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The modeling of speech production often relies on a source-filter approach. Although methods parameterizing the filter have nowadays reached a certain maturity, there is still a lot to be gained for several speech processing applications in finding an appropriate excitation model. This manuscript presents a Deterministic plus Stochastic Model (DSM) of the residual signal. The DSM consists of two contributions acting in two distinct spectral bands delimited by a maximum voiced frequency. Both components are extracted from an analysis performed on a speaker-dependent dataset of pitch-synchronous residual frames. The deterministic part models the low-frequency contents and arises from an orthonormal decomposition of these frames. As for the stochastic component, it is a high-frequency noise modulated both in time and frequency. Some interesting phonetic and computational properties of the DSM are also highlighted. The applicability of the DSM in two fields of speech processing is then studied. First, it is shown that incorporating the DSM vocoder in HMM-based speech synthesis enhances the delivered quality. The proposed approach turns out to significantly outperform the traditional pulse excitation and provides a quality equivalent to STRAIGHT. In a second application, the potential of glottal signatures derived from the proposed DSM is investigated for speaker identification purpose. Interestingly, these signatures are shown to lead to better recognition rates than other glottal-based methods.