interpolators.lib

A library to handle interpolation. Its official prefix is it.

This library provides several basic interpolation functions, as well as interpolators taking a gen circuit of N outputs producing values to be interpolated, triggered by a idv read index signal. Two points and four points interpolations are implemented.

The idv parameter is to be used as a read index. In -single (= singleprecision) mode, a technique based on 2 signals with the pure integer index and a fractional part in the [0,1] range is used to avoid accumulating errors. In -double (= doubleprecision) or -quad (= quadprecision) modes, a standard implementation with a single fractional index signal is used. Three functions int_part, frac_part and mak_idv are available to manipulate the read index signal.

Here is a use-case with waveform. Here the signal given to interpolator_XXX uses the idv model.

waveform_interpolator(wf, step, interp) = interp(gen, idv)
with {
   gen(idx) = wf, (idx:max(0):min(size-1)) : rdtable with { size = wf:(_,!); };   /* waveform size */
   index = (+(step)~_)-step;  /* starting from 0 */
   idv = it.make_idv(index);  /* build the signal for interpolation in a generic way */
};

waveform_linear(wf, step) = waveform_interpolator(wf, step, it.interpolator_linear);
waveform_cosine(wf, step) = waveform_interpolator(wf, step, it.interpolator_cosine);
waveform_cubic(wf, step) = waveform_interpolator(wf, step, it.interpolator_cubic);

waveform_interp(wf, step, selector) = waveform_interpolator(wf, step, interp_select(selector))
with {
   /* adapts the argument order */
   interp_select(sel, gen, idv) = it.interpolator_select(gen, idv, sel);
};

waveform and index 
waveform_interpolator1(wf, idv, interp) = interp(gen, idv)
with {
   gen(idx) = wf, (idx:max(0):min(size-1)) : rdtable with { size = wf:(_,!); };   /* waveform size */
};

waveform_linear1(wf, idv) = waveform_interpolator1(wf, idv, it.interpolator_linear);
waveform_cosine1(wf, idv) = waveform_interpolator1(wf, idv, it.interpolator_cosine);
waveform_cubic1(wf, idv) = waveform_interpolator1(wf, idv, it.interpolator_cubic);

waveform_interp1(wf, idv, selector) = waveform_interpolator1(wf, idv, interp_select(selector))
with {
   /* adapts the argument order */
   interp_select(sel, gen, idv) = it.interpolator_select(gen, idv, sel);
};

Some tests here:

wf = waveform {0.0, 10.0, 20.0, 30.0, 40.0, 50.0, 60.0, 50.0, 40.0, 30.0, 20.0, 10.0, 0.0};

process = waveform_linear(wf, step), waveform_cosine(wf, step), waveform_cubic(wf, step) with { step = 0.25; };

process = waveform_interp(wf, 0.25, nentry("algo", 0, 0, 3, 1));

process = waveform_interp1(wf, idv, nentry("algo", 0, 0, 3, 1))
with {
   step = 0.1;
   idv_aux = (+(step)~_)-step;  /* starting from 0 */
   idv = it.make_idv(idv_aux);  /* build the signal for interpolation in a generic way */
};

/* Test linear interpolation between 2 samples with a `(idx,dv)` signal built using a waveform */
linear_test = (idx,dv), it.interpolator_linear(gen, (idx,dv))
with {
   /* signal to interpolate (only 2 points here) */
   gen(id) = waveform {3.0, -1.0}, (id:max(0)) : rdtable;
   dv = waveform {0.0, 0.25, 0.50, 0.75, 1.0}, index : rdtable;
   idx = 0; 
   /* test index signal */
   index = (+(1)~_)-1;   /* starting from 0 */
};

/* Test cosine interpolation between 2 samples with a `(idx,dv)` signal built using a waveform */
cosine_test = (idx,dv), it.interpolator_cosine(gen, (idx,dv))
with {
   /* signal to interpolate (only 2 points here) */
   gen(id) = waveform {3.0, -1.0}, (id:max(0)) : rdtable;
   dv = waveform {0.0, 0.25, 0.50, 0.75, 1.0}, index : rdtable;
   idx = 0;
   /* test index signal */
   index = (+(1)~_)-1;   /* starting from 0 */
};

/* Test cubic interpolation between 4 samples with a `(idx,dv)` signal built using a waveform */
cubic_test = (idx,dv), it.interpolator_cubic(gen, (idx,dv))
with {
   /* signal to interpolate (only 4 points here) */
   gen(id) = waveform {-1.0, 2.0, 1.0, 4.0}, (id:max(0)) : rdtable;
   dv = waveform {0.0, 0.25, 0.50, 0.75, 1.0}, index : rdtable;
   idx = 0;
   /* test index signal */
   index = (+(1)~_)-1;   /* starting from 0 */
};

References

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

Two points interpolation functions

`(it.)interpolate_linear`

Linear interpolation between 2 values.

Usage

interpolate_linear(dv,v0,v1) : _

Where:

dv: in the fractional value in [0..1] range
v0: is the first value
v1: is the second value

Reference:

https://github.com/jamoma/JamomaCore/blob/master/Foundation/library/includes/TTInterpolate.h

`(it.)interpolate_cosine`

Cosine interpolation between 2 values.

Usage

interpolate_cosine(dv,v0,v1) : _

Where:

dv: in the fractional value in [0..1] range
v0: is the first value
v1: is the second value

Reference:

https://github.com/jamoma/JamomaCore/blob/master/Foundation/library/includes/TTInterpolate.h

Four points interpolation functions

`(it.)interpolate_cubic`

Cubic interpolation between 4 values.

Usage

interpolate_cubic(dv,v0,v1,v2,v3) : _

Where:

dv: in the fractional value in [0..1] range
v0: is the first value
v1: is the second value
v2: is the third value
v3: is the fourth value

Reference:

https://www.paulinternet.nl/?page=bicubic

Two points interpolators

`(it.)interpolator_two_points`

Generic interpolator on two points (current and next index), assuming an increasing index.

Usage

interpolator_two_points(gen, idv, interpolate_two_points) : si.bus(outputs(gen))

Where:

gen: a circuit with an 'idv' reader input that produces N outputs
idv: a fractional read index expressed as a float value, or a (int,frac) pair
interpolate_two_points: a two points interpolation function

`(it.)interpolator_linear`

Linear interpolator for a 'gen' circuit triggered by an 'idv' input to generate values.

Usage

interpolator_linear(gen, idv) : si.bus(outputs(gen))

Where:

gen: a circuit with an 'idv' reader input that produces N outputs
idv: a fractional read index expressed as a float value, or a (int,frac) pair

`(it.)interpolator_cosine`

Cosine interpolator for a 'gen' circuit triggered by an 'idv' input to generate values.

Usage

interpolator_cosine(gen, idv) : si.bus(outputs(gen))

Where:

gen: a circuit with an 'idv' reader input that produces N outputs
idv: a fractional read index expressed as a float value, or a (int,frac) pair

Four points interpolators

`(it.)interpolator_four_points`

Generic interpolator on interpolator_four_points points (previous, current and two next indexes), assuming an increasing index.

Usage

interpolator_four_points(gen, idv, interpolate_four_points) : si.bus(outputs(gen))

Where:

gen: a circuit with an 'idv' reader input that produces N outputs
idv: a fractional read index expressed as a float value, or a (int,frac) pair
interpolate_four_points: a four points interpolation function

`(it.)interpolator_cubic`

Cubic interpolator for a 'gen' circuit triggered by an 'idv' input to generate values

Usage

interpolator_cubic(gen, idv) : si.bus(outputs(gen))

Where:

gen: a circuit with an 'idv' reader input that produces N outputs
idv: a fractional read index expressed as a float value, or a (int,frac) pair

`(it.)interpolator_select`

Generic configurable interpolator (with selector between in [0..3]). The value 3 is used for no interpolation.

Usage

interpolator_select(gen, idv, sel) : _,_... (equal to N = outputs(gen))

Where:

gen: a circuit with an 'idv' reader input that produces N outputs
idv: a fractional read index expressed as a float value, or a (int,frac) pair
sel: an interpolation algorithm selector in [0..3] (0 = linear, 1 = cosine, 2 = cubic, 3 = nointerp)

Lagrange based interpolators

`(it.)lagrangeCoeffs(N, xCoordsList)`

This is a function to generate N + 1 coefficients for an Nth-order Lagrange basis polynomial with arbitrary spacing of the points.

Usage

lagrangeCoeffs(N, xCoordsList, x) : si.bus(N + 1)

Where:

N: order of the interpolation filter, known at compile-time
xCoordsList: a list of N + 1 elements determining the x-axis coordinates of N + 1 values, known at compile-time
x: a fractional position on the x-axis to obtain the interpolated y-value

Reference

`(it.)lagrangeInterpolation(N, xCoordsList)`

Nth-order Lagrange interpolator to interpolate between a set of arbitrarily spaced N + 1 points.

Usage

x , yCoords : lagrangeInterpolation(N, xCoordsList) : _

Where:

N: order of the interpolator, known at compile-time
xCoordsList: a list of N + 1 elements determining the x-axis spacing of the points, known at compile-time
x: an x-axis position to interpolate between the y-values
yCoords: N + 1 elements determining the values of the interpolation points

Example: find the centre position of a four-point set using an order-3 Lagrange function fitting the equally-spaced points [2, 5, -1, 3]:

N = 3;
xCoordsList = (0, 1, 2, 3);
x = N / 2.0;
yCoords = 2, 5, -1, 3;
process = x, yCoords : lagrangeInterpolation(N, xCoordsList);

which outputs ~1.938.

Example: output the dashed curve showed on the Wikipedia page (top figure, https://en.wikipedia.org/wiki/Lagrange_polynomial):

N = 3;
xCoordsList = (-9, -4, -1, 7);
x = os.phasor(16, 1) - 9;
yCoords = 5, 2, -2, 9;
process = x, yCoords : lagrangeInterpolation(N, xCoordsList);

Reference

https://ccrma.stanford.edu/~jos/pasp/Lagrange_Interpolation.html Sanfilippo and Parker 2021, "Combining zeroth and first‐order analysis with Lagrange polynomials to reduce artefacts in live concatenative granular processing." Proceedings of the DAFx conference 2021, Vienna, Austria.
https://dafx2020.mdw.ac.at/proceedings/papers/DAFx20in21_paper_38.pdf

`(it.)frdtable(N, S)`

Look-up circular table with Nth-order Lagrange interpolation for fractional indexes. The index is wrapped-around and the table is cycles for an index span of size S, which is the table size in samples.

Usage

frdtable(N, S, init, idx) : _

Where:

N: Lagrange interpolation order, known at compile-time
S: table size in samples, known at compile-time
init: signal for table initialisation
idx: fractional index wrapped-around 0 and S

Example test program

Test the effectiveness of the 5th-order interpolation scheme by creating a table look-up oscillator using only 16 points of a sinewave; compare the result with a non-interpolated version:

N = 5;
S = 16;
index = os.phasor(S, 1000);
process = rdtable(S, os.sinwaveform(S), int(index)) ,
          it.frdtable(N, S, os.sinwaveform(S), index);

`(it.)frwtable(N, S)`

Look-up updatable circular table with Nth-order Lagrange interpolation for fractional indexes. The index is wrapped-around and the table is circular indexes ranging from 0 to S, which is the table size in samples.

Usage

frwtable(N, S, init, w_idx, x, r_idx) : _

Where:

N: Lagrange interpolation order, known at compile-time
S: table size in samples, known at compile-time
init: constant for table initialisation, known at compile-time
w_idx: it should be an INT between 0 and S - 1
x: input signal written on the w_idx positions
r_idx: fractional index wrapped-around 0 and S

Example test program

Test the effectiveness of the 5th-order interpolation scheme by creating a table look-up oscillator using only 16 points of a sinewave; compare the result with a non-interpolated version:

N = 5;
S = 16;
rIdx = os.phasor(S, 300);
wIdx = ba.period(S);
process = rwtable(S, os.sinwaveform(S), wIdx, os.sinwaveform(S), int(rIdx)) ,
          frwtable(N, S, os.sinwaveform(S), wIdx, os.sinwaveform(S), rIdx);

Misc functions

`(it.)remap`

Linearly map from an input domain to an output range.

Usage

_ : remap(from1, from2, to1, to2) : _

Where:

from1: the domain's lower bound.
from2: the domain's upper bound.
to1: the range's lower bound.
to2: the range's upper bound.

Note that having from1 == from2 in the mapping will cause a division by zero that has to be taken in account.

Example: An oscillator remapped from [-1., 1.] to [100., 1000.]

os.osc(440) : it.remap(-1., 1., 100., 1000.)

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