Transforms and Applications Handbook, Third EditionTRANSFORMS APPLICATIONS H A N D B O O K
T H I R D E D I T I O N
A N D
Series Editor Richard C. Dorf University of California, Davis
Titles Included in the Series
The Avionics Handbook, Second Edition, Cary R. Spitzer The Biomedical Engineering Handbook, Third Edition, Joseph D. Bronzino The Circuits and Filters Handbook, Third Edition, Wai-Kai Chen The Communications Handbook, Second Edition, Jerry Gibson The Computer Engineering Handbook, Vojin G. Oklobdzija The Control Handbook, William S. Levine CRC Handbook of Engineering Tables, Richard C. Dorf Digital Avionics Handbook, Second Edition, Cary R. Spitzer The Digital Signal Processing Handbook, Vijay K. Madisetti and Douglas Williams The Electrical Engineering Handbook, Third Edition, Richard C. Dorf The Electric Power Engineering Handbook, Second Edition, Leonard L. Grigsby The Electronics Handbook, Second Edition, Jerry C. Whitaker The Engineering Handbook, Third Edition, Richard C. Dorf The Handbook of Ad Hoc Wireless Networks, Mohammad Ilyas The Handbook of Formulas and Tables for Signal Processing, Alexander D. Poularikas Handbook of Nanoscience, Engineering, and Technology, Second Edition, William A. Goddard, III, Donald W. Brenner, Sergey E. Lyshevski, and Gerald J. Iafrate The Handbook of Optical Communication Networks, Mohammad Ilyas and Hussein T. Mouftah The Industrial Electronics Handbook, J. David Irwin The Measurement, Instrumentation, and Sensors Handbook, John G. Webster The Mechanical Systems Design Handbook, Osita D.I. Nwokah and Yidirim Hurmuzlu The Mechatronics Handbook, Second Edition, Robert H. Bishop The Mobile Communications Handbook, Second Edition, Jerry D. Gibson The Ocean Engineering Handbook, Ferial El-Hawary The RF and Microwave Handbook, Second Edition, Mike Golio The Technology Management Handbook, Richard C. Dorf Transforms and Applications Handbook, Third Edition, Alexander D. Poularikas The VLSI Handbook, Second Edition, Wai-Kai Chen
TRANSFORMS APPLICATIONS
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Library of Congress Cataloging-in-Publication Data
Transforms and applications handbook / editor, Alexander D. Poularikas. -- 3rd ed. p. cm. -- (Electrical engineering handbook ; 43)
Includes bibliographical references and index. ISBN-13: 978-1-4200-6652-4 ISBN-10: 1-4200-6652-8 1. Transformations (Mathematics)--Handbooks, manuals, etc. I. Poularikas, Alexander D., 1933- II. Title. III. Series.
QA601.T73 2011 515’.723--dc22 2009018410
Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com
and the CRC Press Web site at http://www.crcpress.com
Contents
Editor .................................................................................................................................................................................................. ix
Contributors ..................................................................................................................................................................................... xi
2 Fourier Transforms ............................................................................................................................................................ 2-1 Kenneth B. Howell
3 Sine and Cosine Transforms ............................................................................................................................................ 3-1 Pat Yip
4 Hartley Transform ............................................................................................................................................................. 4-1 Kraig J. Olejniczak
5 Laplace Transforms ............................................................................................................................................................ 5-1 Alexander D. Poularikas and Samuel Seely
6 Z-Transform ........................................................................................................................................................................ 6-1 Alexander D. Poularikas
7 Hilbert Transforms ............................................................................................................................................................ 7-1 Stefan L. Hahn
8 Radon and Abel Transforms ........................................................................................................................................... 8-1 Stanley R. Deans
9 Hankel Transform .............................................................................................................................................................. 9-1 Robert Piessens
10 Wavelet Transform .......................................................................................................................................................... 10-1 Yulong Sheng
11 Finite Hankel Transforms, Legendre Transforms, Jacobi and Gegenbauer Transforms, and Laguerre and Hermite Transforms ...................................................................................................................... 11-1 Lokenath Debnath
12 Mellin Transform ............................................................................................................................................................. 12-1 Jacqueline Bertrand, Pierre Bertrand, and Jean-Philippe Ovarlez
13 Mixed Time–Frequency Signal Transformations ...................................................................................................... 13-1 G. Fay Boudreaux-Bartels
14 Fractional Fourier Transform ........................................................................................................................................ 14-1 Haldun M. Ozaktas, M. Alper Kutay, and Çagatay Candan
v
15 Lapped Transforms .......................................................................................................................................................... 15-1 Ricardo L. de Queiroz
16 Zak Transform .................................................................................................................................................................. 16-1 Mark E. Oxley and Bruce W. Suter
17 Discrete Time and Discrete Fourier Transforms ...................................................................................................... 17-1 Alexander D. Poularikas
18 Discrete Chirp-Fourier Transform ............................................................................................................................... 18-1 Xiang-Gen Xia
19 Multidimensional Discrete Unitary Transforms ....................................................................................................... 19-1 Artyom M. Grigoryan
20 Empirical Mode Decomposition and the Hilbert–Huang Transform ................................................................. 20-1 Albert Ayenu-Prah, Nii Attoh-Okine, and Norden E. Huang
Appendix A: Functions of a Complex Variable ................................................................................................................ A-1
Appendix B: Series and Summations ....................................................................................................................................B-1
Appendix C: Definite Integrals ...............................................................................................................................................C-1
Appendix E: Vector Analysis ................................................................................................................................................... E-1
Appendix F: Algebra Formulas and Coordinate Systems ............................................................................................... F-1
Index ............................................................................................................................................................................................. IN-1
vi Contents
Preface to the Third Edition
The third edition of Transforms and Applications Handbook follows a similar approach to that of the second edition. The new edition builds upon the previous one by presenting additional important transforms valuable to engineers and scientists. Numerous examples and different types of applications are included in each chapter so that readers from different backgrounds will have the opportunity to become familiar with a wide spectrum of applications of these transforms. In this edition, we have added the following important transforms:
1. Finite Hankel transforms, Legendre transforms, Jacobi and Gengenbauer transforms, and Laguerre and Hermite transforms 2. Fraction Fourier transforms 3. Zak transforms 4. Continuous and discrete Chirp–Fourier transforms 5. Multidimensional discrete unitary transforms 6. Hilbert–Huang transforms
I would like to thank Richard Dorf, the series editor, for his help. Special thanks also go to Nora Konopka, the acquisitions editor for engineering books, for her relentless drive to finish the project.
Alexander D. Poularikas
MATLAB1 is a registered trademark of The MathWorks, Inc. For product information, please contact:
The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508 647 7000 Fax: 508-647-7001 E-mail: info@mathworks.com Web: www.mathworks.com
vii
Editor
Alexander D. Poularikas received his PhD from the University of Arkansas, Fayetteville, Arkansas, and became a professor at the University of Rhode Island, Kingston, Rhode Island. He became the chairman of the engineering department at the University of Denver, Denver, Colorado, and then became the chairman of the electrical and computer engineering department at the University of Alabama in Huntsville, Huntsville, Alabama.
Dr. Poularikas has published seven books and has edited two. He has served as the editor in chief of the Signal Processing series (1993–1997) with Artech House and is now the editor in chief of the Electrical Engineering and Applied Signal Processing series as well as the Engineering and Science Primer series (1998 to present) with Taylor & Francis. He is a Fulbright scholar, a lifelong senior member of the IEEE, and a member of Tau Beta Pi, Sigma Nu, and Sigma Pi. In 1990 and in 1996, he received the Outstanding Educators Award of the IEEE, Huntsville Section. He is now a professor emeritus at the University of Alabama in Huntsville.
Dr. Poularikas has authored, coauthored, and edited the following books:
Electromagnetics, Marcel Dekker, New York, 1979. Electrical Engineering: Introduction and Concepts, Matrix Publishers, Beaverton, OR, 1982. Workbook, Matrix Publishers, Beaverton, OR, 1982. Signals and Systems, Brooks=Cole, Boston, MA, 1985. Elements of Signals and Systems, PWS-Kent, Boston, MA, 1988. Signals and Systems, 2nd edn., PWS-Kent, Boston, MA, 1992. The Transforms and Applications Handbook, CRC Press, Boca Raton, FL, 1995. The Handbook for Formulas and Tables for Signal Processing, CRC Press, Boca Raton, FL, 1998, 2nd edn. (2000), 3rd edn. (2009). Adaptive Filtering Primer with MATLAB, Taylor & Francis, Boca Raton, FL, 2006. Signals and Systems Primer with MATLAB, Taylor & Francis, Boca Raton, FL, 2007. Discrete Random Signal Processing and Filtering Primer with MATLAB, Taylor & Francis, Boca Raton, FL, 2009.
ix
Contributors
Jacqueline Bertrand National Center for Scientific Research University of Paris Paris, France
Pierre Bertrand Department of Electromagnetism and
Radar French National Aerospace Research
Establishment (ONERA) Palaiseau, France
G. Fay Boudreaux-Bartels University of Rhode Island Kingston, Rhode Island
Çagatay Candan Department of Electrical and Electronics
Engineering Middle East Technical University Ankara, Turkey
Stanley R. Deans University of South Florida Tampa, Florida
Lokenath Debnath Department of Mathematics University of Texas-Pan American Edinburg, Texas
Artyom M. Grigoryan Department of Electrical and Computer
Engineering The University of Texas San Antonio, Texas
Stefan L. Hahn Warsaw University of Technology Warsaw, Poland
Kenneth B. Howell University of Alabama in Huntsville Huntsville, Alabama
Norden E. Huang Research Center for Adaptive Data
Analysis National Central University Chungli, Taiwan
M. Alper Kutay The Scientific and Technological Research
Council of Turkey National Research Institute of Electronics
and Cryptology Ankara, Turkey
Jean-Philippe Ovarlez Department of Electromagnetism and
Radar French National Aerospace Research
Establishment (ONERA) Palaiseau, France
Mark E. Oxley Department of Mathematics and Statistics Graduate School of Engineering and
Management Air Force Institute of Technology Wright-Patterson Air Force Base, Ohio
Haldun M. Ozaktas Department of Electrical Engineering Bilkent University Ankara, Turkey
Robert Piessens Catholic University of Leuven Leuven, Belgium
Alexander D. Poularikas University of Alabama in Huntsville Huntsville, Alabama
Ricardo L. de Queiroz Xerox Corporation Webster, New York
Samuel Seely (deceased) Westbrook, Connecticut
Yulong Sheng Department of Physics, Physical
Engineering and Optics Laval University Quebec, Canada
Bruce W. Suter Air Force Research Laboratory Information Directorate Rome, New York
xi
Engineering University of Delaware Newark, Delware
Pat Yip McMaster University Hamilton, Ontario, Canada
xii Contributors
Alexander D. Poularikas University of Alabama in Huntsville
1.1 Introduction to Signals ............................................................................................................... 1-1 Functions (Signals), Variables, and Point Sets . Limits and Continuous Functions . Energy and Power Signals
1.2 Distributions, Delta Function.................................................................................................... 1-4 Introduction . Testing Functions . Definition of Distributions . The Delta Function .
The Gamma and Beta Functions 1.3 Convolution and Correlation .................................................................................................. 1-13
Convolution . Convolution Properties 1.4 Correlation................................................................................................................................... 1-19 1.5 Orthogonality of Signals ........................................................................................................... 1-19
Chebyshev Polynomials . Bessel Functions . Zernike Polynomials 1.6 Sampling of Signals.................................................................................................................... 1-47
The Sampling Theorem . Extensions of the Sampling Theorem 1.7 Asymptotic Series ....................................................................................................................... 1-52
Asymptotic Sequence . Poincaré Sense Asymptotic Sequence . Asymptotic Approximation .
Asymptotic Power Series . Operation of Asymptotic Power Series References ................................................................................................................................................ 1-55
1.1 Introduction to Signals
A knowledge of a broad range of signals is of practical import- ance in describing human experience. In engineering systems, signals may carry information or energy. The signals with which we are concerned may be the cause of an event or the conse- quence of an action.
The characteristics of a signal may be of a broad range of shapes, amplitudes, time duration, and perhaps other physical properties. In many cases, the signal will be expressed in analytic form; in other cases, the signal may be given only in graphical form.
It is the purpose of this chapter to introduce the mathematical representation of signals, their properties, and some of their applications. These representations are in different formats depending on whether the signals are periodic or truncated, or whether they are deduced from graphical representations.
Signals may be classified as follows:
1. Phenomenological classification is based on the evolution type of signal, that is, a perfectly predictable evolution defines a deterministic signal and a signal with unpredict- able behavior is called a random signal.
2. Energy classification separates signals into energy signals, those having finite energy, and power signals, those with a finite average power and infinite energy.
3. Morphological classification is based on whether signals are continuous, quantitized, sampled, or digital signals.
4. Dimensional classification is based on the number of inde- pendent variables.
5. Spectral classification is based on the shape of the fre- quency distribution of the signal spectrum.
1.1.1 Functions (Signals), Variables, and Point Sets
The rule of correspondence from a set Sx of real or complex number x to a real or complex number
y ¼ f (x) (1:1)
is called a function of the argument x. Equation 1.1 specifies a value (or values) y of the variable y (set of values in Y) corre- sponding to each suitable value of x in X. In Equation 1.1 x is the independent variable and y is the dependent variable.
A function of n variables x1, x2, . . . , xn associates values
y ¼ f (x1, x2, . . . , xn) (1:2)
of a dependent variable y with ordered sets of values of the independent variables x1, x2, . . . , xn.
The set Sx of the values of x (or sets of values of x1, x2, . . . , xn) for which the relationships (1.1) and (1.2) are defined constitutes the domain of the function. The corresponding set of Sy of values of y is the Sx range of the function.
1-1
A single-valued function produces a single value of the dependent variable for each value of the argument. A multiple-valued function attains two or more values for each value of the argument.
The function y(x) has an inverse function x(y) if y¼ y(x) implies x¼ x(y).
A function y¼ f (x) is algebraic of x if and only if x and y satisfy a relation of the form F(x, y)¼ 0, where F(x, y) is a polynomial in x and y. The function y¼ f (x) is rational if f (x) is a polynomial or is a quotient of two polynomials.
A real or complex function y¼ f(x) is bounded on a set Sx if and only if the corresponding set Sy of values y is bounded. Furthermore, a real function y¼ f(x) has an upper bound, least upper bound (l.u.b.), lower bound, greatest lower bound (g.l.b.), maximum, or minimum on Sx if this is also true for the corresponding set Sy.
1.1.1.1 Neighborhood
Given any finite real number a, an open neighborhood of the point a is the set of all points {x} such that jx aj< d for any positive real number d.
An open neighborhood of the point (a1, a2, . . . , an), where all ai are finite, is the set of all points (x1, x2, . . . , xn) such that jx1 a1j< d, jx2 a2j< d, . . . , and jxn anj< d for some positive real number d.
1.1.1.2 Open and Closed Sets
A point P is a limit point (accumulation point) of the point set S if and only if every neighborhood of P has a neighborhood contained entirely in S, other than P itself.
A limit point P is an interior point of S if and only if P has a neighborhood contained entirely in S. Otherwise P is a boundary point.
A point P is an isolated point of S if an only if P has a neighborhood in which P is the only point belonging to S.
A point set is open if and only if it contains only interior points. A point set is closed if and only if it contains all its limit points;
a finite set is closed.
1.1.2 Limits and Continuous Functions
1. A single-value function f(x) has a limit
lim x!a
f (x) ¼ L, L ¼ finite
as x! a{ f(x)! L as x! a} if and only if for each positive real number e there exists a real number d such that 0< jx aj< d implies that f (x) is defined and j f (x) Lj< e.
2. A single-valued function f(x) has a limit
lim x!1 f (x) ¼ L, L ¼ finite
as x ! 1 if and only if for each positive real number e there exists a real number N such that x>N implies that f (x) is defined and j f(x) Lj< e.
1.1.2.1 Operations with Limits
1.1.2.2 Asymptotic Relations between Two Functions
Given two real or complex functions f(x), g(x) of a real or complex variable x, we write
1. f(x)¼O[g(x)]; f(x) is of the order g(x) as x ! a if and only if there is a neighborhood of x¼ a such that j f(x)=g(x)j is bounded.
2. f(x) g(x); f(x) is asymptotically proportional to g(x) as x! a if and only if limx!a[ f(x)=g(x)] exists and it is not zero.
3. f(x) ffi g(x); f(x) is asymptotically equal to g(x) as x ! a if and only if
lim x!a
[f (x)=g(x)] ¼ 1:
4. f(x)¼ o[g(x)]; f(x) becomes negligible compared with g(x) if and only if
lim x!a
[f (x)=g(x)] ¼ 0:
5. f (x) ¼ w(x)þ O[g(x)] if f (x) w(x) ¼ O[g(x)]
f (x) ¼ w(x)þ o[g(x)] if f (x) w(x) ¼ o[g(x)]
1.1.2.3 Uniform Convergence
1. A single-valued function f(x1, x2) converges uniformly on a set S of values of x2, limx1!a f (x1, x2) ¼ L(x2) if and only if for each positive real number e there exists a real number d such that 0< jx1 aj< d implies that f(x1, x2) is defined and j f(x1, x2) L(x2)j< e for all x2 in S (d is independent of x2).
2. A single-valued function f(x1, x2) converges uniformly on a set S of values of x2 limx1!1 f (x1, x2) ¼ L(x2) if and only if for each positive real number e there exists a real number N such that for x1>N implies that f(x1, x2) is defined and j f(x1, x2) L(x2)j< e for all x2 in S.
3. A sequence of functions f1(x), f2(x), . . . converges uni- formly on a set S of values of x to a finite and unique function
lim x!1 fn(x) ¼ f (x)
TABLE 1.1 Operations with Limits
limx!a [ f (x)þ g(x)] ¼ limx!a f (x)þ limx!a g(x)
limx!a [bf (x)] ¼ b limx!a f (x)
limx!a [ f (x)g(x)] ¼ limx!a f (x) limx!a g(x)
limx!a f (x) g(x)
(limx!a g(x) 6¼ 0)
a may be finite or infinite.
1-2 Transforms and Applications Handbook
if and only if for each positive real number e there exists a real integer N such that for n>N implies that j fn(x) f (x)j< e for all n in S.
1.1.2.4 Continuous Functions
1. A single-valued function f (x) defined in the neighborhood of x¼ a is continuous at x¼ a if and only if for every positive real number e there exists a real number d such that jx aj< d implies j f(x) f(a)j< e.
2. A function is continuous on a series of points (interval or region) if and only if it is continuous at each point of the set.
3. A real function continuous on a bounded closed interval [a, b] is bounded on [a, b] and assumes every value between and including its g.l.b. and its l.u.b. at least once on [a, b].
4. A function f(x) is uniformly continuous on a set S and only if for each positive real number e there exists a real number d such that jxXj< d implies j f(x) f(X)j< e for all X in S.
If a function is continuous in a bounded closed interval [a, b], it is uniformly continuous on [a, b]. If f(x) and g(x) are continuous at a point, so are the functions f(x)þ g(x) and f(x) f(x).
1.1.2.5 Limits
1. A function f(x) of a real variable x has the right-hand limit limx!aþ f(x)¼ f(aþ)¼ Lþ at x¼ a if and only if for each positive real number e there exists a real number d such that 0< x a< d implies that f(x) is defined and j f(x) Lþj< e.
2. A function f(x) of a real variable x has the left-hand limit limx!a f(x)¼ f(a)¼ L at x¼ a if and only if for each positive real number e there exists a real number d such that 0< a< d implies that f(x) is defined and j f(x) Lj< e.
3. If limx!a f(x) exists, then limx!aþ f(x)¼ limx!a f(x)¼ limx!a f(x). Consequently, limx!a f(x)¼ limx!aþ f(x) implies the existence of limx!a f(x).
4. The function f(x) is right continuous at x¼ a if f(aþ)¼ f(a). 5. The function f(x) is left continuous at x¼ a if f(a)¼ f(a). 6. A real function f(x) has a discontinuity of the first kind
at point x¼ a if f(aþ) and f(a) exist. The greatest differ- ence between two of these number f(a), f(aþ), f(a) is the saltus of f(x) at the discontinuity. The discontinuities of the first kind of f(x) constitute a discrete and countable set.
7. A real function f(x) is piecewise continuous in an interval I if and only if f(x) is continuous throughout I except for a finite number of discontinuities of the first kind.
1.1.2.6 Monotonicity
1. A real function f(x) of a real variable x is a strongly mono- tonic in the open interval (a, b) if f(x) increases as x increases in (a, b) or if f(x) decreases as x decreases in (a, b).
2. A function f(x) is weakly monotonic in (a, b) if f(x) does not decrease, or if f(x) does not increase in (a, b). Analo- gous definitions apply to monotonic sequences.
3. A real function of a real variable x is of bounded variation in the interval (a, b) if and only if there exists a real number of M such that
Xm i¼1
j f (xi) f (xi1)j < M for all partitions
a ¼ x0 < x1 < x2 < < xm ¼ b
of the interval (a, b). If f(x) and g(x) are of bounded variation in (a, b), then f(x)þ g(x) and f(x)g(x) are of bounded variation also. The function f(x) is of bounded variation in every finite open interval where f(x) is bounded and has a finite number of relative maxima and minima and discontinuities (Dirichlet conditions).
A function of bounded variation in (a, b) is bounded in (a, b) and its discontinuities are only of the first kind.
Table 1.2 presents some useful mathematical functions.
TABLE 1.2 Some Useful Mathematical Functions
1. Signum function
(
2. Step function
u(t) ¼ 1 2 þ 1 2 sgn(t) ¼ 1 t > 0
0 t < 0
4. Pulse function
pa(t) ¼ u(t þ a) u(t a) ¼ 1 jtj < a 0 jtj > a
5. Triangular pulse
jtj < a
ga(t) ¼ eat2 , 1 < t < 1 8. Error function
erf(t) ¼ 2ffiffiffiffi p
t2dt ¼ 2ffiffiffiffi p
(1)nt2nþ1
n!(2nþ 1) Properties:
erf (1) ¼ 1, erf (0) ¼ 0, erf (t) ¼ erf (t)
erfc(t) ¼ complementary error function
¼ 1 erf (t) ¼ 2ffiffiffiffi p
p Ð1 t et2dt
9. Exponential function
10. Double exponential
f (t) ¼ eajtj, 1 < t < 1 11. Lognormal function
f (t) ¼ 1 t e‘n2t=2, 0 < t < 1
12. Rayleigh function
Signals and Systems 1-3
1.1.3.1 Energy Signals
If we consider any signal f(t) as denoting a voltage that exists across a 1 V resistor, then
f 2(t) 1
Therefore, the integral
E ¼ ðb a
f 2(t)dt joule (1:3)
representing the energy dissipated in the resistor during the time interval (a, b). A signal is called energy signal if
ð1 1
0 lim T!1
f 2(t)dt < 1 (1:5)
For complex-valued signals, we must introduce j f(t)j2 instead of f 2(t).
We may represent the energy in a finite interval in terms of the coefficients of the basis function wi; that is, we write the energy integral in the form
E ¼ ðb a
f (t) X1 n¼0
cnwn(t)dt
cn
c2n kwn(t)k2 (1:6)
w2 n(t)dt ¼ cn kwn(t)k2
Because the square of the norm kwn(t)k2 is the energy associated with the nth orthogonal function, Equation 1.6 shows that the energy of the signal is the sum of the energies of its individual orthogonal components weighted by cn. Note that this is the Parseval theorem. This equation shows that the set {wn(t)} forms an orthogonal (complete) set, and the signal energy can be calculated from this representation.
Example
(a) Ð1 0 u2(t) dt ¼ Ð10 dt ¼ 1; limT!1 1
2T
2T
2T tjT0
¼ 1 2 < 1.
This implies that u(t) is a power signal. (b) The signal eat u(t), a> 0 is an energy signal.
1.2 Distributions, Delta Function
1.2.1 Introduction
The delta function d(t) often called the impulse or Dirac delta function, occupies a central place in signal analysis. Many physical phenomena such as point sources, point charges, concentrated loads on structures, and voltage or current sources, acting for very short times, can be modeled as delta functions.
Strictly speaking, delta functions are not functions in the accepted mathematical sense, and they cannot be treated with rigor within the framework of classical analysis. However, if dis- tributions are introduced, then the concept of a delta function and operations on delta functions can be given a precise meaning.
1.2.2 Testing Functions
A distribution is a generalization of a function. Within the framework of distributions, any function encountered in appli- cations, such as unit-step functions and pulses, may be differen- tiated as many times as we desire, and any convergent series of functions may be differentiated term by term.
A testing function w(t) is a real-valued function of the real variable that can be differentiated an arbitrary number of times, and which is identical to zero outside a finite interval.
Example
a2t2 jtj < a 0 jtj a
( (1:7)
2. If f(t) is zero outside a finite interval
c(t) ¼ ð1
¼ testing function
1-4 Transforms and Applications Handbook
3. A sequence of testing functions, {wn} 1 n<1, converges to zero if all wn are identically zero outside some interval independent of n and each wn, as well as all of its deriva- tives, tends uniformly to zero.
Example
w(t)
4. Testing functions belong to a set D, where D is a linear vector space, and if w1 2 D and w2 2 D, then w1þw2 2 D and aw1 2 D for any number a.
1.2.3 Definition of Distributions
A distribution (or generalized function) g(t) is a process of assigning to an arbitrary test function w(t) a number Ng[w(t)]. A distribution is also a functional.
Example
ð1 1
f (t)w(t)dt ¼ Nf [w(t)] (1:8)
exists for every test function w(t) in the set. For example, if f(t)¼ u(t) then
ð1 1
w(t)dt (1:9)
The function u(t) is s distribution that assigns to w(t) a number equal to its area from zero to infinity.
1.2.3.1 Properties of Distributions
ð1 1
for all test functions and all numbers ai. 2. Summation
ð1 1
1 g1(t)w(t)dt þ
4. Scaling
ð1 1
6. Odd distribution
7. Derivative
ð1 1
dg(t) dt
1 g(t)
dw(t) dt
¼ ð1
1 g(t)
dw(t) dt
dt (1:16)
where the integrated term is equal to zero in view of the properties of testing functions.
8. The nth derivative
ð1 1
1 g(t)[f (t)w(t)]dt (1:18)
provided that f(t)w(t) belongs to the set of test functions. 10. Convolution
ð1 1
ð1 1
g1(t)g2(t t)dt
2 4
3 5w(t)dt
¼ ð1
1 g1(t)
ð1 1
g2(t t)w(t)dt
2 4
Signals and Systems 1-5
1.2.3.2 Definition
A sequence of distributions {gn(t)} 1 1 is said to converge to the
distribution g(t) if
for all w belonging to the set of test functions.
11. Every distribution is the limit, in the sense of distributions, of sequence of infinitely differentiable functions.
12. If gn(t)! g(t) and rn(t)! r(t) (r is a distribution), and the numbers an ! a, then
d dt
angn(t) ! ag(t) (1:21)
13. Any distribution g(t) may be differentiated as many times as desired. That is, the derivative of any distribution always exists and it is a distribution.
1.2.4 The Delta Function
1.2.4.1 Properties
Based on the distribution properties, the properties of the delta function are given below.
1. The delta function is a distribution assigning to the func- tion w(t) the number w(0); thus
ð1 1
3. Scaled
ð1 1
d(at) ¼ 1 jaj d(t)
and hence (a¼1)
d(t) ¼ d(t) ¼ even (1:24)
4. Multiplication by continuous function
ð1 1
1 d(t)[f (t)w(t)]dt ¼ f (0)w(0)
If f(t) is continuous at 0, then
f (t)d(t) ¼ f (0)d(t) (1:25)
and
(1:28)
df (0) dt
(1:30)
t dd(t) dt
¼ d(t) (1:31)
Set f(t)¼w(t)¼ 1 in Equation 1.29 to find the relation
ð1 1
dd(t) dt
is an odd function
(1)k n!
dnkd(t) dtnk
1 u(t)
dw(t) dt
and comparing with Equation 1.22 we find that
d(t) ¼ du(t) dt
(1:34)
Therefore, the generalized derivatives of discontinuous function contain impulses. An is the jump at the discon- tinuity point t¼ tn of the expression An w(t tn). Also
dd(t) dt
Hence
(1:36)
If r(t) has a finite or countably infinite number of zeros at tn on the entire t axis and these points r(t) have a continu- ous derivative dr(tn)=dt 6¼ 0, then
d[r(t)] ¼ X n
d(t tn) dr(tn) dt
2 d(t þ 1) (1:38)
d(sin t) ¼ X1
In addition, the following relation in also true:
dd[r(t)] dt
¼ X n
dd(ttn) dt
for all t0
¼ ð1
1 d(t t1)d(t t t2)dt ¼ d[t (t1 þ t2)] (1:42)
f (t) * d(t) ¼ ð1
1 f (t t)d(t)dt ¼ f (t 0) ¼ f (t) (1:43)
1.2.4.2 Distributions as Generalized Limits
We can define a distribution as a generalized limit of a sequence fn(t) of ordinary function. If there exists a sequence fn(t) such that the limit
lim n!1
ð1 1
fn(t)w(t)dt (1:44)
exists for every test function in the set, then the result is a number depending on w(t). Hence, we may define a distribution g(t) as
g(t) ¼ lim fn(t) (1:45)
and, therefore, equivalently
d(t) ¼ lim fn(t) (1:46)
Consider the two sequences shown in Figure 1.1a and b. The rectangular pulse sequence is given by
pe(t) ¼ u(t) u(t e) e
and has area unity whatever the value of e. Because w(t) is continuous, it follows that
lim e!0
e!0
ep p
ð1 1
it follows that
d(t) ¼ lim v!1
ð1 1
ðV V
2 sin Vt t
¼ 2pd(t) (1:51)
Figure 1.1c shows the derivatives of the sequence Equation 1.48. The following examples will elucidate some of the delta proper- ties and the use of the delta function in Table 1.3.
Example
Equivalence of expressions involving the delta functions:
(a) (cos tþ sin t)d(t)¼ d(t) (b) cos 2tþ sin td(t)¼ cos 2t (c) 1þ 2etd(t 1)¼ 1þ 2e1d(t 1)
Example
ð1 1
1
d(t k)dt ¼ Xn k¼1
k2 ¼ 1 6 [n(nþ 1)(2nþ 1)]
t
4
2
1
√πε
(b)
dfε(t) dt
Example
d dt (2u(t þ 1)þ u(1 t)) ¼ d
dt (2u(t þ 1)þ u[ (t 1)])
¼ 2d(t þ 1) d(t 1)
d dt ([2 u(t)] cos t) ¼ d
dt (2 cos t u(t) cos t)
¼ 2 sin t d(t) cos t þ u(t) sin t
¼ (u(t) 2) sin t d(t)
d dt
2
2
1. d(at) ¼ 1 jaj d(t)
2. d t t0 a
¼ jajd(t t0)
a
5. d(t) ¼ d(t); d(t) ¼ even function
6. Ð1 1 d(t)f (t)dt ¼ f (0)
7. Ð1 1 d(t t0)f (t) ¼ f (t0)
8. f (t)d(t) ¼ f (0)d(t)
9. f (t)d(t t0) ¼ f (t0)d(t t0)
10. td(t) ¼ 0
11. Ð1 1 Ad(t)dt ¼ Ð11 Ad(t t0)dt ¼ A
12. f (t)*d(t) ¼ convolution ¼ Ð11 f (t t)d(t)dt ¼ f (t)
13. d(t t1)*d(t t2) ¼ Ð1 1 d(t t1)d(t t t2)dt ¼ d[t (t1 þ t2)]
14. PN
n¼N d(t nT)* PN
n¼N d(t nT)¼P2N n¼2N (2N þ 1 jnj)d(t nT)
15. Ð1 1
16. Ð1 1
dd(t t0) dt
17. Ð1 1
18. f (t) dd(t) dt
¼ df (0) dt
19. t dd(t) dt
(1)n m!
, m > n
¼ df (t) dt
(1)k n!
dnkd(t) dtnk
26. dnd( t)
dtn ¼ (1)n
dnd(t) dtn

¼ ad(t)
30. d(t t0) ¼ du(t t0) dt
31. dsgn(t)
d(t tn) dr(tn) dt
6¼ 0
33. dd[r(t)]
dt ¼ X
6¼ 0, dr(t) dt
6¼ 0
34. d( sin t) ¼P1 n¼1 d(t np)
35. d(t2 1) ¼ 1 2 d(t 1)þ 1
2 d(t þ 1)
[d(t þ a)þ d(t a)]
37. d(t) ¼ lime!0 et2=effiffiffiffiffiffi ep
p
e t2 þ e2
41.
¼ td(t)þ u(t) (t 1)d(t 1) u(t 1)d(t 1)
42. combT (t)¼ P1
n¼1 d(tnT), f (t) combT (t)¼ P1
n¼1 f (nT)d(tnT)
combv0 (v) ¼ ^{combT (t)} ¼ v0
X1 n¼1 d(v nv0),v0 ¼ 2p
T
ð1 1
dt¼ (1)2 d2
dt2 [e2t sin 4 t]jt¼0 ¼ 2 2 4¼ 16
ð1 1
dt þ 2
d2d(t 2) dt2
dd(t 1) dt
dtþ 2 ð1
d2d(t 2) dt2
¼ (1)(3t2 þ 2)jt¼1 þ (1)22(6t)jt¼2
¼5þ 24¼ 19
Example
ð4 0
dt ¼ 1
e4td[ (2t 3)]dt ¼ ð4 0
e4td(2t 3)dt ¼ 1 2 e6
ð1 1
d(t np) (1)n
ð2p 2p
2p
¼ 1 2p
cosh u d uþ p
2
sin p
The gamma function is defined by the formula
G(z) ¼ ð1 0
et tz1dt, Re{z} > 0 (1:52)
We shall mainly concentrate on the positive values of z and we shall take the following relationship as the basic definition of the gamma function:
G(x) ¼ ð1 0
et tx1dt, x > 0 (1:53)
The gamma function converges for all positive values of x are shown in Figure 1.2.
The incomplete gamma function is given by
g(x, t) ¼ ðt 0
tx1et dt, x > 0, t > 0 (1:54)
The beta function is a function of two arguments and is given by
B(x, y) ¼ ð1 0
tx1(1 t)ytdt, x > 0, y > 0 (1:55)
(x) 6
1-10 Transforms and Applications Handbook
The beta function is related to the gamma function as follows:
B(x, y) ¼ G(x)G(y) G(x þ y)
(1:56)
1.2.5.1 Integral Expressions of G(x)
If we set u¼ et in Equation 1.54, then 1=u¼ et, loge(1=u)¼ t, (1=u)du¼ dt, and [loge(1=u)]
x1¼ tx1, for the limits t¼ 0 u¼ 1, and t¼1 u¼ 0. Hence
G(x) ¼ ð1 0
loge 1 u
x1
du (1:57)
Starting from the definitions and setting t¼m2 (dt¼ 2m dm) we obtain (limits are the same)
G(x) ¼ ð1 0
m2(x1)em2 2m dm
¼ 2 ð1 0
1.2.5.2 Properties and Specific Evaluations of G(x)
Setting xþ 1 in place of x we obtain
G(x þ 1) ¼ ð1 0
txþ11 et dt ¼ ð1 0
tx et dt
¼ ð1 0
tx d(et) ¼ tx et j10 þ ð1 0
xtx1 et dt
¼ xG(x) (1:59)
G(x) ¼ G(x þ 1) x
(1:60)
, x 6¼ 0, 1, 2, . . . (1:62)
From Equation 1.53 with x¼ 1, we find that G(1)¼ 1. Using Equation 1.59 we obtain
G(2) ¼ G(1þ 1) ¼ 1G(1) ¼ 1 1 ¼ 1,
G(3) ¼ G(2þ 1) ¼ 2G(2) ¼ 2 1, G(4) ¼ G(3þ 1) ¼ 3G(3) ¼ 3 2 1:
Hence we obtain
G(nþ 1) ¼ nG(n) ¼ n(n 1)! ¼ n!, n ¼ 0, 1, 2, . . . (1:63)
G(n) ¼ (n 1)!, n ¼ 1, 2, . . . (1:64)
To find G 1 2
we first set t¼ u2
G 1 2
¼ ð1 0
2eu2 du, (t ¼ u2)
Hence its square value is
G2 1 2
ðp=2 0
and thus
(1:65)
Next let us find the expression for G nþ 1 2
for integer positive
G nþ 1 2
If we proceed to apply Equation 1.61, we finally obtain
G nþ 1 2
ffiffiffiffi p
p 2n
ffiffiffiffi p
p 2nþ1
ffiffiffiffi p
Example
To find the ratio G(xþ n)=G(x n) where n is a positive integer and x n 6¼ 0, 1, 2, . . . , we proceed as follows [see Equa- tion 1.61]:
G(xþ n) G(x n)
¼ (xþ n 1)(xþ n 2)G(xþ n 2) G(x n)
¼
¼ (xþ n 1)(xþ n 2)(xþ n 3) (xþ n 2n)G(xþ n 2n) G(x n)
¼ (xþ n 1)(xþ n 2) (x n) (1:69)
Example
2nG(nþ 1)¼ 2nnG(n)¼ 2nn(n 1)G(n 1)
¼ ¼ 2nn(n 1)(n 2) 2 1 ¼ 2nn!¼ (2 1)(2 2)(2 3) (2 n)¼ 2 4 6 2n
(1:70)
If n 1 is substituted in place of n, we obtain
2 4 6 (2n 2) ¼ 2n1G(n) (1:71)
Example
G(2n) G(n)
ffiffiffiffi p
=
G nþ 1 2
G(n)2nG(nþ 1)
G(n)2 4 6 2n
(see previous example). But
1 3 5 (2n 1) ¼ 1 2 3 4 5 (2n 2)(2n 1) 2 4 (2n 2)
¼ G(2n) 2n1G(n)
p G(nþ 1)
¼ 1 3 5 (2n 1) 2 4 6 2n (1:74)
1.2.5.3 Remarks on Gamma Function
1. The gamma function is continuous at every x except 0 and the negative integers.
2. The second derivative is positive for every x> 0, and this indicates that the curve y¼G(x) is concave upward for all x> 0.
3. G(x) !þ1 as x ! 0þ through positive values and as x !þ1.
4. G(x) becomes, alternatively, negatively infinite and posi- tively infinite at negative integers.
5. G(x) attains a single minimum for 0< x<1 and is located between x¼ 1 and x¼ 2.
The beta function is defined by
B(x, y) ¼ ð1 0
tx1(1 t)y1dt, x > 0, y > 0 (1:75)
From the above definition we write
B(y, x) ¼ ð1 0
ty1(1 t)x1dt ¼ ð0 1
(1 s)y1sx1ds
sx1(1 s)y1 ds ¼ B(x, y) (1:76)
where we set 1 t¼ s. If we set t¼ sin2u, dt¼ 2sin u cos u du and the limits of u are 0
and p=2, then
2 sin2x1 u cos2y1 u du (1:77)
The integral representation of the beta function is given by
B(x, y) ¼ ð1 0
ux1 du
(uþ 1)xþy , x > 0, y > 0 (1:78)
Set t¼ pt in Equation 1.52 and find the relation
ð1 0
, Re{p} > 0 (1:79)
Next set p¼ 1þ u and z¼ xþ y in the above equation to find that
1
1-12 Transforms and Applications Handbook
Substituting Equation 1.80 in Equation 1.78, we obtain
B(x, y) ¼ 1 G(x þ y)
ð1 0
eut ux1 du
¼ G(x) G(x þ y)
ð1 0
et ty1 dt ¼ G(x)G(y) G(x þ y)
(1:81)
sin pp , 0 < p < 1 (1:82)
From the identities G(xþ 1)¼ xG(x), G(x)¼G(1 x)=(x), B(x, y)¼G(x)G(y)=G(xþ y) together with Equation 1.82, we obtain
G(p)G(1 p) ¼ p
Example
, n > 0, a > 1
ð1 0
y aþ 1
¼ (aþ 1)n ð1 0
yn1x ey dy ¼ G(n) (aþ 1)n
Example
To evaluate the integral Ð1 0 ex2dx, we write it in the form
ð1 0
x0ex2dx
which, if compared with the integral in Table 1.4, we have the correspondence a¼ 0, b¼ 1, c¼ 2. Hence we obtain
ð1 0
cb(aþ1)=c
1.3.1 Convolution
Convolution of functions, although a mathematical relation, is extremely important to engineers. If the impulse response of a system is known, that is, the response of the system to a delta function input, the output of the system is the convolution of the
TABLE 1.4 Gamma and Beta Function Relations
G(x) ¼ Ð10 et tx1dt x> 0
G(x) ¼ Ð10 2u2x1eu2du x> 0
G(x) ¼ Ð 10 log 1 r
x1
G(x) ¼ G(x þ 1) x
x 6¼ 0, 1, 2, . . .
G(x) ¼ (x 1)G(x 1) x 6¼ 0, 1, 2, . . .
G(x) ¼ G(1 x) x
x 6¼ 0, 1, 2, . . .
G(n) ¼ (n 1)! n¼ 1, 2, 3, . . . , 0!¼ 1
G 1 2
ffiffiffiffi p
p 2n
p p
G n 1 2
p p
2n1 n¼ 1, 2, . . .
G(nþ 1) ¼ 2 4 6 2n 2n n¼ 1, 2, . . .
G(2n) ¼ 1 3 5 (2n 1)G(n)21n n¼ 1, 2, . . .
G(2n) G(n)
ffiffiffiffi p
n! ¼ n e
< h n!
c
a>1, b> 0, c> 0
B(x, y) ¼ Ð 10 tx1(1 t)y1 dt x> 0, y> 0
B(x, y) ¼ Ð p=20 2 sin2x1 u cos2y1 u du x> 0, y> 0
B(x, y) ¼ Ð10 ux1
(uþ 1)xþy du x> 0, y> 0
B(x, y) ¼ G(x)G(y) G(x þ y)
B(x, y)¼B(y, x)
B(x, 1 x) ¼ p
sin xp 0< x< 1
B(x, y) ¼ B(x þ 1, y)þ B(x, y þ 1) x> 0, y> 0
B(x, nþ 1) ¼ 1 2 n x(x þ 1) (x þ n)
x> 0
Signals and Systems 1-13
input and its impulse response. The convolution of two functions is given by
g(t) ¼: f (t)*h(t) ¼ ð1
1 f (t)h(t t)dt (1:84)
Proof Let f (t) be written as a sum of elementary fi(t). The output g(t) is also given by the sum of the outputs gi(t) due to each elementary function fi(t). (Table 1.5) Hence
f (t) ¼ X i
fi(t), g(t) ¼ X i
gi(t) (1:85)
If Dt is sufficiently small, the area of fi(t) equals f(ti) Dt
(see Figure 1.3). Hence, the output is approximately f(ti) Dt h (t ti) because fi(t) is concentrated near the point ti. As Dt ! 0, we thus conclude that
X i
h(t) ¼ 0, t < 0 (1:86)
and, therefore, the output of the system becomes
g(t) ¼ ðt
ð1 0
f (t t)h(t)dt (1:87)
If, also, f (t)¼ 0 for t< 0, then g(t)¼ 0 for t< 0; for t> 0 we obtain
g(t) ¼ ðt 0
f (t t)h(t)dt (1:88)
The convolution does not exist for all functions. The sufficient conditions are
1. Both f (t) and h(t) must be absolutely integrable in the interval (1, 0].
2. Both f (t) and h(t) must be absolutely integrable in the interval [0, 1).
3. Either f (t) or h(t) (or both) must be absolutely integrable in the interval (1, 1).
For example, the convolution cos v0t * cos v0t does not exist.
Example
If the functions to be convoluted are
f (t) ¼ 1, 0 < t < 1, h(t) ¼ etu(t)
then the output is given by
g(t) ¼ ð1
The ranges are
1. 1< t< 0. No overlap of f(t) and h(t) takes place. Hence, g(t)¼ 0.
2. 0< t< 1. Overlap occurs from 0 to t. Hence
g(t) ¼ ðt 0
etdt ¼ 1 et
TABLE 1.5 G(x), 1 x 1.99
x 0 1 2 3 4 5 6 7 8 9
1.0 1.0000 .9943 .9888 .9835 .9784 .9735 .9687 .9642 .9597 .9555
.1 .9514 .9474 .9436 .9399 .9364 .9330 .9298 .9267 .9237 .9209
.2 .9182 .9156 .9131 .9108 .9085 .9064 .9044 .9025 .9007 .8990
.3 .8975 .8960 .8946 .8934 .8922 .8912 .8902 .8893 .8885 .8879
.4 .8873 .8868 .8864 .8860 .8858 .8857 .8856 .8856 .8857 .8859
.5 .8862 .8866 .8870 .8876 .8882 .8889 .8896 .8905 .8914 .8924
.6 .8935 .8947 .8959 .8972 .8986 .9001 .9017 .9033 .9050 .9068
.7 .9086 .9106 .9126 .9147 .9168 .9191 .9214 .9238 .9262 .9288
.8 .9314 .9341 .9368 .9397 .9426 .9456 .9487 .9518 .9551 .9584
.9 .9618 .9652 .9688 .9724 .9761 .9799 .9837 .9877 .9917 .9958
t
Δτ
1-14 Transforms and Applications Handbook
3. 1< t<1, Overlap occurs from 0 to 1. Hence
g(t) ¼ ð1 0
1.3.1.1 Definition: Convolution Systems
The convolution of any continuous and discrete system is given respectively by
y(t) ¼ ð1
m¼1 h(n,m)x(m) (1:90)
If the systems are time invariant, the kernels h() are functions of the difference of their argument. Hence
h(n,m) ¼ h(nm), h(t, t) ¼ h(t t)
and therefore
1.3.1.2 Definition: Impulse Response
The impulse response h(t) of a system is the result of a delta function input to the system. Its value at t is the response to a delta function at t¼ 0.
Example
The voltage yc(t) across the capacitor of an RC circuit in series with an input voltage source y(t) is given by
dyc(t) dt
y(t)
For a given initial condition yc(t0) at time t¼ t0 the solution is
yc(t) ¼ e(tt0)=RCyc(t0)þ 1 RC
ðt t0
e(tt)=RCy(t)dt, t t0
For a finite initial condition and t0 !1, the above equation is written in the form
yc(t) ¼ 1 RC
et=RC
h(t) ¼ 1 RC
Example
A discrete system that smooths the input signal x(n) is described by the difference equation
y(n) ¼ ay(n 1)þ (1 a)x(n), n ¼ 0, 1, 2, . . .
By repeated substitution and assuming zero initial condition y(1)¼ 0, the output of the system is given by
y(n) ¼ (1 a) Xn m¼0
anmx(m), n ¼ 0, 1, 2, . . . (1:93)
If we define the impulse response of the system by
h(n) ¼ (1 a)an , n ¼ 0, 1, 2, . . .
the system has an input–output relation
y(n) ¼ X1
which indicates that the system is a convolution one.
Example
y(t) ¼ ð1
which shows that its impulse response is h(t)¼ d(t t0).
1.3.1.3 Definition: Nonanticipative Convolution System
A system, discrete or continuous, is nonanticipative if and only if its impulse response is
h(t) ¼ 0, t < 0
with t ranging over the range in which the system is defined. If the delay t0 of a pure delay system is positive, then the
system in nonanticipative; and if it is negative, the system is anticipative.
Signals and Systems 1-15
ð1 1
f (t t)h(t)dt
Set t t¼ t0 in the first integral, and then rename the dummy variable t0 to t.
Distributive
g(t) ¼ f (t)*[h1(t)þ h2(t)] ¼ f (t)*h1(t)þ f (t)*h2(t)
This property follows directly as a result of the linear property of integration.
Associative
Shift invariance
g(t t0) ¼ f (t t0)*h(t) ¼ ð1
1 f (t t0)h(t t)dt
Write g(t) in its integral form, substitute t t0 for t, set tþ t0¼ t0, and then rename the dummy variable.
Area property
mf ¼ ð1
Kf ¼ mf
The convolution g(t)¼ f (t) * h(t) leads to
Ag ¼ Af Ah
Proof
¼ ð1
Scaline property
*h
g(t) ¼ f (t)*h(t) ¼ [fr(t)þ jfi(t)]*[hr(t)þ jfhi(t)]
¼ [fr(t)*hr(t) fi(t)*hi(t)]þ j[fr(t)*hi(t)þ fi(t)*hr(t)]
Derivative of delta function
¼ ð1
Moment expansion
Expand f(t t) in Taylor series about the point t¼ 0
f (t t)¼ f (t) tf (1)(t)þ t2
2! f (2)(t)þ þ ( t)n1
(n 1)! f (n1)(t)þ en
1-16 Transforms and Applications Handbook
Insert into convolution integral
1 h(t)dt f (1)
(1)n1 ð1
2! f (2)(t)
where bracketed numbers in exponents indicate order of differ- entiation.
Truncation Error
En ¼ 1 n!
(t)nf (n)(t t1)h(t)dt
Because t1 depends on t, the function f (n)(t t1) cannot be taken outside the integral. However, if f (n)(t) is continuous and tnh(t) 0, then
En ¼ 1 n! f (n)(t t0)
ð1 1
where t0 is some constant in the interval of integration.
Fourier transform
Proof
¼ ð1
Inverse Fourier transform
Band-limited function
If f(t) is s-band limited, then the output of a system is
g(t) ¼ ð1
X1 n¼1
g(t) ¼ f (t)*hs(t) ¼ X1
n¼1 Tf (nT)d(t nT)
" # *hs(t)
The convolution properties are given in Table 1.6.
1.3.2.1 Stability of Convolution Systems
1.3.2.1.1 Definition: Bounded Input Bounded Output (BIBO) Stability
A discrete or continuous convolution system with impulse response h is BIBO stable if and only if the impulse satisfies the inequality, Snjhj < 1 or
Ð R jh(t)jdt < 1. If the system is BIBO
stable, then
jh(t)jdt supjx(t)j, t 2 R
for every finite amplitude input x(t) (y is the input of the system).
Example
If the impulse response of a discrete system is h(n)¼ abn, n¼ 0, 1, 2, . . . , then
X1 n¼0
jajjbjn ¼ jaj 1 1jbj jb j< 1
1 jb j1
Signals and Systems 1-17
The above indicates that for jbj< 1 the system is BIBO and for jbj 1 the system is unstable.
Example
If h(t)¼ u(t) then jh(t)j ¼ Ð10 ju(t)jdt ¼ 1, which indicates the system is not BIBO stable.
1.3.2.1.2 Harmonic Inputs
If the input function is of complex exponential order ejvt, then its output is
y(t) ¼ ð1
ð1 1
h(t)ejvt dt ¼ H(v)ejvt
The above equation indicates that the output is the same as the input ejvt with its amplitude modified by jH(v)j and its phase by tan1 (Hi(v)=Hr(v)) where Hr(v)¼Re{H(v)} and Hi(v)¼ Im {Hi(v)}.
For the discrete case we have the relation
y(n) ¼ ejvnH(ejv)
TABLE 1.6 Convolution Properties
1. Commutative g(t) ¼ Ð11 f (t)h(t t)dt ¼ Ð11 f (t t)h(t)dt
2. Distributive g(t) ¼ f (t)*[h1(t)þ h2(t)] ¼ f (t)*h1(t)þ f (t)*h2(t)
3. Associative [[f (t)*h1(t)]*h2(t)] ¼ f (t)*[h1(t)*h2(t)]
4. Shift invariance g(t) ¼ f (t)*h(t)
g(t t0) ¼ f (t t0)*h(t) ¼ Ð1 1 f (t t0)h(t t)dt
5. Area property Af ¼ area of f(t),
mf ¼ Ð1 1 tf (t)dt ¼ first moment
Kf ¼ mf
Ag¼Af Ah, Kg¼KfþKh
6. Scaling g(t)¼ f(t) * h(t)
f t a
a
7. Complex-valued functions g(t) ¼ f (t)*h(t) ¼ [fr(t)*hr(t) fi(t)*hi(t)]þ j[fr(t)*hi(t)þ fi(t)*hr(t)]
8. Derivative g(t) ¼ f (t)*
dd(t) dt
2! f (1)(t)þ þ (1)n1
n 1! mh(n1)f
En ¼ ( 1)nmhn
n! f (n)(t t0), t0 ¼ constant in the interval of integration
10. Fourier transform F{f (t)*h(t)} ¼ F(v)H(v)
11. Inverse Fourier transform 1 2p
Ð1 1 F(v)H(v)ejvt dv ¼ Ð11 f (t)h(t t)dt
12. Band-limited function g(t) ¼ Ð11 f (t)h(t t)dt ¼P1 n¼1 Tf (nT)hs(t nT)
hs(t) ¼ 1 2p
Ðs s
H(v)ejvtdv, f (t) ¼ sband limited ¼ 0, jtj > s
13. Cyclical convolution x(n) y(n) ¼PN1 m¼0 x((nm)mod N)y(m)
14. Discrete-time x(n)*y(n) ¼P1 m¼1 x(nm)y(m)
15. Sampled x(nT)*y(nT) ¼ T P1
m¼1 x(nT mT)y(mT)
1-18 Transforms and Applications Handbook
1.4 Correlation
The cross-correlation of two different functions is defined by the relation
Rfh(t) ¼: f (t) } h(t) ¼ ð1
1 f (t)h(t t)dt ¼
ð1 1
(1:95)
When f (t)¼ h(t) the correlation operation is called autocorrela- tion.
Rff (t) ¼: f (t) } f (t) ¼ ð1
1 f (t)f (t t)dt ¼
ð1 1
Rfh(t) ¼: f (t) } h*(t) ¼ ð1
1 f (t)h*(t t)dt (1:97)
Rff (t) ¼: f (t) } f *(t) ¼ ð1
1 f (t)f *(t t)dt (1:98)
The two basic properties of correlation are
f (t) } h(t) 6¼ h(t) } f (t) (1:99)
jRff (t)j ¼: jf (t) } f *(t)j ¼ ð1
1 f (t)f *(t t)dt
ð1
Example
The cross-correlation of the following two functions, f(t)¼ p(t) and h(t)¼ e(t3) u(t 3), is given by
Rfh(t) ¼ ð1
The ranges of t are
1. t>2: Rfh(t)¼ 0 (no overlap of function)
2. 4< t<2: Rfh(t) ¼ Ð 1 3þt e
(tt3)dt ¼ 1 e2et
3. 1< t<4: Rfh(t) ¼ Ð 1 1 e
(tt3)dt ¼ ete2(e2 1)
The discrete form of correlation is given by
x(n) } y(n) ¼ X1
(1:101)
(1:102)
m¼1 x(mT nT )y*(mT )
sampled cross-correlation (1:103)
1.5.1 Introduction
Modern analysis regards some classes of functions as multidimen- sional vectors introducing the definition of inner products and expansion in term of orthogonal functions (base functions). In this section, functionsF(t), f(t), F(x), . . . symbolize either functions of one independent variable t, or, for brevity, a function of a set n independent variables t1, t2, . . . , tn. Hence, dt¼ dt1 . . . dtn.
A real or complex function f(t) defined on the measurable set E of elements {r} is quadratically integrable on E if and only if
ð E
jf (t)j2dt
exists in the sense of Lebesque. The class L2 of all real or complex functions is quadratically integrable on a given interval if one regards the functions f(t), h(t), . . . as vectors and defines
Vector sum of f (t) and h(t) as f (t)þ h(t)
Product of f (t) by a scalar a as af (t)
The inner product of f(t) and h(t) is defined as
hf , hi ¼: ð I
g(t)f *(t)h(t)dt (1:104)
where g(t) is a real nonnative function (weighting function) quadratically integrable on I.
Signals and Systems 1-19
The norm is L2 is the quantity
k f k¼ [hf , f i]1=2 ¼: ð I
g(t)jf (t)j2dt 2 4
3 5 1=2
(1:105)
If k f k exists and is different from zero, the function is normalizable.
Normalization
Inequalities
If f(t), h(t), and the nonnegative weighting function g(t) are quadratically integrable on I, then
Cauchy–Schwarz inequality
g(t)f *hdt
(1:106)
gjf þ hj2dt 0 @
Convergence in mean
dhf , hi¼: k f hk ¼: ð I
g(t)jf (t) h(t)j2dt 2 4
3 5 1=2
(1:108)
The root-mean-square difference of the above equation between the two functions f(t) and h(t) is equal to zero if and only if f(t)¼ h(t) for almost all t in I.
Every sequence in I of functions r0(t), r1(t), r2(t), . . . converges in the mean to the limit r(t) if and only if
d2hrn, ri¼: krn rk2 ¼: ð I
g(t)jrn(t) r(t)j2dt! 0 as n!1
(1:109)
lim:n!1rn(t) ¼ r(t) (1:110)
Convergence in the mean does not necessarily imply convergence of the sequence at every point, nor does convergence of all points on I imply convergence in the mean.
Riess–Fischer Theorem
The L2 space with a given interval I is complete; every sequence of quadratically integrable functions r0(t), r1(t), r2(t), . . . such that lim.m!1,n!1jrm rnj ¼ 0 (Cauchy sequence), converges in the mean to a quadratically integrable function r(t) and defines r(t) uniquely for almost all t in I.
Orthogonality
Two quadratically integrable functions f (t), h(t) are orthogonal on I if and only if
hf , hi ¼ ð I
g(t)f *(t)h(t)dt ¼ 0 (1:111)
Orthonormal
A set of function ri(t), i¼ 1, 2, . . . is an orthonormal set if and only if
hri, rji¼: ð I
g(t)ri*(t)rj(t)dt¼ dij ¼ 0 if i 6¼ j 1 if i¼ j
(i, j¼ 1, 2, . . . )
(1:112)
Every set of normalizable mutually orthogonal functions is lin- early independents.
Bessel’s inequalities
Given a finite or infinite orthonormal set w1(t), w2(t), w3(t), . . . and any function f (t) quadratically integrable over I
X i
jhwif ij2 hf , f i (1:113)
The equal sign applies if and only if f(t) belongs to the space spanned by all wi(t).
1-20 Transforms and Applications Handbook
Complete orthonormal set of functions (orthonormal bases)
A set of functions {wi(t)}, i¼ 1, 2, . . . , in L2 is a complete ortho- normal set if and only if the set satisfies the following conditions:
1. Every quadratically integrable function f(t) can be expanded in the form
f (t) ¼ hf ,w1iw1 þ hf ,w2iw2 þ þ hf ,wiiwi þ , i ¼ 1, 2, . . .
2. If (1) above is true, then
hf , f i ¼ jhf ,w1ij2 þ jhf ,w2ij2 þ
which is the completeness relation (Parseval’s identity).
3. For any pair of functions f(t) and h(t) in L2, the relation holds
hf , hi ¼ hf ,w1ihh,w1i þ hf ,w2ihh,w2i þ
4. The orthonormal set w1(t), w2(t), w3(t), . . . is not contained in any other orthonormal set in L2.
The above conditions imply the following: given a complete orthonormal set {wi(t)}, i¼ 1, 2, . . . in L2 and a set of complex numbers hf ,w1i, hf ,w2i þ such that
P1 i¼1 jhf ,wiij2 < 1,
there exists a quadratically integrable function f(t) such that hf ,w1iw1 þ hf ,w2iw2 þ converges in the mean of f(t).
Gram–Schmidt orthonormalization process Given any countable (finite or infinite) set of linear independent functions r1(t), r2(t), . . . normalizable in I, there exists an orthog- onal set w1(t), w2(t), . . . spanning the same space of functions. Hence
w1 ¼ r1,w2 ¼ r2 Ð I w1r2 dtÐ I w
2 1 dt
2 1 dt
2 2 dt
w2, etc: (1:114)
wi(t) ¼ yi(t)
kyi(t)k ¼ yi(t)
y1(t) ¼ r1(t), yiþ1(t) ¼ riþ1(t) Xi k¼1
hwk, riþ1iwk(t), i ¼ 1, 2, . . .
(1:115)
ð I
jfn(t) f (t)j2 dt
yields the least mean square error. The set {wi(t)}, i¼ 1, 2, . . . is orthonormal and the approximation to f(t) is
fn(t) ¼ a1w1(t)þ a2w2(t)þ þ anwn(t), n ¼ 1, 2, . . .
(1:116)
1.5.2.1 Relations of Legendre Polynomials
Legendre polynomials are closely associated with physical phe- nomena for which spherical geometry is important. The polyno- mials Pn(t) are called Legendre polynomials in honor of their discoverer, and they are given by
Pn(t) ¼ X[n=2] k¼0
(1)k(2n 2k)!tn2k
2nk!(n k)!(n 2k)! (1:117)

Pn(t)sn jsj < 1
8>>< >>:
(1:117a)
Table 1.7 gives the first eight Legendre polynomials. Figure 1.4 shows the first six Legendre polynomials.
TABLE 1.7 Legendre Polynomials
2
2 t
8 t2 þ 3
8 t3 þ 15
dn
Recursive formulas
(nþ 1)Pnþ1(t) (2nþ 1)tPn(t)þ nPn1(t) ¼ 0, n ¼ 1, 2, . . .
(1:119)
n ¼ 0, 1, 2, . . . (1:120)
tP0 n(t) P0
P0 nþ1(t) P0
n1(t) ¼ (2nþ 1)Pn(t) n ¼ 1, 2, . . . (1:122)
(t2 1)P0 n(t) ¼ ntPn(t) nPn1(t) (1:123)
P0(t) ¼ 1, P1(t) ¼ t (1:124)
Example
From Equation 1.117, when n is even, implies, Pn(t)¼ Pn(t) and when n is odd, Pn(t)¼Pn(t). Therefore
Pn(t) ¼ (1)nPn(t) (1:125)
Example
From Equation 1.123 t¼ 1 implies 0¼ nPn1(1) nPn1(1) or Pn(1)¼ Pn1(1). For n¼ 1 it implies P1(1)¼ P0(1)¼ 1. For n¼ 2 P2(1)¼ P1(1)¼ 1, and so forth. Hence, Pn(1)¼ 1. From Equation 1.125 Pn(1)n. Hence
Pn(1) ¼ 1, Pn(1) ¼ (1)n (1:126)
Pn(t) < 1 for1 < t < 1 (1:127)
Example
d dt
Use Equation 1.121 to find
d dt
or
(1 t2)Pn 00(t) 2tP0n(t)þ n(nþ 1)Pn(t) ¼ 0 (1:128)
We have deduced the Legendre polynomials y¼ Pn(t) (n¼ 0, 1, 2, . . . ) as the solution of the linear second-order ordinary differential equation
(1 t2)y 00(t) 2ty0(t)þ n(nþ 1)y(t) ¼ 0 (1:128a)
called the Legendre differential equation. If we let x¼ cos w then the above equation transforms to
the trigonometric form
y 00 þ (cotw)y0 þ n(nþ 1)y ¼ 0 (1:128b)
It can be shown than Equation 1.128a has solutions of a first kind
y ¼ C0 1 n(nþ 1) 2!
t2 þ n(nþ 1)(n 2)(nþ 3) 4!
t4
t3 þ (n 1)(nþ 2)(n 3)(nþ 4) 5!
t5
(1:128c)
valid for jtj< 1, C0 and C1 being arbitrary constants.
Schläfli’s integral formula
Pn(t) ¼ 1 2pj
where C is any regular, simple, closed curve surrounding t.
1.5.2.2 Complete Orthonormal System, 1 2 (2nþ 1) 1=2Pn(t) n o
The Legendre polynomials are orthogonal in [1, 1]
ð1 1
–1 –0.5
and therefore the set
2
is orthonormal.
f (t) ¼ X1 n¼0
anPn(t) 1 < t < 1 (1:132a)
an ¼ 2nþ 1 2
ð1 1
f (t)Pn(t)dt n ¼ 0, 1, 2, . . . (1:132b)
For even f(t), the series will contain term Pn(t) of even index; if f(t) is odd, the term of odd index only.
If the real function f(t) is piecewise smooth in (1, 1) and if it is square integrable in (1, 1), then the series Equation 1.132a converges of f(t) at every continuity point of f(t).
1.5.2.2.2 Change of Range
If a function f(t) is defined in [a, b], it is sometimes necessary in the application to expand the function in a series of orthogonal polynomials in this interval. Clearly the substitution
t ¼ 2 b a
x bþ a 2
2 t þ bþ a
2
(1:133)
transform the interval [a, b] of the x-axis into the interval [1, 1] of the t-axis. It is, therefore, sufficient to consider the expansion in series of Legendre polynomials of
f b a 2
anPn(t) (1:134a)
ð1 1
Pn(t)dt (1:134b)
f (t) ¼ X1 n¼0
anXn(t) (1:135a)
dn(t a)n(t b)n
ðb a
Suppose f(t) is given by

ð1 a
Using Equation 1.122, and noting that Pn(1)¼ 1, we obtain
an ¼ 1 2 [Pnþ1(a) Pn1(a)], a0 ¼ 1
2 (1 a)
f (t) ffi 1 2 (1 a) 1
2
Example
Suppose f(t) is given by

The function is an odd function and, therefore, f(t)Pn(t) is an odd function of Pn(t) with even index. Hence, an are zero for n¼ 0, 2, 4, . . . For odd index n, the product f(t)Pn(t) is even and hence
an ¼ nþ 1 2
ð1 1
ð1 0
Pn(t)dt, n¼ 1,3,5, . . .
Using Equation 1.122 and setting n¼ 2kþ 1, k¼ 0, 1, 2, . . . we obtain
a2kþ1 ¼ (4k þ 3) ð1 0
P2kþ1(t)dt ¼ ð1 0
¼ [P2kþ2(t) P2k(t)]j10 ¼ P2k(0) P2kþ2(0)
where we have used the property Pn(1)¼ 1 for all n. But
P2n(0) ¼ 1 2
2kþ 2
(1)k(2k)!(4k þ 3) 22kþ1k!(k þ 1)!
P2kþ1(t), 1 t 1 (1:137)
1.5.2.3 Associated Legendre Polynomials
If m is a positive integer and 1 t 1, then
Pm n (t) ¼ (1 t2)m=2 d
mPn(t) dtm
, m ¼ 1, 2, . . . , n (1:138)
where Pm n (t) is known as the associated Legendre function or
Ferrer’s functions.
dtnþm (t2 1)n, m ¼ 1, 2, . . . n; nþm 0
(1:139)
Properties
(nm)! (nþm)!
(nmþ 1)Pm nþ1(t) (2nþ 1)tPm
n (t)þ (nþm)Pm n1(t) ¼ 0
(1:142)
1 2nþ 1
1 2nþ 1
nþ1 (t)
n (t) [n(nþ 1)m(m 1)]Pm1 n (t)
(1:145)
m k (t)dt ¼ 0, k 6¼ n (1:146)
ð1 1
(1:147)
Example
mPn(t)dt, we use the Rodrigues formula and proceed as follows:
ð1 1
ð1 1
dtn
¼ 1 2n n!
[tm Dn1(t21)n]jt¼1 t¼1m
ð1 1
2 4
3 5
where integration by parts was used. The left expression is zero because of the presence of the expression (t2 1)n.
(a) For m< n and after m integrations by parts we obtain
ð1 1
¼ (1)mm!
t¼1¼ 0,
m < n
(b) m n. Integrate n times by parts to find the following expression:
ð1 1
Cmn ¼ (1)mm(m 1)(m 2) (m [n 1]) 2nn!
Multiplying numerator and denominator by (m n)! and incorporating the (1)n in the integrand, we obtain
ð1 1
tmn(1 t2)ndt, m n
If m n is odd the integrand is an odd function and hence is equal to zero. If m n is even then the integrand is even and hence
ð1 1
ð1 0
2n1(m n)!(mþ nþ 1)G mþnþ1
2
If m¼ n
2
¼ n!2n2nn! 2n1(2nþ 1)(2n)(2n 1)(2n 2)(2n 3) (3)(2)(1)
¼ 2nþ1(n!)2
(2nþ 1)!
2ð Þ 2n1(mn)!(mþnþ1)G mþnþ1
2ð Þ m> n,mn is even
2nþ1(n!)2
P2n(t) ¼ (1)n
t2k
P2n(0) ¼ (1)n(2n 1)! 22n1(n 1)!n!
¼ (1)n2n[(2n 1)!] 22nn[(n 1)!]n!
¼ (1)n(2n)!
22n(n!)2
Example
To evaluate Ð 1 0 Pm(t)dt for m 6¼ 0, we must consider the two
cases: m being odd and m being even.
(a) m is even and m 6¼ 0
ð1 0
The result is due to the orthogonality principle.
(b) m is odd and m 6¼ 0. From the relation (see Table 1.8)
ð1 t
[Pm1(t) Pmþ1(t)]
ð1 0
[Pm1(0) Pmþ1(0)]
ð1 0
(1) m1 2 (m 1)!
2m1 m1 2
!
2mþ1 mþ1 2
(2mþ 1)2mþ1 mþ1 2
2m mþ1 2
Example
One hemisphere of a homogeneous spherical solid is main- tained at 3008C while the other half is kept at 758C. To find the temperature distribution we must use the equation for heat conduction
qT qt

where T is the temperature t is the time k is the thermal conductivity r is the density c is the specific heat qQ=qt is the rate of heat generation
Because of the steady-state condition of the problem, qT=qt¼ qQ=qt¼ 0. Hence, the equation becomes
r2T ¼ q2T qx2
¼ 0
where T is independent of u. Assuming a solution of the form
T ¼ FG ¼ f (r)g(w)
we obtain
qT qr
1. 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X1 n¼0
k¼0
(1)k(2n 2k)!tn2k
2 , n is even;
[n=2] ¼ (n 1)=2, n is odd 3. P0(t)¼ 1
4. P2n(0) ¼ 1 2
n
5. P2nþ 1(0)¼ 0, n¼ 0, 1, 2, . . .
6. P2n(t)¼P2n(t), P2nþ 1(t)¼P2nþ 1(t), n¼ 0, 1, 2, . . .
7. Pn(t)¼ (1)nPn(t), n¼ 0, 1, 2, . . .
8. Pn(1)¼ 1, n¼ 0, 1, 2, . . . ;
Pn(1)¼ (1)n, n¼ 0, 1, 2, . . .
9. Pn(t) ¼ 1 2nn!
dn
dtn (t2 1)n ¼ Rodrigues formula, n¼ 0, 1, 2, . . .
10. (nþ 1)Pnþ 1(t) (2nþ 1)tPn(t)þ nPn1(t)¼ 0, n¼ 1, 2, . . .
11. P0 nþ1(t) 2tP0
n(t)þ P0 n1(t) Pn(t) ¼ 0, n¼ 1, 2, . . .
12. P0 n1(t) ¼ Pn(t)þ 2tP0
n(t) P0 nþ1(t) n¼ 1, 2, . . .
13. P0 nþ1(t) ¼ Pn(t)þ 2tP0
n(t) P0 n1(t) n¼ 1, 2, . . .
14. P0 nþ1(t) tP0
15. tP0 n(t) P0
16. P0 nþ1(t) P0
n1(t) ¼ (2nþ 1)Pn(t) n¼ 1, 2, . . .
17. (1 t2)P0 n(t) ¼ nPn1(t) ntPn(t) n¼ 1, 2, . . .
18. jPn(t)j< 1, 1< t< 1
19. P2n(t) ¼ (1)n
t2k, n¼ 0, 1, 2, . . .
20. (1 t2)P0 n(t) ¼ (nþ 1)[tPn(t) Pnþ1(t)], n¼ 0, 1, 2, . . .
21. Ð 1 1 Pn(t)dt ¼ 0, n¼ 1, 2, . . .
22. jPn(t)j 1, jtj 1
23. Ð 1 1 Pn(t)Pm(t)dt ¼ 0, n 6¼ m
24. Ð 1 1 [Pn(t)]
2dt ¼ 2 2nþ 1
, n¼ 0, 1, 2, . . .
25. 1 2
Ð 1 1 t
mPs(t)dt ¼ m(m 2) (m sþ 2) (mþ sþ 1)(mþ s 1) (mþ 1)
, m, s are even
Ð 1 1 t
mPs(t)dt ¼ (m 1)(m 3) (m sþ 2) (mþ sþ 1)(mþ s 1) (mþ 2)
, m, s are odd
4n2 1 , n¼ 1, 2, . . .
28. Ð 1 1 Pn(t)P
0 nþ1(t)dt ¼ 2, n¼ 0, 1, 2, . . .
29. Ð 1 1 tP
0 n(t)Pn(t)dt ¼
2n 2nþ 1
30. Ð 1 1 (1 t2)P0
n(t)P 0 k(t)dt ¼ 0, k 6¼ n
31. Ð 1 1 (1 t)1=2Pn(t)dt ¼ 2
ffiffiffi 2
32. Ð 1 1 t
2Pnþ1(t)Pn1(t)dt ¼ 2n(nþ 1) (4n2 1)(2nþ 3)
, n¼ 1, 2, . . .
2 1)Pnþ1(t)P0 n(t)dt ¼
, n¼ 1, 2, . . .
nPn(t)dt ¼ 2nþ1(n!)2
35. Ð 1 1 t
2[Pn(t)] 2dt ¼ 2
(2nþ 1)2 (nþ 1)2
2nþ 3 þ n2
Introducing these relations in the Laplacian, we obtain
2rG dF dr
¼ 0
dr2
dr2
G
Setting the above ratios equal to positive constant k2, k 6¼ 0, we obtain
r2 d2F dr2
þ k2G ¼ 0
TABLE 1.8 (continued) Properties of Legendre and Associate Legendre Functions
36. Pm n (t) ¼ (1 t2)m=2 dm
dtm Pn(t), m> 0
1 2nn!
38. Pm n (t) ¼ (1)m
(nm)! (nþm)!
Pm n (t)
40. (nmþ 1)Pm nþ1(t) (2nþ 1)tPm
n (t)þ (nþm)Pm n1(t) ¼ 0
41. (1 t2)1=2Pm n (t) ¼
1 2nþ 1
1 2nþ 1
(nmþ 1)(nmþ 2)Pm1 nþ1 (t)]
43. Pmþ1 n (t) ¼ 2mt(1 t2)1=2Pm
n (t) [n(nþ 1)m(m 1)]Pm1 n (t)
44. Ð 1 1 P
m n (t)P
k 6¼ n
n (t) 2
(nþm)! (nm)!
n (t)
48. P1 2n(0) ¼ 0, P1
2nþ1(0) ¼ (1)n(2nþ 1)!
Pm n (0) ¼ (1)(nm)=2 (nþm)!
2n[(nm)=2]![(nþm)=2]! , nþm is even
50. Ð 1 1 P
m n (t)P
k n(t)(1 t2)1dt ¼ 0, k 6¼ m
51. Ð 1 1 (1 t2)1=2P2m(t)dt ¼ G 1
2þmð Þ m!
G 1 2 þm
G 3 2 þm
1 2nþ 1
[Pn1(t) Pnþ1(t)]
k¼0
, q>1
1=2Pn(t)dt ¼ 2(1)n=2
2nþ 1 n is even
2(1)(n1)=2
8>>< >>:
1=2Pn(t)dt ¼ 2(1)(nþ2)=2
(2n 1)(2nþ 3) n is even
2(1)(nþ3)=2
8>>< >>:
Signals and Systems 1-27
For k2¼ n(nþ 1), we recognize that the above equation is the Legendre equation with G playing the role of y. Thus, a particular solution is
G ¼ CnPn(cos w)
where Cn is an arbitrary constant. With k2¼ n(nþ 1) the general solution for F is given by
F ¼ Snr n þ Bn
rnþ1
where Sn and Bn are arbitrary constants. Because for r¼ 0 the second term becomes infinity, we set Bn¼ 0. Hence, the product solution is
T ¼ FG ¼ SnCnr nPn(cos w) ¼ Dnr
nPn(cos w)
Because Legendre polynomials are continuous we must create a procedure to alleviate this problem. We denote the excess of the temperature T on the upper half of the surface over that of T on the lower half. On the bounding great circle between these halves, we arbitrarily set it equal to (300 75)=2. We then have
TE(w) ¼ 225 0 w < p=2
0 p=2 < w p
225=2 w ¼ p=2
8>>< >>:
If we let x¼ cos w, then TE(w) becomes f(x)
f (x) ¼ 225 0 < x 1
0 1 x < 0
8>< >:
anPn(x), an ¼ 2nþ 1 2
ð1 0
f (x)Pn(x)dx
16 P3(x)þ 11

Setting Dn¼ an=R n, where an is the coefficient of Pn(x) and R is
the radius of the solid, the solution is given by
T(r,w) ¼ 75þ X1 n¼0
an r R
n Pn(cos w)
r R
Table 1.8 gives relationships of Legendre and associated Legendre functions.
1.5.3 Hermite Polynomials
1.5.3.1 Generating Function
If we define the Hermite polynomial by the Rodrigues formula
Hn(t) ¼ (1)net 2 dnet2
dtn , n ¼ 0, 1, 2, . . . , 1 < t < 1
(1:148)
H0(t) ¼ 1,
H1(t) ¼ 2t,
and therefore
(1)kn! k!(n 2k)!
(2t)n2k (1:149)
[n=2] largest integer n=2
The Hermite polynomials are orthogonal with weight g(t) ¼ et2
on the interval (1, 1). The relation between Hermite polynomial and the generating
function is
w(t, x) ¼ e2txx2 ¼ X1 n¼0
Hn(t) n!
xn, jxj < 1 (1:150)
Because w(t, x) is the entire function in x it can be expanded in Taylor’s series at x¼ 0 with jxj<1.
Hence the derivatives of the expansion are
qnw qxn
dun
1-28 Transforms and Applications Handbook
Example
Let t¼ 0 in Equation 1.150 and expand ex2 in power series. Comparing equal powers of both sides we find that
H2n(0) ¼ (1)n (2n)! n!
Hermite polynomials are even for even n and odd for n odd. Hence,
Hn(t) ¼ (1)nHn(t) (1:151)
1.5.3.2 Recurrence Relation
If we substitute w(t, x) of Equation 1.150 into identity
qw qx
Hn(t) n!
Hn(t) n!
n! þ H1(t) 2tH0(t) ¼ 0
But H1(t) 2tH0(t)¼ 0 and hence
Hnþ1(t) 2tHn(t)þ 2nHn1(t) ¼ 0, n ¼ 1, 2, . . . (1:152)
If we use
H0 n(t) ¼ 2nHn1(t), n ¼ 1, 2, . . . (1:153)
Eliminating Hn1(t) from Equations 1.153 and 1.152, we obtain
Hnþ1(t) 2tHn(t)þ H0 n(t) ¼ 0, n ¼ 0, 1, 2, . . . (1:154)
Differentiate Equation 1.153, combine with Equation 1.152, and use the relation H0
nþ1 ¼ 2(nþ 1)H(nþ1)1, we obtain
H 00 n 2tH0
n(t)þ 2nHn(t) ¼ 0, n ¼ 0, 1, 2, . . . (1:155)
From the above equation, with y¼Hn(t) (n¼ 0, 1, 2, . . . ), we observe that the Hermite polynomials are the solution to the second-order ordinary differential equation known as the Her- mite equation
y00 2ty0 þ 2ny ¼ 0 (1:156)
1.5.3.3 Integral Representation and Integral Equation
The integral representation of Hermite polynomials is given by
Hn(t) ¼ (j)n2net 2ffiffiffiffi
The integral equation satisfied by the Hermite polynomials is
et2=2Hn(t) ¼ 1
jn ffiffiffiffiffiffi 2p
(1:157a)
Also, because H2m(t) is an even function and H2mþ 1(t) is an odd function, then the above equation implies the following two integrals:
et2=2H2m(t) ¼ (1)m ffiffiffiffi 2 p
r ð1 0
et2=2H2mþ1(t) ¼ (1)m ffiffiffiffi 2 p
r ð1 0
1.5.3.4 Orthogonality Relation: Hermite Series
The orthogonality property of the Hermite polynomials is given by
ð1 1
et2Hm(t)Hn(t)dt ¼ 0 if m 6¼ n (1:159)
and
ffiffiffiffi p
200
150
100
50
H3(t)
H5(t)
H2(t)
H1(t)
–100
–150
–200
wn(t) ¼ 2nn! ffiffiffiffi p
p 1=2 et2=2Hn(t), n ¼ 0, 1, 2, . . . ,
1 < t < 1 (1:161)
THEOREM 1.1
If f(t) is piecewise smooth in every finite interval [a, a] and
ð1 1
then the Hermite series
CnHn(t), 1 < t < 1 (1:162)
Cn ¼ 1 2nn!
ffiffiffiffi p
p ð1
1 et2 f (t)Hn(t)dt n ¼ 0, 1, 2, . . . (1:163)
converges pointwise to f(t) at every continuity point and con- verges at [ f(tþ) f(t)]=2 at points of discontinuity.
Example
The function f(t)¼ t2p, p¼ 1, 2, . . . satisfies Theorem 1.1 and it is even. Hence,
t2p ¼ Xp n¼0
C2nH2n(t)
where

to find C2n, integration by parts was performed n times.
Example
The function eat, where a is an arbitrary number, satisfies Theorem 1.1. Hence
eat ¼ X1 n¼0
CnHn(t)
where
2nn! ffiffiffiffi p
2nn! ea2=4
Example
The sgn(t) function is odd and hence its expansion takes the form
sgn(t) ¼ X1 n¼0
C2nþ1H2nþ1(t)
where
ffiffiffiffi p
p ð1
which results from Equations 1.152 and 1.153, to find that
C2nþ1 ¼ H2n(0) 22n(2nþ 1)!
ffiffiffiffi p
1.5.4 Laguerre Polynomials
The generating function for the Laguerre polynomials is given by
w(t, x) ¼ (1 x)1 exp tx 1 x
h i ¼ X1 n¼0
Ln(t)x n,
By expressing the exponential function in a series, realizing that
k 1 m

and finally making the change of index m¼ n k, Equation 1.164 leads to
Ln(t) ¼ Xn k¼0
(1)kn!tk
(k!)2(n k)! n ¼ 0, 1, 2, . . . , 0 t < 1
(1:165)
The Rodrigues formula for creating Laguerre polynomials is given by
Ln(t) ¼ et
which can be verified by application of the Leibniz formula
dn
, n ¼ 1, 2, . . . (1:167)
For a real a>1 the general Laguerre polynomials are defined by the formula
Lan(t) ¼ et ta
n! dn
dtn (et tnþa), n ¼ 0, 1, 2, . . . (1:168a)
Using Leibniz’s formula
(t)k
k!(n k)! (1:168b)
Table 1.10 gives a few Laguerre polynomials. Figure 1.6 shows several Laguerre polynomials.
1.5.4.2 Recurrence Relations
The generating function w(t, x), Equation 1.164 satisfies the identity
(1 x2) qw qx
þ (t 1)w ¼ 0 (1:169)
Substituting Equation 1.164 in Equation 1.169 and equating the coefficients of xn to zero, we obtain
(nþ 1)Lnþ1(t)þ (t 1 2n)Ln(t)þnLn1(t)¼ 0, n¼ 1,2, . . .
(1:170)
dtn
k¼0
(1)kn! k!(n 2k)!
3. e2txx2 ¼P1 n¼0 Hn(t)
xn
n!
5. H2nþ1(0) ¼ 0, H0 2n(0) ¼ 0, H0
2nþ1(0)
6. Hn(t)¼ (1)n Hn(t)
7. H2n(t) are even functions, H2nþ1 (t) are odd functions
8. Hnþ1(t) 2tHn(t)þ 2nHn1(t)¼ 0, n¼ 1, 2, . . .
9. H0 n(t) ¼ 2nHn1(t), n¼ 1, 2, . . .
10. Hnþ1(t) 2tHn(t)þ H0 n(t) ¼ 0 n¼ 0, 1, 2, . . .
11. H 00 n(t) 2tH0
n(t)þ 2nHn(t) ¼ 0 n¼ 0, 1, 2, . . .
12. Hn(t) ¼ (j)n2net 2ffiffiffiffi
p p Ð1
13. et2=2Hn(t) ¼ 1
jn ffiffiffiffiffiffi 2p
¼ integral equation
r Ð1 0 ey2=2
H2m(y) cos ty dy
r Ð1 0 ey2=2
H2mþ1(y) sin ty dy
16. Ð1 1 et2Hm(t)Hn(t) dt ¼ 0, if m 6¼ n
17. Ð1 1 et2H2
n(t) dt ¼ 2nn! ffiffiffiffi p
p n¼ 0, 1, 2, . . .
18. f (t) ¼ P1 n¼0
CnHn(t) 1< t<1
Cn ¼ 1 2nn!
p Ð1 1 et2 f (t)Hn(t)dt
19. Ð1 1 tket2Hn(t)dt ¼ 0, k ¼ 0, 1, 2, . . . , n 1
20. Ð1 1 t2et2H2
n(t) dt ¼ ffiffiffiffi p
p 2nn! nþ 1
ffiffiffiffi p
p n!
2 Pn(t)
n(t)dt ¼ 2n 1 2G nþ 1
2
n!
L3(t) ¼ 1 3! (t3 þ 9t2 18t þ 6)
L4(t) ¼ 1 4! (t4 16t3 þ 72t2 96t þ 24)
Signals and Systems 1-31
(1 x) qw qt
we obtain the relation
L0n(t) L0n1(t)þ Ln1(t) ¼ 0, n ¼ 1, 2, . . . (1:172)
From this we obtain
L0n1(t) ¼ L0n(t)þ Ln1(t) (1:174)
From Equation 1.170 by differentiation we find
(nþ 1)L0nþ1(t)þ (t 1 2n)L0n(t)þ Ln(t)þ nL0n1(t) ¼ 0
(1:175)
Eliminating L0nþ1(t) and L0n1(t) by using Equations 1.173 through 1.175, we obtain
tL0n(t) ¼ nLn(t) nLn1(t) (1:176)
By differentiating Equation 1.176 and using Equation 1.172, we obtain
tLn 00(t)þ L0n(t) ¼ nLn1(t)
Next, eliminating Ln1(t) using Equation 1.176 we obtain
tLn 00(t)þ (1 t)L0n(t)þ nLn(t) ¼ 0 (1:177)
Setting y¼ Ln(t) (n¼ 0, 1, 2, . . . ), we conclude that all Ln(t) are the solution to the Laguerre equation
ty00 þ (1 t)y0 þ ny ¼ 0 (1:178)
1.5.4.3 Orthogonality, Laguerre Series
ð1 0
etLn(t) Lm(t)dt ¼ 0, n 6¼ m (1:179)
ð1 0
n! ¼ 1, n ¼ 0, 1, 2, . . . (1:180)
For the generalized Laguerre polynomials, the orthogonality rela- tions
ð1 0
et taLam(t)L a n(t)dt ¼ 0, n 6¼ m, a > 1
ð1 0
dt ¼ G(nþ aþ 1) n!
, a > 1, n ¼ 0, 1, 2, . . .
(1:181)
The orthogonal system for the generalized polynomials on the interval 0 t<1 is
wa n(t) ¼
n! G(nþ aþ 1)
1=2 et=2ta=2Lan(t), n ¼ 0, 1, 2, . . .
(1:182)
f (t) ¼ X1 n¼0
CnLn(t), 0 t < 1 (1:183)
where
et f (t)Ln(t)dt, n ¼ 0, 1, 2, . . . (1:184)
THEOREM 1.2
If f(t) is piecewise smooth in every finite interval t1 t t2, 0< t1< t2<1 and
ð1 0
et f 2(t)dt < 1
then the Laguerre series converges pointwise to f(t) at every continuity point of f(t), and at the points of discontinuity the series converges to [ f(tþ) f(t)]=2.
20
15
10
5
–5
–10
–15
–25
1-32 Transforms and Applications Handbook
If we set a¼m¼ integer (m¼ 0, 1, 2, . . . ), then Equation 1.168b becomes
Lmn (t) ¼ Xn k¼0
(1)k(nþm)!tk
The Rodrigues formula is
dtn (et tnþm) (1:186)
Example
tb ¼ X1 n¼