Digital delay line

The standard delay line with integer delay is derived from the Z-transform of a discrete-time signal ${\displaystyle x}$ delayed by ${\displaystyle M}$ samples^[4]:

${\displaystyle y[n]=x[n-M]}$ ${\displaystyle {\xrightarrow[{}]{\mathcal {Z}}}}$ ${\displaystyle Y(z)=\overbrace {z^{-M}} ^{H_{M}(z)}X(z).}$

In this case, ${\displaystyle z^{-M}=H_{M}(z)}$ is the integer delay filter with:

${\displaystyle {\begin{cases}|\centerdot |=1=0dB,&{\text{zero dB gain}}\\\measuredangle =-\omega M,&{\text{linear phase with }}\omega =2\pi fT_{s}{\text{ where }}T_{s}{\text{ is the sampling period in seconds }}[s].\end{cases}}}$

The discrete-time domain filter for integer delay ${\displaystyle M}$ as the inverse zeta transform of ${\displaystyle H_{M}(z)}$ is trivial, since it is an impulse shifted by ${\displaystyle M}$^[5]:

${\displaystyle h_{m}[n]={\begin{cases}{\text{1}},&{\text{for }}n=M\\0,&{\text{for }}n\neq M.\end{cases}}}$

Working in the discrete-time domain with fractional delays is less trivial. In its most general theoretical form, a delay line with arbitrary fractional delay is defined as a standard delay line with delay ${\displaystyle D\in \mathbb {R} }$, which can be modelled as the sum of an integer component ${\displaystyle M\in \mathbb {Z} }$ and a fractional component ${\displaystyle d\in \mathbb {R} }$ which is smaller than one sample:

(Fractional) Delay Line - ${\displaystyle {\mathcal {Z}}}$ Domain

${\displaystyle H_{D}(z)=z^{-D}\;\;\;\;\;{\text{where}}\;\;\;\;\;D=\overbrace {\lfloor D\rfloor } ^{M}+\overbrace {(D-\lfloor D\rfloor )} ^{d}}$

Def. 1

This is the ${\displaystyle {\mathcal {Z}}}$ 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 ${\displaystyle H_{D}(z)}$.^[2]

The conceptually easiest solution is obtained by sampling the continuous-time domain solution, which is trivial for any delay value. Given a continuous-time signal ${\displaystyle x}$ delayed by ${\displaystyle D\in \mathbb {R} }$ samples, or ${\displaystyle \tau =DT_{s}}$ seconds^[6]:

${\displaystyle y(t)=x(t-D)}$ ${\displaystyle {\xrightarrow[{}]{\mathcal {F}}}}$ ${\displaystyle Y(\omega )=\overbrace {e^{-j\omega D}} ^{H_{ideal}(\omega )}X(\omega ).}$

In this case, ${\displaystyle e^{-j\omega D}=H_{ideal}(\omega )}$ is the continuous-time domain fractional delay filter with:

${\displaystyle {\begin{cases}|\centerdot |=1=0dB,&{\text{zero dB gain}}\\\measuredangle =-\omega D,&{\text{linear phase}}\\\tau _{gr}=-{d\measuredangle \over {d\omega }}=D,&{\text{constant group delay}}\\\tau _{ph}=-{\measuredangle \over {\omega }}=-D,&{\text{constant phase delay.}}\end{cases}}}$

The naive solution for the sampled filter ${\displaystyle h_{ideal}[n]}$ is the sampled inverse Fourier transform of ${\displaystyle H_{ideal}(\omega )}$, which produces a non-causal IIR filter shaped as a Cardinal Sine ${\displaystyle sinc()}$ shifted by ${\displaystyle D}$^[6]:

${\displaystyle h_{ideal}[n]={\mathcal {F}}^{-1}[H_{ideal}(\omega )]={1 \over {2\pi }}\int \limits _{-\pi }^{+\pi }e^{j\omega D}e^{j\omega n}d\omega =sinc(n-D)={sin(\pi (n-D)) \over {\pi (n-D)}}}$

The continuous-time domain ${\displaystyle sinc}$ is shifted by the fractional delay while the sampling is always aligned to the cartesian plane, therefore:

• when the delay is an integer number of samples ${\displaystyle D\in \mathbb {N} }$, the sampled shifted ${\displaystyle sinc}$ degenerates to a shifted impulse just like in the theoretical solution.
• when the delay is a fractional number of samples ${\displaystyle D\in \mathbb {R} }$, the sampled shifted ${\displaystyle sinc}$ produces a non-causal IIR filter, which is not implementable in practice.

The conceptually easiest implementable solution is the causal truncation of the naive solution above.^[7]

${\displaystyle h_{\tau }[n]={\begin{cases}sinc(n-D)&{\text{for }}0\leq n\leq N\\0&{\text{otherwise}}\end{cases}}\;\;\;\;\;{\text{where}}\;\;\;\;\;{N-1 \over {2}}<D<{N+1 \over {2}}\;\;\;\;\;{\text{and}}\;\;\;\;\;N\;{\text{is the order of the filter.}}}$

Truncating the impulse response might however cause instability, which can be mitigated in a few ways:

• Windowing the truncated impulse response, therefore smoothing it. Note that in this case we have to add a further shift ${\displaystyle L}$ in order to align the window and the ${\displaystyle sinc()}$ and provide symmetric filtering^[7][8].

  ${\displaystyle h_{\tau }[n]={\begin{cases}w(n-D)sinc(n-D)&{\text{for }}L\leq n\leq L+N\\0&{\text{otherwise}}\end{cases}}\;\;\;\;\;{\text{where}}\;\;\;\;\;L={\begin{cases}round(D)-{N \over {2}}&{\text{for even }}N\\\lfloor D\rfloor -{N-1 \over {2}}&{\text{for odd }}N\end{cases}}}$

• General Least Square (GLS) Method:^[2] iteratively adjusts the frequency response by windowing a Least Square Integral Error design, which minimises the square integral error between ideal and truncated frequency responses of the filter, defined as:

${\displaystyle E_{LS}={1 \over {2\pi }}\int \limits _{-\alpha \pi }^{\alpha \pi }w(\omega )|H_{D}^{truncated}(e^{j\omega })-H_{D}^{id}(e^{j\omega })|^{2}d\omega \;\;\;\;\;{\text{where }}0<\alpha \leq 1{\text{ is the passband width parameter}}}$

• Lagrange Interpolator (Maximally Flat Fractional Delay Filter):^[9] adds "flatness" constraints to the first N derivatives of the Least Square Integral Error. This method is of particular interest because it has a closed form solution:

${\displaystyle h_{D}[n]=\prod _{k=0,\;k\neq n}^{N}{D-k \over {n-k}}\;\;\;\;\;{\text{where}}\;\;\;\;\;0\leq n\leq N}$

What follows is an expansion of the formula above displaying the resulting filters of order up to ${\displaystyle N=3}$:

Lagrange Interpolator Formula Expansion[7]
	${\displaystyle h_{\tau }[0]}$	${\displaystyle h_{\tau }[1]}$	${\displaystyle h_{\tau }[2]}$	${\displaystyle h_{\tau }[3]}$
N = 1	${\displaystyle 1-D}$	${\displaystyle D}$	-	-
N = 2	${\displaystyle {(D-1)(D-2) \over {2}}}$	${\displaystyle -D(D-2)}$	${\displaystyle {D(D-1) \over {2}}}$	-
N = 3	${\displaystyle -{(D-1)(D-2)(D-3) \over {6}}}$	${\displaystyle {D(D-2)(D-3) \over {2}}}$	${\displaystyle -{D(D-1)(D-3) \over {2}}}$	${\displaystyle {D(D-1)(D-2) \over {6}}}$

Another approach is designing an IIR filter of order ${\displaystyle N}$ with a Z-transform structure that forces it to be an all-pass while still approximating a ${\displaystyle D}$ delay^[7]:

${\displaystyle H_{D}(z)={z^{-N}A(z) \over {A(z^{-1})}}={a_{N}+a_{N-1}z^{-1}+...+a_{1}z^{-(N-1)}+z^{-N} \over {1+a_{1}z^{-1}+...+a_{N-1}z^{-(N-1)}+a_{N}z^{-N}}}\;\;\;\;\;{\text{which has}}\;\;\;\;\;{\begin{cases}|\centerdot |=1=0dB&0dB{\text{ gain}}\\\measuredangle _{H_{D}(z)}=-N\omega +2\measuredangle _{A(z)}=-D\omega &{\text{desired value for delay }}D\end{cases}}}$

The reciprocally placed zeros and poles of ${\displaystyle A(z){\text{ and }}A(z^{-1})}$ respectively flatten the frequency ${\displaystyle |\centerdot |}$ response, while the phase is function of the phase of ${\displaystyle A(z)}$. Therefore, the problem becomes designing the FIR filter ${\displaystyle A(z)}$, that is finding its coefficients ${\displaystyle a_{k}}$ as a function of D (note that ${\displaystyle a_{0}=1}$ always), so that the phase approximates best the desired value ${\displaystyle \measuredangle _{H_{D}(z)}=-D\omega }$.^[7]

The main solutions are:

• Iterative minimization of Least Square Phase Error,^[2] which is defined as:

${\displaystyle E_{LS}={1 \over {2\pi }}\int \limits _{-\pi }^{\pi }w(\omega )|\underbrace {\underbrace {-D\omega } _{\measuredangle _{ID}}-\underbrace {(-N\omega +2\measuredangle _{A(z)})} _{\measuredangle _{H}}} _{\Delta \measuredangle _{H_{D}}}|^{2}d\omega }$

• Iterative minimization of Least Square Phase Delay Error,^[2] which is defined as:

${\displaystyle E_{LS}={1 \over {2\pi }}\int \limits _{-\pi }^{\pi }w(\omega )|{{\Delta \measuredangle _{H_{D}}} \over {\omega }}|^{2}}$

• Thiran All-Pole Low-Pass Filter with Maximally Flat Group Delay.^[11] This yields a closed solution for finding the coefficients ${\displaystyle a_{k}}$ for positive delay ${\displaystyle D>0}$:

${\displaystyle a_{k}=(-1)^{k}{\binom {N}{k}}\prod _{l=0}^{N}{D+l \over {D+k+l}}\;\;\;\;\;{\text{where}}\;\;\;\;\;{\binom {n}{k}}={N! \over {k!(N-k)!}}}$

What follows is an expansion of the formula above displaying the resulting coefficients of order up to ${\displaystyle N=3}$:

Thiran All-Pole Low-Pass Filter Coefficients Formula Expansion[7]
	${\displaystyle a_{0}}$	${\displaystyle a_{1}}$	${\displaystyle a_{2}}$	${\displaystyle a_{3}}$
N = 1	1	${\displaystyle -{D-1 \over {D+1}}}$	-	-
N = 2	1	${\displaystyle -2{D-2 \over {D+1}}}$	${\displaystyle {(D-1)(D-2) \over {(D+1)(D+2)}}}$	-
N = 3	1	${\displaystyle -3{D-3 \over {D+1}}}$	${\displaystyle 3{(D-2)(D-3) \over {(D+1)(D+2)}}}$	${\displaystyle -{(D-1)(D-2)(D-3) \over {(D+1)(D+2)(D+3)}}}$

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 $28,565 in 2024).^[12][13]

• Analog delay line
• Delay-line memory
• Interpolation