Abstract

The linear canonical transform (LCT) describes the effect of quadratic phase systems on a wavefield and generalizes many optical transforms. In this paper, the computation method for the discrete LCT using the adaptive least-mean-square (LMS) algorithm is presented. The computation approaches of the block-based discrete LCT and the stream-based discrete LCT using the LMS algorithm are derived, and the implementation structures of these approaches by the adaptive filter system are considered. The proposed computation approaches have the inherent parallel structures which make them suitable for efficient VLSI implementations, and are robust to the propagation of possible errors in the computation process.

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References

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  1. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  2. J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett.33(3), 228–230 (2008).
    [CrossRef] [PubMed]
  3. A. Stern, “Why is the linear canonical transform so little known?” Proc. AIP860,225–234 (2006).
    [CrossRef]
  4. D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun.126(4-6), 207–212 (1996).
    [CrossRef]
  5. A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process.56(6), 2383–2394 (2008).
    [CrossRef]
  6. S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process.50(1), 11–26 (2002).
    [CrossRef]
  7. B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F49(5), 592–603 (2006).
    [CrossRef]
  8. L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1631–1646 (2007).
    [CrossRef]
  9. A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975)
  10. L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, 1975).
  11. S. C. Pei and J. J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process.48(5), 1338–1353 (2000).
    [CrossRef]
  12. F. S. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett.16(8), 727–730 (2009).
    [CrossRef]
  13. J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process.88(11), 2825–2832 (2008).
    [CrossRef]
  14. J. Zhao, R. Tao, Y. L. Li, and Y. Wang, “Uncertainty principles for linear canonical transform,” IEEE Trans. Signal Process.57(7), 2856–2858 (2009).
    [CrossRef]
  15. B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
    [CrossRef]
  16. B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson., “Stationary and nonstationary learning characteristics of the LMS adaptive filter,” Proc. IEEE64(8), 1151–1162 (1976).
    [CrossRef]
  17. B. Widrow and S. D. Stearns, Adaptive Signal Processing (Prentice-Hall, 1985).
  18. B. Widrow, J. M. McCool, and M. Ball, “The complex LMS algorithm,” Proc. IEEE63(4), 719–720 (1975).
    [CrossRef]
  19. B. Widrow, P. Baudrenghien, M. Vetterli, and P. F. Titchener, “Fundamental relations between the LMS algorithm and the DFT,” IEEE Trans. Circ. Syst.34(7), 814–820 (1987).
    [CrossRef]

2009 (2)

F. S. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett.16(8), 727–730 (2009).
[CrossRef]

J. Zhao, R. Tao, Y. L. Li, and Y. Wang, “Uncertainty principles for linear canonical transform,” IEEE Trans. Signal Process.57(7), 2856–2858 (2009).
[CrossRef]

2008 (3)

J. J. Healy and J. T. Sheridan, “Cases where the linear canonical transform of a signal has compact support or is band-limited,” Opt. Lett.33(3), 228–230 (2008).
[CrossRef] [PubMed]

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process.88(11), 2825–2832 (2008).
[CrossRef]

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process.56(6), 2383–2394 (2008).
[CrossRef]

2007 (1)

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1631–1646 (2007).
[CrossRef]

2006 (1)

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F49(5), 592–603 (2006).
[CrossRef]

2002 (1)

S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process.50(1), 11–26 (2002).
[CrossRef]

2000 (1)

S. C. Pei and J. J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process.48(5), 1338–1353 (2000).
[CrossRef]

1996 (1)

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun.126(4-6), 207–212 (1996).
[CrossRef]

1987 (1)

B. Widrow, P. Baudrenghien, M. Vetterli, and P. F. Titchener, “Fundamental relations between the LMS algorithm and the DFT,” IEEE Trans. Circ. Syst.34(7), 814–820 (1987).
[CrossRef]

1976 (1)

B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson., “Stationary and nonstationary learning characteristics of the LMS adaptive filter,” Proc. IEEE64(8), 1151–1162 (1976).
[CrossRef]

1975 (2)

B. Widrow, J. M. McCool, and M. Ball, “The complex LMS algorithm,” Proc. IEEE63(4), 719–720 (1975).
[CrossRef]

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

Agarwal, G. S.

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun.126(4-6), 207–212 (1996).
[CrossRef]

Ball, M.

B. Widrow, J. M. McCool, and M. Ball, “The complex LMS algorithm,” Proc. IEEE63(4), 719–720 (1975).
[CrossRef]

Baudrenghien, P.

B. Widrow, P. Baudrenghien, M. Vetterli, and P. F. Titchener, “Fundamental relations between the LMS algorithm and the DFT,” IEEE Trans. Circ. Syst.34(7), 814–820 (1987).
[CrossRef]

Candan, C.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process.56(6), 2383–2394 (2008).
[CrossRef]

Deng, B.

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F49(5), 592–603 (2006).
[CrossRef]

Ding, J. J.

S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process.50(1), 11–26 (2002).
[CrossRef]

S. C. Pei and J. J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process.48(5), 1338–1353 (2000).
[CrossRef]

Dong, E.

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

Glover, J. R.

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

Goodlin, R. C.

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

Gotchev, A.

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1631–1646 (2007).
[CrossRef]

Healy, J. J.

Hearn, R. H.

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

James, D. F. V.

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun.126(4-6), 207–212 (1996).
[CrossRef]

Johnson, C. R.

B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson., “Stationary and nonstationary learning characteristics of the LMS adaptive filter,” Proc. IEEE64(8), 1151–1162 (1976).
[CrossRef]

Kaunitz, J.

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

Koc, A.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process.56(6), 2383–2394 (2008).
[CrossRef]

Kutay, M. A.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process.56(6), 2383–2394 (2008).
[CrossRef]

Larimore, M. G.

B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson., “Stationary and nonstationary learning characteristics of the LMS adaptive filter,” Proc. IEEE64(8), 1151–1162 (1976).
[CrossRef]

Li, Y. L.

J. Zhao, R. Tao, Y. L. Li, and Y. Wang, “Uncertainty principles for linear canonical transform,” IEEE Trans. Signal Process.57(7), 2856–2858 (2009).
[CrossRef]

McCool, J. M.

B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson., “Stationary and nonstationary learning characteristics of the LMS adaptive filter,” Proc. IEEE64(8), 1151–1162 (1976).
[CrossRef]

B. Widrow, J. M. McCool, and M. Ball, “The complex LMS algorithm,” Proc. IEEE63(4), 719–720 (1975).
[CrossRef]

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

Oktem, F. S.

F. S. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett.16(8), 727–730 (2009).
[CrossRef]

Onural, L.

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1631–1646 (2007).
[CrossRef]

Ozaktas, H. M.

F. S. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett.16(8), 727–730 (2009).
[CrossRef]

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process.56(6), 2383–2394 (2008).
[CrossRef]

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1631–1646 (2007).
[CrossRef]

Pei, S. C.

S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process.50(1), 11–26 (2002).
[CrossRef]

S. C. Pei and J. J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process.48(5), 1338–1353 (2000).
[CrossRef]

Sheridan, J. T.

Stern, A.

A. Stern, “Why is the linear canonical transform so little known?” Proc. AIP860,225–234 (2006).
[CrossRef]

Stoykova, E.

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1631–1646 (2007).
[CrossRef]

Tao, R.

J. Zhao, R. Tao, Y. L. Li, and Y. Wang, “Uncertainty principles for linear canonical transform,” IEEE Trans. Signal Process.57(7), 2856–2858 (2009).
[CrossRef]

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process.88(11), 2825–2832 (2008).
[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F49(5), 592–603 (2006).
[CrossRef]

Titchener, P. F.

B. Widrow, P. Baudrenghien, M. Vetterli, and P. F. Titchener, “Fundamental relations between the LMS algorithm and the DFT,” IEEE Trans. Circ. Syst.34(7), 814–820 (1987).
[CrossRef]

Vetterli, M.

B. Widrow, P. Baudrenghien, M. Vetterli, and P. F. Titchener, “Fundamental relations between the LMS algorithm and the DFT,” IEEE Trans. Circ. Syst.34(7), 814–820 (1987).
[CrossRef]

Wang, Y.

J. Zhao, R. Tao, Y. L. Li, and Y. Wang, “Uncertainty principles for linear canonical transform,” IEEE Trans. Signal Process.57(7), 2856–2858 (2009).
[CrossRef]

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process.88(11), 2825–2832 (2008).
[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F49(5), 592–603 (2006).
[CrossRef]

Widrow, B.

B. Widrow, P. Baudrenghien, M. Vetterli, and P. F. Titchener, “Fundamental relations between the LMS algorithm and the DFT,” IEEE Trans. Circ. Syst.34(7), 814–820 (1987).
[CrossRef]

B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson., “Stationary and nonstationary learning characteristics of the LMS adaptive filter,” Proc. IEEE64(8), 1151–1162 (1976).
[CrossRef]

B. Widrow, J. M. McCool, and M. Ball, “The complex LMS algorithm,” Proc. IEEE63(4), 719–720 (1975).
[CrossRef]

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

Williams, C. S.

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

Zeidler, J. R.

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

Zhao, J.

J. Zhao, R. Tao, Y. L. Li, and Y. Wang, “Uncertainty principles for linear canonical transform,” IEEE Trans. Signal Process.57(7), 2856–2858 (2009).
[CrossRef]

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process.88(11), 2825–2832 (2008).
[CrossRef]

IEEE Signal Process. Lett. (1)

F. S. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett.16(8), 727–730 (2009).
[CrossRef]

IEEE Trans. Circ. Syst. (1)

B. Widrow, P. Baudrenghien, M. Vetterli, and P. F. Titchener, “Fundamental relations between the LMS algorithm and the DFT,” IEEE Trans. Circ. Syst.34(7), 814–820 (1987).
[CrossRef]

IEEE Trans. Circ. Syst. Video Tech. (1)

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1631–1646 (2007).
[CrossRef]

IEEE Trans. Signal Process. (4)

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process.56(6), 2383–2394 (2008).
[CrossRef]

S. C. Pei and J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process.50(1), 11–26 (2002).
[CrossRef]

S. C. Pei and J. J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process.48(5), 1338–1353 (2000).
[CrossRef]

J. Zhao, R. Tao, Y. L. Li, and Y. Wang, “Uncertainty principles for linear canonical transform,” IEEE Trans. Signal Process.57(7), 2856–2858 (2009).
[CrossRef]

Opt. Commun. (1)

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun.126(4-6), 207–212 (1996).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (3)

B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, “Adaptive noise canceling: Principles and applications,” Proc. IEEE63(12), 1692–1716 (1975).
[CrossRef]

B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson., “Stationary and nonstationary learning characteristics of the LMS adaptive filter,” Proc. IEEE64(8), 1151–1162 (1976).
[CrossRef]

B. Widrow, J. M. McCool, and M. Ball, “The complex LMS algorithm,” Proc. IEEE63(4), 719–720 (1975).
[CrossRef]

Science in China Ser. F (1)

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F49(5), 592–603 (2006).
[CrossRef]

Signal Process. (1)

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process.88(11), 2825–2832 (2008).
[CrossRef]

Other (5)

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975)

L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, 1975).

A. Stern, “Why is the linear canonical transform so little known?” Proc. AIP860,225–234 (2006).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

B. Widrow and S. D. Stearns, Adaptive Signal Processing (Prentice-Hall, 1985).

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Figures (6)

Fig. 1
Fig. 1

The model of the adaptive filter.

Fig. 2
Fig. 2

Block discrete LCT computation by LMS adaptive filter.

Fig. 3
Fig. 3

(a) Sliding discrete LCT computation by LMS adaptive filter and (b) details of time-variant phase network.

Fig. 4
Fig. 4

(a) The continuous signal and (b) its discrete version.

Fig. 5
Fig. 5

(a) The amplitude and (b) the real part results of the continuous LCT and the discrete LCT with parameters (a,b,c,d)=( 3,1,1/2,1/2 ) .

Fig. 6
Fig. 6

(a) The amplitude and (b) the real part results of the continuous LCT and the discrete LCT with parameters (a,b,c,d)=( cosα,sinα,sinα,cosα ) by α=3π/20 .

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

f M (u)= L M [f](u)={ 1 j2πb exp( j d 2b u 2 ) f(t)exp( j a 2b t 2 j 1 b ut )dt , b0 d exp( j cd 2 u 2 )f(du), b=0
f ˜ M (u)= L M [ n= f(nΔt)δ(tnΔt) ](u) = 1 j2πb exp( j d 2b u 2 ) n= f(nΔt)exp( j a 2b n 2 Δ t 2 j 1 b unΔt ) . = 1 Δt exp( j d 2b u 2 ) n= f M (un 2πb Δt )exp[ j d 2b ( un 2πb Δt ) 2 ]
f M [ m ]= 1/N exp( j d 2b m 2 Δ u 2 ) n N f[ n ]exp( j a 2b n 2 Δ t 2 j 2π N mn )
ξ=E{ | e[ n ] | 2 }=E{ e[ n ] e [ n ] } =E{ ( d[ n ] X T [ n ]W[ n ] )( d [ n ] X H [ n ] W [ n ] ) }
W[ n+1 ]=W[ n ]+μ( [ n ] )
W[ n+1 ]=W[ n ]+μ( ^ [ n ] ) =W[ n ]+2μe[ n ] X [ n ] =( I2μ X [ n ] X T [ n ] )W[ n ]+2μd[ n ] X [ n ]
X[ n ]= 1 N exp( j a 2b n 2 Δ t 2 )× [ 1 exp( j d 2b 1 2 Δ u 2 +j 2π N 1n ) exp( j d 2b 2 2 Δ u 2 +j 2π N 2n ) exp( j d 2b (N1) 2 Δ u 2 +j 2π N (N1)n ) ] T
W[ 1 ]=( I2μ X [ 0 ] X T [ 0 ] )W[ 0 ]+2μd[ 0 ] X [ 0 ]=2μd[ 0 ] X [ 0 ],
W[ 2 ]=( I2μ X [ 1 ] X T [ 1 ] )W[ 1 ]+2μd[ 1 ] X [ 1 ] =4 μ 2 d[ 0 ] X [ 1 ] X T [ 1 ] X [ 0 ]+2μ( d[ 0 ] X [ 0 ]+d[ 1 ] X [ 1 ] ).
X T [ 1 ] X [ 0 ]= 1 N exp[ j a 2b ( 1 2 0 2 )Δ t 2 ] k=0 N1 exp[ j 2π N (10)k ] =0,
W[ 2 ]=2μ( d[ 0 ] X [ 0 ]+d[ 1 ] X [ 1 ] ).
W[ n ]=2μ m=0 n1 d[ m ] X [ m ] , n=1,,N.
W[ N ]= 2μ N [ m=0 N1 d[ m ]exp( j a 2b m 2 Δ t 2 ) exp( j d 2b 1 2 Δ u 2 ) m=0 N1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N 1m ) exp( j d 2b 2 2 Δ u 2 ) m=0 N1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N 2m ) exp( j d 2b (N1) 2 Δ u 2 ) m=0 N1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N (N1)m ) ]
W[ N+1 ]=( I2μ X [ N ] X T [ N ] )W[ N ]+2μd[ N ] X [ N ] =2μ m=0 N d[ m ] X [ m ] 4 μ 2 X [ N ]( X T [ N ] m=0 N1 d[ m ] X [ m ] ).
X T [ N ] X [ l ]={ exp( j a 2b N 2 Δ t 2 ), l=0 0, l=1,,(N1) .
W[ N+1 ]=2μ m=1 N d[ m ] X [ m ] 4 μ 2 X [ N ]d[ 0 ]exp( j a 2b N 2 Δ t 2 )+2μd[ 0 ] X [ 0 ].
W[ N+1 ]=2μ m=1 N d[ m ] X [ m ] +2μ(12μ)d[ 0 ] X [ 0 ].
W[ N+2 ]=2μ m=2 N+1 d[ m ] X [ m ] +2μ(12μ)( X [ 1 ]d[ 1 ]+ X [ 0 ]d[ 0 ] ).
W[ n ]=2μ m=nN n1 d[ m ] X [ m ] +2μ(12μ) m=0 nN1 d[ m ] X [ m ] , n=(N+1),,2N.
W[ n ]=2μ m=nN n1 d[ m ] X [ m ] +2μ(12μ) m=n2N nN1 d[ m ] X [ m ] + 2μ ) 2 m=n3N n2N1 d[ m ] X [ m ] + .
W[ lN ]= m=lNN lN1 d[ m ] X [ m ] = 1 N [ m=lNN lN1 d[ m ]exp( j a 2b m 2 Δ t 2 ) exp( j d 2b 1 2 Δ u 2 ) m=lNN lN1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N 1m ) exp( j d 2b 2 2 Δ u 2 ) m=lNN lN1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N 2m ) exp( j d 2b (N1) 2 Δ u 2 ) m=lNN lN1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N (N1)m ) ].
w k [ lN ]= 1 N exp( j d 2b k 2 Δ u 2 ) m=lNN lN1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N mk ) = 1 N exp( j d 2b k 2 Δ u 2 ) m=lNN lN1 d[ m ]exp{ j a 2b [ m(lNN) ] 2 Δ t 2 j 2π N mk }× exp{ j a 2b [ (lNN) 2 +2m(lNN) ]Δ t 2 } = 1 N exp( j d 2b k 2 Δ u 2 ) m=0 N1 d[ m+lNN ]exp( j a 2b m 2 Δ t 2 j 2π N mk )× exp{ j a 2b [ 2m(lNN)+ (lNN) 2 ]Δ t 2 } .
LCT[ n ]=[ LCT 0 [ n ] LCT 1 [ n ] LCT 2 [ n ] LCT N1 [ n ] ]= 1 N [ m=0 N1 d[ n(N1)+m ]exp( j a 2b m 2 Δ t 2 ) exp( j d 2b 1 2 Δ u 2 ) m=0 N1 d[ n(N1)+m ]exp( j a 2b m 2 Δ t 2 j 2π N 1m ) exp( j d 2b 2 2 Δ u 2 ) m=0 N1 d[ n(N1)+m ]exp( j a 2b m 2 Δ t 2 j 2π N 2m ) exp( j d 2b (N1) 2 Δ u 2 ) m=0 N1 d[ n(N1)+m ]exp( j a 2b m 2 Δ t 2 j 2π N (N1)m ) ].
W[ n ]= m=nN n1 d[ m ] X [ m ] = 1 N [ m=nN n1 d[ m ]exp( j a 2b m 2 Δ t 2 ) exp( j d 2b 1 2 Δ u 2 ) m=nN n1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N 1m ) exp( j d 2b 2 2 Δ u 2 ) m=nN n1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N 2m ) exp( j d 2b (N1) 2 Δ u 2 ) m=nN n1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N (N1)m ) ].
w k [ n ]= 1 N exp( j d 2b k 2 Δ u 2 ) m=nN n1 d[ m ]exp( j a 2b m 2 Δ t 2 j 2π N mk ) = 1 N exp( j d 2b k 2 Δ u 2 ) m=0 N1 d[ m+nN ]× exp[ j a 2b (m+nN) 2 Δ t 2 j 2π N (m+nN)k ] = 1 N exp( j d 2b k 2 Δ u 2 j 2π N nk ) m=0 N1 d[ m+nN ]× . exp[ j a 2b (m+nN) 2 Δ t 2 j 2π N mk ] = 1 N exp( j d 2b k 2 Δ u 2 j 2π N nk+j a 2b (nN) 2 Δ t 2 ) m=0 N1 d[ m+nN ]× exp{ j a 2b [ m 2 Δ t 2 +2m(nN)Δ t 2 ]j 2π N mk }
LCT k [ n1 ]= 1 N exp( j d 2b k 2 Δ u 2 ) m=0 N1 d[ nN+m ]exp( j a 2b m 2 Δ t 2 j 2π N km ) .

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