Abstract

For certain types of digitally generated holographic memories, it is important that the Fourier components to be stored have a constant, or nearly constant, modulus. It is shown that by interlacing two sequences of complex numbers, one a data sequence representing the information to be stored and the second a parity sequence, derivable from the data sequence, the discrete Fourier transform of the combined sequence can be made to have a constant modulus. A slight variation of the method leads to a spectrum with varying modulus but only a discrete set of phases. The sequences may be one-dimensional or multidimensional.

© 1972 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. B. Lesem, P. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
    [CrossRef]
  2. A. W. Lohman, D. F. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef]
  3. W. H. Lee, Appl. Opt. 9, 639(1970).
    [CrossRef] [PubMed]
  4. C. B. Burkekliardt, Appl. Opt. 9, 1949 (1970).
  5. J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1970).
    [CrossRef]
  6. W. J. Dallas, Appl. Opt. 10, 673 (1971).
    [CrossRef] [PubMed]
  7. W. J. Dallas, Appl. Opt. 10, 674 (1971).
    [CrossRef] [PubMed]
  8. G. B. Anderson, T. S. Huang, Pattern Recognition 3, 185 (1971).
    [CrossRef]
  9. The Hamming weight of a binary sequence is simply the number of binary ones it contains.
  10. For completeness we point out that the N-point spectra {Fm} and {Pm} may be recovered from the 2N-point spectrum {Gm} by the prescriptions Fm = (1/2)(Gm + Gm+N), Pm = (1/2)(Gm − Gm+N), m = 0, 1,…,N − 1.
  11. M. R. Schroeder, IEEE Trans. Inform. Theory IT-16, 85 (1970).
    [CrossRef]

1971

1970

M. R. Schroeder, IEEE Trans. Inform. Theory IT-16, 85 (1970).
[CrossRef]

W. H. Lee, Appl. Opt. 9, 639(1970).
[CrossRef] [PubMed]

C. B. Burkekliardt, Appl. Opt. 9, 1949 (1970).

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1970).
[CrossRef]

1969

L. B. Lesem, P. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

1967

Anderson, G. B.

G. B. Anderson, T. S. Huang, Pattern Recognition 3, 185 (1971).
[CrossRef]

Burkekliardt, C. B.

Dallas, W. J.

Goodman, J. W.

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1970).
[CrossRef]

Hirsch, P.

L. B. Lesem, P. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Huang, T. S.

G. B. Anderson, T. S. Huang, Pattern Recognition 3, 185 (1971).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Lee, W. H.

Lesem, L. B.

L. B. Lesem, P. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Lohman, A. W.

Paris, D. F.

Schroeder, M. R.

M. R. Schroeder, IEEE Trans. Inform. Theory IT-16, 85 (1970).
[CrossRef]

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1970).
[CrossRef]

Appl. Opt.

IBM J. Res. Dev.

L. B. Lesem, P. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1970).
[CrossRef]

IEEE Trans. Inform. Theory

M. R. Schroeder, IEEE Trans. Inform. Theory IT-16, 85 (1970).
[CrossRef]

Pattern Recognition

G. B. Anderson, T. S. Huang, Pattern Recognition 3, 185 (1971).
[CrossRef]

Other

The Hamming weight of a binary sequence is simply the number of binary ones it contains.

For completeness we point out that the N-point spectra {Fm} and {Pm} may be recovered from the 2N-point spectrum {Gm} by the prescriptions Fm = (1/2)(Gm + Gm+N), Pm = (1/2)(Gm − Gm+N), m = 0, 1,…,N − 1.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Geometric representation of the original DFT component Fm and the parity DFT component Pm chosen for spectrum leveling.

Fig. 2
Fig. 2

Location of the vectors Fm′, Pm′, Gm, Fm+N′, Pm+N′, and Gm+N on the complex plane. This arrangement corresponds to a choice of − for ∓ and + for ± in Eqs. (12) and (14).

Fig. 3
Fig. 3

Location of vectors in the complex plane under phase quantization. The final DFT components Gm and Gm+N lie along the preselected lines at angles ψh and ψk.

Fig. 4
Fig. 4

Block diagram for obtaining a level spectrum from the data sequence. Phase coding is optional (but recommended), as is the check.

Fig. 5
Fig. 5

(a) Magnitude and phase of the interlaced data-parity sequence that levels the spectrum of the data sequence {1101111011110110}. (b) Modulus and phase of the leveled spectrum.

Fig. 6
Fig. 6

(a) Magnitude and phase of the interlaced data-parity sequence that levels the spectrum of the data sequence {1000010011000010}. (b) Modulus and phase of the leveled spectrum.

Fig. 7
Fig. 7

(a) Magnitude and phase of the interlaced data-parity sequence that quantizes the phase of the spectrum of the data sequence {1101111011110110}.(b) Modulus and phase of the phase-quantized spectrum.

Equations (58)

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

F m = n = 0 N - 1 f n exp ( - i 2 π m n N ) ,
f n = 1 N m = 0 N - 1 F m exp ( i 2 π m n N ) .
F m = { F m m = 0 , 1 , , N - 1 , F m - N m = N , N + 1 , , 2 N - 1.
f n = { f n / 2 ( n even ) 0             ( n odd )             n = 0 , 1 , , 2 N - 1.
P m = { P m m = 0 , 1 , 2 , , N - 1 , - P m - N m = N , N + 1 , , 2 N - 1.
p n = { 0 n even p ( n - 1 ) / 2 n odd n = 0 , 1 , , 2 N - 1.
G m = F m + P m             m = 0 , 1 , , 2 N - 1.
g n = f n + p n             n = 0 , 1 , , 2 N - 1.
g n = { f n / 2 n even p ( n - 1 ) / 2 n odd n = 0 , 1 , , 2 N - 1.
G m = A             m = 0 , 1 , , N - 1 ,
A max m F m             ( m = 0 , 1 , , N - 1 ) .
P m = ± i ( F m / F m ) ( A 2 - F m 2 ) 1 2 ,
G m = { A exp { i [ arg ( F m ) ± cos - 1 F m A ] } m = 0 , 1 , , N - 1 , A exp { i [ arg ( F m - N ) cos = 1 F m - N A ] } m = N , N + 1 , , 2 N - 1.
F m = R m + i X m , β m = ( A 2 - R m 2 - X m 2 R m 2 + X m 2 ) 1 2             m = 0 , 1 , , N - 1 ,
G m = { [ ( R m X m β m ) + i ( X m ± R m β m ) ] m = 0 , 1 , , N - 1 , [ ( R m - N ± X m - N β m - N ) + i ( X m - N R m - N β m - N ) ] m = N , N + 1 , , 2 N - 1.
G m = { A ( cos γ m + i sin γ m ) m = 0 , 1 , , N - 1 , - A ( cos γ m + i sin γ m ) m = N , N + 1 , , 2 N - 1.
ψ k < ψ k + 1 ,             0 ψ k < 2 π , 0 < ψ k + 1 - ψ k < π ,             and             ψ K ψ 0 + 2 π .
G m = { 2 F m sin ( θ m - ψ k ) sin ( ψ h - ψ k ) exp ( i ψ h ) m = 0 , 1 , , N - 1 , 2 F m - N sin ( ψ h - θ m - N ) sin ( ψ h - ψ k ) exp ( i ψ k ) m = N , N + 1 , , 2 N = 1.
P m = [ F m sin ( ψ h - θ m ) ] / sin α , arg { P m } = ψ h + α - ( π / 2 ) ± ( π / 2 ) ,
0 < α = cot - 1 [ sin ( θ m - ψ k ) sin ( ψ h - ψ k ) sin ( ψ h - θ m ) - cot ( ψ h - ψ k ) ] π - ( ψ h - ψ k ) .
F m = R m + i X m , G m = 2 i X m , G m + N = 2 R m .
F m , r = n = 0 N - 1 q = 0 Q - 1 f n , q exp [ - i 2 π ( n m N + q r Q ) ] n = 0 , 1 , , N - 1 , q = 0 , 1 , , Q - 1.
F m , r = { F m , R m = 0 , 1 , , N - 1 , r = 0 , 1 , , Q - 1 , m = 0 , 1 , , N - 1 , F m , r - Q r = Q , Q + 1 , , 2 Q - 1.
P m , r = { P m , r r = 0 , 1 , , Q - 1 , - P m , r - Q r = Q , Q + 1 , , 2 Q - 1 ,
f n , q = { f n , ( q / 2 ) q even , 0 q odd ,
p n , q = { 0             q even 1 N Q m = 0 N - 1 r = 0 Q - 1 P m , r exp { i 2 π ( n m N + r q 2 Q ) } q odd
P m , r = ± i ( F m , r / F m , r ) ( A 2 - F m , r 2 ) 1 2 .
A max m , r F m , r .
A max m F m .
ϕ n = { 0 n = 0 - 2 π j = 0 n - 1 ( n - j ) p j n = 1 , 2 , , N - 1 , N odd - 2 π j = 0 n - 1 ( n - j ) p j + π N n n = 1 , 2 , , N - 1 , N even
p j = f j 2 / n = 0 N - 1 f n 2 .
ϕ n = - π [ n ( n + 1 ) / N ]             n = 0 , 1 , , N - 1
ϕ n = - ( π / N ) n 2             n = 0 , 1 , , N - 1.
{ U } = { 1101111011110110 } , { V } = { 1000010011000010 } .
n = 1 2 N m = 0 2 N - 1 F m exp ( i 2 π 2 N m n ) = 1 2 N m = 0 N - 1 F m exp ( i 2 π 2 N m n ) + 1 2 N m = N 2 N - 1 F m - N exp ( i 2 π 2 N m n ) ,
f n = 1 2 N m = 0 N - 1 F m exp ( i 2 π 2 N m n ) + 1 2 N m = 0 N - 1 F m exp [ i 2 π 2 N ( m + N ) n ] = 1 2 N [ 1 + exp ( i π n ) ] m = 0 N - 1 F m exp ( i 2 π N m n 2 ) .
f n = { f n / 2 n even , 0 n odd .
p n = 1 2 N [ 1 - exp ( i n π ) ] m = 0 N - 1 P m exp ( i 2 π N m n 2 ) = 1 2 N [ 1 - exp ( i n π ) ] m = 0 N - 1 [ P m exp ( i π m N ) ] × exp [ i 2 π N m ( n - 1 2 ) ] .
p n = { 0 n even , p n - 1 / 2 n odd ,
p n 1 N m = 0 N - 1 [ P m exp ( i π m N ) ] exp ( i 2 π N m n ) .
f n , q = 1 2 N Q m = 0 N - 1 r = 0 2 Q - 1 F m , r exp [ i 2 π ( m n N + r q 2 Q ) ] .
f n , q = 1 2 [ 1 + exp ( i π q ) ] 1 N Q m = 0 N - 1 r = 0 Q - 1 × F m , r exp [ i 2 π ( m n N + r q 2 Q ) ] .
f n , q = { f n , ( q / 2 ) q even , 0 q odd ,
p n , q = 1 2 N Q m = 0 N - 1 r = 0 2 Q - 1 P m , r exp [ i 2 π ( m n N + r q 2 Q ) ] .
p n , q = 1 2 [ 1 - exp ( i π q ) ] 1 N Q m = 0 N - 1 r = 0 Q - 1 × P m , r exp [ i 2 π ( m n N + r q 2 Q ) ] .
p n , q = { 0 q even , p n , ( q - 1 ) / 1 q odd ,
p n , q 1 N Q m = 0 N - 1 r = 0 Q - 1 [ P m , r exp ( i π r Q ) ] exp [ i 2 π ( m n N + r q Q ) ] .
f n = exp [ - i ( π / N ) n 2 ]             n = 0 , 1 , , N - 1.
F m = n = 0 N - 1 exp ( - i π N n 2 ) exp ( - i 2 π N m n ) ,
F m = exp ( i π N m 2 ) n = 0 N - 1 exp [ - i π N ( n + m ) 2 ] = exp ( i π N m 2 ) H ( m ) ,
H ( m ) n = 0 N - 1 exp [ - i π N ( n + m ) 2 ] .
H ( m ) - H ( m - 1 ) = n = 0 N - 1 exp [ - i π N ( n + m ) 2 ] - n = 0 N - 1 exp [ - i π N ( n + m - 1 ) 2 ] = exp [ - i π N ( N + m - 1 ) 2 ] - exp [ - i π N ( m - 1 ) 2 ] = { exp ( - i π N ) exp [ - i 2 π ( m - 1 ) ] - 1 } × exp [ - i π N ( m - 1 ) 2 ] .
H ( m ) = H ( m - 1 ) .
f n = exp [ - i ( π / N ) n ( n + 1 ) ] n = 0 , 1 , , N - 1.
F m = n = 0 N - 1 exp [ - i π N n ( n + 1 ) ] exp ( - i 2 π N n m ) = exp [ i π N ( m + 1 2 ) 2 ] n = 0 N - 1 exp [ - i π N ( n + m + 1 2 ) 2 ] .
K ( m ) n = 0 N - 1 exp [ - i π N ( n + m + 1 2 ) 2 ]
K ( m ) - K ( m - 1 ) = exp [ - i π N ( N + m - 1 2 ) 2 ] - exp [ - i π N ( m - 1 2 ) 2 ] = { exp ( - i π N ) exp [ - i 2 π ( m - 1 2 ) ] - 1 } exp [ - i ( π / N ) ( m - 1 2 ) 2 ] .
K ( m ) = K ( m - 1 ) .

Metrics