Result of XSLT numerical search of cogwheel phase cycling parameters

We present the data of the 5-column table from XSLT numerical search in another MS Excel table (and its PDF file) suitable for analyses.

Recall that the winding number $w_A$ for the first pulse A has been chosen to be zero for simplicity. That of the second pulse $w_B$ (header of the table) can take any value from 1 to 22.

For a given value of $w_B$, a non-zero value for the winding number $w_C$ of the third pulse C in the same column means that the associated coherence transfer pathway {0Q -> $p_{\mathrm{AB}}$Q -> $p_{\mathrm{BC}}$Q -> -1Q} is also observed by the cogwheel phase cycling. Conversely, if a value for the winding number of the third pulse C does not appear in this column, all the coherence transfer pathways, except for the antiecho coherence transfer pathway, are filtered by the cogwheel phase cycling.

It is easy to see that the missing value for the winding number of the third pulse is in the following table:

The winding number for the receiver phase is defined with the formula:

$w_{\mathrm{Rec}}$ = -3 $w_A$ + 2 $w_B$ + 2 $w_C$ mod 23.

It is deduced from the formula for the receiver phase:

${\phi }_{\mathrm{Rec}}$ = -3 ${\phi }_A$ + 2 ${\phi }_B$ + 2 ${\phi }_C$ mod 23.

Levitt and coworkers denote this phase cycling by: CogN( $w_A$, $w_B$, $w_C$, $w_{\mathrm{Rec}}$). There are 22 sets of winding numbers for selecting the antiecho coherence transfer pathway.

For example, one of them is Cog23(0, 3, 4, 14) from the third column of the above table. Since the receiver phase increment is not a multiple of 90°, Levitt and coworkers show that the winding number of the receiver can be subtracted from all the winding numbers. In other words, we use in practice:

$w_A$ = -14, $w_B$ = -11, $w_C$ = -10, and $w_{\mathrm{Rec}}$ = 0.