# Statistical Characterization of the Magnetic Field in Space during Magnetic Storms
## 4 Power Spectrum of Each Component of the Magnetic Field during a Magnetic Storm  
The power spectrum is an important numerical feature of a stochastic process. It is the Fourier transform of the autocorrelation function, indicating the amount of power contributed by each frequency component of the stochastic process. Power spectrum estimation is widely used in cognitive radio analysis, spectrum multiplexing, transmit power control, and interference suppression [35; 36; 37]. In wireless communication systems, power spectrum measurements are crucial for techniques such as delay estimation, channel equalization, and speech enhancement [38; 39; 40].  
For a stationary stochastic process, the power spectrum is the Fourier transform of the autocorrelation function:  
\[S(\omega)=\int_{-\infty}^{+\infty}R(\tau)\mathrm{e}^{-\mathrm{j}\omega\tau} \mathrm{d}\tau \tag{6}\]  
The analysis in Section 3 shows that the magnetic field components exhibit non-stationary properties under magnetic storm conditions, so it is necessary to introduce a time variable in addition to frequency; i.e., for non-stationary processes, the power spectrum is a binary function of both time and frequency, indicating the magnitude of the power contributed by a frequency component at a given observation time. The result is referred to as the dynamic power spectrum [41]. The dynamic power spectrum and the observed time series satisfy  
\[\begin{array}{l}\int_{-\infty}^{+\infty}S(t,\omega)\mathrm{d}\omega=\left|x (t)\right|^{2}\\ \int_{-\infty}^{+\infty}S(t,\omega)\mathrm{d}t=\left|X(\omega)\right|^{2} \end{array} \tag{7}\]  
where \(S(t,\omega)\) is the dynamic power spectrum, \(x(t)\) is the observed value, and \(X(\omega)\) is the Fourier transform of \(x(t)\).  
According to the conditions in (7), the dynamic power spectrum can be expressed as described in [41] for a non-stationary stochastic process:  
\[S(t,\omega)=\int_{-\infty}^{+\infty}E(x(t)x(t+\tau))\mathrm{e}^{-\mathrm{j} \omega\tau}\mathrm{d}\tau=\int_{-\infty}^{+\infty}R(t,\tau)\mathrm{e}^{- \mathrm{j}\omega\tau}\mathrm{d}\tau \tag{8}\]  
By substituting the binary autocorrelation function obtained in Section 3 into the above equation, the dynamic power spectrum of each component of the magnetic field during the six magnetic storms can be obtained, as shown in Figure 4.  
Figure 3: The magnitude of each component of the magnetic field. (**a**) 0:00 on 6 March 2016, to 23:00 on 7 March 2016; (**b**) 0:00 on 13 October 2016, to 23:00 on 14 October 2016; (**c**) 0:00 on 27 May 2017, to 23:00 on 28 May 2017; (**d**) 00:00 on 8 September 2017, to 23:00 on 9 September 2017; (**e**) 0:00 on 27 September 2020, to 23:00 on 28 September 2020; (**f**) 00:00 on 4 November 2021, to 23:00 on 5 November 2021.  
For a nonstationary magnetic field during magnetic storms, the dynamic power spectrum in each observation period is mainly concentrated in the very-low-frequency band, with an attenuation of up to 100 dB or more as the frequency rises. This result confirms the conclusion that the fluctuations caused by magnetic storm excitation occur in the very-low-frequency band [42; 43].