User:Primoh/sandbox 1



The Dattorro industry scheme is a digital system for implementing a wide range of delay based audio effects for digital signals proposed by Jon Dattorro. The common nature of these effects allows to produce the output signal as linear combinations of (dynamically modulated) delayed replicas of the input signal. The scheme proposed allows to implement such effects in a compact form using just a set of three parameters to control the type of effect. This system is based on digital delay lines and to ensure a proper resolution in time domain, it must leverage fractional delay lines to avoid discontinuities. 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.

Delay line interpolation
Consider continuous time signal $$ y(t) $$. The $$\tau$$-delayed version of such signal is $$y(t) = x(t-\tau)$$. Going to the discrte time domain, i.e., sampling the signal with sampling frequency $$F_s$$ at $$n = tF_s $$, we obtain $$y[n] = x[n-M] $$. The delay line is described in terms of the Z-transform of the discrete time signal: $$Y(z) = \mathcal Z \{y \}(n) = z^{-M}X(z) = H_M(z) X(z) $$ which, brought back in the time domain with the inverse Z-transform, corrsponds to $$h_m[n] = \delta[n-M] $$, where $$\delta[\cdot]$$ is the Dirac delta. It should be note that the former equation holds for $$D\in\mathbb N$$ but for the implementation we need a fractional delay, meaning $$D =\lfloor D\rfloor + (D-\lfloor D\rfloor) = M+d,\quad d\in\mathbb R, \ 0<d<1$$. In this case, interpolation is required to reconstruct the value of the signal that lies between two samples. One can resort to the preferred interpolation technique, e.g., Lagrange interpolation, or all-pass interpolation.



Block diagram
The output of the delay block $$ H_{M}(z) = z^{-M}$$ 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 $$

Setting the coefficients according to the desired effect results in a change in topology of the filter. Setting the feedback to 0, for example results in a FIR filter whereas if the coefficients are set to be the same we approximate an all-pass filter



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.

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. 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 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).

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. This effect is very similar to chorus but main difference is due to the delay range. Chorus usually has longer delay, larger depth and lower modulating frequency.

White Chorus
White chorus is a modification to the standard chorus effect aimed at reducing the abberations introduced by the forward path. The change consists of adding a negative feedback path with a different and fixed tap point in order to obtain an approximation of an all-pass configuration.

Doubling
When the system is used without feedback we achieve 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 trebble loudspeakers .

Pseudocode
The filter can be implemented in software or hardware. 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·$$\pi $$·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

Further Readings

 * $H^{\infty}$-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),