## 2 System Description
### Filters
#### 2.3.1 Receiver Filter  
There are several requirements that should be taken into account when designing the receiving filters: (i) filter order: higher performance for higher orders, but difficult to implement; (ii) filter cut-off frequency: if closer to the band limit, it could attenuate the desired signal; (iii) frequency response: it should attenuate the undesirable bands, including image bands from the mixing process, while having low and flat attenuation in the pass band; (iv) impulse response: the output signal should not be severely distorted under the risk of receiving errors. This last aspect may be determinant of a successful demodulation, since the filters need to remove high-frequency components while maintaining the baseband signal integrity. Hence, \(p(t)\) and \(g(t)\) need to be selected to have a constant group delay in the baseband bandwidth, i.e., the filter pass band. This property ensures the digital data, which are PAM pulses of the in-phase and quadrature components of the QAM symbols, do not suffer different delays for different frequencies. Note that the frequency spectrum of a PAM signal is a _sinc_waveform, which has an infinite bandwidth; however, most of the energy is concentrated in the central lobe [51]. Therefore, considering the absence of a receiver equalizer in IoT devices (for the sake of low complexity and cost), one of the main requirements for the receiving filter is a constant group delay.  
Two filter types were considered for the receiver filter, \(g(t)\): (i) Bessel and (ii) Butterworth. Bessel filters are known for having a flat group delay and are commonly used in digital communication systems as a transmission filter. Nevertheless, their frequency response performance is poor when compared to other filter types such as Butterworth or Chebyshev filters, particularly concerning the attenuation decay after the cut-off frequency. Figure 5a presents the normalized filters' frequency response, when all are designed to have a magnitude of \(-3\) dB at the cut-off frequency of \(1\) Hz, Figure 5c,d shows the phase and the group delay for the Bessel, Butterworth, and Chebyshev filter types, with respect to the normalized frequency. It is observable that Bessel has the slowest decay, while Chebyshev has the fastest (although with a ripple in the pass band). Regarding the group delay, the Bessel filter has the flattest pass band response, while Chebyshev has a great amount of variation. The Butterworth filter presents a compromise between Bessel and Chebyshev, valid for both magnitude and group delay response. Due to its wide group delay variations, Chebyshev filters were not considered in further analysis, for the proposed system.  
Unlike other filters, the Bessel filter \(-3\) dB cut-off frequency is not constant with the filter order, leading to an attenuation higher than \(3\) dB at the desired cut-off frequency. The desire cut-off frequency can be determined asymptomatically by analyzing the high-frequency behavior [52]. In contrast with Figure 5, Figure 6 shows a Bessel filter where a high-frequency response is asymptotically coincident with the response of a Butterworth filter, both order three. This operation is achieved by normalizing the filter poles and gain, as described in [52]. An asymptotic analysis of the high-frequency response of the Bessel filter reveals a normalized cut-off frequency of \(1\) Hz, equal to the Butterworth filter. However, the magnitude at \(1\) Hz is \(-3\) dB and \(-6.24\) dB for the Butterworth and Bessel filters, respectively. The attenuation suffered by the Bessel filters in the asymptotic cut-off frequency increases as the filter order increases, which results in an increasingly non-flat amplitude response in the pass band, with increasing filter order. Both filters present non-flat group delay. Nevertheless, the Butterworth filter presents non-flat group delay at lower frequencies.