Abstract

Many multiplexing instruments utilize the fast Hadamard transform (FHT) to demultiplex the signal. In the past, the FHT includes the π1 and π2 transformations to reorder vectors before and after a Sylvester-type Hadamard transform. Although the computational effort involved in the π1 and the Sylvester-type Hadamard transform scales as n log2n, calculating the π2 transformation (which only has to be done once) scales as n2. Recently Gunson (1980) has suggested a method by which the π transformations are symmetric, that is π2 = π1. We have calculated a complete set of symmetric π transformations for FHT of sizes 23 to 230. Special emphasis has been placed on the phase of the π transformation so as to have the correct phase in the demultiplexed signal.

© 1983 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

1981

1978

1977

S. Miyamoto, Space Sci. Instrum. 3, 473 (1977).

1973

A. Rosencwaig, Opt. Commun. 7, 305 (1973).
[CrossRef]

1971

1970

1969

1962

E. J. Watson, Math. Comput. 16, 368 (1962).

Cannon, T. M.

Fenimore, E. E.

Fine, T.

Fredman, M. L.

Gunson, J.

J. Gunson, “A Fast Colvolution Transform for M-Sequences,” preprint, U. Birmingham, U.K. (1980).

Harwit, M.

Harwit, M. O.

M. O. Harwit, Appl. Opt. 10, 1415 (1971).
[CrossRef] [PubMed]

M. O. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

Miyamoto, S.

S. Miyamoto, Space Sci. Instrum. 3, 473 (1977).

Nelson, E. D.

Phillips, P. G.

Rosencwaig, A.

A. Rosencwaig, Opt. Commun. 7, 305 (1973).
[CrossRef]

Sloane, N. J. A.

N. J. A. Sloane, T. Fine, P. G. Phillips, M. Harwit, Appl. Opt. 8, 2103 (1969).
[CrossRef] [PubMed]

M. O. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

Watson, E. J.

E. J. Watson, Math. Comput. 16, 368 (1962).

Weston, G. S.

Appl. Opt.

J. Opt. Soc. Am.

Math. Comput.

E. J. Watson, Math. Comput. 16, 368 (1962).

Opt. Commun.

A. Rosencwaig, Opt. Commun. 7, 305 (1973).
[CrossRef]

Space Sci. Instrum.

S. Miyamoto, Space Sci. Instrum. 3, 473 (1977).

Other

M. O. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

J. Gunson, “A Fast Colvolution Transform for M-Sequences,” preprint, U. Birmingham, U.K. (1980).

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Figures (3)

Fig. 1
Fig. 1

Generation of an m-sequence for m = 4. The polynomial and seed are from Table I. The polynomial gives the rule for generating successive binary patterns from the seed, and the first bit gives the m-sequence.

Fig. 2
Fig. 2

Example of SB·η. The middle row of SB corresponds to the response of the instrument to a source in the center of the ψ. The minus signs stand for −1.

Fig. 3
Fig. 3

Generation of π transformations from the SB of Fig. 2: (a) π1 generation by Harwit and Sloane utilize the first m columns of S. (b) πs utilizes columns of S so that π1 = π2 and ψ′ has the correct phase.

Tables (2)

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Table I Parameters for Generating m-Sequence in Their Natural Shift

Tables Icon

Table II Columns to Generate πs

Equations (6)

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η j = i = 0 n 1 ψ i s i + j ,
η = S · ψ .
s i B 1 if s i = 1 , s i B 1 if s i = 0 , S i j B = s i + j B ,
ψ = 1 2 m 1 S B · η .
ψ = 1 2 m 1 π 2 H s π 1 η .
s c i N = 1 , s c i + c j N = 0

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