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Time-Varying Delay-Line Reads
If
denotes the input to a time-varying delay, the output can be
written as
where

denotes the time-varying delay in seconds. In
discrete-time implementations, when

is not an integer multiple
of the
sampling interval,

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
to a complex sinusoid at frequency
:
The output is now
The instantaneous phase of this
signal is
which can be differentiated to give the instantaneous frequency
 |
(4.8) |
where

denotes the output frequency, and

denotes the time derivative
of the delay

. Thus,
the delay growth-rate,

, equals the
relative frequency downshift: