Vector Formulation
Denote the sound-source velocity by
where
is time. Similarly,
let
denote the velocity of the listener, if any. The
position of source and listener are denoted
and
, respectively, where
is 3D
position. We have velocity related to position by
Consider a Fourier component of the source at frequency
![$ \omega_s $](http://www.dsprelated.com/josimages_new/pasp/img1249.png)
![$ \omega_l $](http://www.dsprelated.com/josimages_new/pasp/img1250.png)
The Doppler effect depends only on velocity components along the line
connecting the source and listener [349, p. 453]. We may
therefore orthogonally project the source and listener
velocities onto the vector
pointing from the source
to the listener. (See Fig.5.8 for a specific example.)
The orthogonal projection of a vector
onto a vector
is given by [451]
![$\displaystyle {\cal P}_{\underline{y}}(\underline{x}) = \frac{\left<\underline{...
...derline{x}^T{\underline{y}}}{{\underline{y}}^T{\underline{y}}}{\underline{y}}.
$](http://www.dsprelated.com/josimages_new/pasp/img1261.png)
In the far field (listener far away), Eq.
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
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Doppler Simulation via Delay Lines
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Summary of Flanging