# An Assessment of Environmental RF Noise Due to IoT Deployment
## 3 Radio Noise Surveying System  
The study of [12] was a precursor to this current paper, and introduced the concept of this RNSS. It demonstrated that it was technically possible to survey ambient radio noise using simplified, cost-effective SDR technology. Unlike the current paper, the previous experiment only explored the \"quiet\" environments unimpacted by IoT emissions. The ING devices in this paper introduce IoT radio activities' influence. Figure 1 shows the system block diagram. A USRP2 motherboard integrated with an SBX daughterboard (400 MHz to 4.4 GHz) and controlled by Gnu Radio software version 3.8.1.0 formed the hardware branch. Based on the frequency, the RNSS used three different terminal antennas. Figure 2 is the GNU Radio companion (GRC) flowgraph for the RF receiver architecture. The system extracted only the real part of the complex data. This study did not require the imaginary part. A QT sink block constantly monitored the operation underway [12]. The theory of complex envelope in RF and digital signal processing (DSP) associates noise power with the real part of a complex signal. The analysis of discrete Fourier transform (DFT) in [25] provides the complex notations for signals in the frequency domain. Equations (1) and (2) transform to the frequency domain by multiplying time domain samples by sine or cosine functions. The DSP terms call it correlating the input signal with the basis function. \[ReX[k]=\sum_{i=0}^{N-1}x[i]\cos(2\pi k/N) \tag{1}\]  
\[ImX[k]=-\sum_{i=0}^{N-1}x[i]\sin(2\pi k/N) \tag{2}\]  
where \(x[i]\) is the signal in the time domain, the index \(i\) increments from 0 to \(N-1\), and the index \(k\) from 0 to \(N/2\). The real part is the cosine wave and the imaginary sine wave, both at the same angular frequency. The sum of the cosine and sine signals results in a cosine wave (of different amplitude and phase shift). Instead of a rectangular representation, the polar form is practical in frequency domain analysis. Equations (3) and (4) introduce two  
Figure 1: RNSS signal chain. Figure 2: GNU Radio receiver configuration. components for the polar form: the magnitude \(MagX[k]\) and the phase \(PhaseX[x]\) of the cosine waves. \[MagX[k]=\sqrt{ReX[k]^{2}+ImX[k]^{2}} \tag{3}\]  
\[PhaseX[x]=\arctan(\
icefrac{{ImX[k]}}{{ReX[k]}}) \tag{4}\]  
Equations (5) and (6) rewrite Equations (3) and (4) to a rectangular form and represents the real and imaginary parts dependent on the magnitude and phase. The polar form does not use a sine wave to describe signals because a sine wave cannot have the DC component of the signal, and the sine wave is nil at 0 Hz. \[ReX[k]=MagX[k]\cos(PhaseX[x]) \tag{5}\]  
\[ImX[k]=MagX[k]\sin(PhaseX[x]) \tag{6}\]  
The theory of signals and bandpass systems in [26] explains that the Fourier transform \(X(f)\) of an original signal \(x(t)\) includes negative and positive components symmetrically around 0 Hz. The Fourier transform is a complex signal where the even function represents the real part, and the odd function is the imaginary part. Filtering out the right-hand side of the \(X(f)\) is necessary to reconstruct the original signal \(x(t)\). As shown in Equations (7) and (8), a linear system retrieves the real part from the complex analytic signal \(\dot{x}(t)\) to recover the original baseband \(x(t)\), where \(\widetilde{x}(t)\) is referred to as the complex envelope. \(f_{0}\) denotes the frequency at the baseband (0 Hz). \[\dot{x}(t)=\widetilde{x}(t)e^{j2\pi f_{0}t} \tag{7}\]  
\[x(t)=Re\big{[}\dot{x}(t)\big{]}=Re[\widetilde{x}(t)e^{j2\pi f_{0}t}] \tag{8}\]  
Equations (9) and (10) show the instantaneous envelope and instantaneous phase of \(x(t)\), where \(x_{c}\) and \(x_{s}\) are the quadrature components of \(\widetilde{x}(t)\). \[ImX[k]=-\sum_{i=0}^{N-1}x[i]\sin(2\pi k/N) \tag{2}\]0  
\[ImX[k]=-\sum_{i=0}^{N-1}x[i]\sin(2\pi k/N) \tag{2}\]1  
Similarly, the study in [27] elaborates that the positive frequencies of the Fourier transform relate to the Hilbert transform at a negative phase shift of \(-90\) degrees.