9.4.2 Discrete Fourier Transforms

In order to measure the oscillation frequency of the magnetization more accurately, the discrete Fourier transform of the discretely sampled data has been calculated. The different components of the magnetization distribution have been sampled at regular time intervals at different spots on and in the nanodot.

Thus, we have obtained $N$ consecutive sampled values at a time interval $\Delta$ during the measurement time $\tau = N \cdot \Delta$. The accuracy of the discrete Fourier transform is determined by two fundamental numbers:

The Nyquist frequency [133]

$$f_N=\frac{1}{2 \cdot \Delta} \quad, \quad T_N = \frac{1}{f_N} = 2 \cdot \Delta$$

describes the maximum frequency, that can be measured, if the sampling interval $\Delta$ is given. The critical sampling of a sine wave requires two sampling points per cycle, the first at the positive peak and the second at its negative trough.

The minimum frequency, which also determines the ``resolution'' of the discrete Fourier transform (the minimum difference in frequency, which can be distinguished)

$$f_r = \frac{1}{T_r} = \frac{1}{N \cdot \Delta} = \frac{1}{\tau}$$

is given by the fact, that the maximum period, which fits into the measurement interval $\tau$, is at most $T_r = \tau$.

Therefore, the discrete Fourier transform of a real valued function with $N$ measurement samples each taken after a constant time interval $\Delta$ delivers $N+1$ values for the amplitudes for a frequency spectrum from 0 (constant offset) to the Nyquist frequency $f_N$ at frequency intervals $f_r$.