Abstract

Analytic signal and Hilbert transform associated with linear canonical transform (LCT) have been developed [Opt. Commun. 281, 1468 (2008)]. However, the aforementioned paper has some drawbacks; for example, this kind of analytic signal cannot preserve the positive frequency bands of the original signal under LCT operations. In this paper, these drawbacks are pointed out, and a better definition for analytic signal associated with LCT is proposed. In addition, a discrete implementation for the analytic signal associated with discrete LCT is also provided. Several numerical examples, shown with both time and Wigner time-frequency domains, are demonstrated to illustrate the efficiency and accuracy of the proposed LCT analytic signal.

© 2013 Optical Society of America

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  1. Y. X. Fu and L. Q. Li, “Generalized analytic signal associated with linear canonical transform,” Opt. Commun. 281, 1468–1472 (2008).
    [CrossRef]
  2. S. M. Kay, “Maximum entropy spectral estimation using the analytical signal,” IEEE Trans. Acoust. Speech Signal Process. 26, 467–469 (1978).
    [CrossRef]
  3. B. Boashash, Time-Frequency Signal Analysis and Processing: A Comprehensive Reference (Elsevier, 2003).
  4. S. L. Marple, “Estimating group delay and phase delay via discrete-time analytic cross-correlation,” IEEE Trans. Signal Process. 47, 2604–2607 (1999).
    [CrossRef]
  5. F. W. King, Hilbert Transforms (Cambridge University, 2009).
  6. D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).
    [CrossRef]
  7. L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas. Propag. 29, 386–391 (1981).
    [CrossRef]
  8. G. M. Livadas and A. G. Constantinides, “Image edge detection and segmentation based on the Hilbert transform,” in Proceedings of the 1988 International Conference On Acoustics, Speech, and Signal Processing (IEEE, 1988), pp. 1152–1155.
  9. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
    [CrossRef]
  10. A. I. Zayed, “Hilbert transform associated with the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 206–208 (1998).
    [CrossRef]
  11. A. Cusmariu, “Fractional analytic signals,” Signal Process. 82, 267–272 (2002).
    [CrossRef]
  12. S. C. Pei and M. H. Yeh, “Discrete fractional Hilbert transform,” IEEE Trans. Circuits Syst. II 47, 1307–1311 (2000).
    [CrossRef]
  13. C. C. Tseng and S. C. Pei, “Design and application of discrete-time fractional Hilbert transformer,” IEEE Trans. Circuits Syst. II 47, 1529–1533 (2000).
    [CrossRef]
  14. R. Tao, X. M. Li, and Y. Wang, “Generalization of the fractional Hilbert transform,” IEEE Signal Process. Lett. 15, 365–368 (2008).
    [CrossRef]
  15. K. B. Wolf, “Construction and properties of canonical transforms,” in Integral Transforms in Science and Engineering (Plenum, 1979), Chap. 9, pp. 381–416.
  16. A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
    [CrossRef]
  17. S. C. Pei and Y. C. Lai, “Discrete linear canonical transforms based on dilated Hermite functions,” J. Opt. Soc. Am. A 28, 1695–1708 (2011).
    [CrossRef]
  18. E. P. Wigner, “On the quantum correlation for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  19. S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications (Prentice-Hall, 1996).

2011 (1)

2008 (3)

R. Tao, X. M. Li, and Y. Wang, “Generalization of the fractional Hilbert transform,” IEEE Signal Process. Lett. 15, 365–368 (2008).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Y. X. Fu and L. Q. Li, “Generalized analytic signal associated with linear canonical transform,” Opt. Commun. 281, 1468–1472 (2008).
[CrossRef]

2002 (1)

A. Cusmariu, “Fractional analytic signals,” Signal Process. 82, 267–272 (2002).
[CrossRef]

2000 (2)

S. C. Pei and M. H. Yeh, “Discrete fractional Hilbert transform,” IEEE Trans. Circuits Syst. II 47, 1307–1311 (2000).
[CrossRef]

C. C. Tseng and S. C. Pei, “Design and application of discrete-time fractional Hilbert transformer,” IEEE Trans. Circuits Syst. II 47, 1529–1533 (2000).
[CrossRef]

1999 (1)

S. L. Marple, “Estimating group delay and phase delay via discrete-time analytic cross-correlation,” IEEE Trans. Signal Process. 47, 2604–2607 (1999).
[CrossRef]

1998 (1)

A. I. Zayed, “Hilbert transform associated with the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 206–208 (1998).
[CrossRef]

1996 (1)

1981 (1)

L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas. Propag. 29, 386–391 (1981).
[CrossRef]

1978 (1)

S. M. Kay, “Maximum entropy spectral estimation using the analytical signal,” IEEE Trans. Acoust. Speech Signal Process. 26, 467–469 (1978).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).
[CrossRef]

1932 (1)

E. P. Wigner, “On the quantum correlation for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Boashash, B.

B. Boashash, Time-Frequency Signal Analysis and Processing: A Comprehensive Reference (Elsevier, 2003).

Candan, C.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Chen, D.

S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications (Prentice-Hall, 1996).

Constantinides, A. G.

G. M. Livadas and A. G. Constantinides, “Image edge detection and segmentation based on the Hilbert transform,” in Proceedings of the 1988 International Conference On Acoustics, Speech, and Signal Processing (IEEE, 1988), pp. 1152–1155.

Cusmariu, A.

A. Cusmariu, “Fractional analytic signals,” Signal Process. 82, 267–272 (2002).
[CrossRef]

Fu, Y. X.

Y. X. Fu and L. Q. Li, “Generalized analytic signal associated with linear canonical transform,” Opt. Commun. 281, 1468–1472 (2008).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).
[CrossRef]

Kay, S. M.

S. M. Kay, “Maximum entropy spectral estimation using the analytical signal,” IEEE Trans. Acoust. Speech Signal Process. 26, 467–469 (1978).
[CrossRef]

King, F. W.

F. W. King, Hilbert Transforms (Cambridge University, 2009).

Koç, A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Kutay, M. A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Lai, Y. C.

Li, L. Q.

Y. X. Fu and L. Q. Li, “Generalized analytic signal associated with linear canonical transform,” Opt. Commun. 281, 1468–1472 (2008).
[CrossRef]

Li, X. M.

R. Tao, X. M. Li, and Y. Wang, “Generalization of the fractional Hilbert transform,” IEEE Signal Process. Lett. 15, 365–368 (2008).
[CrossRef]

Livadas, G. M.

G. M. Livadas and A. G. Constantinides, “Image edge detection and segmentation based on the Hilbert transform,” in Proceedings of the 1988 International Conference On Acoustics, Speech, and Signal Processing (IEEE, 1988), pp. 1152–1155.

Lohmann, A. W.

Marple, S. L.

S. L. Marple, “Estimating group delay and phase delay via discrete-time analytic cross-correlation,” IEEE Trans. Signal Process. 47, 2604–2607 (1999).
[CrossRef]

Mendlovic, D.

Ozaktas, H. M.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Pei, S. C.

S. C. Pei and Y. C. Lai, “Discrete linear canonical transforms based on dilated Hermite functions,” J. Opt. Soc. Am. A 28, 1695–1708 (2011).
[CrossRef]

C. C. Tseng and S. C. Pei, “Design and application of discrete-time fractional Hilbert transformer,” IEEE Trans. Circuits Syst. II 47, 1529–1533 (2000).
[CrossRef]

S. C. Pei and M. H. Yeh, “Discrete fractional Hilbert transform,” IEEE Trans. Circuits Syst. II 47, 1307–1311 (2000).
[CrossRef]

Qian, S.

S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications (Prentice-Hall, 1996).

Tao, R.

R. Tao, X. M. Li, and Y. Wang, “Generalization of the fractional Hilbert transform,” IEEE Signal Process. Lett. 15, 365–368 (2008).
[CrossRef]

Taylor, L. S.

L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas. Propag. 29, 386–391 (1981).
[CrossRef]

Tseng, C. C.

C. C. Tseng and S. C. Pei, “Design and application of discrete-time fractional Hilbert transformer,” IEEE Trans. Circuits Syst. II 47, 1529–1533 (2000).
[CrossRef]

Wang, Y.

R. Tao, X. M. Li, and Y. Wang, “Generalization of the fractional Hilbert transform,” IEEE Signal Process. Lett. 15, 365–368 (2008).
[CrossRef]

Wigner, E. P.

E. P. Wigner, “On the quantum correlation for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, K. B.

K. B. Wolf, “Construction and properties of canonical transforms,” in Integral Transforms in Science and Engineering (Plenum, 1979), Chap. 9, pp. 381–416.

Yeh, M. H.

S. C. Pei and M. H. Yeh, “Discrete fractional Hilbert transform,” IEEE Trans. Circuits Syst. II 47, 1307–1311 (2000).
[CrossRef]

Zalevsky, Z.

Zayed, A. I.

A. I. Zayed, “Hilbert transform associated with the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 206–208 (1998).
[CrossRef]

IEEE Signal Process. Lett. (2)

A. I. Zayed, “Hilbert transform associated with the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 206–208 (1998).
[CrossRef]

R. Tao, X. M. Li, and Y. Wang, “Generalization of the fractional Hilbert transform,” IEEE Signal Process. Lett. 15, 365–368 (2008).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

S. M. Kay, “Maximum entropy spectral estimation using the analytical signal,” IEEE Trans. Acoust. Speech Signal Process. 26, 467–469 (1978).
[CrossRef]

IEEE Trans. Antennas. Propag. (1)

L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas. Propag. 29, 386–391 (1981).
[CrossRef]

IEEE Trans. Circuits Syst. II (2)

S. C. Pei and M. H. Yeh, “Discrete fractional Hilbert transform,” IEEE Trans. Circuits Syst. II 47, 1307–1311 (2000).
[CrossRef]

C. C. Tseng and S. C. Pei, “Design and application of discrete-time fractional Hilbert transformer,” IEEE Trans. Circuits Syst. II 47, 1529–1533 (2000).
[CrossRef]

IEEE Trans. Signal Process. (2)

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

S. L. Marple, “Estimating group delay and phase delay via discrete-time analytic cross-correlation,” IEEE Trans. Signal Process. 47, 2604–2607 (1999).
[CrossRef]

J. Inst. Elect. Eng. (1)

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Y. X. Fu and L. Q. Li, “Generalized analytic signal associated with linear canonical transform,” Opt. Commun. 281, 1468–1472 (2008).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

E. P. Wigner, “On the quantum correlation for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Signal Process. (1)

A. Cusmariu, “Fractional analytic signals,” Signal Process. 82, 267–272 (2002).
[CrossRef]

Other (5)

S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications (Prentice-Hall, 1996).

G. M. Livadas and A. G. Constantinides, “Image edge detection and segmentation based on the Hilbert transform,” in Proceedings of the 1988 International Conference On Acoustics, Speech, and Signal Processing (IEEE, 1988), pp. 1152–1155.

F. W. King, Hilbert Transforms (Cambridge University, 2009).

B. Boashash, Time-Frequency Signal Analysis and Processing: A Comprehensive Reference (Elsevier, 2003).

K. B. Wolf, “Construction and properties of canonical transforms,” in Integral Transforms in Science and Engineering (Plenum, 1979), Chap. 9, pp. 381–416.

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Figures (8)

Fig. 1.
Fig. 1.

WDFs of (a) original signal and (b) the transformed signal after LCT operation.

Fig. 2.
Fig. 2.

Realization block diagram for generalized LCT analytic signal in [1] using Eq. (10) and their corresponding WDFs.

Fig. 3.
Fig. 3.

Realization block diagram for proposed LCT CAS using Eq. (11) and their corresponding WDFs.

Fig. 4.
Fig. 4.

Discrete generalized analytic signal realization [1]: (a) input signal x1[n]; (b) after DLCT with M1; (c) after unit step function; (d) after IDLCT with M1, and their corresponding WDFs with LCT parameter matrix M1=(3/2,1/2,1/2,3/2) using Eq. (17). The WDF of generalized analytic signal [1] in (d) is partially fragmented and broken.

Fig. 5.
Fig. 5.

Discrete generalized analytic signal realization [1]: (a) input signal x1[n]; (b) after DFT; (c) after unit step function; (d) after IDLCT with M1, and their corresponding WDFs with LCT parameter matrix M1=(3/2,1/2,1/2,3/2) using Eq. (17). The WDF of CAS in (d) contains the complete positive frequency bands without distortion.

Fig. 6.
Fig. 6.

Traditional analytic signal recovery (a) from discrete CAS x+M(t) and (b) after DLCT with complementary matrix M˜1 using Eq. (14). The recovered traditional analytic signal in (b) contains the positive frequencies of the signal.

Fig. 7.
Fig. 7.

Scalable edge detection of square pulse function for b=0.7, 1.0, and 1.3 in (a), (b), and (c), respectively. The ripple effect comes from the high frequency of square pulse function due to the well-known Gibbs phenomenon.

Fig. 8.
Fig. 8.

(a) Secure SSB communication system block. (b) Block diagram of the SSB modulator. (c) Block diagram of the SSB demodulator. The LCT parameters (a, b, c, d) are used as the secret keys for secure SSB demodulation.

Equations (20)

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X(w)=X*(w),whereX(w)F{x(t)},
x+(t)=F1{X+(w)}=F1{2U(w)X(w)},whereU(w)={0,ifw<01,ifw0,
H{x(t)}=F1{isgn(w)X(w)},wheresgn(w)={1,ifw<00,ifw=01,ifw>0,
x+(t)=x(t)+iH{x(t)}.
whenb0OLCTM{x(t)}=1ib·exp[iπ(dbu22but+abt2)]x(t)dt;whenb=0OLCTM{x(t)}=d·exp(iπdct2)·x(dt),whereM=[abcd]andadbc=1.
Wx(t),y(t)(t,w)x(t+τ/2)·y*(tτ/2)exp(i2πwτ)dτ,
Wx(t),x(t)(t,w)=x(t+τ/2)·x*(tτ/2)exp(i2πwτ)dτ,
ifx(t)=αg(t)+βs(t)Wx(t),x(t)(t,w)=|α|2Wg(t),g(t)(t,w)+|β|2Ws(t),s(t)(t,w)+αβ*Wg(t),s(t)(t,w)+α*βWs(t),g(t)(t,w)=|α|2Wg(t),g(t)(t,w)+|β|2Ws(t),s(t)(t,w)+CrossTermCrossTermαβ*Wg(t),s(t)(t,w)+α*βWs(t),g(t)(t,w).
WOLCTM{x(t)},OLCTM{x(t)}(u,v)=WOLCTM{x(t)},OLCTM{x(t)}(at+bw,ct+dw)where[uv]=[abcd][tw]=[at+bwct+dw].
A(a,b)(t)=OLCTM1{2U(u)XM(u)},whereXM(u)OLCTM{x(t)},U(u)={0,ifu<01,ifu0.
x+M(t)=OLCTM1{2U(w)X(w)},whereX(w)F{x(t)}.
x+M(t)=xM(t)+iHM{x(t)},wherexM(t)OLCTM˜1{x(t)},HM{x(t)}OLCTM˜1{H{x(t)}}.
M˜=[0110]M,andM˜1=M1[0110],
x+M(t)=OLCTM˜1{x(t)}+iOLCTM˜1{H{x(t)}}=OLCTM˜1{x(t)+iH{x(t)}}=OLCTM˜1{x+(t)},or,x+(t)=OLCTM˜{x+M(t)}=x(t)+iH{x(t)}.
HM{x(t)}=OLCTM˜1{H{x(t)}}=OLCTM1F{H{x(t)}}=OLCTM1{isgn(w)X(w)}.
x1(t)=cos(2πt)·exp((t0.3)2)+sin(3πt)·exp(t2).
M1=[cosαsinαsinαcosα]=[32121232].
M˜1=[0110][32121232]=[12323212].
M2=[0b1/b0].
x+M(t)=OLCTM1{[1+sgn(w)]X(w)}=OLCTM1{X(w)}+OLCTM1{sgn(w)X(w)},M1=M˜1[0110],where[0110]is the inverse FT matrix,OLCTM1{·}=OLCTM˜1F1{·}=OLCTM˜1F1{X(w)}+OLCTM˜1F1{sgn(w)X(w)}=OLCTM˜1{x(t)}+OLCTM˜1{iH{x(t)}},LetxM(t)OLCTM˜1{x(t)},HM{x(t)}OLCTM˜1{H{x(t)}}=xM(t)+iHM{x(t)}.

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