## 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, taking 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:  
\[STFT(t,\upomega)=\int_{-\infty}^{+\infty}y_{a}(\uptau)h(\uptau-t)e^{-j\upomega \uptau}\text{d}\tau \tag{12}\]  
where \(h(t)\) is the analysis window function, which is a real and even window function centered on zero, and the commonly-used window functions include Hamming, Hanning, Chebyshev, Kaiser and Gaussian windows; \(y_{a}(t)\) is the defined analytical signal to be transformed, and it is fed through the analysis bandpass filter with center frequency \(\omega\) whose bandwidth is equal to that of the analysis window \(h\); and \(STFT(t,\omega)\) is a function of \(t\) and \(\omega\), which is linear and depends on the chosen window \(h\). This operation in Equation (12) differs from the classical Fourier transform only by the presence of a window function \(h(t)\). The STFT is generated by taking the Fourier transform of smaller durations of the original signal. Alternatively, the STFT can be interpreted as the projection of the analytic signal \(y_{a}(\uptau)\) onto a set of bases \(h^{*}(\uptau-t)\text{exp}\{j\omega\uptau\}\) with parameters \(t\) and \(\omega\). Since the bases are no longer of infinite extent in time, it is possible to monitor how the signal frequency spectrum varies as a function of time. This is accomplished by the translation of the window as a function of time instant \(t\), resulting in a two-dimensional joint TFR \(STFT(t,\omega)\) of the original time signal. 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:  
\[STFT(t,\omega)=\frac{1}{2\uppi}e^{-j\omega t}\int_{-\infty}^{+\infty}Y_{a}( \omega^{\prime})H(\omega^{\prime}-\omega)e^{j\omega^{\prime}t}\text{d}\omega^ {\prime} \tag{13}\]  
where \(H(\omega)\) 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 resolutions 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 \(H(\omega)\). As a result, the choice of a window to represent the signal by the STFT imposes a compromise between the conservation of temporal localization and that of frequency localization.