## Tuesday, 1 July 2008

In my previous post I discussed about NMR FID/spectra having two channels, the Real and Imaginary parts and that in general, only the Real part is displayed. In fact, we could use the term Stereo-FID in the same fashion as we use Stereo-Sound (remember that NMR spectra span audio frequencies).
If we think about these stereo signals as coming from two receiver coils in quadrature and assuming simultaneous detection, we could formulate the following scenario:

It’s a common practise to acquire several (hundred or thousand) FIDs which are then added together (signal averaging). Because NMR responses build up in proportion to the number of signals recorded, N, whilst the noise varies randomly from one measurement to the next and thus, adds up more slowly, as sqrt(N), there is an overall improvement in sensitivity of sqrt(N).
If the noise in the Real and Imaginary channels were totally independent, it should be possible to add to the Real channel the Imaginary channel (after a 90º phase shift to make it phase coherent with regard to the Real part) so that we could achieve a further improvement of sensitive of sqrt(2)!. Would this be possible? This will be the subject of this post.

Of course, this ‘trick’ would only work if the noise in the two channels were totally independent. In the NMR field it is assumed that the experimental noise is stationary and white (i.e. the correlation between two consecutive points is zero). But what about the correlation between the noise in the two different channels? Are they correlated? This is very easy to analyze experimentally with Mnova by writing a very simple script which calculates the Correlation Coefficient (Pearson). For example, consider the spectrum below. We could use this script to calculate the Pearson Correlation Coefficient using the points between 5000 & 10000 which is a signal free region

As shown in the figure below, the correlation coefficient between the noise in both channels is almost zero. You can repeat this operation with different spectra and you will arrive to similar results

So it appears as if the noise in both channels is statistically uncorrelated, something which should be intuitively expected as the 2 acquisition channels are orthogonal. Does this mean that the noise in both channels is independent?
We can make a very simple experiment: We could phase a spectrum in order to make the real part perfectly in phase and then change the phase of the imaginary part in such a way that the peaks in both channels become perfectly phase-coherent. Next, both channels can be added so that the signals will increase 2 fold. On the other hand, if the noise in both channels were phase-incoherent, it will increase more slowly and therefore the overall S/N should increase by sqrt(2).

We can carry out this experiment very easily by applying a 90º phase shift into the imaginary channel and then summing up both channels. This is done with the following script:

Before applying this script, we need to calculate the SNR of the original spectrum. We can use the central peak of the Chloroform multiplet as a reference peak to estimate the SNR as depicted in the figure below:

Next, we can apply the script above (sumReImPhased) to apply a 90º phase shift to the imaginary part followed by the addition of the imaginary channel to the real one. If the SNR of this new spectrum is calculated we get the following result:

We can appreciate that the Chloroform peak is now twice the height, but the standard deviation has also increased two-fold, so the SNR remains constant. What a disappointment!

Conclusions:

We have first found out that the noise in the Real and Imaginary channels is statistically uncorrelated. However, this does not mean that they are independent. Just as sin(x) and cos(x) functions are orthogonal "uncorrelated", but not independent (sin^2(x) = 1 – cos^2(x)).
And we have shown that noise in the Re and Im channels is certainly not independent: the SNR does not improve at all.
The essence is that if the spectrometer has just one coil (which is, to the best of my knowledge, always the case in spectroscopy), then the noise in the two orthogonal channels would not be independent and thus the sqrt(2) sensitivity enhancement is not possible. It will be necessary to have two separate coils (and two receivers) in quadrature to have uncorrelated real and imaginary noise. Why is this not possible? I don’t really know, most likely because of lack of space …

References:

Experimental Noise in Data Acquisition and Evaluation III. Exponential Multiplication, Discrete Sampling, and Truncation Effects in FT Spectroscopy. Dr. S. Sýkora.

Patty said...

Hi Carlos,

This is a very interesting post. Your conclusion is of course not surprising as phase correcting the imaginary spectrum effectively makes it identical to the real spectrum.

Glenn

Carlos Cobas said...

Thanks Glenn!
The key idea in my post was that if quadrature detection was carried out using two (physically) different detectors, the noise in the two channles should be independent and therefore, summing the values from both parts (after a 90 phase shift of the imaginary part) should increase the SNR as sqrt(2).

In other words, phase correction of the imaginary part should not make it identical to the real spectrum. However, as only one detector is used, phase correction of the imaginary part makes it identical to the real one.
Cheers,
Carlos

kluwer said...

Sure, interesting concept and again the identical Re-spectrum is obvious since the Im and Re spectra are correlated by a rotation matrix

[x' ] [cos Phi sin Phi][x]
[y'] = [-sin Phi cos Phi][y]

'phasing' the Im-channel can only give the Re-channel including the noise!

greetings

Dr. Kluwer

Carlos Cobas said...

Thanks Dr. Kluwer for your comment.
Do you think that the noise in both channels will not be statiscally independent if the two components of the FID are recorded using two different devices?