1. Recall that the Nyquist-Shannon theorem states that the minimum sampling rate fs necessary to faithfully resolve a signal of interest with frequency f is given by: fs 2f (a) Consider the following 1D 13C NMR spectrum, given as the Fourier transform in the frequency domain. Note that the y axis units are arbitrary, and the x axis represents the frequency in kHz, increasing right to left. Identify the highest frequency signal in this dataset, and determine whether or not the actual sampling frequency (59.524kHz) was sufficient to faithfully render all resonances in the spectrum. What is the maximum frequency of a resonance that can be faithfully rendered at this sampling rate? 25 25 - 21442.075 - 17560.276 15423.241 9718.133 9686.150 9654.128 9303.159 7129.056 7060.676 6292.448 5321.362 5303.375 4997.101 4970.318 4794.607 4653.003 4601.200 4551.400 4502.794 4011.554 4007.107 3550.974 3523.137 3504.844 3492.921 3054.328 2998.199 2870.221 2838.189 2694.008 20 20 15 2 10 10 2645.336 2428.483 15 20 [rel] 5 0 [kHz] 10 10 55 0 (b) Assuming the ADC for the spectrometer used to acquire this data operates at 16 bit, what is the minimum data transfer rate demanded of this ADC for the above dataset, if the data is generated in 1.101s and must be converted in the same time? (c) Consider the case in which the sampling rate is 30.0kHz. Which of the signals in the above spectrum would not be faithfully rendered, and at what frequencies would their aliases appear? (Hint: Drawing out a Nyquist zone diagram can be helpful). 3