The condition described above of convergence almost everywhere is satisfied only in the limit as an infinite number H(el)l of terms are included in the Fourier series expansion. If the infinite series expansion of the Fourier series is truncated to UNCATED a finite number of terms, as it must always be in practical pplications, then the approximation will exhibit an oscilla- tory behavior around the discontinuity, known as the Gibbs [ Van Valkenburg, 1974]. Let sN(t) deno truncated Fourier series approximation of s(t), where only the terms in Eq (145a)from n=-N to n= Nare included if the complex Fourier series representation is used or where only the terms in Eq (146a)from n=0 to n= Nare included FIGURE 14.5 Gibbs phenomenon in a low-pass if the trigonometric form of the Fourier series is used. It is digital filter caused by truncating the impulse well known that in the vicinity of a discontinuity at to the response to N terms. Gibbs phenomenon causes sN(t)to be a poor approximation to s(t). The peak magnitude of the Gibbs oscillation is 13% of the size of the jump discontinuity s(to)-s(t+) regardless of the number of terms used in the approximation. As N increases, the region which contains the oscillation becomes more concentrated in the neighborhood of the discontinuity, until, in the limit as N approaches infinity, the Gibbs oscillation is squeezed into a single point of mismatch at to. The Gibbs phenom non is illustrated in Fig. 14.5, where an ideal low-pass frequency response is approximated by an impulse response function that has been limited to having only N nonzero coefficients, and hence the Fourier series expansion contains only a finite number of terms If s(t)in Eq. (14.7)is replaced by s,(r)it is important to understand the behavior of the error msex as a function of n, wher men ()-s()dt (14.8) An important property of the Fourier series is that the exponential basis functions e/mu '(or sin(no,t)and constitute an orthonormal set; i. e, tnt=1 for n=k, and tk=0 for n+k, where for the trigonometric form) cos(mo) for the trigonometric form)forn=0,±1,±2,…(orn=0,1,2, (149) As terms are added to the Fourier series expansion, the orthogonality of the basis functions guarantees that ne error decreases monotonically in the mean-squared sense, i. e, that mseN monotonically decreases as Nis increased. Therefore, when applying Fourier series analysis, including more terms always improves the accuracy Fourier transform of periodic ct signal For a periodic signal s(r) the CT Fourier transform can then be applied to the Fourier series expansion of s(r) ion for the "line spectrum"that 2元 8 (14.10) The spectrum is shown in Fig. 14.6. Note the similarity be plot of the Fourier coefficients in Fig 14. 2, which was heuristically interpreted as a line spectrum. Figures 14.2 e 2000 by CRC Press LLC© 2000 by CRC Press LLC The condition described above of convergence almost everywhere is satisfied only in the limit as an infinite number of terms are included in the Fourier series expansion. If the infinite series expansion of the Fourier series is truncated to a finite number of terms, as it must always be in practical applications, then the approximation will exhibit an oscilla￾tory behavior around the discontinuity, known as the Gibbs phenomenon [Van Valkenburg, 1974]. Let sN¢(t) denote a truncated Fourier series approximation of s(t), where only the terms in Eq. (14.5a) from n = –N to n = N are included if the complex Fourier series representation is used or where only the terms in Eq. (14.6a) from n = 0 to n = N are included if the trigonometric form of the Fourier series is used. It is well known that in the vicinity of a discontinuity at t0 the Gibbs phenomenon causes sN¢(t)to be a poor approximation to s(t). The peak magnitude of the Gibbs oscillation is 13% of the size of the jump discontinuity s (t 0 – ) – s(t 0 +) regardless of the number of terms used in the approximation. As N increases, the region which contains the oscillation becomes more concentrated in the neighborhood of the discontinuity, until, in the limit as N approaches infinity, the Gibbs oscillation is squeezed into a single point of mismatch at t0. The Gibbs phenom￾enon is illustrated in Fig. 14.5, where an ideal low-pass frequency response is approximated by an impulse response function that has been limited to having only N nonzero coefficients, and hence the Fourier series expansion contains only a finite number of terms. If s¢(t) in Eq. (14.7) is replaced by s N ¢(t) it is important to understand the behavior of the error mseN as a function of N, where (14.8) An important property of the Fourier series is that the exponential basis functions ejnwot (or sin(nwot) and cos(nwot) for the trigonometric form) for n = 0, ±1, ±2, … (or n = 0, 1, 2, … for the trigonometric form) constitute an orthonormal set; i.e., tnk = 1 for n = k, and tnk = 0 for n ¹ k, where (14.9) As terms are added to the Fourier series expansion, the orthogonality of the basis functions guarantees that the error decreases monotonically in the mean-squared sense, i.e., that mseN monotonically decreases as N is increased. Therefore, when applying Fourier series analysis, including more terms always improves the accuracy of the signal representation. Fourier Transform of Periodic CT Signals For a periodic signal s(t) the CT Fourier transform can then be applied to the Fourier series expansion of s(t) to produce a mathematical expression for the “line spectrum” that is characteristic of periodic signals: (14.10) The spectrum is shown in Fig. 14.6. Note the similarity between the spectral representation of Fig. 14.6 and the plot of the Fourier coefficients in Fig. 14.2, which was heuristically interpreted as a line spectrum. Figures 14.2 and FIGURE 14.5 Gibbs phenomenon in a low-pass digital filter caused by truncating the impulse response to N terms. mseN = ( ) - ¢ ( ) Ú- s t s t dt N T T 2 2 2 t T e e dt nk jn t jn t T T o o = ( ) ( )( ) - Ú- 1 2 2 w w F s t F a e a n n jn t n n o n o { } ( ) = Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ Ô = - ( ) = • • = -• • Â Â w 2p d w w