Digital delay line
Delay lines are commonly used to delay audio signals feeding loudspeakers to compensate for the speed of sound in air, and to align video signals with accompanying audio, called audio-to-video synchronization.
Delay lines may compensate for electronic processing latency so that multiple signals leave a device simultaneously despite having different pathways.
Digital delay lines are widely used building blocks in methods to simulate room acoustics, musical instruments and effects units.
Digital waveguide synthesis shows how digital delay lines can be used as sound synthesis methods for various musical instruments such as string instruments and wind instruments.
[2] The Dattorro scheme is an industry standard implementation of digital filters using fractional delay lines.
{\displaystyle {\begin{cases}|\centerdot |=1=0dB,&{\text{zero dB gain}}\\\measuredangle =-\omega M,&{\text{linear phase with }}\omega =2\pi fT_{s}{\text{ where }}T_{s}{\text{ is the sampling period in seconds }}[s].\end{cases}}}
The discrete-time domain filter for integer delay
is trivial, since it is an impulse shifted by
Working in the discrete-time domain with fractional delays is less trivial.
, which can be modelled as the sum of an integer component
domain representation of a non-trivial digital filter design problem: the solution is an any time-domain filter that represents or approximates the inverse Z-transform of
The conceptually easiest solution is obtained by sampling the continuous-time domain solution, which is trivial for any delay value.
is the continuous-time domain fractional delay filter with:
{\displaystyle {\begin{cases}|\centerdot |=1=0dB,&{\text{zero dB gain}}\\\measuredangle =-\omega D,&{\text{linear phase}}\\\tau _{gr}=-{d\measuredangle \over {d\omega }}=D,&{\text{constant group delay}}\\\tau _{ph}=-{\measuredangle \over {\omega }}=-D,&{\text{constant phase delay.
The naive solution for the sampled filter
is the sampled inverse Fourier transform of
, which produces a non-causal IIR filter shaped as a Cardinal Sine
is shifted by the fractional delay while the sampling is always aligned to the cartesian plane, therefore:
Truncating the impulse response might however cause instability, which can be mitigated in a few ways:
{\displaystyle E_{LS}={1 \over {2\pi }}\int \limits _{-\alpha \pi }^{\alpha \pi }w(\omega )|H_{D}^{truncated}(e^{j\omega })-H_{D}^{id}(e^{j\omega })|^{2}d\omega \;\;\;\;\;{\text{where }}0<\alpha \leq 1{\text{ is the passband width parameter}}}
What follows is an expansion of the formula above displaying the resulting filters of order up to
Another approach is designing an IIR filter of order
with a Z-transform structure that forces it to be an all-pass while still approximating a
Therefore, the problem becomes designing the FIR filter
What follows is an expansion of the formula above displaying the resulting coefficients of order up to
: Digital delay lines were first used to compensate for the speed of sound in air in 1973 to provide appropriate delay times for the distant speaker towers at the Summer Jam at Watkins Glen rock festival in New York, with 600,000 people in the audience.
New York City–based company Eventide Clock Works provided digital delay devices each capable of 200 milliseconds of delay.
Four speaker towers were placed 200 feet (60 m) from the stage, their signal delayed 175 ms to compensate for the speed of sound between the main stage speakers and the delay towers.
Six more speaker towers were placed 400 feet from the stage, requiring 350 ms of delay, and a further six towers were placed 600 feet away from the stage, fed with 525 ms of delay.
Each Eventide DDL 1745 module contained one hundred 1000-bit shift register chips and a bespoke digital-to-analog converter, and cost $3,800 (equivalent to $27,679 in 2023).
