A method is presented for restoration of noisy bandlimited archived speech records. Speech is modeled with a formant-tracking linear prediction (FTLP) model of the spectral envelope and a harmonic noise model (HNM) of the excitation. The time-varying trajectories of the parameters of the LP and HNM models are tracked with Viterbi classifiers and denoised with Kalman filters. A frequency domain pitch estimation is proposed, which searches for the peak SNRs at the harmonics. The LP-HNM model is used to deconstruct noisy speech, de-noise its LP and HNM models and then reconstitute the cleaned speech. The missing spectrum at lower and higher frequency bands are reconstructed through spectral extrapolation of the LP-HNM model. Comparative evaluations show the performance gains obtained from the proposed method.