The nature of Fourier transform NMR

Before discussing Fourier transform NMR, the nature of Fourier transforms shall be explained. We have already noted that a representation in the time domain can be transformed into an equivalent representation in the frequency domain and vice versa. Consider the two different representations 

Now suppose that we have two such sine waves superimposed with different amplitudes and different frequencies. In the next figure, the superposition of two sine waves of different frequencies is represented. The result is an interference pattern with beats in it. In the representation of amplitude vs. frequency there are two spikes with horizontal coordinates corresponding to the appropriate frequencies. One can continue to add sinusoidal waves to both diagrams and see that the pattern in the amplitude vs. time diagram becomes increasingly complex, while in the amplitude vs. frequency diagram one simply adds more spikes to the curve. 

These two representations, amplitude vs. time and amplitude vs. frequency, both contain the same information. One representation can be transformed into the other by a mathematical technique called a Fourier transform represented by 

In the transformation from the frequency to the time domain (the top equation), the amplitude of a wave A(uj) is multiplied by a sinusoidal wave of unit amplitude, exp(-iu)t), and this product is added over all frequencies at time t. The translation from time space to frequency space is the inverse of this process. Consider now the application to 

One advantage of Fourier transform NMR becomes immediately obvious. Consider an analogy Suppose there were a 

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