Example: MFCC

In section 5.3.3, we showed empirically that MCMC-UMS for ACF constraints is related to the Burg maximum entropy spectral estimation. We now ask if the method can be extended to other linear constraints (other than ACF) and if this also results in good spectral estimators. We designed matrix ${\bf A}$ to compute the MEL band analysis (Section 5.2.6).

In Figure 5.14, we see results of MCMC-UMS, similar to Figure 5.11. Rather than using synthetic data, we captured a segment of 768 samples of human speech at 12kHz, then calculated the MFCC features ${\bf z}$. With this feature value fixed, we used MCMC-UMS to produce random spectra. On the top graph, we see one random MCMC-UMS sample. On the bottom, we show the LP solution, the MaxEnt solution $\hat{\mbox{\boldmath $\lambda$}}({\bf z}^*)$, and the average of 10000 full MCMC-UMS iterations. Once again we can conclude that $\hat{\mbox{\boldmath $\lambda$}}({\bf z}^*)$ precisely predicts the mean of the MCMC-UMS generated spectra. Important to note is that $\hat{\mbox{\boldmath $\lambda$}}({\bf z}^*)$ is a very smooth spectral estimate that is, at least visually, very satisfactory. Our proposed method may be preferred to open-ended MFCC synthesis methods [45] since the resulting spectrum is feature-reproducing and has optimal smoothness as a result of satisfying the maximum entropy rule. This is because maximizing the entropy (5.14) is the same as maximizing the spectral flatness [42,31].

Figure: Top: One sample spectrum from MCMC-UMS. Bottom graph: LP solution (line with dots), $\hat{\mbox{\boldmath $\lambda$}}({\bf z}^*)$ (circles), average of 10000 samples (curve through circles).