Discrete Wavelet Transform: From Frames to Fast Wavelet Transform
Agostino Abbate, Casimer M. DeCusatis, Pankaj Kumar Das · Birkhäuser Boston eBooks · 2002
Both the short-time Fourier transform and the continuous wavelet transform can be seen as operators that project the signal s(t) from the one-dimensional time domain into the two-dimensional time-frequency plane. In the case of the continuous wavelet transform the scaling a and delay b are assumed to be continuous in value; that is, it is said that the CWT is defined in the (ℝ + ) 2 plane where the parameters a and b are continuous in value: ( a ∈ℝ + and b ∈ℝ ). Since no new information can be created by this transform, the same information contained in the signal s(t) with t ∈ ℝ is available with the CWT. The increase in complexity from t 6 ℝ to (a, b) ∈ (ℝ + ) 2 results only in a redundant representation of the signal. This redundancy can be reduced by discretizing the transform parameters ( a, b ). Care must be taken so that we can still achieve reconstruction without any loss of information. Thus, the first question we must answer is how do we sample the parameters ( a,b )? These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.