## 2 Theoretical Development of the Time-Frequency Analysis  
For a stationary signal characterized by time-invariant statistic properties, it can be adequately analyzed using well-known spectral techniques based on the Fourier transform (FT) [9]. For example, the Fourier filters can be employed to separate the noise from the real signal when the processes are linear and the noises have distinct time or frequency scales, which are different from those of the true signal [16]. However, the Fourier methods might fall short when the processes are either nonlinear or non-stationary. In fact, the GPS satellite signal with low received power might be \"transient\" in the receiver, and it therefore can be defined as a non-stationary signal since a non-stationary signal is \"transient\" in nature with duration generally shorter than the observation interval [9, 16]. A concept of time-frequency representation had been proposed since 1950s [7], and it could be treated as an extension to Fourier analysis that is applicable to characterize signals whose spectral characteristics change with time. Current time-frequency representation tools, such as the Fourier spectrum, the short time Fourier transform (STFT or Gabor), the wavelet (Morlet) transform, or many high order schemes are generally implemented to the time-frequency analyses [7, 17, 18], require lower computational power and might be suitable for real-time applications, however the spectrogram generated might have poor time-frequency localization properties [11]. Another time-frequency analysis was represented by the Wigner-Ville distribution, which better localizes the signal on the time-frequency plane, but has the possible presence of cross-terms [11]. In general, a fast Fourier transform (FFT) algorithm could represent the spectrum of a signal in frequency domain. However, the resulting spectrum might have an exponentially decayed envelope if the signal is with a non-sinusoidal part [19]. As a result, several time-frequency tools, which are based on FT, cannot provide a precise spectrogram. This paper therefore implements a time-frequency analysis based on the Fourier sine spectrum to represent the information on a spectrogram with small error [19]. It could present almost all the frequency components of the received GPS signal on the corresponding spectrogram. The procedure of generating a spectrogram with small error by the Fourier sine spectrum is depicted in Figure 1[19]. Assume that a discrete signal can be approximated by [20]:  
\[y_{j}=\sum_{l=0}^{\infty}\Biggl{[}b_{l}\cos\frac{2\pi t_{j}}{\lambda_{l}}+c_{ l}\sin\frac{2\pi t_{j}}{\lambda_{l}}\Biggr{]}+\sum_{n=0}^{N}a_{n}t_{j}^{n},\ \ \ -\infty<j<\infty \tag{1}\]  
where the second summation represents the non-sinusoidal part and \(N\) is referred to as the largest power for which \(a_{n}\to 0\) for all \(n>N\). Note that the non-sinusoidal part is interpreted as the sum of monotonic part and all the Fourier modes whose wavelength is longer than the expansion interval. The collected signal is input to an iterative Gaussian smoothing filter, which can remove the non-sinusoidal components for grasping pure spectral information of the input signal [20]. After the iterative Gaussian smoothing method by \(m\) times, where \(m>N/2\), the corresponding results can be rearranged in the following [19,20]:  
\[\dot{y_{m}}=\sum_{l=0}^{\infty}\left[1-\exp\left(\frac{-2\pi^{2}\sigma^{2}}{ \lambda_{l}^{2}}\right)\right]^{m}\left[b_{l}\cos\frac{2\pi t}{\lambda_{l}}+c_ {l}\sin\frac{2\pi t}{\lambda_{l}}\right] \tag{2}\]  
\[\overline{y}(m)=y-\dot{y_{m}}=\sum_{l=0}^{N}A_{l,m,\sigma}\left[b_{l}\cos\frac {2\pi t}{\lambda_{l}}+c_{l}\sin\frac{2\pi t}{\lambda_{l}}\right] \tag{3}\]  
where \(\dot{y_{m}}\) and \(\overline{y}_{m}\) are the high frequency and smoothed parts at the \(m^{\text{th}}\) smoothing step, respectively, and \(\sigma\) is the smoothing factor. As a result, the non-sinusoidal part is removed completely.