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Chapters

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

Chapter Contents:

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Frequency Envelopes

We normally work in practice with instantaneous frequency deviation instead of phase:

$\displaystyle \Delta \omega_k(t) \isdefs \frac{d}{dt} \phi_k(t)$

Since the th channel of an -channel uniform filter-bank has nominal bandwidth given by , the frequency deviation usually does not exceed $\pm f_s/(2N)$ .

Note that is a narrowband signal centered about the channel frequency $\omega_k$ . As detailed in Chapter 8, it is typical to heterodyne the channel signals to ``base band'' by shifting the input spectrum by $-\omega_k$ so that the channel bandwidth is centered about frequency zero (dc). This may be expressed by modulating the analytic signal by $\exp(-j\omega_k t)$ to get

$\displaystyle x_k^b(t) \isdefs e^{-j\omega_k t} x_k^a(t) = a_k(t) e^{j\phi_k(t)}$

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:

$\displaystyle \Delta\omega_k(t) \isdefs \frac{d}{dt} \angle x_k^b(t) = \dot{\phi}_k(t)$

where

$\displaystyle \dot{\phi}_k(t) \isdef \frac{d}{dt} \phi_k(t)$

denotes the time derivative of $\phi_k(t)$ . For notational simplicity, let $x(t) \isdeftext \mbox{re\ensuremath{\left\{x_k^b(t)\right\}}}$ and $y(t)\isdeftext \mbox{im\ensuremath{\left\{ x_k^b(t)\right\}}}$ . Then we have

$\displaystyle \dot{\phi}_k(t) \eqsp \frac{d}{dt}\tan^{-1}\left(\frac{y}{x}\righ... ...c{ \frac{d}{dt}{(y/x)}}{ 1+(y/x)^2} \eqsp \frac{x\dot{y}-y\dot{x}}{x^2+y^2} .$

For discrete time, we replace by to obtain [175]

$\displaystyle \Delta\omega_k(n) \isdefs \dot{\phi}_k(n) \eqsp \frac{x(n)\,\dot{y}(n)-y(n)\,\dot{x}(n)}{x^2(n)+y^2(n)} \protect$

(H.4)

Initially, the sliding FFT was used (hop size in the notation of Chapter 7 and Chapter 8). 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 [278].

Using (H.3) and (H.4) to compute the instantaneous amplitude and frequency for each subband, we obtain data such as shown qualitatively in Fig.H.15. A matlab algorithm for phase unwrapping is given in §G.4.1.

$\begin{psfrags} % latex2html id marker 42030\psfrag{ak} []{ \LARGE$ a_k(t)$\ }... ...ude envelope (top) and frequency envelope (bottom).} \end{figure} \end{psfrags}$

Previous: Computing Vocoder Parameters
Next: Envelope Compression

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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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