# Bottlenose Dolphins' Clicks Comply with Three Laws of Efficient Communication
## 2 Materials and Methods
### Signal Analysis  
The raw 5 min long recordings containing vocalisation streams were denoised via a digital 4 th-order IIR Butterworth high-pass filter with a cut-off frequency of 100 Hz. The sequences of clicks were extracted from the records using the short-time (frames of 100 msec.) median frequency: any signal frame with a median frequency above 5 kHz and a maximum level above 0.05 V was considered active, as extracted sequences were amplitude-normalised. Individual clicks were identified via envelope detection using the following routine: (i) filtration was conducted with a 2 nd-order IIR Butterworth high-pass filter with a cut-off frequency of 5 kHz; (ii) sequence segmentation was carried out with a frame duration of 2.5 msec; (iii) coarse envelope detection was performed via short-time kurtosis computation--every frame with kurtosis above 30 was marked as one containing an individual click; (iv) finally, all individual peaks were processed further to reduce reverberation from the aquarium walls (cf. Figures 1 and 2). Refer to Appendix A for details. All steps of the signal analysis were carried out in the MATLAB R2017b environment (the software code is publicly available at [[https://www.mathworks.com/matlabcentral/fileexchange/68476-data-acquisition-and-logging-with-matlab](https://www.mathworks.com/matlabcentral/fileexchange/68476-data-acquisition-and-logging-with-matlab)]([https://www.mathworks.com/matlabcentral/fileexchange/68476-data-acquisition-and-logging-with-matlab](https://www.mathworks.com/matlabcentral/fileexchange/68476-data-acquisition-and-logging-with-matlab))). We focused on the amplitude of the clicks as the principal acoustic feature because we hypothesised that it could encode communicative information. The rationale was the following: In spoken language, listeners rely on several low-level acoustic dimensions, namely, fundamental frequency (pitch), spectral shape (formants), temporal extent, and amplitude (intensity), to recognise phonemic contrasts and to recover higher-level prosody. Amplitude modulations, in particular, signal lexical stress (e.g., the louder syllable in 'PERmit' vs. 'perMIT'), cue rhythmic grouping for speech segmentation, and combine with pitch and duration to convey sentence focus and discourse prominence. Because intensity can carry meaning while remaining largely independent of spectral content, we reasoned that dolphins might also exploit the energy envelopes of their broadband clicks as a communicative variable. Moreover, amplitude is robust to orientation-dependent frequency shifts under water and straightforward to measure at the sub-millisecond scale at which clicks occur, making it both biologically accessible to dolphins and analytically tractable. For these theoretical and practical reasons, amplitude was selected as a key parameter in our study of dolphin click communication. We opted for a 148 \(\upmu\)s window (37 \(\times\) 4 \(\upmu\)s) for principled, data- and literature-driven reasons [19; 38; 39]. Using the same window and temporal resolution prevents the introduction of artefacts due to resampling, while being large enough to encompass most of the envelopes in our dataset (Appendix A), yet short enough to exclude low-amplitude reverberation tails. Fixed-length amplitude vectors of similar dimensionality are also typical in automated click-classification work [19], because they place clicks of slightly different duration in a common acoustic space without time-warping. Correspondingly, each click  
Figure 1: A sequence of nine clicks. The pulses in the sequence follow a broadly similar structure. The green arrow points to a possible echo from a reflected pulse. Figure 2: The structure of a typical pulse in the sequence **(left)** and the distribution of its frequency spectrum **(right)**. from the dataset is represented as a \(1\times 37\) vector of amplitude values, which can have either positive or negative values.