#### Frequency Envelopes

It is convenient in practice to work with instantaneous frequency deviation instead of phase:

 (G.9)

Since the th channel of an -channel uniform filter-bank has nominal bandwidth given by , the frequency deviation usually does not exceed .

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 . For notational simplicity, let and . Then we have

 (G.13)

For discrete time, we replace by to obtain [186]

 (G.14)

Initially, the sliding FFT was used (hop size in the notation of Chapters 8 and 9). Larger hop sizes can result in phase ambiguities, i.e., it can be ambiguous exactly how many cycles of a quasi-sinusoidal component occurred during the hop within a given channel, especially for high-frequency channels. In many applications, this is not a serious problem, as it is only necessary to recreate a psychoacoustically equivalent peak trajectory in the short-time spectrum. For related discussion, see [299].

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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