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{{Short description|Delay-based audio effects system}}
[[File:Dattorro block scheme.svg|thumb|Dattorro block diagram|420x420px]]

The '''Dattorro industry scheme''' is a digital system used to implement a wide range of delay-based audio effects for digital signals. It was proposed by Jon Dattorro.<ref name="dattorro">{{cite journal|last=Dattorro|first=Jon|title=Effect Design - Part 2: Delay-Line Modulation and Chorus|year = 1997|url=http://www.aes.org/e-lib/browse.cfm?elib=10159 |journal=Journal of the Audio Engineering Society|volume=45 |issue=10 |pages=764–788 }}</ref> The common nature of these effects allows to produce an output signal as the [[linear combination]] of (dynamically modulated) delayed replicas of the input signal. The proposed scheme allows to implement such effects in a compact form, only using a set of three parameters to control the type of effect.

The Dattorro industry scheme is based on [[digital delay line]]s and to ensure a proper resolution in the time domain, it leverages fractional delay lines, thus avoiding discontinuities.<ref>{{Citation |last1=Smith |first1=Julius O. |title=Computational Acoustic Modeling with Digital Delay |date=June 5, 2008 |url=http://ccrma.stanford.edu/realsimple/Delay/Delay_lines.html |website=Center for Computer Research in Music and Acoustics |access-date=2007-08-21 |last2=Lee |first2=Nelson}}</ref>

The effects that this scheme is able to produce are: echo, chorus, vibrato, flanger, doubling, white chorus. These effects are characterized by a nominal delay, a modulating function for the delay and the depth of modulation.<ref>{{Cite book |last=Zolzer |first=Udo |url=https://picture.iczhiku.com/resource/eetop/sykfGloQfYJRZBcv.pdf |title=DAFX: Digital Audio Effects |publisher=John Wiley and Sons |year=2011}}</ref>

== Delay line interpolation ==
Consider continuous time signal <math> y(t) </math>. The <math>\tau</math>-delayed version of such signal is <math>y(t) = x(t-\tau)</math>. Considering the signal in the discrete time domain (i.e., sampling it with [[Sampling frequencies|sampling frequency]] <math>F_s</math> at <math>n = tF_s </math>), we obtain <math>y[n] = x[n-M] </math>. The delay line can then be described in terms of the [[Z-transform]] of the discrete time signal as 
 
<math>Y(z) = \mathcal Z \{y \}(n) = z^{-M}X(z)  = H_M(z) X(z) </math>. 
 
When going back in the time domain exploiting the inverse Z-transform, this corresponds to <math>h_m[n] = \delta[n-M] </math>, where <math>\delta[\cdot]</math> is the [[Dirac delta]]. The former equation holds for <math>D\in\mathbb N</math> but for the implementation we need a fractional delay, meaning <math>D =\lfloor D\rfloor + (D-\lfloor D\rfloor) = M+d,\quad d\in\mathbb R, \ 0<d<1</math>. In this case, [[interpolation]] is required to reconstruct the value of the signal that lies between two samples.<ref>{{Cite journal |last1=Laakso |last2=Valimaki |last3=Karjalainen |year=1996 |title=Splitting the unit delay [FIR/all pass filters design] |url=https://ieeexplore.ieee.org/document/482137 |journal=IEEE Signal Processing Magazine |volume=13 |issue=1 |pages=30–60 |doi=10.1109/79.482137|bibcode=1996ISPM...13...30L }}</ref> One can resort to the preferred interpolation technique, e.g., [[Lagrange polynomial|Lagrange interpolation]],<ref>{{cite journal |last=Rocchesso |first=Davide |title=Fractionally addressed delay lines |journal=IEEE Transactions on Speech and Audio Processing |year=2000 |volume=8 |issue=6 |pages=717–727 |url=https://ieeexplore.ieee.org/document/876310 |doi=10.1109/89.876310|arxiv=cs/0005022 |s2cid=9626440 }}</ref> or all-pass interpolation.<ref>{{Cite book |last1=Hacthabiboglu |first1=H. |last2=Gunel |first2=B. |last3=Kondoz |first3=A.M. |title=2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings |chapter=Interpolated Allpass Fractional-Delay Filters Using Root Displacement |date=2006 |chapter-url=https://ieeexplore.ieee.org/document/1660791 |publisher=IEEE |volume=3 |pages=III–864–III-867 |doi=10.1109/ICASSP.2006.1660791 |isbn=978-1-4244-0469-8|s2cid=234579 |url=https://openresearch.surrey.ac.uk/view/delivery/44SUR_INST/12139482700002346/13140452040002346 }}</ref>

[[File:Linear_interpolator.svg|thumb|Linear interpolator block diagram]]
[[File:All_pass_interpolator.svg|thumb|All-pass interpolator block diagram]]

== Block diagram ==

The output of the delay block <math> H_{M}(z) = z^{-M}</math> is noted above and below as rarely it is taken from the last sample of the tap but instead it changes dynamically. Considering the end to end structure, we can write the filter as: 
{{Equation box 1|border|indent=:|title=Dattorro scheme |equation=<math display="block">Y(z)  = H(z)X(z) = \frac{b+f_f z^{-D (t) } } {1+z^{-D(t)} f_b }X(z)</math>|border colour=#0073CF|background colour=#F5FFFA}}

Setting the coefficients according to the desired effect results in a change of the filter topology. For example, setting the feedback to 0 results in a [[FIR filter]], whereas, if the coefficients are set to be equal, we approximate an [[all-pass filter]].<ref name="dattorro"/>

[[File:Guitar string E2.ogg|thumb|Guitar string <math>E_2 </math> sound]]
[[File:Echo effect.ogg|thumb|Echo]]
[[File:Vibrato effect.ogg|thumb|Vibrato]]
[[File:Flanger effect.ogg|thumb|Flanger]]
[[File:Doubling_effect.ogg|thumb|Doubling]]
[[File:Chorus effect.ogg|thumb|Chorus effect]]
[[File:White chorus effect.ogg|thumb|White chorus]]

== Effects ==

All the effects can be obtained by changing the parameters of the Dattorro system and by setting the delay ranges according to the following table. Delay values are expressed in milliseconds.<ref name="dattorro"/>

{| class="wikitable"
!Effect
!Onset
!Nominal
!Range End
|-
|[[Vibrato]]
|0
|Minimal
|5
|-
|[[Flanging|Flange]]
|0
|1
|10
|-
|[[Chorus effect|Chorus]]
|1
|5
|30
|-
|[[Voicing (music)#Doubling|Doubling]]
|10
|20
|100
|-
|[[Echo]]
|50
|80
|<math> \infty</math>
|}

=== Vibrato ===
Vibrato is a small quasi-periodic change in pitch of a tone. It's more of a technique than an effect per se but can be added to any audio signal.<ref name="McPherson">{{Cite book |last=McPherson |first=Joshua D. Reiss, Andrew |url=https://www.taylorfrancis.com/books/mono/10.1201/b17593/audio-effects-joshua-reiss-andrew-mcpherson |title=Audio Effects: Theory, Implementation and Application |date=2015-01-13 |publisher=CRC Press |isbn=978-0-429-09723-2 |location=Boca Raton |doi=10.1201/b17593}}</ref> The delay is modulated with a low frequency sinusoidal function and no mix of the direct path of the signal is considered.

=== Chorus ===
The [[Chorus (audio effect)|chorus]] is an effect which tries to emulate multiple independent voices playing in unison.  This effect is made as a linear combination of the input signal (dry signal) and a dynamically delayed version of the input (wet signal).<ref>{{cite web|last= Smith|first = Julius O.|url=https://ccrma.stanford.edu/~jos/pasp/Chorus_Effect.html| title =  Chorus effect}}</ref>

=== Flanging ===
The flanging effect originated with tape machines. This effect was created by mixing two tape machines set to play the same track but one of them is slowed down. This produces a lowering in pitch and a delay of the slow track. The process is then repeated with the other track reabsorbing the accumulated delay.<ref>{{cite web|last= Smith|first = Julius O.|url=https://ccrma.stanford.edu/~jos/pasp/Flanging.html#6405| title = Flanging }}</ref> 
This effect is very similar to chorus and the main difference is due to the delay range. Chorus usually has longer delay, larger depth and lower modulating frequency.<ref name="McPherson"/>

=== White chorus ===
White chorus is a modification to the standard chorus effect aimed at reducing the aberrations introduced by the forward path. The change consists in adding a negative feedback path with a different and fixed tap point in order to obtain an approximation of an all-pass configuration.<ref name="dattorro"/>

=== Doubling ===
When the system is used without feedback we achieve [[Voicing (music)#Doubling|doubling]]. This effect is analogous to that of the [[Leslie speaker]], a particular kind of speaker consisting of a rotating chamber in front of the bass loudspeaker and rotating cones above the treble loudspeakers
.<ref>{{cite web|last=Henricksen|first=Clifford A. |title=Unearthing The Mysteries Of The Leslie Cabinet|url=http://www.theatreorgans.com/hammond/faq/mystery/mystery.html}}</ref>

== Knob settings ==
{| class="wikitable"
!Effect
!Blend <math>b</math>
!Feedforward <math>f_f</math>
!Feedback <math>f_b</math>
!Modulation type D(t)
|-
|[[Vibrato]]
|0.0
|1.0
|0.0
|sinusoid
|-
|[[Flanging|Flanger]]
|0.7071
|0.7071
|<math>-</math>0.7071
|sinusoid
|-
|White chorus
|0.7071
|1.0
|0.7071
|low pass noise
|-
|[[Chorus effect|Chorus]]
|1.0
|0.7071
|0.0
|low pass noise
|-
|[[Voicing (music)#Doubling|Doubling]]
|0.7071
|0.7071
|0.0
|low pass noise
|-
|[[Echo]]
|1.0
|<math> \ll</math>1.0
|<1.0
|none
|}

== Pseudocode ==
The filter can be implemented in software or hardware. In the following is the pseudocode for a software Dattorro system.

 '''function''' dattorro '''is'''
     '''input:''' ''x'',                                   {{color|green|# input signal}}
            ''Fs'',                                  {{color|green|# sampling frequency}}
            ''depth,''                               {{color|green|# modulation depth}}
            ''freq,''                                {{color|green|# LFO frequency}}
            ''b'',                                   {{color|green|# blend knob}}
            ''ff'',                                  {{color|green|# feedforward knob}}
            ''fb'',                                  {{color|green|# feedback knob}}
            ''mod''                                  {{color|green|# modulation type}}
     '''output:''' ''y''                                   {{color|green|# signal with FX applied}}
 
     ''depth_samples = depth·Fs''                    {{color|green|# compute delay in samples}}
     '''if''' ''mod'' is sinusoid                          {{color|green|# compute delay sequence}}
         ''delay_seq = depth_samples*(1 + sin(2·<math>\pi </math>·freq*t))''
     '''else''' ''mod'' is noise
         lowpass_noise = noise_gen()             {{color|green|# generate white noise and low-pass filter it}}
         ''delay_seq = depth_samples + depth_samples·lowpass_noise''
     
     '''for each''' ''n'' of the ''N'' samples
 
         ''d_int = floor(delay_seq(n))''             {{color|green|# integer delay}}
         ''d_frac = delay_seq(n) - d_int           {{color|green|# fractional delay}}''        
         ''h0 = (d_frac - 1)·(d_frac - 2)/2''        {{color|green|# first FIR filter coefficient}}
         ''h1 = d_frac·(2 - d_frac)''                {{color|green|# second FIR filter coefficient}}
         ''h2 = d_frac/2·(d_frac - 1)''              {{color|green|# third FIR filter coefficient}}
         
         ''x_d = delay_line(d_int + 1)·h0 + delay_line(d_int + 2)·h1 + delay_line(d_int + 3)·h2''  {{color|green|# delay input}}                                       
         
         ''f_b = fb·x_d''                            {{color|green|# delayed input feedback}}
         ''delay_line(2:L) = delay_line(1:L-1)''     {{color|green|# delay line update}}
         ''delay_line(1) = x(i) - fb_comp''
         
         ''f_f = ff·x_d''                            {{color|green|# feedforward component}}
         ''blend_comp = b·delay_line(1)''            {{color|green|# blend component}}
 
         ''y(n) = ff_comp + blend_comp''             {{color|green|# compute output sample}}
   
     '''return''' ''y''

== See also ==
* [[Digital delay line]]
* [[Effects unit]]
* [[Filter design]]

== References ==
{{reflist}}

== Further reading ==
* ''[https://arxiv.org/pdf/1308.0797.pdf <math>H^{\infty}</math>-optimal delay filters]'' by Masaaki Nagahara and Yutaka Yamamoto
* ''[https://fmipa.umri.ac.id/wp-content/uploads/2016/03/Udo-Zolzer-digital-audio-signal-processing.9780470997857.40435.pdf Digital Audio Signal Processing]'' by Udo Zolzer
* Dattorro, Jon, ed. 1988; ''[http://www.aes.org/e-lib/browse.cfm?elib=5125 The Implementation of Recursive Digital Filters for High-Fidelity Audio]''; (JAES Volume 36 Issue 11),

== External links ==
* ''[https://ccrma.stanford.edu/~jos/filters/ Introduction to Digital Filters]'' by Julius Smith
* ''[http://users.spa.aalto.fi/vpv/publications/vesan_vaitos/ Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters]'' by Valimaki Vesa
* ''[https://ccrma.stanford.edu/~jos/pasp/Time_Varying_Delay_Effects.html Time varying delay effects]'' by Julius Smith

{{Electronic filters}}

[[Category:Audio effects]]
[[Category:Digital signal processing]]
[[Category:Effects units]]
[[Category:Instrument amplifiers]]
[[Category:Loudspeakers]]