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We experimentally investigate the optical noise induced by stimulated Brillouin scattering (SBS) in single-mode fibers. The noise, which is caused by the random nature of spontaneous Brillouin scattering, is induced in the transmitted wave as well as in the backward Brillouin wave. In fibers where the Brillouin gain spectra consist of several resonances, the induced intensity noise of the transmitted wave has a wide spectrum and it may be the dominant noise in the receiver for a broad radio frequency band (0-1 GHz). This unexpected result is explained by a multiple scattering process which is caused by stimulated and by thermally excited phonons. The SBS induced noise may cause a performance degradation in analog optical systems which require high optical powers and low noise. Using fibers with a single Brillouin resonance may limit the noise to lower frequencies.