / / Steven T. Karris
Signals and systems [[electronic resource] ] : with MATLAB applications / / Steven T. Karris
Autore Karris Steven T
Edizione [3rd ed.]
Pubbl/distr/stampa Fremont, CA, : Orchard Publications, c2007
Descrizione fisica x, (650) p. : ill
Disciplina 621.382/2
Soggetto topico Signal processing - Mathematics
System analysis
ISBN 1-280-75117-7
9786610751174
1-934404-00-4
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Title -- Preface -- Contents -- Chapter 01 -- Chapter 1 -- Elementary Signals -- his chapter begins with a discussion of elementary signals that may be applied to electric networks. The unit step, unit ramp, and delta functions are then introduced. The sampling and sifting properties of the delta function are defined and ... -- 1.1 Signals Described in Math Form -- Consider the network of Figure 1.1 where the switch is closed at time . -- Figure 1.1. A switched network with open terminals -- We wish to describe in a math form for the time interval . To do this, it is conve nient to divide the time interval into two parts, , and . -- For the time interval , the switch is open and therefore, the output voltage is zero. In other words, -- (1.1) -- For the time interval , the switch is closed. Then, the input voltage appears at the output, i.e., -- (1.2) -- Combining (1.1) and (1.2) into a single relationship, we obtain -- (1.3) -- We can express (1.3) by the waveform shown in Figure 1.2. -- Figure 1.2. Waveform for as defined in relation (1.3) -- The waveform of Figure 1.2 is an example of a discontinuous function. A function is said to be dis continuous if it exhibits points of discontinuity, that is, the function jumps from one value to another without taking on any intermediate values. -- 1.2 The Unit Step Function -- A well known discontinuous function is the unit step function which is defined as -- (1.4) -- It is also represented by the waveform of Figure 1.3. -- Figure 1.3. Waveform for -- In the waveform of Figure 1.3, the unit step function changes abruptly from to at . But if it changes at instead, it is denoted as . In this case, its waveform and definition are as shown in Figure 1.4 and relation (1.5) respectively. -- Figure 1.4. Waveform for -- (1.5).
If the unit step function changes abruptly from to at , it is denoted as . In this case, its waveform and definition are as shown in Figure 1.5 and relation (1.6) respectively. -- Figure 1.5. Waveform for -- (1.6) -- Example 1.1 -- Consider the network of Figure 1.6, where the switch is closed at time . -- Figure 1.6. Network for Example 1.1 -- Express the output voltage as a function of the unit step function, and sketch the appropriate waveform. -- Solution: -- For this example, the output voltage for , and for . Therefore, -- (1.7) -- and the waveform is shown in Figure 1.7. -- Figure 1.7. Waveform for Example 1.1 -- Other forms of the unit step function are shown in Figure 1.8. -- Figure 1.8. Other forms of the unit step function -- Unit step functions can be used to represent other time-varying functions such as the rectangular pulse shown in Figure 1.9. -- Figure 1.9. A rectangular pulse expressed as the sum of two unit step functions -- Thus, the pulse of Figure 1.9(a) is the sum of the unit step functions of Figures 1.9(b) and 1.9(c) and it is represented as . -- The unit step function offers a convenient method of describing the sudden application of a volt age or current source. For example, a constant voltage source of applied at , can be denoted as . Likewise, a sinusoidal voltage source that is a... -- Example 1.2 -- Express the square waveform of Figure 1.10 as a sum of unit step functions. The vertical dotted lines indicate the discontinuities at , and so on. -- Figure 1.10. Square waveform for Example 1.2 -- Solution: -- Line segment has height , starts at , and terminates at . Then, as in Example 1.1, this segment is expressed as -- (1.8) -- Line segment has height , starts at and terminates at . This segment is expressed as -- (1.9) -- Line segment has height , starts at and terminates at . This segment is expressed as -- (1.10).
Line segment has height , starts at , and terminates at . It is expressed as -- (1.11) -- Thus, the square waveform of Figure 1.10 can be expressed as the summation of (1.8) through (1.11), that is, -- (1.12) -- Combining like terms, we obtain -- (1.13) -- Example 1.3 -- Express the symmetric rectangular pulse of Figure 1.11 as a sum of unit step functions. -- Figure 1.11. Symmetric rectangular pulse for Example 1.3 -- Solution: -- This pulse has height , starts at , and terminates at . Therefore, with refer ence to Figures 1.5 and 1.8 (b), we obtain -- (1.14) -- Example 1.4 -- Express the symmetric triangular waveform of Figure 1.12 as a sum of unit step functions. -- Figure 1.12. Symmetric triangular waveform for Example 1.4 -- Solution: -- We first derive the equations for the linear segments and shown in Figure 1.13. -- Figure 1.13. Equations for the linear segments of Figure 1.12 -- For line segment , -- (1.15) -- and for line segment , -- (1.16) -- Combining (1.15) and (1.16), we obtain -- (1.17) -- Example 1.5 -- Express the waveform of Figure 1.14 as a sum of unit step functions. -- Figure 1.14. Waveform for Example 1.5 -- Solution: -- As in the previous example, we first find the equations of the linear segments linear segments and shown in Figure 1.15. -- Figure 1.15. Equations for the linear segments of Figure 1.14 -- Following the same procedure as in the previous examples, we obtain -- Multiplying the values in parentheses by the values in the brackets, we obtain -- and combining terms inside the brackets, we obtain -- (1.18) -- Two other functions of interest are the unit ramp function, and the unit impulse or delta function. We will introduce them with the examples that follow. -- Example 1.6.
In the network of Figure 1.16 is a constant current source and the switch is closed at time . Express the capacitor voltage as a function of the unit step. -- Figure 1.16. Network for Example 1.6 -- Solution: -- The current through the capacitor is , and the capacitor voltage is -- (1.19) -- where is a dummy variable. -- Since the switch closes at , we can express the current as -- (1.20) -- and assuming that for , we can write (1.19) as -- (1.21) -- or -- (1.22) -- Therefore, we see that when a capacitor is charged with a constant current, the voltage across it is a linear function and forms a ramp with slope as shown in Figure 1.17. -- Figure 1.17. Voltage across a capacitor when charged with a constant current source -- 1.3 The Unit Ramp Function -- The unit ramp function, denoted as , is defined as -- (1.23) -- where is a dummy variable. -- We can evaluate the integral of (1.23) by considering the area under the unit step function from as shown in Figure 1.18. -- Figure 1.18. Area under the unit step function from -- Therefore, we define as -- (1.24) -- Since is the integral of , then must be the derivative of , i.e., -- (1.25) -- Higher order functions of can be generated by repeated integration of the unit step function. For example, integrating twice and multiplying by , we define as -- (1.26) -- Similarly, -- (1.27) -- and in general, -- (1.28) -- Also, -- (1.29) -- Example 1.7 -- In the network of Figure 1.19, the switch is closed at time and for . Express the inductor voltage in terms of the unit step function. -- Figure 1.19. Network for Example 1.7 -- Solution: -- The voltage across the inductor is -- (1.30) -- and since the switch closes at , -- (1.31) -- Therefore, we can write (1.30) as -- (1.32).
But, as we know, is constant ( or ) for all time except at where it is discontinuous. Since the derivative of any constant is zero, the derivative of the unit step has a non-zero value only at . The derivative of the unit step function is def... -- 1.4 The Delta Function -- The unit impulse or delta function, denoted as , is the derivative of the unit step . It is also defined as -- (1.33) -- and -- (1.34) -- To better understand the delta function , let us represent the unit step as shown in Fig ure 1.20 (a). -- Figure 1.20. Representation of the unit step as a limit -- The function of Figure 1.20 (a) becomes the unit step as . Figure 1.20 (b) is the derivative of Figure 1.20 (a), where we see that as , becomes unbounded, but the area of the rect angle remains . Therefore, in the limit, we can think of as a... -- Two useful properties of the delta function are the sampling property and the sifting property. -- 1.4.1 The Sampling Property of the Delta Function -- The sampling property of the delta function states that -- (1.35) -- or, when , -- (1.36) -- that is, multiplication of any function by the delta function results in sampling the func tion at the time instants where the delta function is not zero. The study of discrete-time systems is based on this property. -- Proof: -- Since then, -- (1.37) -- We rewrite as -- (1.38) -- Integrating (1.37) over the interval and using (1.38), we obtain -- (1.39) -- The first integral on the right side of (1.39) contains the constant term -- this can be written outside the integral, that is, -- (1.40) -- The second integral of the right side of (1.39) is always zero because -- and -- Therefore, (1.39) reduces to -- (1.41) -- Differentiating both sides of (1.41), and replacing with , we obtain -- (1.42) -- 1.4.2 The Sifting Property of the Delta Function.
The sifting property of the delta function states that.
Record Nr. UNINA-9910784213103321
Karris Steven T  
Fremont, CA, : Orchard Publications, c2007
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Signals and systems [[electronic resource] ] : with MATLAB applications / / Steven T. Karris
Signals and systems [[electronic resource] ] : with MATLAB applications / / Steven T. Karris
Autore Karris Steven T
Edizione [3rd ed.]
Pubbl/distr/stampa Fremont, CA, : Orchard Publications, c2007
Descrizione fisica x, (650) p. : ill
Disciplina 621.382/2
Soggetto topico Signal processing - Mathematics
System analysis
ISBN 1-280-75117-7
9786610751174
1-934404-00-4
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Title -- Preface -- Contents -- Chapter 01 -- Chapter 1 -- Elementary Signals -- his chapter begins with a discussion of elementary signals that may be applied to electric networks. The unit step, unit ramp, and delta functions are then introduced. The sampling and sifting properties of the delta function are defined and ... -- 1.1 Signals Described in Math Form -- Consider the network of Figure 1.1 where the switch is closed at time . -- Figure 1.1. A switched network with open terminals -- We wish to describe in a math form for the time interval . To do this, it is conve nient to divide the time interval into two parts, , and . -- For the time interval , the switch is open and therefore, the output voltage is zero. In other words, -- (1.1) -- For the time interval , the switch is closed. Then, the input voltage appears at the output, i.e., -- (1.2) -- Combining (1.1) and (1.2) into a single relationship, we obtain -- (1.3) -- We can express (1.3) by the waveform shown in Figure 1.2. -- Figure 1.2. Waveform for as defined in relation (1.3) -- The waveform of Figure 1.2 is an example of a discontinuous function. A function is said to be dis continuous if it exhibits points of discontinuity, that is, the function jumps from one value to another without taking on any intermediate values. -- 1.2 The Unit Step Function -- A well known discontinuous function is the unit step function which is defined as -- (1.4) -- It is also represented by the waveform of Figure 1.3. -- Figure 1.3. Waveform for -- In the waveform of Figure 1.3, the unit step function changes abruptly from to at . But if it changes at instead, it is denoted as . In this case, its waveform and definition are as shown in Figure 1.4 and relation (1.5) respectively. -- Figure 1.4. Waveform for -- (1.5).
If the unit step function changes abruptly from to at , it is denoted as . In this case, its waveform and definition are as shown in Figure 1.5 and relation (1.6) respectively. -- Figure 1.5. Waveform for -- (1.6) -- Example 1.1 -- Consider the network of Figure 1.6, where the switch is closed at time . -- Figure 1.6. Network for Example 1.1 -- Express the output voltage as a function of the unit step function, and sketch the appropriate waveform. -- Solution: -- For this example, the output voltage for , and for . Therefore, -- (1.7) -- and the waveform is shown in Figure 1.7. -- Figure 1.7. Waveform for Example 1.1 -- Other forms of the unit step function are shown in Figure 1.8. -- Figure 1.8. Other forms of the unit step function -- Unit step functions can be used to represent other time-varying functions such as the rectangular pulse shown in Figure 1.9. -- Figure 1.9. A rectangular pulse expressed as the sum of two unit step functions -- Thus, the pulse of Figure 1.9(a) is the sum of the unit step functions of Figures 1.9(b) and 1.9(c) and it is represented as . -- The unit step function offers a convenient method of describing the sudden application of a volt age or current source. For example, a constant voltage source of applied at , can be denoted as . Likewise, a sinusoidal voltage source that is a... -- Example 1.2 -- Express the square waveform of Figure 1.10 as a sum of unit step functions. The vertical dotted lines indicate the discontinuities at , and so on. -- Figure 1.10. Square waveform for Example 1.2 -- Solution: -- Line segment has height , starts at , and terminates at . Then, as in Example 1.1, this segment is expressed as -- (1.8) -- Line segment has height , starts at and terminates at . This segment is expressed as -- (1.9) -- Line segment has height , starts at and terminates at . This segment is expressed as -- (1.10).
Line segment has height , starts at , and terminates at . It is expressed as -- (1.11) -- Thus, the square waveform of Figure 1.10 can be expressed as the summation of (1.8) through (1.11), that is, -- (1.12) -- Combining like terms, we obtain -- (1.13) -- Example 1.3 -- Express the symmetric rectangular pulse of Figure 1.11 as a sum of unit step functions. -- Figure 1.11. Symmetric rectangular pulse for Example 1.3 -- Solution: -- This pulse has height , starts at , and terminates at . Therefore, with refer ence to Figures 1.5 and 1.8 (b), we obtain -- (1.14) -- Example 1.4 -- Express the symmetric triangular waveform of Figure 1.12 as a sum of unit step functions. -- Figure 1.12. Symmetric triangular waveform for Example 1.4 -- Solution: -- We first derive the equations for the linear segments and shown in Figure 1.13. -- Figure 1.13. Equations for the linear segments of Figure 1.12 -- For line segment , -- (1.15) -- and for line segment , -- (1.16) -- Combining (1.15) and (1.16), we obtain -- (1.17) -- Example 1.5 -- Express the waveform of Figure 1.14 as a sum of unit step functions. -- Figure 1.14. Waveform for Example 1.5 -- Solution: -- As in the previous example, we first find the equations of the linear segments linear segments and shown in Figure 1.15. -- Figure 1.15. Equations for the linear segments of Figure 1.14 -- Following the same procedure as in the previous examples, we obtain -- Multiplying the values in parentheses by the values in the brackets, we obtain -- and combining terms inside the brackets, we obtain -- (1.18) -- Two other functions of interest are the unit ramp function, and the unit impulse or delta function. We will introduce them with the examples that follow. -- Example 1.6.
In the network of Figure 1.16 is a constant current source and the switch is closed at time . Express the capacitor voltage as a function of the unit step. -- Figure 1.16. Network for Example 1.6 -- Solution: -- The current through the capacitor is , and the capacitor voltage is -- (1.19) -- where is a dummy variable. -- Since the switch closes at , we can express the current as -- (1.20) -- and assuming that for , we can write (1.19) as -- (1.21) -- or -- (1.22) -- Therefore, we see that when a capacitor is charged with a constant current, the voltage across it is a linear function and forms a ramp with slope as shown in Figure 1.17. -- Figure 1.17. Voltage across a capacitor when charged with a constant current source -- 1.3 The Unit Ramp Function -- The unit ramp function, denoted as , is defined as -- (1.23) -- where is a dummy variable. -- We can evaluate the integral of (1.23) by considering the area under the unit step function from as shown in Figure 1.18. -- Figure 1.18. Area under the unit step function from -- Therefore, we define as -- (1.24) -- Since is the integral of , then must be the derivative of , i.e., -- (1.25) -- Higher order functions of can be generated by repeated integration of the unit step function. For example, integrating twice and multiplying by , we define as -- (1.26) -- Similarly, -- (1.27) -- and in general, -- (1.28) -- Also, -- (1.29) -- Example 1.7 -- In the network of Figure 1.19, the switch is closed at time and for . Express the inductor voltage in terms of the unit step function. -- Figure 1.19. Network for Example 1.7 -- Solution: -- The voltage across the inductor is -- (1.30) -- and since the switch closes at , -- (1.31) -- Therefore, we can write (1.30) as -- (1.32).
But, as we know, is constant ( or ) for all time except at where it is discontinuous. Since the derivative of any constant is zero, the derivative of the unit step has a non-zero value only at . The derivative of the unit step function is def... -- 1.4 The Delta Function -- The unit impulse or delta function, denoted as , is the derivative of the unit step . It is also defined as -- (1.33) -- and -- (1.34) -- To better understand the delta function , let us represent the unit step as shown in Fig ure 1.20 (a). -- Figure 1.20. Representation of the unit step as a limit -- The function of Figure 1.20 (a) becomes the unit step as . Figure 1.20 (b) is the derivative of Figure 1.20 (a), where we see that as , becomes unbounded, but the area of the rect angle remains . Therefore, in the limit, we can think of as a... -- Two useful properties of the delta function are the sampling property and the sifting property. -- 1.4.1 The Sampling Property of the Delta Function -- The sampling property of the delta function states that -- (1.35) -- or, when , -- (1.36) -- that is, multiplication of any function by the delta function results in sampling the func tion at the time instants where the delta function is not zero. The study of discrete-time systems is based on this property. -- Proof: -- Since then, -- (1.37) -- We rewrite as -- (1.38) -- Integrating (1.37) over the interval and using (1.38), we obtain -- (1.39) -- The first integral on the right side of (1.39) contains the constant term -- this can be written outside the integral, that is, -- (1.40) -- The second integral of the right side of (1.39) is always zero because -- and -- Therefore, (1.39) reduces to -- (1.41) -- Differentiating both sides of (1.41), and replacing with , we obtain -- (1.42) -- 1.4.2 The Sifting Property of the Delta Function.
The sifting property of the delta function states that.
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Karris Steven T  
Fremont, CA, : Orchard Publications, c2007
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