Practical view : continuous wavelet transforms can be done mathematically but numerically to get it from matlab or equivalent library you have to discretize into wavelets of fixed frequency (satisfying the equivalency of the Shannon sampling rule to not degrade the original signal content)
When we talk about continuous representations those are mathematical representations of the signals (theoretical).
But to work with signals in reality you need to sample them (which means they will be in discrete form). And that is why for all signal processing techniques, we have both continuous and discrete representations.
In very rough terms.Recall convolution.Convolution means etymologically
”mid 16th century: from medieval Latin convolutio(n- ), from convolvere ‘roll together’ (see convolve).”
The idea is to slide a known function over a some data curve to determine how much of the known function is contained in the data curve. As an example if you
slide a sinusoid of a particular harmonic frequency over a periodic curve then you get the amplitude of the known sinusoid contained in the given data curve.This is the familiar Fourier coefficient.Here we do not know the location of the sinusoid
in the data curve only it’s magnitude.
Now in case of wavelet we first define two functions ,the average A and the detail D which is wavy. These are equipped with the property that they can be located at any point in time say T and also expanded at that location ie the scale or frequency or period . This is quite powerful because on a plot of freq/ scale and time we can locate the D in time and determine the freq content at that location.The A gives the remaining part after D is removed at that location.
Finally we can get an intuitive idea of continuous and discrete wavelet transform
by deciding how we slide the Detail and Approx functions.If we slide it in a smooth and continuous manner we get continuous wavelet transform. if we take discrete steps in time to locate the wavelets we get discrete wavelet transform.
Of course this discretisation is different from digital signal processing DSP.
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