The Bell Labs Theory
Duration limiting, or time limiting, refers to restricting a signal by setting its values equal to zero outside of a finite time interval or, more generally, outside of a compact set. Bandwidth limiting, or band limiting, refers to restricting a signal by setting its amplitudes equal to zero outside of a finite frequency interval or again, more generally, outside of a compact set. This book addresses primarily the theory of time and band limiting whose core was developed by Landau, Pollak, and Slepian in a series of papers [195, 196, 303, 309] appearing in the Bell System Technical Journal in the early to middle part of the 1960s, and a broader body of work that grew slowly but steadily out of that core up until around 1980, with a resurgence since 2000, due in large part to the importance of time and band limiting in wireless communications. The 1960s Bell Labs theory of time and band limiting is but one aspect of the Bell Labs information theory. The foundations of this encompassing theory were laid, in large part, in Nyquist’s fundamental papers “Certain Topics in Telegraph Transmission Theory” , which appeared in the Transactions of the American Institute of Electrical Engineers in 1928, and “Certain Factors Affecting Telegraph Speed,” published in April 1924 in the Bell System Technical Journal, along with Hartley’s paper “Transmission of Information,” which also appeared in the Bell System Technical Journal in 1928 . These papers quantified general ideas that were in the air, though certain specific versions were attributed to Kelvin and Wiener among others. Of course, Claude Shannon’s seminal work, “A Mathematical Theory of Communication,” which appeared in the Bell System Technical Journal in July and October 1948 , is often cited as providing the basis for much of modern communications theory. His sampling theory plays a central role in Chap. 5 of this monograph. The works of Nyquist and Hartley however remain, in some ways, more germane to the study at hand.
KeywordsLegendre Polynomial Trigonometric Polynomial Liouville Theory Hermite Function Bell System Technical Journal
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