delays - Faust Libraries

# delays.lib

This library contains a collection of delay functions. Its official prefix is `de`.

#### References

- https://github.com/grame-cncm/faustlibraries/blob/master/delays.lib

## Basic Delay Functions

### `(de.)delay`

Simple `d` samples delay where `n` is the maximum delay length as a number of samples. Unlike the `@` delay operator, here the delay signal `d` is explicitly bounded to the interval [0..n]. The consequence is that delay will compile even if the interval of d can't be computed by the compiler. `delay` is a standard Faust function.

#### Usage

    _ : delay(n,d) : _

Where:

- `n`: the max delay length in samples
- `d`: the delay length in samples (integer)

### `(de.)fdelay`

Simple `d` samples fractional delay based on 2 interpolated delay lines where `n` is the maximum delay length as a number of samples.

`fdelay` is a standard Faust function.

#### Usage

    _ : fdelay(n,d) : _

Where:

- `n`: the max delay length in samples
- `d`: the delay length in samples (float)

### `(de.)sdelay`

s(mooth)delay: a mono delay that doesn't click and doesn't transpose when the delay time is changed.

#### Usage

    _ : sdelay(n,it,d) : _

Where:

- `n`: the max delay length in samples
- `it`: interpolation time (in samples), for example 1024
- `d`: the delay length in samples (float)

### `(de.)prime_power_delays`

Prime Power Delay Line Lengths.

#### Usage

    si.bus(N) : prime_power_delays(N,pathmin,pathmax) : si.bus(N);

Where:

- `N`: positive integer up to 16 (for higher powers of 2, extend 'primes' array below)
- `pathmin`: minimum acoustic ray length in the reverberator (in meters)
- `pathmax`: maximum acoustic ray length (meters) - think "room size"

#### Reference

- https://ccrma.stanford.edu/~jos/pasp/Prime_Power_Delay_Line.html

## Lagrange Interpolation

### `(de.)fdelaylti` and `(de.)fdelayltv`

Fractional delay line using Lagrange interpolation.

#### Usage

    _ : fdelaylt[i|v](N, n, d) : _

Where:

- `N=1,2,3,...` is the order of the Lagrange interpolation polynomial (constant numerical expression)
- `n`: the max delay length in samples
- `d`: the delay length in samples

`fdelaylti` is most efficient, but designed for constant/slowly-varying delay. `fdelayltv` is more expensive and more robust when the delay varies rapidly.

Note: the requested delay should not be less than `(N-1)/2`.

#### References

- https://ccrma.stanford.edu/~jos/pasp/Lagrange_Interpolation.html
- fixed-delay case
- variable-delay case
- Timo I. Laakso et al., "Splitting the Unit Delay - Tools for Fractional Delay Filter Design", IEEE Signal Processing Magazine, vol. 13, no. 1, pp. 30-60, Jan 1996.
- Philippe Depalle and Stephan Tassart, "Fractional Delay Lines using Lagrange Interpolators", ICMC Proceedings, pp. 341-343, 1996.

### `(de.)fdelay[N]`

For convenience, `fdelay1`, `fdelay2`, `fdelay3`, `fdelay4`, `fdelay5` are also available where `N` is the order of the interpolation, built using `fdelayltv`.

## Thiran Allpass Interpolation

Thiran Allpass Interpolation.

#### Reference

- https://ccrma.stanford.edu/~jos/pasp/Thiran_Allpass_Interpolators.html

### `(de.)fdelay[N]a`

Delay lines interpolated using Thiran allpass interpolation.

#### Usage

    _ : fdelay[N]a(n, d) : _

(exactly like `fdelay`)

Where:

- `N=1,2,3, or 4` is the order of the Thiran interpolation filter (constant numerical expression), and the delay argument is at least `N-1/2`. First-order: `d` at least 0.5, second-order: `d` at least 1.5, third-order: `d` at least 2.5, fourth-order: `d` at least 3.5.
- `n`: the max delay length in samples
- `d`: the delay length in samples

#### Note

The interpolated delay should not be less than `N-1/2`. (The allpass delay ranges from `N-1/2` to `N+1/2`). This constraint can be alleviated by altering the code, but be aware that allpass filters approach zero delay by means of pole-zero cancellations.

Delay arguments too small will produce an UNSTABLE allpass!

Because allpass interpolation is recursive, it is not as robust as Lagrange interpolation under time-varying conditions (you may hear clicks when changing the delay rapidly).

## Others

### `(de.)multiTapSincDelay`

Variable delay line using multi-tap sinc interpolation.

This function implements a continuously variable delay line by superposing (2K+2) auxiliary delayed signals whose positions and gains are determined by a sinc-based interpolation method. It extends the traditional crossfade delay technique to significantly reduce spectral coloration artifacts, which are problematic in applications like Wave Field Synthesis (WFS) and auralization.

Operation:

- If tau1 and tau2 are very close (|tau2 - tau1| ≈ 0), a simple fixed fractional delay is applied

- Otherwise, a variable delay is synthesized by:

  - Computing (2K+2) taps symmetrically distributed around tau1 and tau2
  - Applying sinc-based weighting to each tap, based on its offset from the target interpolated delay tau
  - Summing all the weighted taps to produce the output

Features:

- Smooth delay variation without introducing Doppler pitch shifts
- Significant reduction of comb-filter coloration compared to classical crossfading
- Switching between fixed and variable delay modes to ensure stability

#### Usage

     _ : multiTapSincDelay(K, MaxDelay, tau1, tau2, alpha) : _

Where:

- `K (integer)`: number of auxiliary tap pairs (a constant numerical expression). Total number of taps = 2*K + 2
- `MaxDelay`: maximum allowable delay in samples (buffer size)
- `tau1`: initial delay in samples (can be fractional)
- `tau2`: target delay in samples (can be fractional)
- `alpha`: interpolation factor between tau1 and tau2 (in [0,1] with 0 = tau1, 1 = tau2)

#### Reference

- T. Carpentier, "Implementation of a continuously variable delay line by crossfading between several tap delays", 2024: https://hal.science/hal-04646939

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