## 2 System Description
### Filters
#### 2.3.2 Emitter Filter  
Contrary to the receiver, the emitter has higher flexibility concerning the filter design, since it allows digitally setting the filter impulse response. The digital filter is designed according to Equations (2) and (3), with particular attention to the pulse shaping filter, \(p(t)\). This section describes the two considered approaches to design \(p(t)\): (a) the raised cosine filter and (b) the filter matched to the analog receiver filter (designated here as the matched filter approach). (a) Raised cosine filters:  
A common pulse shaping filter used in digital communication systems is the raised cosine filter, which is able to mitigate Intersymbol Interference (ISI) in communication systems. Its coefficients are computed as follows:  
\[p(t)=\frac{sin(\frac{\pi t}{T})}{\frac{\pi t}{T}}\frac{cos(\frac{\pi\beta t}{T })}{1-\frac{(2\beta t}{T})^{2}} \tag{19}\]  
Figure 5: Analog third-order filter types’ comparison: (**a**) magnitude, (**b**) phase, and (**c**) group delay. where \(T=1/R\) is the data symbol period and \(\beta\in[0,1]\) is the filter roll-off factor. The filter bandwidth is given by:  
\[B^{\prime}=\frac{R}{2}\left(\beta+1\right) \tag{20}\]  
From Equation (20), using a lower \(\beta\) value, a higher data rate, \(R\), can be transmitted using the same bandwidth, \(B^{\prime}\), up to two-times higher, improving spectral efficiency. From the literature, many authors studied the impact of changing \(\beta\), trying to optimize the systems for a high data rate by increasing \(R\) for the same \(B^{\prime}\). However, a high data rate is not a priority in most of the IoT systems, in particular for the proposed system, the goal of which is to demodulate an m-CAP signal with low resources. Hence, fixing \(R\) while varying \(\beta\) allows a bandwidth compression: for the same \(R\), as \(\beta\) decreases, \(B^{\prime}\) also decreases. There are two main reasons to fix \(R\) while varying \(\beta\):  
* Bearing in mind the receiver architecture's simplicity, the synchronism mechanism can be simplified if there is no constellation rotation between each symbol. CAP modulation is known to have a deterministic phase shift between each symbol [53]. Although the phase shift between modulated symbols is deterministic, the receiver would require a complex device synchronization mechanism to accommodate the signal periodic phase shift. In other words, using \(\beta\
eq 1\) would require the receiver to continuously adjust the LO phase for each received symbol. This can be avoided if \(R=B/2\), since according to Equation (4), the \(m^{th}\) carrier has a period that is a multiple of \(1/R\). * Analog demodulation with a continuous spectrum may negatively impact the data recovery due to the non-ideal frequency response of analog filters, i.e., it has a finite value decay. Therefore, by fixing \(R=B/2\) and varying \(\beta\), guard bands are created as shown in Figure 7.  
Figure 6: Comparison between Bessel and Butterworth third order filters, considering high-frequency asymptote normalization: (**a**) magnitude and (**b**) group delay. (b) Matched filters:  
As previously mentioned, a matched filter is commonly used to improve the SNR in communication systems [51]. A matched filter is the optimal linear filter that allows SNR maximization and is derived from the original filter as follows:  
\[g(t)=p(\tau-t) \tag{21}\]  
where \(g(t)\) is a matched filter for the \(p(t)\) filter and \(\tau\) is a time delay. In a fully digital communication system, the matched filter is chosen for both the emitter and receiver bearing in mind the desired impulse response for the convolution of both filters. For instance, the square-root raised cosine filter is commonly used, both in the emitter and receiver. The convolution between both gives the raised cosine filter impulse response, resulting in an ISI-free communication [51]. However, as said, the raised cosine filter is not realizable in the analog domain.