Spectrum analysis of speech signals is important for their detection, recognition, and separation. Speech signals are nonstationary with time-varying frequencies which, when analyzed by Fourier analysis over a short time window, exhibit harmonic spectra, i.e., the fundamental frequencies are accompanied by multiple associated harmonic frequencies. With proper modeling, such harmonic signal components can be cast as group sparse and solved using group sparse signal reconstruction methods. In this case, all harmonic components contribute to effective signal detection and fundamental frequency estimation with improved reliability and spectrum resolution. The estimation of the fundamental frequency signature is implemented using the block sparse Bayesian learning technique, which is known to provide high-resolution spectrum estimations. Simulation results confirm the superiority of the proposed technique when compared to the conventional STFT-based methods.