Detection of damage rate in structures caused by low cycle fatigue under dynamic loading such as seismic or wind excitations is of great coocern in the structural community. A health monitoring system using wavelet transform has been implemented for detecting the fatigue damage which is modeled as a random impulse in the input signal. In this work, a linear single-degree-of-freedom oscillator is used as a model for the structure. The occurrence time of the impulses has been detected for several signal-to-noise ratios. The detectability of the impulse signal was observed to be affected by the sampling rate the signal-to-noise ratio, and the vanishing moments of the wavelets used. Additional noise was introduced in the output response of the system and the detectability of the occurrence time of impulses was again investigated. It turned out that the detectability in generel was greatly deteriorated for even very small noise amplitudes specially at high sampling frequencies and was almost lost when the noise level was 10 percent of the output signal. Sampling frequency plays an important and reverse role in the detection of impulsive singularities in the presence or absence of output measurement noise. There is a tradeoff between the sampling rate and detectability depending on the level of signal noise. For low noise level, detectability is improved while increasing the sampling frequency. However, in the case of high output noise, a low sampling rate is suggested.