## 4 Separation Mechanisms of Networks
### Proximity in Simultaneous Grouping
The excellent time-frequency resolution characteristics in these networks can be attributed to the encoder-separator-decoder framework, which convolves with mixtures in the time-domain to obtain the 2-D intermediate representations instead of STFT. For the separator, the introduction of dilated convolution with increasing dilation factors in ConvTasNet ensures a large enough receptive field. Furthermore, the dual-path structure in DPRNN and DPTNet brings a dynamic receptive field, which retains more time-domain information while having higher frequency resolution. Figure 5: SI-SDRi (dB) as a function of the frequency of tone A (kHz) and \(\Delta\)F (kHz), where (**a**–**i**) represent results of three well-trained networks (ConvTasNet, DPRNN, and DPTNet) with three different parameters, respectively. The dashed white curve is the \(\Delta\)F separable threshold at each frequency in the human auditory system. Another phenomenon worth attention is an inseparable blue line with a slope of 1 in the bottom left corner in almost every subgraph. It indicates that when \(\Delta\)F is equal to the frequency of tone A, the separation of two pure concurrent tones is more difficult. This phenomenon is consistent with the harmonic constraints in the human auditory system, which is also a critical principle for auditory scene analysis [34]. When components have a harmonic relationship (the frequency of one component is twice that of another component), they will be regarded as coming from the same sound source and inseparable in perception. Furthermore, the harmonicity effect line occurs at low frequencies for all networks. For the human auditory system, higher-order harmonics also cannot be resolved and results in less harmonic constraints at high frequencies. In addition, the harmonic effect line lies in the inseparable range, which is under the white dashed curve. It indicates that although the surrounding points cannot be separated due to the proximity principle, these cases that are in harmonic will be more challenging to separate. It reflects the interaction of proximity and harmonicity principles. The significance of the harmonic effect line in each subgraph reflects the network's weights impacted on proximity and harmonicity principles. For ConvTasNet, the harmonic effect line is not significant enough. It may be because the network is more sensitive to the proximity principle, while for DPRNN and DPTNet, the harmonic effect lines can be seen effortlessly. For DPRNN with a 4-sample window, there is even another inseparable line with a 3:1 frequency ratio. In summary, there are three contributions from this experiment. First, regardless of network structures and parameters, the proximity principle in frequency is learned spontaneously by these networks. There is no correlation between these networks because their structure and parameters are different. After training on the same dataset in Section 3, all networks showed similar performance in this proximity experiment, which is also consistent with the performance of the human auditory system (the white dashed curve). It implies that the learned principles are not accidental but the optimal strategies selected by networks when facing the same separation task. Second, the comparison of networks with different parameters helps us understand the resolution ability of networks. It illustrates that a network with a small window size has a better frequency resolution and explains why networks with smaller window sizes perform better for universal source separation in Section 3. It provides experimental support for parameter selection and subsequent network optimization and benefits the explanation of the \"black box\" of networks. Third, in addition to the proximity principle, other factors such as harmonicity also contribute to the segregation or integration of components. The interaction between different cues is also worthy of our attention.