FIGURE 14.1 Periodic CT signal used in Fourier series example FIGURE 14.2 Magnitude of the Fourier coefficients for the example in Fig. 14.3. i.e., the spectrum contains only integer multiples of the fundamental frequency @o Therefore, the equation pair given by Eq (145a)and(145b)can be interpreted as a transform pair that is similar to the CT Fourier transform for periodic signals. This leads to the observation that the classical Fourier series can be interprete a special transform that provides a one-to-one invertible mapping between the discrete-spectral domain and the ct domain Trigonometric Fourier Series Although the complex form of the Fourier series expansion is useful for complex periodic signals, the Fourier series can be more easily expressed in terms of real-valued sine and cosine functions for real-valued periodic signals In the following discussion it will be assumed that the signal s(r) is real valued for the sake of simplifying the discussion. When s(r) is periodic and real valued it is convenient to replace the complex exponential form of the Fourier series with a trigonometric expansion that contains sin(o r)and cos (o t)terms with corre sponding real-valued coefficients [Van Valkenburg, 1974. The trigonometric form of the Fourier series for a real-valued signal s(r)is given by 4)=∑boo)+∑csn(o 146a) where O,=2T/T. The b, and c, terms are real-valued Fourier coefficients determined by h=()4() b,=(27)4)om0)dt,n=1,2 (2/T) s()sin(no). dt l,2, (146b) e 2000 by CRC Press LLC© 2000 by CRC Press LLC i.e., the spectrum contains only integer multiples of the fundamental frequency wo. Therefore, the equation pair given by Eq. (14.5a) and (14.5b) can be interpreted as a transform pair that is similar to the CT Fourier transform for periodic signals. This leads to the observation that the classical Fourier series can be interpreted as a special transform that provides a one-to-one invertible mapping between the discrete-spectral domain and the CT domain. Trigonometric Fourier Series Although the complex form of the Fourier series expansion is useful for complex periodic signals, the Fourier series can be more easily expressed in terms of real-valued sine and cosine functions for real-valued periodic signals. In the following discussion it will be assumed that the signal s(t) is real valued for the sake of simplifying the discussion. When s(t) is periodic and real valued it is convenient to replace the complex exponential form of the Fourier series with a trigonometric expansion that contains sin(wot) and cos(wot) terms with corre￾sponding real-valued coefficients [Van Valkenburg, 1974]. The trigonometric form of the Fourier series for a real-valued signal s(t) is given by (14.6a) where wo = 2p/T. The bn and cn terms are real-valued Fourier coefficients determined by and (14.6b) FIGURE 14.1 Periodic CT signal used in Fourier series example. FIGURE 14.2 Magnitude of the Fourier coefficients for the example in Fig. 14.3. s t b n c n n n n n ( ) = ( ) + ( ) = • = • Â Â 0 0 0 1 cos w w sin b T s t dt T T 0 2 2 = (1 ) ( ) Ú- b T s t n t dt n n T T = ( ) ( ) ( ) = º Ú- 2 1 2 0 2 2 cos w , , , c T s t n t dt n n T T = ( ) ( ) ( ) = º Ú- 2 1 2 0 2 2 sin w , ,