## 3 Time Frequency Transforms
This compromise is due to [PERSON]'s uncertainty principle, which states that the time and frequency standard deviations of the corresponding window functions are related by the following inequality:  
\[\sigma_{t}\sigma_{\omega}  
From Equation (14), it is easy to know that the window width in time and the window width in frequency are inversely proportional to each other. Therefore, choosing a small time window leads to good resolution in time and necessarily implies poor resolution in frequency; conversely, a long time window yields poor time resolution, but good frequency resolution can be obtained. The length of the window function plays a fundamental role in this TF compromise.  
The squared magnitude of the STFT, denoted by \(S(t,\omega)\), is called the spectrogram, which can be written as follows:  
\[S(t,\omega)=|STFT(t,\omega)|^{2} \tag{19}\]  
where \(STFT(t,\omega)\) is the STFT defined in Equation (12). The spectrogram of the signal has poor TF localization properties due to the presence of the analysis window function. Here, we illustrate the influence effect on the spectrogram by using different types and lengths of the analysis window functions. As an example, the time domain signal, which contains four Gaussian components, is shown in Figure 1. By using different types of analysis window functions, the spectrogram of this signal is provided in Figure 2. In Figure 2a, the Chebyshev window function is selected, and in Figure 2b, the Kaiser window function is used; in both cases, the window size is 15 samples. It is clear that the spectrogram shows poor TF localization properties, which presents different TF characteristics depending on the selected window type. Figure 1: Time domain trace of four Gaussian components’ signal. Furthermore, in order to evaluate the TF characteristics of the spectrogram by varying the length of the analysis window function, an example of the spectrogram of the aforementioned signal is presented in Figure 3, where the Hamming window is adopted. In Figure 3a, the window size is 15 samples, and it is easy to know that the small time window length implies poor resolution in frequency; in Figure 3b, the window size is increased to 31 samples, and improved resolution in frequency can be achieved when the window length is increased. From Figure 3, we can conclude that if the length of the adopted window function is small, then the localization of the frequency content of the aforementioned signal will be smeared and vice versa. This further verifies the limitation from the uncertainty principle with the spectrogram approach. We can know that the spectrogram of the signal presents the TF resolution trade-off problem due to the presence of the analysis window function. Figure 3: Spectrogram of four Gaussian components’ signals, which has been calculated by using Hamming window function. (**a**) Window size = 15; (**b**) Window size = 31. Figure 2: Spectrogram of four Gaussian components’ signals, which has been calculated by using a window length of 15 samples. (**a**) Window type = Chebyshev; (**b**) Window type = Kaiser. In order to avoid the TF resolution trade-off problem of the spectrogram, WVD is adopted in interference detection for GNSS receivers. In the WVD, a time-dependent instantaneous auto-correlation function is chosen as:  
\[R(t,\tau)=y_{a}(t+\frac{\tau}{2})y_{a}^{*}(t-\frac{\tau}{2}) \tag{20}\]  
The WVD of \(y_{a}(t)\) is then defined as the Fourier transform of this time-dependent instantaneous auto-correlation function [23], written as follows:  
\[WVD(t,\upomega)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}R(t,\uptau)e^{-j\upomega \uptau}\upalpha\uptau=\frac{1}{2\uppi}\int_{-\infty}^{+\infty}y_{a}(t+\frac{ \uptau}{2})y_{a}^{*}(t-\frac{\uptau}{2})e^{-j\upomega\uptau}\upalpha\uptau \tag{21}\]  
where \(y_{a}(\cdot)\) denotes the signal to be analyzed, \(t\) is the time, \(\upomega\) represents the angular frequency, \(\uptau\) is called the lag variable and \((*)\) denotes the complex conjugate.