Abstract
In this chapter, we outline the Signal Theory as it is usually presented in textbooks, where each class of signals is separately developed. “Separately” means that, for each class, definitions and development are presented independently. For convenience, we call this approach the Classical Signal Theory in contrast to the approach of this book, the Unified Signal Theory. The classes of signals we will consider are:
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1.
Aperiodic continuous-time signals,
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2.
Periodic continuous-time signals,
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3.
Periodic discrete-time signals,
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4.
Periodic discrete-time signals.
These classes are introduced to give readers the classical background, before dealing with the Unified Theory.
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Notes
- 1.
- 2.
Strictly speaking, a pulse denotes a signal of “short” duration, but more generally this term is synonymous with aperiodic signal.
- 3.
For real signals, it is customary to call as the band the half of the spectral extension measure.
- 4.
We prefer to reserve the term transfer function to the Laplace transform of the impulse response.
References
R.N. Bracewell, The Fourier Transform and Its Applications, 2nd edn. (McGraw–Hill, New York, 1986)
A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill, New York, 1962)
W. Rudin, Functional Analysis (McGraw–Hill, New York, 1991)
L. Schwartz, Théorie des Distributions (Hermann, Parigi, 1966)
G. Sansone, Orthogonal Functions (Interscience, New York, 1959)
E.C. Titchmars, Introduction to the Theory of Fourier Integrals (Oxford University Press, New York, 1937)
P.M. Woodward, Probability and Information Theory, with Applications to Radar (Pergamon/Macmillan & Co., New York, 1953)
Books on Classical Signal Theory
G. Cariolaro, Teoria dei Segnali Determinati (Patron, Bologna, 1977)
H.S. Carslaw, Introduction to the Theory of Fourier’s Series and Integrals, 3rd edn. (Dover, New York, 1952)
G.R. Cooper, C.D. McGillem, Methods of Signal and System Analysis (Holt, Rinehart and Winston, New York, 1967)
G.R. Cooper, C.D. McGillem, Continuous and Discrete Signal and System Analysis (Holt, Rinehart and Winston, New York, 1974)
J.B. Cruz, M.E. Van Valkenburg, Signals in Linear Circuits (Houghton Mifflin, Boston, 1974)
H. Dym, H.P. McKean, Fourier Series and Integrals (Academic Press, New York, 1972)
L.E. Franks, Signal Theory (Prentice Hall, Englewood Cliffs, 1969)
R.A. Gabel, R.A. Roberts, Signals and Linear Systems (Wiley, New York, 1973)
D. Lindner, Introduction to Signals and Systems (McGraw–Hill, New York, 1999)
A.V. Oppenheim, A.S. Willsky, I.T. Young, Signals and Systems (Prentice Hall, Englewood Cliffs, 1983)
A. Papoulis, Signal Analysis (McGraw–Hill, New York, 1977)
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L.R. Rabiner, C.M. Rader (eds.), Digital Signal Processing (IEEE Press, New York, 1972)
M.J. Roberts, Signals and Systems (McGraw–Hill, New York, 2003)
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J. Sherrick, Concepts in Signals and Systems (Prentice Hall, Englewood Cliffs, 2001)
W.M. Siebert, Circuits, Signals and Systems (McGraw–Hill, New York, 1986)
S. Soliman, M. Srinath, Continuous and Discrete Signals and Systems (Prentice Hall, Englewood Cliffs, 1990)
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Appendix: Fourier Transform of the Signum Signal sgn(t)
Appendix: Fourier Transform of the Signum Signal sgn(t)
The Fourier transform definition (2.55b) yields
These integrals do not exist. However, sgn (t) can be expressed as the inverse Fourier transform of the function 1/(iπf), namely
provided that the integral is interpreted as a Cauchy principal value, i.e.,
Using Euler’s formula, we get
where the integrand (1/i2πf)cos (2πf) in an odd function of f, and therefore the integral is zero. Then
Now, for t=0 we find x(0)=0. For t≠0, letting
we obtain
It remains to evaluate the integral
To this end, we use the rule (2.61) giving for a Fourier pair s(t),S(f)
with s(t)=rect(t), S(f)=sinc(f) (see (2.64a, 2.64b)). Hence, we obtain
Combination of the above results gives x(t)=sgn(t).
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Cariolaro, G. (2011). Classical Signal Theory. In: Unified Signal Theory. Springer, London. https://doi.org/10.1007/978-0-85729-464-7_2
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