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An exact theory describing the intensity PSD of photonic signal processors based on modulated thermal sources with smooth spectrum, first-order dispersion and intensity-noise-limited direct detection is presented. The theory is applied to the evaluation of the signal-to-noise ratio (SNR) of Microwave Photonic Filters (MPF) and of systems based on Incoherent Frequency-to-Time Mapping (IFTM) and Time-Spectrum Convolution (TSC) driven by thermal carriers whose optical spectra (continuous or sliced) is smooth at microwave scales. It is shown that the noise PSD of MPF based on low-index amplitude modulation coincides with the white-noise PSD of unmodulated, continuous wave (cw) polarized thermal light. In turn, both IFTM and TSC show modulation-dependent noise PSDs resulting in an SNR not better than cw, which decreases with pulse spreading. For IFTM the SNR is poor, with typical values of a few dB, whereas the SNR of TSC interpolates between the cw SNR and that of IFTM, thus showing a range of parameters where the SNR of IFTM is outperformed.