Abstract

The method of projections on convex sets is a procedure for signal recovery when partial information about the signal is available in the form of suitable constraints. We consider the use of this method in an inner-product space in which the vector space consists of real sequences and vector addition is defined in terms of the convolution operation. Signals with a prescribed Fourier-transform magnitude constitute a closed and convex set in this vector space, a condition that is not valid in the commonly used l2 (or L2) Hilbert-space framework. This new framework enables us to construct minimum-phase signals from the partial Fourier-transform magnitude and/or phase information.

© 1988 Optical Society of America

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References

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  1. D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
    [CrossRef]
  2. L. B. Bregman, “The method of successive projection for finding a common point of convex sets,” Dokl. USSR 162, 688–692 (1965).
  3. A. Lent, H. Tuy, “An iterative algorithm for extrapolation of band limited functions,”J. Math. Anal. Appl. 83, 554–565 (1981).
    [CrossRef]
  4. M. I. Sezan, H. Stark, “Image restoration by the method of convex projections. Part 2. Applications and numerical results,”IEEE Trans. Med. Imag. MI-1, 95–101 (1982).
    [CrossRef]
  5. H. J. Trussell, M. R. Civanlar, “Feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
    [CrossRef]
  6. A. E. Cetin, R. Ansari, “An iterative procedure for designing two dimensional FIR filters,” Electron. Lett. 23, 131–134. (1987).
    [CrossRef]
  7. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.
  8. B. Bogert, M. Heally, J. W. Tukey, “The quefrency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross cepstrum,” in Symposium on Time Series Analysis, M. Rosenblatt, ed. (Wiley, New York, 1963), pp. 208–243.
  9. E. Krajnik, B. Pondelicek, “Cepstrum as logarithm in a Banach algebra,” in Proceedings of the European Signal Processing Conference, European Association for Signal Processing, ed. (Elsevier, Amsterdam, 1986), pp. 414–416.
  10. W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973).
  11. T. F. Quatieri, J. M. Tribolet, “Computation of the real cepstrum and minimum-phase reconstruction,” in Programs for Digital Signal Processing, Digital Signal Processing Committee, eds. (Institute of Electrical and Electronics Engineers, New York, 1979), Chap. 7.2.
  12. P. Halmos, Introduction to Hilbert Space (Chelsea, New York, 1957).
  13. C. N. Dorny, A Vector Approach to Models and Optimization (Wiley, New York, 1968).
  14. T. J. Berkhout, “On the minimum phase criterion of sampled signals,”IEEE Trans. Geosci. Electron. GE-11, 186–196 (1973).
    [CrossRef]
  15. M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
    [CrossRef]
  16. T. F. Quatieri, A. V. Oppenheim, “Iterative techniques for minimum signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1187–1193 (1981).
    [CrossRef]
  17. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  18. A. Levi, H. Stark, “Image restoration by the method of generalized projections with applications to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  19. C. L. Bryne, M. A. Fiddy, “Estimation of continuous object distributions from limited Fourier magnitude measurements,” J. Opt. Soc. Am. A 4, 112–117 (1987).
    [CrossRef]
  20. D. E. Dudgeon, “The existence of cepstra for two-dimensional polynomials,”IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 115–128 (1975).
  21. D. M. Goodman, “Some properties of the multidimensional complex cepstrum and their relationship to the stability of multidimensional systems,” Circuits Syst. Signal Process. 6, 3–30 (1987).
    [CrossRef]
  22. M. P. Ekstrom, J. W. Woods, “Two-dimensional spectral factorization with applications in recursive digital filtering,”IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 115–128 (1976).
    [CrossRef]
  23. M. A. Fiddy, King’s College, London (personal communication, April1986).
  24. A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
    [CrossRef]
  25. A. E. Cetin, R. Ansari, “A procedure for antenna array pattern synthesis,” in Proceedings of the International IEEE Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 2308–2311.
  26. D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1968).

1987 (3)

A. E. Cetin, R. Ansari, “An iterative procedure for designing two dimensional FIR filters,” Electron. Lett. 23, 131–134. (1987).
[CrossRef]

D. M. Goodman, “Some properties of the multidimensional complex cepstrum and their relationship to the stability of multidimensional systems,” Circuits Syst. Signal Process. 6, 3–30 (1987).
[CrossRef]

C. L. Bryne, M. A. Fiddy, “Estimation of continuous object distributions from limited Fourier magnitude measurements,” J. Opt. Soc. Am. A 4, 112–117 (1987).
[CrossRef]

1984 (3)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

A. Levi, H. Stark, “Image restoration by the method of generalized projections with applications to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

H. J. Trussell, M. R. Civanlar, “Feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

1982 (2)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections. Part 2. Applications and numerical results,”IEEE Trans. Med. Imag. MI-1, 95–101 (1982).
[CrossRef]

1981 (2)

A. Lent, H. Tuy, “An iterative algorithm for extrapolation of band limited functions,”J. Math. Anal. Appl. 83, 554–565 (1981).
[CrossRef]

T. F. Quatieri, A. V. Oppenheim, “Iterative techniques for minimum signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1187–1193 (1981).
[CrossRef]

1980 (1)

M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
[CrossRef]

1978 (1)

1976 (1)

M. P. Ekstrom, J. W. Woods, “Two-dimensional spectral factorization with applications in recursive digital filtering,”IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 115–128 (1976).
[CrossRef]

1975 (1)

D. E. Dudgeon, “The existence of cepstra for two-dimensional polynomials,”IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 115–128 (1975).

1973 (1)

T. J. Berkhout, “On the minimum phase criterion of sampled signals,”IEEE Trans. Geosci. Electron. GE-11, 186–196 (1973).
[CrossRef]

1965 (1)

L. B. Bregman, “The method of successive projection for finding a common point of convex sets,” Dokl. USSR 162, 688–692 (1965).

Ansari, R.

A. E. Cetin, R. Ansari, “An iterative procedure for designing two dimensional FIR filters,” Electron. Lett. 23, 131–134. (1987).
[CrossRef]

A. E. Cetin, R. Ansari, “A procedure for antenna array pattern synthesis,” in Proceedings of the International IEEE Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 2308–2311.

Berkhout, T. J.

T. J. Berkhout, “On the minimum phase criterion of sampled signals,”IEEE Trans. Geosci. Electron. GE-11, 186–196 (1973).
[CrossRef]

Bogert, B.

B. Bogert, M. Heally, J. W. Tukey, “The quefrency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross cepstrum,” in Symposium on Time Series Analysis, M. Rosenblatt, ed. (Wiley, New York, 1963), pp. 208–243.

Bregman, L. B.

L. B. Bregman, “The method of successive projection for finding a common point of convex sets,” Dokl. USSR 162, 688–692 (1965).

Bryne, C. L.

Cetin, A. E.

A. E. Cetin, R. Ansari, “An iterative procedure for designing two dimensional FIR filters,” Electron. Lett. 23, 131–134. (1987).
[CrossRef]

A. E. Cetin, R. Ansari, “A procedure for antenna array pattern synthesis,” in Proceedings of the International IEEE Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 2308–2311.

Civanlar, M. R.

H. J. Trussell, M. R. Civanlar, “Feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

Dorny, C. N.

C. N. Dorny, A Vector Approach to Models and Optimization (Wiley, New York, 1968).

Dudgeon, D. E.

D. E. Dudgeon, “The existence of cepstra for two-dimensional polynomials,”IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 115–128 (1975).

Ekstrom, M. P.

M. P. Ekstrom, J. W. Woods, “Two-dimensional spectral factorization with applications in recursive digital filtering,”IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 115–128 (1976).
[CrossRef]

Fiddy, M. A.

Fienup, J. R.

Goodman, D. M.

D. M. Goodman, “Some properties of the multidimensional complex cepstrum and their relationship to the stability of multidimensional systems,” Circuits Syst. Signal Process. 6, 3–30 (1987).
[CrossRef]

Halmos, P.

P. Halmos, Introduction to Hilbert Space (Chelsea, New York, 1957).

Hayes, M. H.

M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
[CrossRef]

Heally, M.

B. Bogert, M. Heally, J. W. Tukey, “The quefrency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross cepstrum,” in Symposium on Time Series Analysis, M. Rosenblatt, ed. (Wiley, New York, 1963), pp. 208–243.

Krajnik, E.

E. Krajnik, B. Pondelicek, “Cepstrum as logarithm in a Banach algebra,” in Proceedings of the European Signal Processing Conference, European Association for Signal Processing, ed. (Elsevier, Amsterdam, 1986), pp. 414–416.

Lent, A.

A. Lent, H. Tuy, “An iterative algorithm for extrapolation of band limited functions,”J. Math. Anal. Appl. 83, 554–565 (1981).
[CrossRef]

Levi, A.

Lim, J. S.

M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Luenberger, D. G.

D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1968).

Oppenheim, A. V.

T. F. Quatieri, A. V. Oppenheim, “Iterative techniques for minimum signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1187–1193 (1981).
[CrossRef]

M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
[CrossRef]

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.

Pondelicek, B.

E. Krajnik, B. Pondelicek, “Cepstrum as logarithm in a Banach algebra,” in Proceedings of the European Signal Processing Conference, European Association for Signal Processing, ed. (Elsevier, Amsterdam, 1986), pp. 414–416.

Quatieri, T. F.

T. F. Quatieri, A. V. Oppenheim, “Iterative techniques for minimum signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1187–1193 (1981).
[CrossRef]

T. F. Quatieri, J. M. Tribolet, “Computation of the real cepstrum and minimum-phase reconstruction,” in Programs for Digital Signal Processing, Digital Signal Processing Committee, eds. (Institute of Electrical and Electronics Engineers, New York, 1979), Chap. 7.2.

Rudin, W.

W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.

Sezan, M. I.

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections. Part 2. Applications and numerical results,”IEEE Trans. Med. Imag. MI-1, 95–101 (1982).
[CrossRef]

Stark, H.

A. Levi, H. Stark, “Image restoration by the method of generalized projections with applications to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections. Part 2. Applications and numerical results,”IEEE Trans. Med. Imag. MI-1, 95–101 (1982).
[CrossRef]

Tribolet, J. M.

T. F. Quatieri, J. M. Tribolet, “Computation of the real cepstrum and minimum-phase reconstruction,” in Programs for Digital Signal Processing, Digital Signal Processing Committee, eds. (Institute of Electrical and Electronics Engineers, New York, 1979), Chap. 7.2.

Trussell, H. J.

H. J. Trussell, M. R. Civanlar, “Feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

Tukey, J. W.

B. Bogert, M. Heally, J. W. Tukey, “The quefrency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross cepstrum,” in Symposium on Time Series Analysis, M. Rosenblatt, ed. (Wiley, New York, 1963), pp. 208–243.

Tuy, H.

A. Lent, H. Tuy, “An iterative algorithm for extrapolation of band limited functions,”J. Math. Anal. Appl. 83, 554–565 (1981).
[CrossRef]

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

Wirnitzer, B.

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Woods, J. W.

M. P. Ekstrom, J. W. Woods, “Two-dimensional spectral factorization with applications in recursive digital filtering,”IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 115–128 (1976).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

Circuits Syst. Signal Process. (1)

D. M. Goodman, “Some properties of the multidimensional complex cepstrum and their relationship to the stability of multidimensional systems,” Circuits Syst. Signal Process. 6, 3–30 (1987).
[CrossRef]

Dokl. USSR (1)

L. B. Bregman, “The method of successive projection for finding a common point of convex sets,” Dokl. USSR 162, 688–692 (1965).

Electron. Lett. (1)

A. E. Cetin, R. Ansari, “An iterative procedure for designing two dimensional FIR filters,” Electron. Lett. 23, 131–134. (1987).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (5)

H. J. Trussell, M. R. Civanlar, “Feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 672–680 (1980).
[CrossRef]

T. F. Quatieri, A. V. Oppenheim, “Iterative techniques for minimum signal reconstruction from phase or magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1187–1193 (1981).
[CrossRef]

M. P. Ekstrom, J. W. Woods, “Two-dimensional spectral factorization with applications in recursive digital filtering,”IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 115–128 (1976).
[CrossRef]

D. E. Dudgeon, “The existence of cepstra for two-dimensional polynomials,”IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 115–128 (1975).

IEEE Trans. Geosci. Electron. (1)

T. J. Berkhout, “On the minimum phase criterion of sampled signals,”IEEE Trans. Geosci. Electron. GE-11, 186–196 (1973).
[CrossRef]

IEEE Trans. Med. Imag. (2)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections. Part 2. Applications and numerical results,”IEEE Trans. Med. Imag. MI-1, 95–101 (1982).
[CrossRef]

J. Math. Anal. Appl. (1)

A. Lent, H. Tuy, “An iterative algorithm for extrapolation of band limited functions,”J. Math. Anal. Appl. 83, 554–565 (1981).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Proc. IEEE (1)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Other (10)

A. E. Cetin, R. Ansari, “A procedure for antenna array pattern synthesis,” in Proceedings of the International IEEE Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 2308–2311.

D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1968).

M. A. Fiddy, King’s College, London (personal communication, April1986).

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 10.

B. Bogert, M. Heally, J. W. Tukey, “The quefrency analysis of time series for echoes: cepstrum, pseudo-autocovariance, cross cepstrum,” in Symposium on Time Series Analysis, M. Rosenblatt, ed. (Wiley, New York, 1963), pp. 208–243.

E. Krajnik, B. Pondelicek, “Cepstrum as logarithm in a Banach algebra,” in Proceedings of the European Signal Processing Conference, European Association for Signal Processing, ed. (Elsevier, Amsterdam, 1986), pp. 414–416.

W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973).

T. F. Quatieri, J. M. Tribolet, “Computation of the real cepstrum and minimum-phase reconstruction,” in Programs for Digital Signal Processing, Digital Signal Processing Committee, eds. (Institute of Electrical and Electronics Engineers, New York, 1979), Chap. 7.2.

P. Halmos, Introduction to Hilbert Space (Chelsea, New York, 1957).

C. N. Dorny, A Vector Approach to Models and Optimization (Wiley, New York, 1968).

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Figures (6)

Fig. 1
Fig. 1

Flow diagram of the algorithm.

Fig. 2
Fig. 2

Original minimum-phase signal for examples 1–4.

Fig. 3
Fig. 3

The l1 norm of the error versus the number of iteration cycles in example 1.

Fig. 4
Fig. 4

The l1 norm of the error versus the number of iteration cycles in example 4.

Fig. 5
Fig. 5

(a) Original image; (b) image reconstructed from FTM samples, which can be used as an initial estimate in iterative phase-retrieval algorithms.

Fig. 6
Fig. 6

Percent error versus the number of iteration cycles.

Equations (73)

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C 0 = i = 1 I C i
x k + 1 = P I · P I 1 P 1 x k , k 0.
F ( x k ) = X k ( ω ) = n = x ( n ) exp ( j ω n ) , k = 1 , 2 ,
n = | x ( n ) | < , x ( n ) R
X ( ω ) 0 for all ω [ π , π ] , with X ( 0 ) > 0.
X ( ω ) = | X ( ω ) | exp [ j Ψ ( ω ) ] ,
Log [ X ( ω ) ] = log [ | X ( ω ) | ] + j Φ ( ω ) ,
π π | Log [ X ( ω ) ] | 2 d ω <
( x 1 * x 2 ) ( n ) = k = x 1 ( n k ) x 2 ( k ) , n = 0 , ± 1 , ± 2 , .
δ ( n ) = { 1 if n = 0 0 otherwise .
x 1 ( n ) = 1 2 π π π 1 X ( ω ) exp ( j ω n ) d ω , n = 0 , ± 1 , ± 2 .
x 1 * x = δ .
c x = F 1 { exp ( c Log [ X ( ω ) ] ) } ,
( x 1 , x 2 ) = 1 2 π π π Log [ X 1 ( ω ) ] ( Log [ X 2 ( ω ) ] ) * d ω .
x * = ( 1 2 π π π | Log [ X ( ω ) ] | 2 d ω ) 1 / 2 .
X k ( ω ) = A k ( ω ) exp [ j Φ k ( ω ) ] , k = 1 , 2 ,
= ( t x 1 ) * [ ( 1 t ) x 2 ]
X ( ω ) = A ( ω ) exp [ j Φ ( ω ) ] = [ X 1 ( ω ) ] t [ X 2 ( ω ) ] 1 t = [ A 1 ( ω ) ] t [ A 2 ( ω ) ] 1 t × exp ( j [ t Φ 1 ( ω ) + ( 1 t ) Φ 2 ( ω ) ] ) .
X ( ω ) = A 1 ( ω ) exp ( j [ t Φ 1 ( ω ) + ( 1 t ) Φ 2 ( ω ) ] ) , ω B .
A ( ω ) = A 1 ( ω ) , ω B .
C A = { x : | X ( ω ) | = A ( ω ) , ω B [ π , π ] }
S A = { x : L A ( ω ) | X ( ω ) | U A ( ω ) , ω B [ π , π ] } .
L A ( ω ) > 0.
C Φ = { x : arg | X ( ω ) | = Φ ( ω ) , ω B [ π , π ] } .
S Φ = { x : L Φ ( ω ) arg | X ( ω ) | U Φ ( ω ) , ω B [ π , π ] } ,
( 2 π ) x n x * 2 = π π | Log X n ( ω ) Log X ( ω ) | 2 d ω = π π | Log A n ( ω ) Log A ( ω ) | 2 d ω + π π | Φ n ( ω ) Φ ( ω ) | 2 d ω 0 as n ,
X n ( ω ) = A n ( ω ) exp [ j Φ n ( ω ) ]
X ( ω ) = A ( ω ) exp [ j Φ ( ω ) ] .
L A ( ω ) A n ( ω ) U A ( ω ) , ω B [ π , π ] .
L A ( ω ) A ( ω ) U A ( ω ) , ω B [ π , π ] .
( x k , x k ) E , k = 1 , 2 ;
( x , x ) = 1 2 π π π | Log X 1 t ( ω ) + Log X 2 1 t ( ω ) | 2 d ω = 1 2 π π π | t Log X 1 ( ω ) + ( 1 t ) Log X 2 ( ω ) | 2 d ω
t 2 ( x 1 , x 1 ) + ( 1 t ) 2 ( x 2 , x 2 ) + 2 t ( 1 t ) x 1 * · x 2 * E .
x ˆ = F 1 ( Log [ X ( ω ) ] ) ,
C ˆ S = { x : x ˆ ( n ) = 0 , for n P } ,
x ˆ = F 1 ( Log [ X 1 t ( ω ) X 2 1 t ( ω ) ] ) = F 1 ( t Log [ X 1 ( ω ) ] ) + F 1 ( ( 1 t ) Log [ X 2 ( ω ) ] ) = t x ˆ 1 + ( 1 t ) x ˆ 2 .
C i 1 = { x : | X ( ω i 1 ) | = | X d ( ω i 1 ) | and x is minimum phase } , i = 1 , 2 , , I .
C m 2 = { x : Φ ( ω m 2 ) = Φ d ( ω m 2 ) and x is minimum phase } , i = 1 , 2 , , M .
C i 1 = { x : | X ( ω i 1 ) | = | X d ( ω i 1 ) | } { x : x is minimum phase } ,
C i 1 = { x : Re [ X ˆ ( ω i 1 ) ] = log [ X ˆ d ( ω i 1 ) ] and x ˆ ( n ) = 0 for n < 0 } , i = 1 , 2 , , I ,
C m 2 = { x : Im [ X ˆ ( ω m 2 ) ] = Φ 2 ( ω m 2 ) and x ˆ ( n ) = 0 for n < 0 } , m = 1 , 2 , , M .
C 0 = ( i = 1 I C i 1 ) ( m = 1 M C m 2 ) ,
minimize x x 0 * ,
| X ( ω i 1 ) | = | X d ( ω i 1 ) | and x is minimum phase ,
x x 0 * 2 = 1 2 π π π | Log X ( ω ) Log X 0 ( ω ) | 2 d ω = 1 2 π π π | X ˆ ( ω ) X ˆ 0 ( ω ) | 2 d ω .
x x 0 * 2 x ˆ x ˆ 0 2 2 .
minimize x ˆ x ˆ 0 2 ,
Re [ X ˆ ( ω i 1 ) ] = log | X d ( ω i 1 ) | , x ˆ ( n ) = 0 n < 0.
x ˆ p ( n ) = x ˆ 0 ( n ) + λ cos ( ω i 1 n ) , n = 0 , 1 , 2 , , N ,
λ = Log | X d ( ω i 1 ) | Re | X ˆ 0 ( ω i 1 ) | n = 0 N cos 2 ( ω i 1 n ) .
min x ˆ x ˆ 0 2
Im [ X ˆ ( ω m 2 ) ] = Φ d ( ω m 2 ) , x ˆ ( n ) = 0 n < 0.
x ˆ p ( n ) = x ˆ 0 ( n ) + λ sin ( ω m 2 n ) , n = 0 , 1 , 2 , , N , ω m 2 0 ,
λ = Φ d ( ω m 2 ) Im [ X ˆ 0 ( ω m 2 ) ] n = 1 N sin 2 ( ω m 2 n ) .
r ( n ) = x ( n ) * x ( n )
R ( ω ) = | X ( ω ) | 2 = [ X ( ω ) ] [ X * ( ω ) ] ,
2 Log | X ( ω ) | = Log [ X ( ω ) ] + Log [ X * ( ω ) ] .
r ˆ ( n ) = x ˆ ( n ) + x ˆ ( n ) .
x ˆ ( n ) = { ( 1 / 2 ) r ˆ ( 0 ) if n = 0 r ˆ ( n ) if n > 0 0 if m < 0 ;
C m , l = { x : | X ( ω 1 m , ω 2 l ) | = | X d ( ω 1 m , ω 2 l ) | and x ˆ ( n 1 , n 2 ) = 0 , for ( n 1 , n 2 ) P } ,
min x C i 1 x x 0 * .
min x x 0 * ,
| X ( ω i 1 ) | = | X d ( ω i 1 ) | .
min x ˆ x ˆ 0 2 ,
Re [ X ˆ ( ω i 1 ) ] = Re [ X ˆ d ( ω i 1 ) ] = log | X d ( ω i 1 ) | ,
Re [ X ˆ ( ω i 1 ) ] = n = 0 x ˆ ( n ) cos ( ω i 1 n ) .
Re [ X ˆ ( ω i ) ] = n = 0 N x ˆ ( n ) cos ( ω i 1 n ) ,
L = x ˆ x ˆ 0 2 + λ ( n = 0 N x ˆ ( n ) cos ( ω i 1 n ) Re [ X ˆ d ( ω i 1 ) ] ) ,
x ˆ x ˆ 0 2 = n = 0 [ x ˆ ( n ) x ˆ 0 ( n ) ] 2 = n = 0 N [ x ˆ ( n ) x ˆ 0 ( n ) ] 2 .
L x ˆ ( n ) = 0 for n = 0 , 1 , , N
L λ = 0.
x ˆ p ( n ) = x ˆ 0 ( n ) + λ cos ( ω i 1 n ) , n = 0 , 1 , , N .
λ = Log | X d ( ω i 1 ) | Re [ X ˆ 0 ( ω i ) ] n = 0 N cos 2 ( ω i 1 n ) .

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