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diff --git a/doc/codec2.pdf b/doc/codec2.pdf Binary files differindex ac2e63c..7dc2619 100644 --- a/doc/codec2.pdf +++ b/doc/codec2.pdf diff --git a/doc/codec2.tex b/doc/codec2.tex index 27181a2..73d2565 100644 --- a/doc/codec2.tex +++ b/doc/codec2.tex @@ -488,24 +488,39 @@ Voicing is determined using a variation of the MBE voicing algorithm \cite{griff For each band we first estimate the complex harmonic amplitude (magnitude and phase) using \cite{griffin1988multiband}: \begin{equation} +\label{eq:est_amp_mbe1} B_m = \frac{\sum_{k=a_m}^{b_m} S_w(k) W^* (k - \lfloor mr \rceil)}{|\sum_{k=a_m}^{b_m} W (k - \lfloor mr \rceil)|^2} \end{equation} -where $r= \omega_0 N_{dft}/2 \pi$ is a constant that maps the $m$-th harmonic to a DFT bin, and $ \lfloor x \rceil$ is the rounding operator. As $w(n)$ is a real and even, $W(k)$ is real and even so we can write: +where $r= \omega_0 N_{dft}/2 \pi$ is a constant that maps the $m$-th harmonic to a DFT bin, and $ \lfloor x \rceil$ is the rounding operator. To avoid non-zero array indexes we define the shifted window function: +\begin{equation} +U(k) = W(k-N_{dft}/2) +\end{equation} +such that $U(N_{dft}/2)=W(0)$. As $w(n)$ is a real and even, $W(k)$ is real and even so we can write: +\begin{equation} +\begin{split} +W^* (k - \lfloor mr \rceil) &= W(k - \lfloor mr \rceil) \\ + &= U(k - \lfloor mr \rceil + Ndft/2) \\ + &= U(k + l) \\ + l &= Ndft/2 - \lfloor mr \rceil \\ + & = \lfloor Ndft/2 - mr \rceil +\end{split} +\end{equation} +for even $Ndft$. We can therefore write \ref{eq:est_amp_mbe1} as: \begin{equation} \label{eq:est_amp_mbe} -B_m = \frac{\sum_{k=a_m}^{b_m} S_w(k) W (k + \lfloor mr \rceil)}{\sum_{k=a_m}^{b_m} |W (k + \lfloor mr \rceil)|^2} +B_m = \frac{\sum_{k=a_m}^{b_m} S_w(k) U(k + l)}{\sum_{k=a_m}^{b_m} |U (k + l)|^2} \end{equation} Note this procedure is different to the $A_m$ magnitude estimation procedure in (\ref{eq:mag_est}), and is only used locally for the MBE voicing estimation procedure. Unlike (\ref{eq:mag_est}), the MBE amplitude estimation (\ref{eq:est_amp_mbe}) assumes the energy in the band of $S_w(k)$ is from the DFT of a sine wave, and $B_m$ is complex valued. The synthesised frequency domain speech for this band is defined as: \begin{equation} -\hat{S}_w(k) = B_m W(k + \lfloor mr \rceil), \quad k=a_m,...,b_m-1 +\hat{S}_w(k) = B_m U(k + l), \quad k=a_m,...,b_m-1 \end{equation} The error between the input and synthesised speech in this band is then: \begin{equation} \begin{split} E_m &= \sum_{k=a_m}^{b_m-1} |S_w(k) - \hat{S}_w(k)|^2 \\ - &=\sum_{k=a_m}^{b_m-1} |S_w(k) - B_m W(k + \lfloor mr \rceil)|^2 + &=\sum_{k=a_m}^{b_m-1} |S_w(k) - B_m U(k + l)|^2 \end{split} \end{equation} A Signal to Noise Ratio (SNR) ratio is defined as: |