Tuesday, July 01, 2008

Asterisk or pizza?

At the basis of weighting there's the convolution theorem.
FT( f ⋅ g ) ∝ FT(f) * FT(g)
which can also be written as:
FT( f ⋅ g ) ∝ FT(f) ⊗ FT(g)
The choice depends, exclusively, on which symbol you use to indicate the convolution operation. If, after reading the linked Wikipedia pages, you feel like being still at the point you had started from, I can propose a simplified approach, based upon Fourier Pairs and practical examples. If, in time domain, you have an exponential decay, after FT you get a Lorentzian curve. The two functions form a Fourier pair. The Gaussian function pairs with itself. A stationary sinusoid pairs with an infinitely narrow line (a nail pointing upwards). It's a curve with no shape and no width (but it has a frequency). Convolution of two curves yields a third curve with the shape of both ancestors and a patrimony (line-widths) equal to the sum of the widths of the ancestors. You can think at the result curve as an empty and opaque envelope with no shape, with both ancestor curves inside. The envelope adheres to the content, so you can see the cumulative shape.
The natural NMR peak can be described, in time domain, as the product of a stationary sinusoidal wave with an exponential decay. This dumped sinusoid forms a Fourier pair with the convolution of a nail with a Lorentzian curve. This is the convolution theorem (applied to NMR): multiplication in time domain is equivalent to convolution in frequency domain. If you consider the wave as the substrate and the exponential decay as the weighting function, the result takes the frequency from the former and the shape from the latter. That's OK. You can't do the contrary (positive substrate and oscillating weight). If your weighting function oscillates, all the peaks will show an extra splitting.
Let's give, now, some reference formulas. We'll use the symbols:
W = full linewidth at half height, always measured in Hz
ν = frequency (Hz)
ω = 2 π ν = angular frequency
The exponential decay, in time domain, is described by a single parameter λ:
f(t) = exp( - λ t ).
It forms a pair with the Lorentzian curve in frequency domain:
F(ω) = λ / (λ² + ω²).
The latter has linewidth:
W = λ / π.
That's the only parameter I consider, instead of λ, because W can be directly compared to natural signals or to the gaussian broadening.
The Gaussian curve, in time domain, has the formula:
g(t) = exp( - σ² t² / 2 ).
The other component of its Fourier pair is still a gaussian function:
G(ω) = √{ 2π } ⋅ exp[ -ω² / (2σ²) ] / σ.
σ = π W / √{ ln(2) } ≅ 3.77344 W.
The exponential lends itself to both kind of uses (it can increase either the sensitivity or the resolution, it's enough to invert the sign of λ). You can also define a new function:
1 / g(t) = exp( σ² t² / 2 ).
to increase the resolution. It's unmanageable: σ could only have near-to-zero values (otherwise the noise becomes intolerable); to the best of my knowledge, nobody has ever used this abort of weight.
We have arrived at dividing, instead of multiplying. If multiplication in time domain is equivalent to convolution in frequency domain, then division is equivalent to deconvolution...


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