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.

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:

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

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

$$a_k(t) = \vert x_k^a(t) \vert$$

$$\phi_k(t) = \angle x_k^a(t) - \omega_kt$$

$$= \tan^{-1} \left[ \frac{\mbox{im\ensuremath{\left\{x_k^a(t)\right\}}}} {\mbox{re\ensuremath{\left\{x_k^a(t)\right\}}}} \right] - \omega_kt \protect$$ (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 transform (§4.6), as shown in Fig.G.11.

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

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

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

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

as given by (G.6).