By Bernard Widrow
A finished and sensible remedy of adaptive sign processing that includes common use of examples.
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28) = RAf) The imaginary term is zero because the integrand is an odd function. Because cos(27rft) is an even function, then hAt) cos(27rft) = heCt) cos[27r( - f)t] and H e(f) = H e( - f); the frequency function is even. 9 Even Time and Frequency Functions As shown in Fig. 7, the Fourier transform of an even time function is a real and even frequency function; conversely, the inverse Fourier transform of a real and even frequency function is an even function of time. 7 Fourier transform of an even function.
This graphical and analytical catalog is by no means complete, but does contain the most frequently encountered transform pairs. ~ K I K 2Alol hit) hit) hltl hltl Time Domain t;, 1 2f~ 3 _ 0 To To To III < ItI = III > h(t) = K&(t) h(t) = K 0 0 o <0 f sin(21Tf01) 2 f 'IT 01 h(1) - 2A 0 = - 2 A h(t) = A 2 A A = IfI > fo IfI = fo IfI < fo K&(f) =0 - = H(f) = K H(f) H(f) 21TTof H(f) = 2AT" sin(2'ITTof) K K 2ATo Hltl Hltl = K W) Hill H(f) Frequency Domain en N -5T .. -4T -3T -2T -T A 1 2T hIt) A hIt) T hIt) Time Domain 3T 4T 5T ...
1. 14. If h(t) is real, show that I H(f) I is an even function. 15. By making a substitution of a variable in Eq. 16. 17. 16(a) to find the Fourier transform of a pulse waveform given the Fourier transform of a triangular waveform. REFERENCES I. BRACEWELL, R. The Fourier Transform and its Applications, 2d Ed. New York: McGraw-Hill, 1986. 2. PAPOULIS, A. The Fourier Integral and Its Applications, 2d Ed. New York: McGraw-Hill, 1984. 3. CHAMPENEY, D. C. Fourier Transforms and Their Physical Application.
Adaptive Signal Processing by Bernard Widrow