## The investigation into an algorithm based on wavelet basis functions for the spatial and frequency decomposition of arbitrary signals.

##### Abstract

The research was directed toward the viability of an O(n) algorithm which could decompose
an arbitrary signal (sound, vibration etc.) into its time-frequency space. The well known
Fourier Transform uses sine and cosine functions (having infinite support on t) as
orthonormal basis functions to decompose a signal i(t) in the time domain to F(w) in the
frequency . domain, where the Fourier coefficients F(w) are the contributions of each
frequency in the original signal. Due to the non-local support of these basis functions, a
signal containing a sharp localised transient does not have localised coefficients, but rather
coefficients that decay slowly. Another problem is that the coefficients F(w) do not convey
any time information. The windowed Fourier Transform, or short-time Fourier Transform,
does attempt to resolve the latter, but has had limited success.
Wavelets are basis functions, usually mutually orthonormal, having finite support in t and
are therefore spatially local. Using non-orthogonal wavelets, the Dominant Scale
Transform (DST) designed by the author, decomposes a signal into its approximate time-frequency
space. The associated Dominant Scale Algorithm (DSA) has O(n) complexity
and is integer-based. These two characteristics make the DSA extremely efficient. The
thesis also investigates the problem of converting a music signal into it's equivalent music
score. The old problem of speech recognition is also examined. The results obtained from
the DST are shown to be consistent with those of other authors who have utilised other
methods. The resulting DST coefficients are shown to render the DST particularly useful in
speech segmentation (silence regions, voiced speech regions, and frication). Moreover, the
Spectrogram Dominant Scale Transform (SDST), formulated from the DST, was shown to
approximate the Fourier coefficients over fixed time intervals within vowel regions of
human speech.