## 3 Preliminaries
### Morlet and Morse Wavelet Functions  
The analyzing function is always referred to as the mother function. The Morlet and Morse are the two selected mother functions used to generate the CWT in this study. The Morlet function \(\psi\) is denoted by:  
\[\psi(t)=e^{\frac{-\tau^{2}}{2}}cos(5t) \tag{1}\]  
where \(t\) is time in seconds and \(\psi\)\(\in L^{2}(\mathbb{R})\) such that \(L\) is the norm. Figure 1a represents the standard Morlet wavelet with \(0.5\leq+\geq-0.5\). \(\psi\) is a bandwidth and time-limited function with a zero-mean and unit energy [28]. The CWT requires shifted and stretched versions of the analyzing mother wavelet function \(\psi\), which is compared with the signal. The stretched or compressed versions of the Morlet mother wavelet as dilated wavelets can be denoted by \(\psi_{\tau,\gamma}\) which is defined in Equation (2). Figure 1b represents the standard Morse mother wavelet.  
\[\psi_{\tau,\gamma}(t)=\frac{1}{\sqrt{\gamma}}\psi(\frac{t-\tau}{\gamma}) \tag{2}\]  
where \(\gamma\) is the scaling or dilation factor that controls the width of the wavelet and \(\tau\) is the translation parameter controlling the location of the wavelet. Scaling a wavelet means stretching it (if \(\gamma>1\)) or compressing it (if \(\gamma<1\)), and translating it means shifting its position in time. The generalized Morse wavelet is a two-parameter family of wavelets defined in the frequency domain by:  
\[\psi_{\beta,\gamma}(\omega)=K_{\beta,\gamma}H(\omega)\omega^{\beta}e^{-\omega \gamma} \tag{3}\]  
where \(K_{\beta,\gamma}\) is a normalizing constant and \(H(\omega)\) is the Heaviside unit step function. To be a valid wavelet, one must have \(\beta>0\) and \(\$amma>0\). By varying these parameters, the generalized Morse wavelets can be given a broad range of characteristics while remaining exactly analytic.  
Fourier transformations are popular in creating feature representations for deep learning systems. However, under the Fourier transform, the time information of a time series is lost [29]. Due to this loss, it is hard to distinguish transient relations or to identify when the structural changes do occur. Therefore, wavelet analysis was introduced given that it performs the estimation of spectral properties of a time series as a function of time while revealing how different periodic components of the time series change over time. It is for this reason that the CWT was used in the feature set formulation in this study.  
Figure 1: (a) A standard Morlet mother wavelet. (b) A standard Morse mother wavelet.  
We have implemented the generalized Morse and Morlet wavelets so as to capture the general performances of limiting special forms of wavelets in both families. In summary, the Morlet wavelet represents a one-parameter family of functions, while the Morse wavelet represents a two-parameter family of functions. The limiting or special forms of the generalized Morse wavelets can be found if the frequency argument \(\omega\) is rescaled by the peak frequency \(\omega_{\beta},\gamma\), defining a new frequency-domain wavelet [30], as seen in Equation (3). Some special forms of the Morse family include the Couchy wavelet and the Bessel wavelets. Only the generalized Morse and Morlet wavelets were used in this study so as to capture the general performances of limiting special forms of wavelets in both families.  
Using the Morlet or Morse mother wavelets, a CWT of a formulated RSSI vector was obtained. The CWT graphs were adopted as images used as fingerprints for training a CNN to estimate the human user position. Similarly, the CWT numerical data, which were essentially the PSD data--in this text referred to as the CWT-PSD data--were used as fingerprint information to train an ANN to estimate the human user position.