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Problems

< NMR.ThermalNoise | Index | Appendices >

Passband of a π/2 pulse

Assume that the duration of the pulse is T. Show that the passband is a sinc(x) function and calculate the smallest frequency where it vanishes. Hint: calculate the complex Fourier transform of a pulse of width T, centred at t=0

Heterodyne quadrature detection

A mixer produces an output proportional to the product of the two inputs. The NMR signal is offset by the few hundred kHz frequency {$\delta\omega$} from the transmission {$\omega_T$}. Show that by mixing it first with the intermediate frequency {$\omega_0$}, and subsequently with {$\omega_T-\omega_0$}, the result can be down-converted to the audio frequency. Describe the two low pass filters that one needs to place after each mixers and the full spectrum that perfect mixers (exact multipliers) without filters would produce. Hint: calculate the product of the three harmonic functions

Quarter wave reflector

Calculate the amplitude of the total wave at the in end of a quarter wavelength guide, when the other end is a node (ground, in the case of a coaxial cable). Hint: superpose a progressive wave with a reflected wave and impose that the combination satisfies the boundary condition.

Diode bridge

Describe, semiperiod by semiperiod, the fate of a radiofrequency wave that encounters a diode bridge in series with the central conductor of a coaxial cable, in case that its amplitude is a) 3 V, or b) 0.5 V

Fast Fourier Transform

Calculate the maximum frequency that you get in the Fast Fourier Transform spectrum of a digitally sampled signal with clock rate {$\nu$}. Indicate maximum and minimum frequencies of the spectrum in the case of simple detection and in the case of quadrature detection.

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