## 3 Time Frequency Transforms  
The classical method for analyzing a signal with time-varying frequency content is to split the time-domain signal into many segments. The signal to be transformed is multiplied by a window function, which is nonzero for only a short period of time, and then, take the Fourier transform of each segment as the window slid along the time axis, resulting in a two-dimensional representation of the signal. This is known as the short-time Fourier transform (STFT) operation, which is the most widely-used method for analyzing non-stationary signals. Additionally, simply, in the continuous time case, it is defined as:  
\[STFT(t,\upomega)=\int_{-\infty}^{+\infty}y_{a}(\uptau)h(\uptau-t)e^{-j\upomega \uptau}\text{d}\uptau \tag{12}\]  
where \(h(t)\) is the analysis window, which is commonly a real and even window function centered on zero, \(y_{a}(t)\) is the defined analytical signal to be transformed and \(STFT(t,\upomega)\) is a function of \(t\) and \(\upomega\), which is linear and depends on the chosen window \(h\). In order to understand the time properties at a particular frequency, the definition of the STFT can also be expressed in the frequency domain by manipulating Equation (12), obtaining the following result:  
\[STFT(t,\upomega)=\frac{1}{2\uptau}e^{-j\upomega t}\int_{-\infty}^{+\infty}Y_{ a}(\upomega^{\prime})H(\upomega^{\prime}-\upomega)e^{j\upomega^{\prime}t} \text{d}\upomega^{\prime} \tag{13}\]  
where \(H(\upomega)\) is the frequency window function, which is the Fourier transform of \(h(t)\). The dual relationship between Equations (12) and (13) is apparent; the TFR can be generated via a moving window in time or a moving window in frequency. In the STFT analysis, one intends to achieve both high time and frequency resolution if possible. However, the resolution in the time domain is limited by the width of the window function \(h(t)\); similarly, the resolution in the frequency domain is limited by the width of the frequency window \(H(\upomega)\). As a result, the choice of a window to represent the signal by its spectrogram imposes a compromise between the conversation of temporal localization and that of frequency localization. This compromise is due to [PERSON]'s uncertainty principle, which states that the window width in time and the window width in frequency are inversely proportional to each other. Therefore, choosing a small time window leadsto good resolution in time and necessarily implies poor resolution in frequency; conversely, a long time window yields poor time resolution, but good frequency resolution. The length of the window plays a fundamental role in this TF compromise. The squared magnitude of the STFT, denoted by \(SPEC(t,\upomega)\), is called spectrogram, which can be written as follows:  
\[SPEC(t,\upomega)=|STFT(t,\upomega)|^{2} \tag{14}\]  
where \(STFT(t,\upomega)\) is the STFT defined in Equation (12). The spectrogram of the signal has poor TF localization properties due to the presence of analysis window function. 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:  
\[R(t,\tau)=y_{a}(t+\frac{\tau}{2})y_{a}^{*}(t-\frac{\tau}{2}) \tag{15}\]  
The WVD of \(y_{a}(t)\) is then defined as the Fourier transform of this time-dependent instantaneous auto-correlation function [27], written as follows:  
\[WVD(t,\upomega)=\frac{1}{2\uppi}\int_{-\infty}^{+\infty}R(t,\tau)e^{-j\upomega \tau}\text{d}\tau=\frac{1}{2\uppi}\int_{-\infty}^{+\infty}y_{a}(t+\frac{\tau }{2})y_{a}^{*}(t-\frac{\tau}{2})e^{-j\upomega\tau}\text{d}\tau \tag{16}\]  
where \(y_{a}(\cdot)\) denotes the signal to be analyzed, \(t\) is the time, \(\upomega\) represents the angular frequency, \(\tau\) is called the lag variable and \((*)\) denotes the complex conjugate. In addition, this class of bilinear (or quadratic) TF distributions can be most easily understood in terms of the ambiguity function.