By Claude E. Shannon
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Extra resources for A Mathematical Theory of Communication
19. BAND L IMITED E NSEMBLES OF F UNCTIONS If a function of time f t is limited to the band from 0 to W cycles per second it is completely determined 1 by giving its ordinates at a series of discrete points spaced 2W seconds apart in the manner indicated by the 5 following result. Theorem 13: Let f t contain no frequencies over W . Then ∞ f t = ∑ Xn sin 2Wt , n 2W t , n ,∞ where Xn = f n 2W : 4 Communication theory is heavily indebted to Wiener for much of its basic philosophy and theory.
This is not possible if R1 C. The last statement in the theorem follows immediately from the definition of R1 and previous results. If it were not true we could transmit more than C bits per second over a channel of capacity C. The first part of the theorem is proved by a method analogous to that used for Theorem 11. We may, in the first place, divide the x y space into a large number of small cells and represent the situation as a discrete case. This will not change the evaluation function by more than an arbitrarily small amount (when the cells are very small) because of the continuity assumed for x y.
This work, although chiefly concerned with the linear prediction and filtering problem, is an important collateral reference in connection with the present paper. We may also refer here to Wiener’s Cybernetics (Wiley, 1948), dealing with the general problems of communication and control. 5 For a proof of this theorem and further discussion see the author’s paper “Communication in the Presence of Noise” published in the Proceedings of the Institute of Radio Engineers, v. 37, No. , 1949, pp. 10–21.
A Mathematical Theory of Communication by Claude E. Shannon