### Vocoder Analysis

The definitions of phase delay and group delay apply quite naturally
to the analysis of the *vocoder* (``voice coder'')
[21,26,54,76].
The vocoder provides a bank of bandpass filters which decompose the
input signal into narrow spectral ``slices.'' This is the analysis
step. For synthesis (often called *additive synthesis*), a bank
of sinusoidal oscillators is provided, having amplitude and frequency
control inputs. The oscillator frequencies are tuned to the filter
center frequencies, and the amplitude controls are driven by the
amplitude envelopes measured in the filter-bank analysis. (Typically,
some data reduction or envelope modification has taken place in the
amplitude envelope set.) With these oscillators, the band slices are
independently regenerated and summed together to resynthesize the
signal.

Suppose we excite only channel of the vocoder with the input signal

*amplitude modulated sinusoid*. The component can be called the

*carrier wave*, while is the

*amplitude envelope*.

If the phase of each channel filter is linear in frequency within the passband (or at least across the width of the spectrum of ), and if each channel filter has a flat amplitude response in its passband, then the filter output will be, by the analysis of the previous section,

where is the phase delay of the channel filter at frequency , and is the group delay at that frequency. Thus, in vocoder analysis for additive synthesis, the phase delay of the analysis filter bank gives the time delay experienced by the oscillator carrier waves, while the group delay of the analysis filter bank gives the time delay imposed on the estimated oscillator amplitude-envelope functions.

Note that a nonlinear phase response generally results in
, and
for
. As a result, the *dispersive* nature of additive synthesis
reconstruction in this case can be seen in Eq.(7.8).

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Numerical Computation of Group Delay

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Group Delay Examples in Matlab