Computing Vocoder Parameters

To compute the amplitude at the output of the 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.H.13. This produces an approximation of the average power in each subband.

$\begin{psfrags} % latex2html id marker 41940\psfrag{x} []{ \LARGE$ x_k(t)$\ } ... ...ope extraction in continuous-time analog circuits.} \end{figure} \end{psfrags}$

In digital signal processing, we can do 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. For this, we generalize Eq. $\,$ (H.1) to the real-part of the corresponding analytic signal:

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

(H.2)

In general, when both amplitude and phase are needed, we must compute two real signals for each vocoder channel:

$\displaystyle a_k(t)$	$\displaystyle =$	$\displaystyle \vert x_k^a(t) \vert \eqsp$ instantaneous amplitude
$\displaystyle \phi_k(t)$	$\displaystyle =$	$\displaystyle \angle x_k^a(t) - \omega_kt \eqsp$ instantaneous phase
	$\displaystyle =$	$\displaystyle \tan^{-1} \left[ \frac{\mbox{im\ensuremath{\left\{x_k^a(t)\right\}}}} {\mbox{re\ensuremath{\left\{x_k^a(t)\right\}}}} \right] - \omega_kt \protect$	(H.3)

We call the instantaneous amplitude of both and , and similarly for $\phi_k(t)$ . We also call the amplitude envelope of the th channel output.

In order to determine these signals, we need to compute the analytic signal from its real part . Ideally, the imaginary part of the analytic signal is obtained from its real part using the Hilbert transform (§E.5), as shown in Fig.H.14.

$\begin{psfrags} % latex2html id marker 41971\psfrag{x} []{ \LARGE$ x_k(t)$\ } ... ...nal from its real part using the Hilbert transform.} \end{figure} \end{psfrags}$

Practical Hilbert transformers are covered in §E.5.

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

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

To obtain the instantaneous amplitude and phase, we simply convert each complex value of to polar form

$\displaystyle x_k^a(t) = a_k(t)e^{j[ \omega_kt +\phi_k(t)] }$

as given by (H.3).

Subsections

Previous: Phase Vocoder Sinusoidal Modeling
Next: Frequency Envelopes

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
    A History of Spectral Audio Signal Processing
       Phase Vocoder Sinusoidal Modeling
          Computing Vocoder Parameters