#### 3.1. Frequency Drift

[8] It is well known that the Fourier transformation (FT) spectrum is obtained by integral from t = −∞ to ∞, which is stationary or time independent. But, the observational signals f(t) are usually generated from a time-varying system, i.e., the varying properties of this system is described by the spectrum in every instant time or named time-frequency spectrum. The simplest method for FT is to integrate in a short time interval, that is, the windowed FT (FT^{w}):

Here, *W*_{τ} is the window function, , the bracket in the right side denote the inner product. The key problem of the time-frequency spectrum in FT^{w} is that the resolution ability or rate in both of time and frequency domains may be contradicted for a fixed width in the window function in time domain. If the width is big, the resolution rate in time domain is lost, and if it is small, the resolution rate in frequency domain is decreased.

[9] The mother function of the wavelet transformation (WT) is selected as Ψ(*t*) ∈ *L*^{2}(−∞,∞) [*Charles*, 1995],

to satisfy

and

where a and b are parameters of WT. Then, the WT and its inversion are defined by

[10] The main advantages of WT are compared with FT^{w} as follows.

[11] 1. The width of the wavelet in time domain depends adaptively on frequency ω, hence the resolution rate in both time and frequency domains is better than that of FT^{w}.

[12] 2. If N is the number of points in time domain, and the sample frequency be ω_{s}, then Δ*t* = . According to the sampling theorem, discrete F_{f}(ω_{i}) in FT^{w} is a linear distribution in [0, ] divided equally by , i.e., Δω = , or Δω ċ Δ*t* = , which is something like the uncertainty relation. If we want to know its spectrum denser in frequency, then we have to divide time domain sparer, and vice versa. However, there is no such a contrariety in WT theoretically.

[13] 3. In FT^{w}, all the discrete points are linear distribution in both the time and frequency domains. But there are a lot of nonlinearity in the practical observations, which are just the most interesting phenomena that we want to know.

[14] Therefore WT is suitable for analysis of a time series containing non-stationary power at many different frequencies as shown in Figure 1. Figure 2 gives the dynamic spectrum of magnetic and electric fields, and two density fluctuations in Figure 1. The frequency drift from 1 kHz (below the low-hybrid frequency) to 200 Hz (below the proton cyclotron frequency). It is also found an extremely low frequency (ELF) component at 20–30 Hz (close to the oxygen ion cyclotron frequency) without the frequency drift in the density fluctuations. The FT of LF component of magnetic and electric fields are shown in Figure 3, which is comparable with the wavelet analysis in Figure 2, e.g., the frequency range is from 200 Hz to 1 kHz with maximum at about 0.5 kHz in Figures 1 and 2.

#### 3.2. Density Cavity

[15] The further analysis is performed using a digital low or high pass filter and FFT to separate the DC, ELF and LF components in the four time series of Figure 1.

Here, m and n are parameters of the filter. The DC component of the density depletions is well coincident with the LF component of electric and magnetic wave packets in the example (Figure 4), which means that the whistler waves may be trapped in the density cavities. Moreover, the ponderomotive force of the wave packets may deepen the seed density cavities [*Seyler*, 1994].

[16] The time delay is about 0.8 ms from the cross correlation between the time profiles (selected in Figure 1b) of two Langmuir probes (with distance of 21 m) to measure the density fluctuations (Figure 5). Hence the speed of the density cavities is about 25 km/s, the angle θ between the line of two probes and velocity of Freja is 70 degree, and the product of 25 km/s and cosθ is comparable with the speed of Freja satellite [*Dovner et al.*, 1994]. The local density structure with time scale of 0.2 s is about fixed in space with a length of about 500 m in the direction of Freja.

#### 3.4. Propagation

[18] The second example of whistler waves is selected at 04:23:21.2– 21.4 UT with three magnetic and one electric fluctuations in Figure 8 (“elmag” mode of F4 experiment). The time scale and evolution are quite similar to that in Figure 1b. The angle between two vectors of the magnetic fluctuations and the ambient (DC) magnetic field by F2 experiment is between 56–57 degree (Figure 9), which means that the propagation of the whistlers is oblique in respect to the ambient magnetic field, because the wave vector is perpendicular to the vector of magnetic fluctuations. The wave vector of the whistlers may be calculated using the data in Figure 8 and the method of the minimum variance analysis [*Song and Russell*, 1999]. The percent variance of three eigenvalue is 50.98, 45.84 and 3.175. The wave vector is obtained from the calculated eigenvector (177.68, −46.52, −24.32) corresponding to the third eigenvalue of the minimum variance, and the magnetic fluctuations rotate mostly in a plane constructed by the other two eigenvectors (Figure 10). The three components of the averaged ambient magnetic field by F2 experiment in the duration of the second example in Figure 8 are 24062.1, −10972.7, and 5151.57 nT, respectively. Hence the angle between the wave vector and ambient magnetic field is about 21 degree, which is consistent with the prediction of Figure 9. Moreover, if the wave vector is calculated in three time intervals of the example, the angle decreases with time from 31, to 26 and 17 degree, respectively.