## 2 Theoretical Development of the Time-Frequency Analysis
Note that, for a finite data, the result of applying the Gaussian smoothing method is just the zero-order least squares response of the data. For the interior points, the resulting smoothed response is approximately diffusive with a wide transition zone around the cutoff frequency selected to filter a given dataset. The iterative Gaussian smoothing method is therefore used to repeatedly smooth the remaining high frequency part. It is effective to remove a non-sinusoidal polynomial with finite degree [20]. Moreover, the whole procedure can be performed in the spectral domain whose operation count is slightly more than twice of the use of the FFT. Subsequently, the zero points around the two ends are identified by a searching procedure and an interpolation method. The desired data, which is between two zero points, is redistributed by a cubic interpolation method to a data with a length of \(2^{\text{m}}\). Performing an odd function mapping of the redistributed data, the data length is of \(2^{\text{m+1}}\). Consequently, the corresponding Fourier sine spectrum could be therefore obtained by performing the FFT. The resulting Fourier sine spectrum is a slightly better approximate parameter representation of the sinusoidal part than that of the original Fourier spectrum. Specifically, the Fourier sine spectrum is a projection in strong sense while the Fourier spectrum is a projection in weak sense, such that the resolution of minor modes of the former is better than that of the latter [20]. Choosing a band-pass filter to filter out the signal at the desired frequency, the signal in time domain is subsequently obtained by performing the inverse FFT (IFFT). For a sinusoidal signal, \(y(t)\), defined in the range of \(-\infty<t<\infty\), the following Hilbert transform is the corresponding imaginary part [21]:  
\[\tilde{y}(t)=\int\limits_{-\infty}^{\infty}\frac{y(\tau)}{\pi(t-\tau)}d\tau \tag{4}\]  
The amplitude and frequency can be evaluated directly as follows:  
\[ \begin{split} z(t)&=y(t)+\tilde{y}(t)=A(t)e^{i\theta (t)}\\ A(t)&=\sqrt{y^{2}(t)+\tilde{y}^{2}(t)}\\ \theta(t)&=\tan^{-1}(\frac{\tilde{y}(t)}{y(t)})=2 \pi f_{\theta}t\end{split}  \tag{5}\]  
Finally, the corresponding imaginary part, and subsequently the amplitude and the frequency information, can be estimated based on the Hilbert transform [21]. As a result, the spectrogram with small error is obtained. From the resulting spectrogram much information concerning the dominant modes and corresponding harmonics can be captured. Moreover, the associated amplitude and frequency variations are partially reflected by the spectrum scattering around a mode. Thus, the detailed variations can be obtained by examining the spectrogram. In other words, it could provide an effective tool for detecting the low received power GPS signals by means of the two dimensional time-frequency analysis. Figure 2 shows an example of the non-sinusoidal part effect. The thin solid line represents the Fourier spectrum of the original signal with a non-sinusoidal part. The heavy solid line indicates the Fourier spectrum of the sinusoidal part. It shows that the spectrum of the original signal has the error in the low frequency part because of the non-sinusoid part signal. The gradient line depicts the resulting spectrum of the sinusoid part via iterative Gaussian smoothing method with odd function mapping. It is important to note that the odd function mapping in step 4 ensures that the periodicity of the signal is valid up to the highest order derivative of the data [19]. Consequently, the spectrum error can be effectively reduced. Figure 1: The procedure of generating a spectrogram with small error using the Fourier sine spectrum [19].