Space/time/frequency methods in adaptive radar - New Jersey ...

28When a function ƒ(t) is scaled:then it is contracted if a> 1 and expanded if a < 1, The CWT may be written asor, by a change of variable, asEquation 2.25 shows that as the scale increases, the filter impulse response h( t-=2-- a,becomes spread out in time therefore, only responds to long-time behavior. Equation2.26 indicates that as the scale grows, an increasingly contracted version of the signalis seen through a constant length filter. The scale factor a has the same interpretationof the scale in maps; Large scales mean global views and small scales mean detailedviews.Figure 2.13 shows a representation of the tile concentration of the STFT andWT. The related concept of resolution is linked to the signal's frequency content.For example, low-pass filtering a signal keeps its scale, but reduces its resolution.Scale changes of continuous time signals do not alter their resolution since the scalechange can be reversed.2.3.3.2 Wavelet Analysis: The CWT may also be introduced as definingwavelets as basis functions. The basis function has already been shown as an innerproduct of the formwhich measures the similarity between the signal and the basis functions