Download e-book for iPad: An Investigation Of The Laws Of Thought by George Boole

By George Boole

ISBN-10: 1602064520

ISBN-13: 9781602064522

George Boole, the daddy of Boolean algebra, released An research of the legislation of notion, a seminal paintings on algebraic good judgment, in 1854. during this research of the elemental legislation of human reasoning, Boole makes use of the symbolic language of arithmetic to check the character of the human brain.

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From Eqs. 32), the Fourier transform of Eq. 34) H(f) = R(f) + jI(f) = HeCf) + Ho(f) where He(f) = R(f) and Ho(f) = jICf). We show in Chapter 9 that decomposition can increase the speed of computation of the FFT. 11 Exponential Waveform Decomposition To demonstrate the concept of waveform decomposition, consider the exponential function [Fig. 35) e-at Following the developments leading to Eq. 9(b) and (c) illustrate the even and odd decompositions, respectively. 9 Waveform decomposition property.

Sin(21TTof) 1 To 'IT! ) = exp ( P) 2 -: 28 The Fourier Transform Chap. 1. 2. 3. Determine the inverse Fourier transform of each of the following: ( (f) = sin(21rfn cos(2'lTfn a) H 2'lTf (b) H(f) = (1 = 0 p)2 If I < 1 otherwise Chap. 2 29 References (c) H(f) f (p + a)(P + 4a) (d) H(f) = A cos(2-rrfto) REFERENCES 1. ARsAc, J. Fourier Transforms and the Theory ofDistributions. Englewood Cliffs, NJ: Prentice Hall, 1966. 2. BRACEWELL, R. The Fourier Transform and its Applications, 2d Ed. New York: McGraw-Hill, 1986.

1 lists various complex time functions and their respective Fourier transforms. 12 Simultaneous Fourier Transforms We can employ the relationships of Eqs. 46) to simultaneously determine the Fourier transform of two real functions. To illustrate this point, recall the linearity property of Eq. 47) Let x(t) = h(t) and y(t) = jg(t), where both h(t) and g(t) are real functions. It follows that X(f) = H(f) and Y(f) = jG(f). Because x(t) is real, then from Eqs. 48) Similarly, because y(t) is imaginary, then from Eqs.

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