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Real world data or signals frequently exhibit slowly changing trends or oscillations punctuated with transients.

On the other hand, images have smooth regions interrupted by edges or abrupt changes in contrast.

These abrupt changes are often the most interesting parts of the data, both perceptually and in terms of the information they provide.

Therefore, to accurately analyze signals and images that have abrupt changes, we need to use a new class of functions that are well localized in time and frequency: This brings us to the topic of Wavelets.

A wavelet is a rapidly decaying, wave-like oscillation that has zero mean.

Unlike sinusoids, which extend to infinity, a wavelet exists for a finite duration.

The availability of a wide range of wavelets is a key strength of wavelet analysis.

To choose the right wavelet, youll need to consider the application youll use it for.

For now, lets focus on two important wavelet transform concepts: scaling and shifting.

Scaling refers to the process of stretching or shrinking the signal in time, which can be expressed using this equation on screen.

S is the scaling factor, which is a positive value and corresponds to how much a signal is scaled in time.

For example, scaling a sine wave by 2 results in reducing its original frequency by half or by an octave.

For a wavelet, there is a reciprocal relationship between scale and frequency with a constant of proportionality.

This constant of proportionality is called the center frequency of the wavelet.

This is because, unlike the sinewave, the wavelet has a band pass characteristic in the frequency domain.

Mathematically, the equivalent frequency is defined using this equation on screen, where Cf is center frequency of the wavelet, s is the wavelet scale, and delta t is the sampling interval.

Therefore when you scale a wavelet by a factor of 2, it results in reducing the equivalent frequency by an octave.

A larger scale factor results in a stretched wavelet, which corresponds to a lower frequency.

A smaller scale factor results in a shrunken wavelet, which corresponds to a high frequency.

A stretched wavelet helps in capturing the slowly varying changes in a signal while a compressed wavelet helps in capturing abrupt changes.

Shifting a wavelet simply means delaying or advancing the onset of the wavelet along the length of the signal.

A shifted wavelet represented using this notation on screen means that the wavelet is shifted and centered at k.

We need to shift the wavelet to align with the feature we are looking for in a signal.

These transforms differ based on how the wavelets are scaled and shifted.

Wavelet Matlab Code Download