A wavelet representation of speech was used to display the instantaneous amplitude and phase within 14 octave frequency bands, representing the envelope and the carrier within each band. Adding stationary noise alters the wavelet pattern, which can be understood as a combination of three simultaneously occurring subeffects: two effects on the wavelet levels (one systematic and one stochastic) and one effect on the wavelet phases. Specific types of signal processing were applied to speech, which allowed each effect to be either included or excluded. The impact of each effect (and of combinations) on speech intelligibility was measured with CVC's. It appeared that the systematic level effect (i.e., the increase of each speech wavelet intensity with the mean noise intensity) has the most degrading effect on speech intelligibility, which is in accordance with measures such as the modulation transfer function and the speech transmission index. However, also the introduction of stochastic level fluctuations and disturbance of the carrier phase seriously contribute to reduced intelligibility in noise. It is argued that these stochastic effects are responsible for the limited success of spectral subtraction as a means to improve speech intelligibility. Results can provide clues for effective noise suppression with respect to intelligibility.