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Physical Audio Signal Processing
    Time-Varying Delay Effects
       Doppler Effect
          Vector Formulation

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Vector Formulation

Denote the sound-source velocity by $ \underline{v}_s(t)$ where $ t$ is time. Similarly, let $ \underline{v}_l(t)$ denote the velocity of the listener, if any. The position of source and listener are denoted $ \underline{x}_s(t)$ and $ \underline{x}_l(t)$, respectively, where $ \underline{x}\isdef (x_1,x_2,x_3)^T$ is 3D position. We have velocity related to position by

$\displaystyle \underline{v}_s= \frac{d}{dt}\underline{x}_s(t) \qquad \underline{v}_l= \frac{d}{dt}\underline{x}_l(t). \protect$ (6.3)

Consider a Fourier component of the source at frequency $ \omega_s $. We wish to know how this frequency is shifted to $ \omega_l $ at the listener due to the Doppler effect.

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 $ \underline{x}_{sl}=\underline{x}_l-\underline{x}_s$ pointing from the source to the listener. (See Fig.5.8 for a specific example.)

The orthogonal projection of a vector $ \underline{x}$ onto a vector $ {\underline{y}}$ 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}}. $

Therefore, we can write the projected source velocity as

$\displaystyle \underline{v}_{sl}= {\cal P}_{\underline{x}_{sl}}(\underline{v}_s... ...line{x}_s\,\right\Vert^2}\left(\underline{x}_l-\underline{x}_s\right). \protect$ (6.4)

In the far field (listener far away), Eq.$ \,$(5.4) reduces to

$\displaystyle \underline{v}_{sl} \approx \frac{\left<\underline{v}_s,\underline... ...derline{x}_l\,\right\Vert\gg\left\Vert\,\underline{x}_s\,\right\Vert). \protect$ (6.5)

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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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