By H. Boffin, et al., [faint scan]
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Now we consider an asynchronous particle that arrives at the RF cavity a bit later than the reference particle. It is obvious that this particle will experience a higher voltage than the reference particle, leading to a higher energy gain. Therefore, one would expect it to arrive earlier at the cavity the next time (later in this book, we will point out that this is true only below the so-called transition energy). 4 Analogously, a particle that arrives earlier than the reference particle “sees” a lower voltage and therefore gains less energy than the reference particle.
N 1/=2 On the right-hand side, we may now rename l as n again. N 1/=2 to 1 included in Eq. N C 1/=2 to N 1: fk D N X1 cn e j 2 nk=N : nD0 This defines the formula for the inverse DFT (not only for odd N ): xk D N X1 Xn e j 2 nk=N : nD0 Please note that in the literature, the factor 1=N is sometimes not included in the definition of the DFT, but it appears in that of the inverse DFT. Our choice was determined by the close relationship to the Fourier series coefficients discussed above. Apart from the factor 1=N , the DFT and the inverse DFT differ only by the sign in the argument of the exponential function.
J! 2 j. Á 2 C D sin ! t Œ1 C cos. t dt D = Ä D C = dt D C 1 j. / 1 e 2 C 1 C! 2 si 1 dt D C = 1 e j. /t 2 j. / D j. Á 2 !. 2 sin x x 1 ! x/. t/ D h. 2/ ! t Tk / k is a sequence of Dirac pulses. t/ XZ C1 ı. t/ at the locations of the delta pulses. / D pk ı.! 0 /: kD 1 According to Eq. t d! D pk ı.! / D 2 kD 1 C1 X ck ı.! 0 /; kD 1 which is an ordinary Fourier series, as Eq. 1) shows. 0 , we get a sum of Dirac pulses that are multiplied by 2 and the Fourier coefficients. t/ as a periodic sequence of Dirac pulses.
