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Computing Vocoder Parameters

To compute the amplitude $ a_k(t)$ at the output of the $ k$ th subband, we can apply an envelope follower. Classically, such as in the original vocoder, this can be done by full-wave rectification and subsequent low pass filtering, as shown in Fig.G.10. This produces an approximation of the average power in each subband.

% latex2html id marker 43232\psfrag{x} []{ \LARGE$ x_k(t)$\ }\psfrag{xkt} []{ \LARGE$ \tilde{x}_k(t)$\ }\psfrag{xk} []{ \LARGE$ x_k(t)$\ }\psfrag{output} []{ \LARGE$ y_k=h*\tilde{x}_k $\ }\begin{figure}[htbp]
\includegraphics[width=\textwidth ]{eps/envelope}
\caption{Classic method for amplitude envelope
extraction in continuous-time analog circuits.}

In digital signal processing, we can do much better than the classical amplitude-envelope follower: We can measure instead the instantaneous amplitude of the (assumed quasi sinusoidal) signal in each filter band using so-called analytic signal processing (introduced in §4.6). For this, we generalize (G.3) to the real-part of the corresponding analytic signal:

$\displaystyle x_k(t)\eqsp a_k(t)\cos[ \omega_kt + \phi_k(t) ] \eqsp$   $\displaystyle \mbox{re\ensuremath{\left\{a_k(t)e^{j\phi_k(t)} e^{j\omega_k t}\right\}}}$$\displaystyle \isdefs$   $\displaystyle \mbox{re\ensuremath{\left\{x^a_k(t)\right\}}}$$\displaystyle \protect$ (G.5)

In general, when both amplitude and phase are needed, we must compute two real signals for each vocoder channel:
$\displaystyle a_k(t)$ $\displaystyle =$ $\displaystyle \vert x_k^a(t) \vert$   (instantaneous amplitude)  
$\displaystyle \phi_k(t)$ $\displaystyle =$ $\displaystyle \angle x_k^a(t) - \omega_kt$   (instantaneous phase)  
  $\displaystyle =$ $\displaystyle \tan^{-1} \left[ \frac{\mbox{im\ensuremath{\left\{x_k^a(t)\right\}}}}
{\mbox{re\ensuremath{\left\{x_k^a(t)\right\}}}} \right] - \omega_kt
\protect$ (G.6)

We call $ a_k(t)$ the instantaneous amplitude at time $ t$ for both $ x_k(t)$ and $ x_k^a(t)$ . The function $ a_k(\cdot)$ as a whole is called the amplitude envelope of the $ k$ th channel output. The instantaneous phase at time $ t$ is $ \phi_k(t)$ , and its time-derivative is instantaneous frequency.

In order to determine these signals, we need to compute the analytic signal $ x_k^a(t)$ from its real part $ x_k(t)$ . Ideally, the imaginary part of the analytic signal is obtained from its real part using the Hilbert transform4.6), as shown in Fig.G.11.

% latex2html id marker 43268\psfrag{x} []{ \LARGE$ x_k(t)$\ }\psfrag{Rex} []{ \LARGE$ \mbox{re\ensuremath{\left\{x_k^a(t)\right\}}}$\ }\psfrag{Imx} []{ \LARGE$ \mbox{im\ensuremath{\left\{x_k^a(t)\right\}}}$\ }\begin{figure}[htbp]
\caption{Creating an analytic signal
from its real part using the Hilbert transform
(\textit{cf.}{} \sref {hilbert}).}

Using the Hilbert-transform filter, we obtain the analytic signal in ``rectangular'' (Cartesian) form:

$\displaystyle x_k^a(t) =$   $\displaystyle \mbox{re\ensuremath{\left\{x_k^a(t)\right\}}}$$\displaystyle + j\,$$\displaystyle \mbox{im\ensuremath{\left\{x_k^a(t)\right\}}}$ (G.7)

To obtain the instantaneous amplitude and phase, we simply convert each complex value of $ x_k^a(t)$ to polar form

$\displaystyle x_k^a(t) \eqsp a_k(t)\,e^{j[ \omega_kt +\phi_k(t)] }$ (G.8)

as given by (G.6).

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University