This paper is motivated from the need to extract the characteristic time and length scales of strong pulselike ground motions with a mathematically formal, objective, and easily reproducible procedure. The investigation uses wavelet analysis to identify and extract energetic acceleration pulses (not velocity pulses) together with their associated frequency and amplitude. The processing of acceleration records with wavelet analysis is capable of extracting pulses that are not detected visually in the acceleration records, yet they become coherent and distinguishable in the velocity records. Most importantly, the proposed analysis is capable of extracting shorter duration distinguishable pulses (not necessarily of random character) that override the longer near-source pulses that are of significant engineering interest. The study elaborates on the role of the weighting function in the definition of the wavelet transform and concludes that longer pulses are captured when less suppressive weighting functions are implemented in the wavelet transform. We examine the capability of several popular symmetric and antisymmetric wavelets to locally match the energetic acceleration pulse. We conclude that the exercise to identify the best-matching wavelet shall incorporate, in addition to the standard translation and dilation-contraction of the wavelet transform, a phase modulation together with a manipulation of the oscillatory character (addition of cycles) of the wavelet. This need leads to the extension of the wavelet transform to a more general wavelet transform during which the mother wavelet is subjected to the four above-mentioned modulations. The mathematical definition and effectiveness of this extended wavelet transform is presented in this paper. The objective identification of the pulse period, amplitude, phase, and oscillatory character of pulselike ground motions with the extended wavelet transform introduced in this paper makes possible the immediate use of closed-form expressions published by other investigators to estimate the peak response of elastic and inelastic systems.