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

272.3.3 Continuous Wavelet TransformIn the case of the STFT, once a window is chosen for the analysis, the time-frequencyresolution is fixed over the entire time-frequency plane. If the resolution is allowedto vary within the time-frequency plan, a multi-resolution analysis will result. TheContinuous Wavelet Transform (CWT) is a transform where all impulse responsesof a filter bank are defined as scaled versions of the same prototype h(t)where a is a scale factor (the constantis used for energy normalization).The CWT is defined asSince the basic wavelet, h(t) is used for all filter impulse responses, wavelet analysisis self-similar at all scales.To relate this transform to the STFT, the basic wavelet h(t) could be chosenas a modulated window [37]The frequency response of wavelet analysis filters in Equation 2.21 satisfy theequationh(t) may be any band-pass function that meet certain criteria [12].2.3.3.1 Scale and Resolution: The local frequency f aƒ 0 is associated withthe scaling of the Wavelet Transform. This local frequency, that depends on thebasic wavelet is related to time-scalings. It is not linked as it was in the case ofthe STFT to frequency modulation. The filter bank responses are dilated as scaleincreases. Therefore, large scale corresponds to contracted signals while small scalecorresponds to dilated signals.