1 | function A = dft_axis(RC); |
---|
2 | % Private function to supply a default quaternion Fourier transform axis, |
---|
3 | % real or complex depending on the (logical) parameter value. 1 indicates |
---|
4 | % real, 0 complex. |
---|
5 | |
---|
6 | % Copyright © 2005 Stephen J. Sangwine and Nicolas Le Bihan. |
---|
7 | % See the file : Copyright.m for further details. |
---|
8 | |
---|
9 | error(nargchk(1, 1, nargin)), error(nargoutchk(1, 1, nargout)) |
---|
10 | |
---|
11 | if ~islogical(RC) |
---|
12 | error('Argument must be a logical value'); |
---|
13 | end |
---|
14 | |
---|
15 | % The value returned must be a root of -1. Real quaternion roots of -1 were |
---|
16 | % studied by Hamilton and any unit pure real quaternion is a root of -1. |
---|
17 | % The value chosen here is unbiased in any particular direction, since it |
---|
18 | % has equal x, y, z components. Complex quaternion roots of -1 are not so |
---|
19 | % simple. See the following for the theory behind the choice made here: |
---|
20 | % |
---|
21 | % S. J. Sangwine, Biquaternion (Complexified Quaternion) Roots of -1, |
---|
22 | % Advances in Applied Clifford Algebras, 16(1), February 2006, 63-68. |
---|
23 | % DOI:10.1007/s00006-006-0005-8. |
---|
24 | % |
---|
25 | % S. J. Sangwine, Biquaternion (complexified quaternion) roots of -1, |
---|
26 | % June 2005. arXiv:math.RA/0506190, available at: http://arxiv.org/. |
---|
27 | |
---|
28 | if RC |
---|
29 | |
---|
30 | % Real axis. We use the unit function because otherwise the result |
---|
31 | % would not have unit modulus. |
---|
32 | |
---|
33 | A = unit(quaternion(1,1,1)); |
---|
34 | |
---|
35 | else |
---|
36 | |
---|
37 | % Complex axis. This value has unit modulus as it stands and the real |
---|
38 | % and imaginary parts are perpendicular, both requirements for a root |
---|
39 | % of -1. Also, the modulus of the real part squared less the modulus |
---|
40 | % of the imaginary part squared, equals 1. |
---|
41 | |
---|
42 | A = complex(quaternion(1,1,1), quaternion(0,1,-1)); |
---|
43 | |
---|
44 | end |
---|