Since S,(t)is a CT signal, it is appropriate to apply the CT Fourier transform to obtain an expression for the pectrum of the sampled signal T (14.3) Since the expression on the right-hand side of Eq.(14.3)is a function of e/e, it is customary to express the transform as F(ejon= fsa(t)). It will be shown later that if o is replaced with a normalized frequency a o/T, so that-It<o<T, then the right side of Eq. (14. 3)becomes identical to the DT Fourier transform that is defined directly for the sequence s[n]=s(nt). Fourier Series Representation of CT Periodic Signals The classical Fourier series representation of a periodic time domain signal s(r)involves an expansion of s(r) into an infinite series of terms that consist of sinusoidal basis functions, each weighted by a complex constant Fourier coefficient) that provides the proper contribution of that frequency component to the complete waveform. The conditions under which a periodic signal s(r)can be expanded in a Fourier series are known as the Dirichlet conditions. They require that in each period s(r) has a finite number of discontinuities, a finite number of a and minima, and that s(t) satisfies the absolute convergence criterion of Eq(14.4)Van Valkenburg, 1974 s(t)at (144) It is assumed throughout the following discussion that the Dirichlet conditions are satisfied by all functions that will be represented by a Fourier series. The Exponential Fourier Series If s(r)is a Ct periodic signal with period T, then the exponential Fourier series expansion of s(r)is given by 4)=∑qm (14.5a) where O, = 2T/T and where the a, terms are the complex Fourier coefficients given by dt (14.5b) For every value of t where s(r) is continuous the right side of Eq (145a)converges to s(t). At values of t where s(t)has a finite jump discontinuity, the right side of Eq (145a)converges to the average of s(t-)and s(t*), where ([)=lims(t-e) and s(()=lims(t+e) For example, the Fourier series expansion of the sawtooth waveform illustrated in Fig. 14. 1 is characterized by T= 2T,O.=1, 0=0, and an =a-m=A cos(n) /GinT)for n=1, 2,... The coefficients of the exponential ourier series given by Eq (145b)can be interpreted as a spectral representation of s(t), since the a,th coefficient represents the contribution of the(no,)th frequency component to the complete waveform. Since the an terms are complex valued, the Fourier domain(spectral)representation has both magnitude and phase spectra For example, the magnitude of the a, values is plotted in Fig. 14.2 for the sawtooth waveform of Fig. 14. 1. The fact that the a, terms constitute a discrete set is consistent with the fact that a periodic signal has a line spectrum c 2000 by CRC Press LLC© 2000 by CRC Press LLC Since sa (t) is a CT signal, it is appropriate to apply the CT Fourier transform to obtain an expression for the spectrum of the sampled signal: (14.3) Since the expression on the right-hand side of Eq. (14.3) is a function of ejwT , it is customary to express the transform as F(ejwT) = F{sa(t)}. It will be shown later that if w is replaced with a normalized frequency w¢ = w/T, so that –p < w¢ < p, then the right side of Eq. (14.3) becomes identical to the DT Fourier transform that is defined directly for the sequence s[n] = sa(nT). Fourier Series Representation of CT Periodic Signals The classical Fourier series representation of a periodic time domain signal s(t) involves an expansion of s(t) into an infinite series of terms that consist of sinusoidal basis functions, each weighted by a complex constant (Fourier coefficient) that provides the proper contribution of that frequency component to the complete waveform. The conditions under which a periodic signal s(t) can be expanded in a Fourier series are known as the Dirichlet conditions. They require that in each period s(t) has a finite number of discontinuities, a finite number of maxima and minima, and that s(t) satisfies the absolute convergence criterion of Eq. (14.4) [Van Valkenburg, 1974]: (14.4) It is assumed throughout the following discussion that the Dirichlet conditions are satisfied by all functions that will be represented by a Fourier series. The Exponential Fourier Series If s(t) is a CT periodic signal with period T, then the exponential Fourier series expansion of s(t) is given by (14.5a) where wo = 2p/T and where the an terms are the complex Fourier coefficients given by (14.5b) For every value of t where s(t) is continuous the right side of Eq. (14.5a) converges to s(t). At values of t where s(t) has a finite jump discontinuity, the right side of Eq. (14.5a) converges to the average of s(t –) and s(t +), where For example, the Fourier series expansion of the sawtooth waveform illustrated in Fig. 14.1 is characterized by T = 2p, wo = 1, a0 = 0, and an = a–n = A cos(np)/(jnp) for n = 1, 2, …. The coefficients of the exponential Fourier series given by Eq. (14.5b) can be interpreted as a spectral representation of s(t), since the anth coefficient represents the contribution of the (nwo)th frequency component to the complete waveform. Since the an terms are complex valued, the Fourier domain (spectral) representation has both magnitude and phase spectra. For example, the magnitude of the an values is plotted in Fig. 14.2 for the sawtooth waveform of Fig. 14.1. The fact that the an terms constitute a discrete set is consistent with the fact that a periodic signal has a line spectrum; F s t F s nT t nT s nT e a a n a j T n n { } ( ) = ( ) ( - ) Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ Ô = ( )[ ] = -• • - = -• • Â Â d w s t dt T T ( ) < • Ú- 2 2 s t a en jn t n o ( ) = = -• • Â w a T s t e dt n n jn t T T o = ( ) ( ) - • < < • - Ú- 1 2 2 w s t s t s t s t - Æ + Æ ( ) = - lim ( ) ( ) = + lim ( ) e e e e 0 0 and