Dattorro industry scheme

Consider continuous time signal ${\displaystyle y(t)}$. The ${\displaystyle \tau }$-delayed version of such signal is ${\displaystyle y(t)=x(t-\tau )}$. Considering the signal in the discrte time domain (i.e., sampling it with sampling frequency ${\displaystyle F_{s}}$ at ${\displaystyle n=tF_{s}}$), we obtain ${\displaystyle y[n]=x[n-M]}$. The delay line can then be described in terms of the Z-transform of the discrete time signal as

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

When going back in the time domain exploiting the inverse Z-transform, this corrsponds to ${\displaystyle h_{m}[n]=\delta [n-M]}$, where ${\displaystyle \delta [\cdot ]}$ is the Dirac delta. The former equation holds for ${\displaystyle D\in \mathbb {N} }$ but for the implementation we need a fractional delay, meaning ${\displaystyle D=\lfloor D\rfloor +(D-\lfloor D\rfloor )=M+d,\quad d\in \mathbb {R} ,\ 0<d<1}$. In this case, interpolation is required to reconstruct the value of the signal that lies between two samples.[4] One can resort to the preferred interpolation technique, e.g., Lagrange interpolation,[5] or all-pass interpolation.[6]

Linear interpolator block diagram

All-pass interpolator block diagram

The output of the delay block ${\displaystyle H_{M}(z)=z^{-M}}$ 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:

Dattorro scheme

$${\displaystyle Y(z)=H(z)X(z)={\frac {b+f_{f}z^{-D(t)}}{1+z^{-D(t)}f_{b}}}X(z)}$$

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.[1]

Guitar string ${\displaystyle E_{2}}$ sound

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.[1]

Effect	Onset	Nominal	Range End
Vibrato	0	Minimal	5
Flange	0	1	10
Chorus	1	5	30
Doubling	10	20	100
Echo	50	80	${\displaystyle \infty }$

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,                                   # input signal
           Fs,                                  # sampling frequency
           depth,                               # modulation depth
           freq,                                # LFO frequency
           b,                                   # blend knob
           ff,                                  # feedforward knob
           fb,                                  # feedback knob
           mod                                  # modulation type
    output: y                                   # signal with FX applied

    depth_samples = depth·Fs                    # compute delay in samples
    if mod is sinusoid                          # compute delay sequence
        delay_seq = depth_samples*(1 + sin(2·${\displaystyle \pi }$·freq*t))
    else mod is noise
        lowpass_noise = noise_gen()             # 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))             # integer delay
        d_frac = delay_seq(n) - d_int           # fractional delay        
        h0 = (d_frac - 1)·(d_frac - 2)/2        # first FIR filter coefficient
        h1 = d_frac·(2 - d_frac)                # second FIR filter coefficient
        h2 = d_frac/2·(d_frac - 1)              # third FIR filter coefficient
        
        x_d = delay_line(d_int + 1)·h0 + delay_line(d_int + 2)·h1 + delay_line(d_int + 3)·h2  # delay input                                       
        
        f_b = fb·x_d                            # delayed input feedback
        delay_line(2:L) = delay_line(1:L-1)     # delay line update
        delay_line(1) = x(i) - fb_comp
        
        f_f = ff·x_d                            # feedforward component
        blend_comp = b·delay_line(1)            # blend component

        y(n) = ff_comp + blend_comp             # compute output sample
  
    return y

• Digital delay line
• Effects unit
• Filter design

• ${\displaystyle H^{\infty }}$-optimal delay filters by Masaaki Nagahara and Yutaka Yamamoto
• Digital Audio Signal Processing by Udo Zolzer
• Dattorro, Jon, ed. 1988; The Implementation of Recursive Digital Filters for High-Fidelity Audio; (JAES Volume 36 Issue 11),

• Introduction to Digital Filters by Julius Smith
• Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters by Valimaki Vesa
• Time varying delay effects by Julius Smith