FIGURE 14.3 Periodic CT signal used in Fourier series FIGURE 14.4 Fourier coefficients for example of Fig. 14.3. An arbitrary real-valued signal s(t)can be expressed as a sum of even and odd components, s(r)=seven(r)+ Soda(n), where seven(t)=seen(-t)and soda(r)=-Sodd(-n), and where seen (t)=[s(r)+s(-t)/2 and sodd (t)=[s(r) s(-t)/2. For the trigonometric Fourier series, it can be shown that seven(t)is represented by the(even)cosine terms in the infinite series, Soda()is represented by the(odd)sine terms, and bo is the dc level of the signal. Therefore, if it can be determined by inspection that a signal has a dc level, or if it is even or odd, then the correct form of the trigonometric series can be chosen to simplify the analysis. For example, it is easily seen that the signal shown in Fig. 14.3 is an even signal with a zero dc level. Therefore, it can be accurately represented by the cosine series with b,= 2A sin(tn/2)/(In/2), n=1, 2,.., as illustrated in Fig. 14.4. In contrast, note that the sawtooth waveform used in the previous example is an odd signal with zero dc level, so that it can be completely specified by the sine terms of the trigonometric series. This result can be demonstrated by pairing each positive frequency component from the exponential series with its conjugate partner; i.e., c,=sin(not) a, ejme'+ a-ne-jneot, whereby it is found that c= 2A cos( nm) /(m) for this example. In general, it is found that a,=(b-c)/2 for n=1, 2,..., o=bo, and a_=a. The trigonometric Fourier series is common in the ignal processing literature because it replaces complex coefficients with real ones and often results in a simpler and more intuitive interpretation of the results onvergence of the Fourier Series The Fourier series representation of a periodic signal is an approximation that exhibits mean-squared conver- gence to the true signal. If s(t)is a periodic signal of period Tand s(t)denotes the Fourier series approximation of s(r), then s(t)and s(r)are equal in the mean-squared sense if A()-s() (147) Even when Eq (14.7)is satisfied, mean-squared error(mse) convergence does not guarantee that s(r)=s(r) at every value of t. In particular, it is known that at values of t where s(t) is discontinuous the Fourier series onverges to the average of the limiting values to the left and right of the discontinuity. For example, if to is a point of discontinuity, then s(to)=[s(+o)+s(t8)J/2, where s(to)and s(tt)were defined previously (note that at points of continuity, this condition is also satisfied by the very definition of continuity). Since the dirichlet conditions require that s(t)have at most a finite number of points of discontinuity in one period, the set S, such that s(r)*s(r) within one period contains a finite number of points, and S, is a set of measure zero in the formal mathematical sense. Therefore, s(t)and its Fourier series expansion s(t)are equal almost everywhere and s(t)can be considered identical to s(t)for analysis in most practical engineering problems. c 2000 by CRC Press LLC© 2000 by CRC Press LLC An arbitrary real-valued signal s(t) can be expressed as a sum of even and odd components, s(t) = seven(t) + sodd(t), where seven(t) = seven(–t) and sodd(t) = –sodd(–t), and where seven(t) = [s(t) + s(–t)]/2 and sodd(t) = [s(t) – s(–t)]/2 . For the trigonometric Fourier series, it can be shown that seven(t) is represented by the (even) cosine terms in the infinite series, sodd(t) is represented by the (odd) sine terms, and b0 is the dc level of the signal. Therefore, if it can be determined by inspection that a signal has a dc level, or if it is even or odd, then the correct form of the trigonometric series can be chosen to simplify the analysis. For example, it is easily seen that the signal shown in Fig. 14.3 is an even signal with a zero dc level. Therefore, it can be accurately represented by the cosine series with bn = 2A sin(pn/2)/(pn/2), n = 1, 2, …, as illustrated in Fig. 14.4. In contrast, note that the sawtooth waveform used in the previous example is an odd signal with zero dc level, so that it can be completely specified by the sine terms of the trigonometric series. This result can be demonstrated by pairing each positive frequency component from the exponential series with its conjugate partner; i.e., cn = sin(nwot) = anejnwot + a–ne –jnwot , whereby it is found that cn = 2A cos(np)/(np) for this example. In general, it is found that an = (bn – jcn)/2 for n = 1, 2, …, a0 = b0, and a–n = an*. The trigonometric Fourier series is common in the signal processing literature because it replaces complex coefficients with real ones and often results in a simpler and more intuitive interpretation of the results. Convergence of the Fourier Series The Fourier series representation of a periodic signal is an approximation that exhibits mean-squared conver￾gence to the true signal. If s(t) is a periodic signal of period T and s¢(t) denotes the Fourier series approximation of s(t), then s(t) and s¢(t) are equal in the mean-squared sense if (14.7) Even when Eq. (14.7) is satisfied, mean-squared error (mse) convergence does not guarantee that s(t) = s¢(t) at every value of t. In particular, it is known that at values of t where s(t) is discontinuous the Fourier series converges to the average of the limiting values to the left and right of the discontinuity. For example, if t0 is a point of discontinuity, then s¢(t0) = [s(t 0 – ) + s(t 0 +)]/2, where s(t 0 – ) and s(t 0 +)were defined previously (note that at points of continuity, this condition is also satisfied by the very definition of continuity). Since the Dirichlet conditions require that s(t) have at most a finite number of points of discontinuity in one period, the set St such that s(t) ¹ s¢(t) within one period contains a finite number of points, and St is a set of measure zero in the formal mathematical sense. Therefore, s(t) and its Fourier series expansion s¢(t) are equal almost everywhere, and s(t) can be considered identical to s¢(t) for analysis in most practical engineering problems. FIGURE 14.3 Periodic CT signal used in Fourier series example 2. FIGURE 14.4 Fourier coefficients for example of Fig. 14.3. mse = ( ) - ¢( ) = Ú- s t s t dt T T 2 2 2 0