By Andreas Antoniou

ISBN-10: 007145425X

ISBN-13: 9780071454254

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Additional info for Digital Signal Processing

Example text

4 Lowpass ﬁltering applied to the signal depicted in Fig. 3: (a) Time-domain representation, (b) amplitude spectrum, (c) phase spectrum. respectively. The effect of differentiating the signal of Eq. 1) is illustrated in Fig. 8. As can be seen by comparing Fig. 3b and c with Fig. 8b and c, differentiation scales the amplitudes of the different frequency components by a factor that is proportional to frequency and adds a phase angle of 12 π to each value of the phase spectrum. In other words, the amplitudes of low-frequency components are attenuated, whereas those of high-frequency components are enhanced.

5) in the Fourier series of Eq. 3) converges to x˜ (t) at all points where x(t) is continuous. At points where x(t) is discontinuous, the Fourier series converges to the average of the left- and right-hand limits of x(t), namely, x(td ) = 12 [x(td −) + x(td +)] as illustrated in Fig. 1 where the left- and right-hand limits of x(t) at t = td are deﬁned as x(td −) = lim x(td − | |) →0 and x(td +) = lim x(td + | |) →0 Proof (See pp. 225–232 of Ref.

7, pp. 387–403, Aug. 1965. J. F. Kaiser, “Some practical considerations in the realization of digital ﬁlters,” Proc. Third Allerton Conf. on Circuits and Systems, pp. 621–633, Oct. 1965. K. Steiglitz, “The equivalence of digital and analog signal processing,” Information and Control, vol. 8, pp. 455–467, Oct. 1965. R. B. Blackman, Data Smoothing and Prediction, Reading, MA: Addison-Wesley, 1965. F. F. Kuo and J. F. Kaiser, System Analysis by Digital Computer, New York: Wiley, 1966. B. Gold and C.