# delays.lib

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

.

#### References

## 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)

## 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

`(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).