F s(t) FIGURE 14.6 Spectrum of the Fourier representation of a periodic signal 14.6 are different, but equivalent, representations of the Fourier line spectrum that is characteristic of periodic Generalized Complex Fourier Transform The CT Fourier transform characterized by Eqs. (141la)and (1411b)can be generalized by considering the variable jo to be the special case of u=o+ jo with o=0, writing Eqs. (14 11)in terms of u, and interpreting u as a complex frequency variable. The resulting complex Fourier transform pair is given by Eqs. (14 11a)and 4)=(2)syt (14.1la) ∫ (14.11b) The set of all values of u for which the integral of Eq. (1411b)converges is called the region of convergence, denoted ROC. Since the transform S(u) is defined only for values of u within the ROC, the path of integration in Eq (14 11a)must be defined by s so the entire path lies within the ROC. In some literature this transform pair is called the bilateral Laplace transform because it is the same result obtained by including both the negative and positive portions of the time axis in the classical Laplace transform integral. The complex Fourier transfor (bilateral Laplace transform) is not often used in solving practical problems, but its significance lies in the fact Identifying this connection reinforces the observation that Fourier and Laplace transform concepts sk se that it is the most general form that represents the place where Fourier and Laplace transform concepts merge. ommon properties because they are derived by placing different constraints on the same parent tom share DT Fourier transform The DT Fourier transform(DTFT) is obtained directly in terms of the sequence samples s[n] by taking the lationship obtained in Eq.(14.)to be the definition of the DTFT. By letting T= l so that the sampling period is removed from the equations and the frequency variable is replaced with a normalized frequency o oT, the dtFt pair is defined by Eqs.(14.12). In order to simplify notation it is not customary to distinguish on normalized(T= 1)or to the unnormalized(T+ 1)frequency variable. s(-)=∑ =(2)e c 2000 by CRC Press LLC© 2000 by CRC Press LLC 14.6 are different, but equivalent, representations of the Fourier line spectrum that is characteristic of periodic signals. Generalized Complex Fourier Transform The CT Fourier transform characterized by Eqs. (14.11a) and (14.11b) can be generalized by considering the variable jw to be the special case of u = s + jw with s = 0, writing Eqs. (14.11) in terms of u, and interpreting u as a complex frequency variable. The resulting complex Fourier transform pair is given by Eqs. (14.11a) and (14.11b): (14.11a) (14.11b) The set of all values of u for which the integral of Eq. (14.11b) converges is called the region of convergence, denoted ROC. Since the transform S(u) is defined only for values of u within the ROC, the path of integration in Eq. (14.11a) must be defined by s so the entire path lies within the ROC. In some literature this transform pair is called the bilateral Laplace transform because it is the same result obtained by including both the negative and positive portions of the time axis in the classical Laplace transform integral. The complex Fourier transform (bilateral Laplace transform) is not often used in solving practical problems, but its significance lies in the fact that it is the most general form that represents the place where Fourier and Laplace transform concepts merge. Identifying this connection reinforces the observation that Fourier and Laplace transform concepts share common properties because they are derived by placing different constraints on the same parent form. DT Fourier Transform The DT Fourier transform (DTFT) is obtained directly in terms of the sequence samples s[n] by taking the relationship obtained in Eq. (14.3) to be the definition of the DTFT. By letting T = 1 so that the sampling period is removed from the equations and the frequency variable is replaced with a normalized frequency w¢ = wT, the DTFT pair is defined by Eqs. (14.12). In order to simplify notation it is not customary to distinguish between w and w¢, but rather to rely on the context of the discussion to determine whether w refers to the normalized (T = 1) or to the unnormalized (T ¹ 1) frequency variable. (14.12a) (14.12b) FIGURE 14.6 Spectrum of the Fourier representation of a periodic signal. s t j S u e du jut j j ( ) = ( ) ( ) - • + • Ú 1 2p s s s u s t e dt jut ( ) = ( ) - • • Ú– Se sne j jn n ¢ - ¢ = -• • ( ) = Â [ ] w w sn Se e d j jn [ ] = ( ) ( ) ¢ ¢ ¢ Ú- 1 2p w w w p p