iPods and Fourier Transforms

The way people collect and listen to music has changed drastically over the last several years. One's entire music collection, which once occupied several shelves in the living room, can now fit in the palm of one's hand and can be listened to virtually anywhere. The iPod / mp3 technology is also being used to change the way we can learn. Many, if not all of us, have spent tens of thousands of dollars to educate ourselves. If you are like me, there were courses in university you would have liked to take but did not have the time. Now, many universities have made their courses available free of charge online through iTunes University. Now, one can audit all of those expensive university courses for free through their iPod Touch or iPhone on the bus or train on the way to work. One excellent course I have been auditing is one called The Fourier Transform and its Applications, given by Brad Osgood of Stanford University's School of Engineering. As NMR spectroscopists, we use the Fourier Transform daily. This course covers many of the mathematical details not learned in NMR courses and explores many applications (other than NMR spectroscopy) where Fourier Transforms are useful. I highly recommend it and I hope that many more universities make their course material public.I regret to say that Bob Dylan is temporarily taking a back seat on my daily commute to work. Sorry Bob!

Echoes and Fourier Transforms

NMR data are often collected using pulse sequences involving echoes. These sequences are usually of the form:

pulse1 - delay1 - pulse2 - delay 2 - acquire data

In theory the echo occurs when delay1 = delay2. Often, this is not quite correct in practice due to the pulses having finite duration and short hidden pre-acquisition delays before the receiver is turned on. It is a good idea to set delay2 < delay1 so that the entire echo is captured. In such cases it is important to discard the data before the top of the echo before Fourier transformation. Failure to do so will result in improper line shapes (in the case of broad lines) or phasing errors. The figure below shows the ²H quadrupolar echo data for perdeuterated PMMA. The effect on the line shape is shown when the data are Fourier transformed before the echo (top trace) and after the echo (bottom trace). The correct line shape is only obtained when the data are Fourier transformed precisely at the top of the echo (middle trace).

The Fourier Transform of a Single Rotational Echo

The spinning sidebands in MAS NMR spectra are the result of rotational echos in the free induction decay. These echos show up as spikes in the free induction decay (and can be used conveniently to set the magic angle). The intensity envelope of the spinning sideband manifold in the NMR spectrum mimics the static wideline spectrum. One can obtain an NMR spectrum similar to the wideline spectrum from MAS data by Fourier transforming a single rotational echo rather than the entire FID. This is illustrated in the figure below for the ²⁷Al MAS data for kaolinite at 21.1 Tesla.