Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books



Chapters

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

Chapter Contents:

Search Physical Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

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 §K.3) or other techniques for implementation of fractional delay [272,388].

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

$\displaystyle \omega_l = \omega_s ( 1 - {\dot D_t}) \protect$ (4.8)

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: