## Appendix A Tonal and Broadband Noise Separation  
The analysis of the experimental data showed a dependence of the azimuthal directivity on the rotational speed of the rotors. In particular, both for the single and twin-rotorconfiguration, the OASPL curves exhibited a significant variation in curvature as RPM increases. However, especially in the single propeller case, by approximating the noise source as a monopole, one would have expected a uniform trend along the microphone array, which is not the case in the experimental results. To better clarify the reasons behind the nonuniform directivity, a signal decomposition method was used to separate the sound pressure spectrum into the two components of tonal and broadband contributions.  
The method followed for the tonal and broadband noise separation is the one presented by [PERSON] and [PERSON] [70], usually adopted to separate open rotor acoustic data, based on the removal of an average tonal waveform.  
In particular, the technique involves the following steps. First, divide the raw acoustic data into uniform segments, take two consecutive ones, and perform a cross-correlation operation to realign and phase-adjust them. In this way, each segment is composed of a common tone component plus a realization of a stationary random part that can be isolated by canceling the tone portion of the signal, which is coherent between the segments. Doing this for the all raw signal length allows building a pseudo time-series of the broadband signal. However, appending the individual broadband segments will create discontinuity at each junction and, thus, introduce a small amount of white noise into the computed spectrum. This effect can be mitigated using a smoothing window, such as Hanning, with each segment. A long time-series of the broadband signal is reconstructed, having the total length equal to about one-half of that of the original raw data set. An FFT operation is now performed to obtain the broadband spectral values as a function of frequency, which can be subtracted to the averaged total spectrum to obtain the tonal contribution.  
The major drawback of this method is that the broadband spectrum may contain irregular, tone-like spikes, due to unaccounted random phase shifts occurring in the data segment-pairs selected for the cross-correlation operation. In addition, the spectrum of the random component may be slightly amplified as a result of the averaging operation in the frequency domain together with the use of smoothing windows. Despite this, the procedure was found to be particularly suited for the case of dual-rotor systems having approximately equal rotational speed [70; 71]; it allows one to obtain a reasonable approximation of the broadband level. For both configurations, Figure 11 presents an example of the spectra obtained with the separation procedure at the ninth microphone (\(\theta=180^{\circ}\)) of the azimuthal array. The figures refer to a rotational speed of 8000 rpm and to the minimum separation distance (\(t=0.04D\)) of the dual propeller system.  
The OASPL can now be computed by integrating the separated spectra to retrieve the directivities of each contribution. For the co-rotating case, Figure 12 illustrates the OASPL at each microphone for the three relative distances analyzed, \(t=0.04D\), \(t=0.16D\), \(t=0.24D\), and the comparison with the isolated rotor configuration at 8000 rpm.  
The modification in the shape of the curves with increasing rpm is mainly attributed to the behavior of the tonal noise, dominated by the trend of the first BPF, which suggests the presence of considerable installation effects. Being at static thrust conditions, a higher rotational speed results in more air being sucked in by the rotor; asymmetries due to the experimental set up in the upstream lead to a visible variation in the directivity patterns.