The Cosmic Dawn and Epoch of Reionisation, during which collapsed structures produce the first ionising photons and proceed to reionise the intergalactic medium, span a large range in redshift (z~30-6) and time (t_{age} ~ 0.1-1.0~Gyr). Exploration of these epochs using the redshifted 21~cm emission line from neutral hydrogen is currently limited to statistical detection and estimation metrics (e.g., the power spectrum) due to the weakness of the signal. Brightness temperature fluctuations in the line-of-sight (LOS) dimension are probed by observing the emission line at different frequencies, and their structure is used as a primary discriminant between the cosmological signal and contaminating foreground extragalactic and Galactic continuum emission. Evolution of the signal over the observing bandwidth leads to the `line cone effect whereby the HI structures at the start and end of the observing band are not statistically consistent, yielding a biased estimate of the signal power, and potential reduction in signal detectability. We implement a wavelet transform to wide bandwidth radio interferometry experiments to probe the local statistical properties of the signal. We show that use of the wavelet transform yields estimates with improved estimation performance, compared with the standard Fourier Transform over a fixed bandwidth. With the suite of current and future large bandwidth reionisation experiments, such as with the 300~MHz instantaneous bandwidth of the Square Kilometre Array, a transform that retains local information will be important.
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