In this paper, we propose a sub-band basis spectrum model which is a new spectrum representation model based on a linear combination of sub-band basis vectors. We apply sparse coding to the pitch-synchronously analyzed log-spectra. Based on the approximation of the resulting basis, we obtain subband basis vectors with 1-cycle sinusoidal shapes that have mel-scale for lower frequencies and equally spaced scale for higher frequencies. Parameters of the sub-band basis spectrum model representing the log spectrum and the phase spectrum are calculated by fitting the basis to the spectrum. Since the parameters represent the shape of a spectrum, it can be easily used for voice adaptation, interpolation and conversion. Experimental results show that the analysis synthesis speech based on the proposed model is close to original speech and that there is no significant difference between the synthetic speech using analysis-synthesis database and those using original database for unit-fusion based TTS.