Result of XSLT numerical search of cogwheel phase cycling parameters

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## 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.

Winding number of the third pulse C ( ${w}_{C}$) versus vertically the coherence orders ${p}_{\mathrm{AB}}$ and ${p}_{\mathrm{BC}}$ and horizontally the winding number of the second pulse B, ${w}_{B}$; the winding number of the first pulse A, ${w}_{A}$, being zero.
${p}_{\mathrm{AB}}$ ${p}_{\mathrm{BC}}$ ${w}_{B}$
12345 67891011 1213141516 171819202122
-3Q 112210219 208197186 17516415 314213112
-2Q 2221201918 1716151413 12111098 7654321
-1Q 2119171513 1197531 2220181614 12108642
-3Q0Q 18138321 1611611914 94221712 722015105
1Q
2Q 71421512 193101718 152261320 411182916
3Q 48121620 159131721 26101418 2237111519
-3Q 171152216 104211593 20148219 137118126
-2Q 71421512 193101718 152261320 411182916
-1Q 10207174 14111218 18515212 22919616313
-2Q0Q 19151173 2218141062 211713951 20161284
1Q
2Q 6121817 1319281420 3915214 10162251117
3Q 15722146 2113520124 191131810 21791168
-3Q
-2Q 15722146 2113520124 191131810 21791168
-1Q 2221201918 171615141312 11109876 54321
-1Q0Q 201714118 5222191613 1074121 181512963
1Q
2Q 51015202 712172249 14191611 1621381318
3Q 3691215 182114710 131619222 5811141720
-3Q 612181713 192814203 9152141016 2251117
-2Q
-1Q 112210219 208197186 175164153 14213112
0Q0Q 2119171513 1197531 2220181614 12108642
1Q
2Q 48121620 159131721 26101418 2237111519
3Q 14519101 1562011216 721123178 22134189
-3Q 12113214 315416517 6187198 20921102211
-2Q 81619172 101831119 41220513 2161422715
-1Q
1Q0Q 2221201918 171615141312 1110987 654321
1Q
2Q 3691215 182114710 131619222 5811141720
3Q 24681012 1416182022 1357911 1315171921
-3Q 1813832116 11611914 94221712 722015105
-2Q 16921811 4201362215 8117103 1912521147
-1Q 12113 214 315416517 6187198 20921102211
2Q0Q
1Q
2Q 246810 121416182022 1357911 1315171921
3Q 133166199 22122155 18821111 1441772010
-3Q 123456 7891011 1213141516 171819202122
-2Q 123456 7891011 1213141516 171819202122
-1Q 123456 7891011 1213141516 171819202122
3Q0Q 123456 7891011 1213141516 171819202122
1Q TTTTTT TTTTT TTTTTT TTTTT
2Q 123456 7891011 121314151617 1819202122
3Q 12345 67891011 121314151617 1819202122
Empty cell means no winding number is available for ${w}_{C}$. For the desired antiecho coherence transfer pathway (0Q -> 3Q -> 1Q -> -1Q), the letter T means that 22 values from 1 to 22 are available for ${w}_{C}$. This is not surprising, because the receiver phase always follows this desired pathway.

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:

 ${w}_{A}$ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ${w}_{B}$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ${w}_{C}$ 9 18 4 13 22 8 17 3 12 21 7 16 2 11 20 6 15 1 10 19 5 14 ${w}_{\mathrm{Rec}}$ 20 17 14 11 8 5 2 22 19 16 13 10 7 4 1 21 18 15 12 9 6 3

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.

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