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We develop and experimentally validate a method to characterize linearly chirped fiber Bragg gratings (CFBGs) under local temperature perturbations for tunable spectral shaping. The heat distribution along the FBG is modeled by a Gaussian-Lorentzian function. The phase and apodization profiles of the CFBG are characterized by measuring the complex reflection spectrum and subsequently using inverse scattering. Finally, coupled mode theory is used to predict the transmittivity of the CFBG under the local temperature perturbations. As an application, we use our model to spectrally shape the spectrum of a gain-switched laser (GSL) and generate ultra-short, optimally designed pulses for high speed wireless data distribution in indoor environments.