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attempt at correcting (19), thanks Bruce Mackinnon

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@ -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: