Frequency Envelopes
It is convenient in practice to work with instantaneous frequency deviation instead of phase:
![]() |
(G.9) |
Since the




Note that
is a narrow-band signal centered about the channel
frequency
. As detailed in Chapter 9, it is typical
to heterodyne the channel signals to ``base band'' by shifting
the input spectrum by
so that the channel bandwidth is
centered about frequency zero (dc). This may be expressed by
modulating the analytic signal by
to get
![]() |
(G.10) |
The `b' superscript here stands for ``baseband,'' i.e., the channel-filter frequency-response is centered about dc. Working at baseband, we may compute the frequency deviation as simply the time-derivative of the instantaneous phase of the analytic signal:
![]() |
(G.11) |
where
![]() |
(G.12) |
denotes the time derivative of



![]() |
(G.13) |
For discrete time, we replace


Initially, the sliding FFT was used (hop size

Using (G.6) and (G.14) to compute the instantaneous amplitude and frequency for each subband, we obtain data such as shown qualitatively in Fig.G.12. A matlab algorithm for phase unwrapping is given in §F.4.1.
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Further Reading about FM Synthesis