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Extra info for A wavelet-based watermarking algorithm for ownership verification of digital images
Since we have uniform spacing f = fk+1 − fk = 1/T , we may still plot Ak and Bk versus k with the understanding that k is the index of equispaced fk ; of course, one may plot Ak and Bk versus the values of fk if that is desired. 7. REVIEW OF RESULTS AND TECHNIQUES 13 3. A non-commensurate y(t) is not periodic, although all its components are periodic. For example, the function √ y(t) = sin(2πt) + 5 sin(2 3πt) √ is not periodic because f1 = 1 and f2 = 3 are not commensurate. 7 Review of Results and Techniques In the preceding sections we show that a sum of sinusoidal modes can be expressed in a number of ways.
Since the time period T of composite y(t) is the shortest duration over which each sine or cosine component completes an integer number of cycles, we determine T by the LCM (least common multiple) of the individual periods. From fk = kf1 and Tk = 1/fk , we obtain T1 = kTk , so T1 is the LCM of the individual periods. Accordingly, the time period T of the composite y(t) is the reciprocal of the fundamental frequency f1 . Note that f1 is the GCD (greatest common divisor) of the individual frequencies.
3. Although the results derived using analog frequency with unit time spacing ( t = 1) will not be different from those derived using the digital frequency, the explicit incorporation of t in the latter s de n ition provides direct means to interpret and apply the results for values of t other than unity. Therefore in suitable contexts we may use the more convenient digital frequency in the analysis and processing of signals without loss of generality. CHAPTER 2. 3 Using analog frequency versus digital frequency.