CSDM origin offset and reference issue

[deepanshs]

Equations

The CSDM LinearDimension defines coordinates as coordinates = $\Delta x(J_k - Z_k) + b_k$, where $J_k = [0, 1, ... N_k -1]$. When complex_fft=False, $Z_k$ = 0. When complex fft=True, $Z_k = N_k/2$ for even $N_k$ else $(N_k-1)/2$

• coordinates_offset = $b_k$
• increment = $\Delta x_k$
• count = N

For NMR datasets, here are two cases---when origin_offset is defined at the receiver frequency, and origin_offset is defined at the reference frequency.

	originoffset $= w\text{RF}$	originoffset $= w\text{ref}$
complex_fft = True	$w\text{ref} = w\text{RF} - b$	$w\text{RF} = w\text{ref} + b$
complex_fft = False, $N$ is odd	$w\text{ref} = w\text{RF} - (b + \frac{N-1}{2} \Delta x)$	$w\text{RF} = w\text{ref} + (b + \frac{N-1}{2} \Delta x)$
complex_fft = False, $N$ is even, $\Delta x$ > 0	$w\text{ref} = w\text{RF} - (b + \frac{N}{2} \Delta x)$	$w\text{RF} = w\text{ref} + (b + \frac{N}{2} \Delta x)$
complex_fft = False, $N$ is even, $\Delta x$ < 0	$w\text{ref} = w\text{RF} - (b + (\frac{N}{2} - 1) \Delta x)$	$w\text{RF} = w\text{ref} + (b + (\frac{N}{2} - 1) \Delta x)$

The conversion from Hz -> ppm is coordinates / ${w_\text{ref}}$. The difference is that when originoffset is ${w\text{RF}}$, the value of ${w_\text{ref}}$ depends on how the LinearDimension is defined.

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What happens to the origin offset for a data subset?

• If the origin offset is $w\text{ref}$, we retain both frequency and ppm dimension scale for the subset data but lose the information on $w\text{RF}$. However, does $w_\text{RF}$ has meaning for a data subset?

• If the origin offset is $w\text{RF}$, we lose the information on $w\text{ref}$. Therefore, the subset data can only be represented in frequency dimension, not dimensionless ppm scale. We retain $w_\text{RF}$, which again seems meaning less for a data subset.

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Cases

1 When dataset is the full spectrum

property	originoffset $= w\text{ref}$	originoffset $= w\text{RF}$
always known	$w_\text{ref}$	$w_\text{RF}$
can be derived	$w_\text{RF}$	$w_\text{ref}$
information loss	-	-
absolute_coordinates	✅	❌
coordinates	✅	✅
ppm_scale (using above equations)	✅	✅

2 Subset of the full spectrum (keeping the origin_offset the same as the original dataset)

property	originoffset $= w\text{ref}$	originoffset $= w\text{RF}$
always known	$w_\text{ref}$	$w_\text{RF}$
can be derived	-	-
information loss	$w_\text{RF}$	$w_\text{ref}$
absolute_coordinates	✅	❌
coordinates	✅	✅
ppm_scale (using above equations)	✅	❌

3 Subset of the full spectrum (updating the origin_offset to maintain the same ppm scale)

property	originoffset $= w\text{ref}$	originoffset $= w'\text{RF}$
always known	$w_\text{ref}$	$w'_\text{RF}$
can be derived	-	-
information loss	$w_\text{RF}$	$w_\text{RF}$
absolute_coordinates	✅	❌
coordinates	✅	✅
ppm_scale (using above equations)	✅	✅

Scenarios

• When origin_offset is $w_\text{RF}$. Case 1 and Case 2(subset has the same origin_offset as the original)

Test file scenario-1.zip Summary origin_offset is always $w\text{RF}$. The coordinates in Hz are known. Since there is no evidence if the csdm object is a complete or a subset, $w\text{ref}$ and therefore the coordinates in ppm are presumed incorrect.

• When origin_offset is $w_\text{ref}$. Case 1 and Case 2(subset has the same origin_offset as the original)

Test file scenario-2.zip Summary origin_offset is always $w\text{ref}$. The coordinates in Hz and ppm are always known. Since there is no evidence if the csdm object is a complete or a subset, the $w\text{RF}$ is presumed incorrect.

[pjgrandinetti]

The assumption that the carrier frequency is at the center of the spectrum is the only way to encode both wRF and wref into o and b.

[pjgrandinetti]

Correct absolute coordinates require $\Delta oo = - \Delta b$. This appears to be true regardless of the complex_fft flag state.