Digital delay line

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 ideal fractional delay line is obtained by sampling the inverse Fourier transform of the continuous-time domain fractional delay filter. Note how for integer delay value this case degenerates to simple shifted impulses. Delaying a sampled signal with this filter conceptually coincides to resampling its analog source with equal sampling period but sample alignment shifted by ${\displaystyle D\in \mathbb {R} }$. Also note that the image shows only the few samples around zero, but the non-causal IIR is defined for an infinite number of samples in both directions of the x-axis.

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:

A block diagram representation of the Lagrange Interpolator formula.[10]

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

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	${\displaystyle -3{D-3 \over {D+1}}}$	${\displaystyle 3{(D-2)(D-3) \over {(D+1)(D+2)}}}$	${\displaystyle -{(D-1)(D-2)(D-3) \over {(D+1)(D+2)(D+3)}}}$