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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Time-Varying Delay Effects
       Doppler Simulation
          Time-Varying Delay-Line Reads

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Time-Varying Delay-Line Reads

If $ x(t)$ denotes the input to a time-varying delay, the output can be written as

$\displaystyle y(t)=x(t-D_t). $

where $ D_t$ denotes the time-varying delay in seconds. In discrete-time implementations, when $ D_t$ is not an integer multiple of the sampling interval, $ x(t-D_t)$ may be approximated to arbitrary accuracy (in a finite band) using bandlimited interpolation (see §4.4) or other techniques for implementation of fractional delay [267,383].

Let's analyze the frequency shift caused by a time-varying delay by setting $ x(t)$ to a complex sinusoid at frequency $ \omega_s $:

$\displaystyle x(t) = e^{j\omega_s t} $

The output is now

$\displaystyle y(t)= x(t-D_t) = e^{j\omega_s \cdot (t-D_t)}. $

The instantaneous phase of this signal is

$\displaystyle \theta(t)= \angle y(t) = \omega_s \cdot(t-D_t) $

which can be differentiated to give the instantaneous frequency

$\displaystyle \omega_l = \omega_s ( 1 - {\dot D_t}) \protect$ (6.6)

where $ \omega_l $ denotes the output frequency, and $ {\dot D_t}\isdef \frac{d}{dt}D_t$ denotes the time derivative of the delay $ D_t$. Thus, the delay growth-rate, $ {\dot D_t}$, equals the relative frequency downshift:

$\displaystyle {\dot D_t}= \frac{\omega_s -\omega_l }{\omega_s }. $

Comparing Eq.$ \,$(5.6) to Eq.$ \,$(5.2), we find that the time-varying delay most naturally simulates Doppler shift caused by a moving listener, with

$\displaystyle {\dot D_t}= -\frac{v_{ls}}{c}. \protect$ (6.7)

That is, the delay growth-rate, $ {\dot D_t}$, should be set to the speed of the listener away from the source, normalized by sound speed $ c$.

Simulating source motion is also possible, but the relation between delay change and desired frequency shift is more complex, viz., from Eq.$ \,$(5.2) and Eq.$ \,$(5.6),

$\displaystyle {\dot D_t}= - \frac{\frac{v_{ls}}{c} + \frac{v_{sl}}{c}}{1-\frac{v_{sl}}{c}} \approx - \left(\frac{v_{ls}}{c} + \frac{v_{sl}}{c}\right) $

where the approximation is valid for $ v_{sl}\ll c$. In Section 5.7.4, a simplified approach is proposed based on moving the delay input instead of its output.

The time-varying delay line was described in §5.1. As discussed there, to implement a continuously varying delay, we add a ``delay growth parameter'' g to the delayline function in Fig.5.1, and change the line

  rptr += 1; // pointer update
to
  rptr += 1 - g; // pointer update
When g is 0, we have a fixed delay line, corresponding to $ {\dot D_t}=0$ in Eq.$ \,$(5.6). When $ \texttt{g}>0$, the delay grows $ \texttt{g}$ samples per sample, which we may also interpret as seconds per second, i.e., $ {\dot D_t}=\texttt{g}$. By Eq.$ \,$(5.7), we see that we need

$\displaystyle \texttt{g} = -\frac{v_{ls}}{c} $

to simulate a listener traveling toward the source at speed $ v_{ls}$.

Note that when the read- and write-pointers are driven directly from a model of physical propagation-path geometry, they are always separated by predictable minimum and maximum delay intervals. This implies it is unnecessary to worry about the read-pointer passing the write-pointers or vice versa. In generic frequency shifters [275], or in a Doppler simulator not driven by a changing geometry, a pointer cross-fade scheme may be necessary when the read- and write-pointers get too close to each other.

Previous: Doppler Simulation via Delay Lines
Next: Multiple Read Pointers

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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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