The Discrete Hilbert Transform for Non-Periodic Signals

The Discrete Hilb ert Transform f or Non-Periodic S ignals Sumanth Kuma r Reddy Gangasani Abstract This note investigates the size of the guard band for non- periodic discrete Hilbert transform, which has recently been proposed fo r data hiding and secur ity applications. It is shown that a guard band equal to the durati on of the message is sufficient for a variety of analog signals and is, theref ore, likely to be adequate for discrete or digital data. Introduction The discrete Hilbert tran sform (DHT) has a variety of applic ations in signal representation and processing [2-4]. The tran sform comes in a variety of forms based on assumptions on how the signal is derived from a corresponding periodic signal. The form presented by Kak [1] takes the signal to be non-periodic. Several versions of the transform for periodic signals are also known [5], [6], [7], [8], and these have been extensively examined in the literatu re. Here we consider the non-peri odic discrete Hilbert transf orm, which has not received as much of attention as it deserves, and investig ate the question of the size of the guard band that is optimal for its use. Further motivation for this study is that Hilbert transform ation has recently been used for data hiding [9 ]. It can also be employed for waveform scrambling [10], [11] as in the new in vention on m asked audio signals [12]. The guard band could, in principle, be used to hide additional information. Therefore, its use should not be considered as an addition al burden that has no redeem ing value. The Problem The problem we consider in this paper is that of approximation of DHT using a finite number of data points. Specifically, let f(n) be non-zero over (0, n -1), then we wish to mini mize m so that () θ < ′ − ∑ = n n f n f k m m 0 2 ) ( ) ( where ) ( n f m ′ = DHT -1 (g(k)) over (- m, n+m-1) and θ is the threshold value for the e rror. 1 In other words, we are considering a guard band of m points on both sides of the signal. The restrictions on the value of θ for a continuous message would be stringent, but for discrete data (the kind used in cryptography applications ) these would be much less stringent, because the quantiz ation inherent in the data would allow its reconstruction even in the case where there is som e residua l RMS error between the original signal and its DHT transform. Definition of DHT In this section begin with the DHT formula in reference [1]. ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ − − = = ∑ ∑ = = ; ) ( 2 ; ) ( 2 ) ( )} ( { even n odd n n k n f n k n f k g n f DHT π π k even k odd Inverse DHT is given by ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ − − − − = ∑ ∑ = = ; ) ( 2 ; ) ( 2 ) ( even k odd k k n k g k n k g n f π π n even n odd Since DHT is defined over all positive and ne gative integers, one needs to investigate how many extra points are required to be ta ken into consideration in the DHT domain. Since the denominator is linear, the effect of the message points would tend to spread out in the transform dom ain. Results We have performed experiments on a variety of analog signals to find the relationship between RMS error and the size of the guard band. The consideration of analog signals was prompted by the fact tha t this helps one find the constraints much better than consideration of digital signals. Sine, ramp, square, and triangle signals were considered for the experim ents. The signal width was 90 points and we considered guard bands ranging from 0 to 900. In each of the four cases considered, the RMS error became quite close to zero as the guard band approached 90. Expectedly, the worst result was obtained for the square signal because for this case the value of the signal points at th e edges of the signal range is high and, therefore, its effect is spread out further into the guard band. 2 Si n e 0 0. 02 0. 04 0. 06 0. 08 0. 1 0. 12 0. 14 0 100 200 300 400 500 600 700 800 900 1000 Gua rd B a nd Si z e Erro r For the sine signal, the value of the RMS erro r for guard band equal to 90 is 1.02% of RMS error without guard band. Ra m p 0. 000000 5. 000000 10. 000000 15. 000000 20. 000000 25. 000000 30. 000000 35. 000000 40. 000000 45. 000000 0 100 200 300 400 500 600 700 800 900 1000 Gua rd B a nd Si z e Er r or For the ramp signal, the value of the RMS erro r for guard band equal to 90 is 0.62% of RMS without guard band. 3 Squar e 0 0. 05 0. 1 0. 15 0. 2 0. 25 0. 3 0 100 200 300 400 500 600 700 800 900 1000 Gua rd ba nd S iz e Error For the square signal, the valu e of the RMS error for guard band equal to 90 is 1.6% of RMS without guard band. Tr i angl e 0 1 2 3 4 5 6 7 8 9 0 100 200 300 400 500 600 700 800 900 1000 Gua rd B a nd Si ze Err o r For the triangle signal, the value of the RMS error for guard band equal to 90 is 1.08% of RMS without guard band. 4 In each of these cases, the RMS error for guard band equal to the message width is much less than 2%. Conclusions We have shown that the non-periodic discrete Hilbert transform can be effectively used for finite sequences. We have performed expe rim ents and determined that to use a g uard band equal to the width of the signal that is transform domain having a total of 3 times as many points as in the signal domain, works very well. The guard band can be used for hiding of information. This opens up the possibility of using DHT for many date security applications. References 1. S. Kak, The discrete Hilbert transfor m. Proc. IEEE, vol. 58, pp. 585-586, 1970. 2. A.V. Oppenheim and R.W. Schafer, Dig ital Signal Processin g. Prentice-Hall, 1975. 3. S. Kak, Causality and limits on frequency functions. Int. Journal of Electronics, vol. 30, pp. 41-47, 1971. 4. S.K. Padala and K.M.M. Prabhu, Systol ic arrays for the discrete Hilbert transform. Circuits, Devices and Systems, IEE Proceedings, vol. 144, pp. 259- 264, 1997. 5. V. Cizek, Discrete Hilbert transform. IEEE Trans. Audio Electroacoustics, vol. AU-18, pp. 340-343, 1970. 6. S. Kak, Hilbert transformation f or discrete data. Int. Journal of Electronics, vol. 34, pp. 177-183, 1973. 7. S. Kak, The discrete finite Hilbert transform. Indian Journal Pure and Applied Maths., vol. 8, pp. 1385-1390, 1977. 8. S.-C. Pei and S.-B. Jaw, Computation of discrete Hilbert tr ansform through fast Hartley transform. IEEE Trans. On Ci rcuits and Systems, vol. 36, pp. 1251-1252, 1989. 9. D. Lixin, A new approach of data hiding within speech based on hash and Hilbert transform. Int. Conf. on Systems and Networks Comm unication, 2006. 10. S. Kak and N.S. Jayant, Speech encryption using waveform scram bling. Bell System Technical Journal, vol. 56, pp. 781-808, 1977. 5 6 11. S. Kak, An overview of analog encryption. Proceedings IEE, vol. 130, Pt. F, pp. 399-404, 1983. 12. R Khan and W.P. Lafay, Apparatus and method for masking audio signals in a signal distribution. US. Patent 6272226