Optimal signal reconstruction from a constant-Q spectrum

In contrast to other well-known techniques for spectral analysis such as the discrete Fourier transform or the wavelet transform the constant-Q transform (CQT) matches the center frequencies of its sub-band filters to the frequency scale of western music and accounts for the requirement of frequency-dependent bandwidths. However, it does not possess a strict mathematical inverse. Therefore, we derive an optimal reconstruction method to recover a signal from its CQT spectrum. The reconstruction problem is posed as a segmented overdetermined minimization problem where the number of frequency bins is larger than the number of reconstructed signal samples in each segment. The framework also enables the reduction of the input-output latency of the CQT. The method is evaluated for different types of music signals as well as for speech and Gaussian noise. For classical music a reconstruction quality of 89 dB signal-to-noise ratio is achieved.