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<li class="nav-item" data-level="1"><a href="#interpolatorslib" class="nav-link">interpolators.lib</a>
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<li class="nav-item" data-level="2"><a href="#two-points-interpolation-functions" class="nav-link">Two points interpolation functions</a>
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<li class="nav-item" data-level="3"><a href="#itinterpolate_linear" class="nav-link">(it.)interpolate_linear</a>
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<li class="nav-item" data-level="3"><a href="#itinterpolate_cosine" class="nav-link">(it.)interpolate_cosine</a>
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<li class="nav-item" data-level="2"><a href="#four-points-interpolation-functions" class="nav-link">Four points interpolation functions</a>
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<li class="nav-item" data-level="3"><a href="#itinterpolate_cubic" class="nav-link">(it.)interpolate_cubic</a>
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<li class="nav-item" data-level="3"><a href="#itinterpolator_two_points" class="nav-link">(it.)interpolator_two_points</a>
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<li class="nav-item" data-level="3"><a href="#itinterpolator_linear" class="nav-link">(it.)interpolator_linear</a>
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<li class="nav-item" data-level="3"><a href="#itinterpolator_cosine" class="nav-link">(it.)interpolator_cosine</a>
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<li class="nav-item" data-level="3"><a href="#itinterpolator_select" class="nav-link">(it.)interpolator_select</a>
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<li class="nav-item" data-level="2"><a href="#generic-piecewise-linear-interpolation" class="nav-link">Generic piecewise linear interpolation</a>
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<li class="nav-item" data-level="3"><a href="#itlerp" class="nav-link">(it.)lerp</a>
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<li class="nav-item" data-level="3"><a href="#itpiecewise" class="nav-link">(it.)piecewise</a>
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<li class="nav-item" data-level="2"><a href="#lagrange-based-interpolators" class="nav-link">Lagrange based interpolators</a>
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<li class="nav-item" data-level="3"><a href="#itlagrangecoeffs" class="nav-link">(it.)lagrangeCoeffs</a>
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<li class="nav-item" data-level="3"><a href="#itlagrangeinterpolation" class="nav-link">(it.)lagrangeInterpolation</a>
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<li class="nav-item" data-level="3"><a href="#itfrdtable" class="nav-link">(it.)frdtable</a>
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<li class="nav-item" data-level="3"><a href="#itfrwtable" class="nav-link">(it.)frwtable</a>
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</div></div>
<div class="col-md-9 main-container" role="main">
<h1 id="interpolatorslib">interpolators.lib</h1>
<p>A library to handle interpolation. Its official prefix is <code>it</code>.</p>
<p>This library provides several basic interpolation functions, as well as interpolators
taking a <code>gen</code> circuit of N outputs producing values to be interpolated, triggered
by a <code>idv</code> read index signal. Two points and four points interpolations are implemented.</p>
<p>The <code>idv</code> parameter is to be used as a read index. In <code>-single</code> (= 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 <code>-double</code> (= doubleprecision) or <code>-quad</code> (= quadprecision) modes,
a standard implementation with a single fractional index signal is used. Three functions <code>int_part</code>, <code>frac_part</code> and <code>mak_idv</code> are available to manipulate the read index signal.</p>
<p>Here is a use-case with <code>waveform</code>. Here the signal given to <code>interpolator_XXX</code> uses the <code>idv</code> model.</p>
<pre><code>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);
};
</code></pre>
<p>Some tests here:</p>
<pre><code>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 */
};
</code></pre>
<h4 id="references">References</h4>
<ul>
<li><a href="https://github.com/grame-cncm/faustlibraries/blob/master/interpolators.lib">https://github.com/grame-cncm/faustlibraries/blob/master/interpolators.lib</a></li>
</ul>
<h2 id="two-points-interpolation-functions">Two points interpolation functions</h2>
<hr />
<h3 id="itinterpolate_linear"><code>(it.)interpolate_linear</code></h3>
<p>Linear interpolation between 2 values.</p>
<h4 id="usage">Usage</h4>
<pre><code>interpolate_linear(dv,v0,v1) : _
</code></pre>
<p>Where:</p>
<ul>
<li><code>dv</code>: in the fractional value in [0..1] range</li>
<li><code>v0</code>: is the first value</li>
<li><code>v1</code>: is the second value</li>
</ul>
<h4 id="reference">Reference:</h4>
<ul>
<li><a href="https://github.com/jamoma/JamomaCore/blob/master/Foundation/library/includes/TTInterpolate.h">https://github.com/jamoma/JamomaCore/blob/master/Foundation/library/includes/TTInterpolate.h</a></li>
</ul>
<hr />
<h3 id="itinterpolate_cosine"><code>(it.)interpolate_cosine</code></h3>
<p>Cosine interpolation between 2 values.</p>
<h4 id="usage_1">Usage</h4>
<pre><code>interpolate_cosine(dv,v0,v1) : _
</code></pre>
<p>Where:</p>
<ul>
<li><code>dv</code>: in the fractional value in [0..1] range</li>
<li><code>v0</code>: is the first value</li>
<li><code>v1</code>: is the second value</li>
</ul>
<h4 id="reference_1">Reference:</h4>
<ul>
<li><a href="https://github.com/jamoma/JamomaCore/blob/master/Foundation/library/includes/TTInterpolate.h">https://github.com/jamoma/JamomaCore/blob/master/Foundation/library/includes/TTInterpolate.h</a></li>
</ul>
<h2 id="four-points-interpolation-functions">Four points interpolation functions</h2>
<hr />
<h3 id="itinterpolate_cubic"><code>(it.)interpolate_cubic</code></h3>
<p>Cubic interpolation between 4 values.</p>
<h4 id="usage_2">Usage</h4>
<pre><code>interpolate_cubic(dv,v0,v1,v2,v3) : _
</code></pre>
<p>Where:</p>
<ul>
<li><code>dv</code>: in the fractional value in [0..1] range</li>
<li><code>v0</code>: is the first value</li>
<li><code>v1</code>: is the second value</li>
<li><code>v2</code>: is the third value</li>
<li><code>v3</code>: is the fourth value</li>
</ul>
<h4 id="reference_2">Reference:</h4>
<ul>
<li><a href="https://www.paulinternet.nl/?page=bicubic">https://www.paulinternet.nl/?page=bicubic</a></li>
</ul>
<h2 id="two-points-interpolators">Two points interpolators</h2>
<hr />
<h3 id="itinterpolator_two_points"><code>(it.)interpolator_two_points</code></h3>
<p>Generic interpolator on two points (current and next index), assuming an increasing index.</p>
<h4 id="usage_3">Usage</h4>
<pre><code>interpolator_two_points(gen, idv, interpolate_two_points) : si.bus(outputs(gen))
</code></pre>
<p>Where:</p>
<ul>
<li><code>gen</code>: a circuit with an 'idv' reader input that produces N outputs</li>
<li><code>idv</code>: a fractional read index expressed as a float value, or a (int,frac) pair</li>
<li><code>interpolate_two_points</code>: a two points interpolation function</li>
</ul>
<hr />
<h3 id="itinterpolator_linear"><code>(it.)interpolator_linear</code></h3>
<p>Linear interpolator for a 'gen' circuit triggered by an 'idv' input to generate values.</p>
<h4 id="usage_4">Usage</h4>
<pre><code>interpolator_linear(gen, idv) : si.bus(outputs(gen))
</code></pre>
<p>Where:</p>
<ul>
<li><code>gen</code>: a circuit with an 'idv' reader input that produces N outputs</li>
<li><code>idv</code>: a fractional read index expressed as a float value, or a (int,frac) pair</li>
</ul>
<hr />
<h3 id="itinterpolator_cosine"><code>(it.)interpolator_cosine</code></h3>
<p>Cosine interpolator for a 'gen' circuit triggered by an 'idv' input to generate values.</p>
<h4 id="usage_5">Usage</h4>
<pre><code>interpolator_cosine(gen, idv) : si.bus(outputs(gen))
</code></pre>
<p>Where:</p>
<ul>
<li><code>gen</code>: a circuit with an 'idv' reader input that produces N outputs</li>
<li><code>idv</code>: a fractional read index expressed as a float value, or a (int,frac) pair</li>
</ul>
<h2 id="four-points-interpolators">Four points interpolators</h2>
<hr />
<h3 id="itinterpolator_four_points"><code>(it.)interpolator_four_points</code></h3>
<p>Generic interpolator on interpolator_four_points points (previous, current and two next indexes), assuming an increasing index.</p>
<h4 id="usage_6">Usage</h4>
<pre><code>interpolator_four_points(gen, idv, interpolate_four_points) : si.bus(outputs(gen))
</code></pre>
<p>Where:</p>
<ul>
<li><code>gen</code>: a circuit with an 'idv' reader input that produces N outputs</li>
<li><code>idv</code>: a fractional read index expressed as a float value, or a (int,frac) pair</li>
<li><code>interpolate_four_points</code>: a four points interpolation function</li>
</ul>
<hr />
<h3 id="itinterpolator_cubic"><code>(it.)interpolator_cubic</code></h3>
<p>Cubic interpolator for a 'gen' circuit triggered by an 'idv' input to generate values.</p>
<h4 id="usage_7">Usage</h4>
<pre><code>interpolator_cubic(gen, idv) : si.bus(outputs(gen))
</code></pre>
<p>Where:</p>
<ul>
<li><code>gen</code>: a circuit with an 'idv' reader input that produces N outputs</li>
<li><code>idv</code>: a fractional read index expressed as a float value, or a (int,frac) pair</li>
</ul>
<hr />
<h3 id="itinterpolator_select"><code>(it.)interpolator_select</code></h3>
<p>Generic configurable interpolator (with selector between in [0..3]). The value 3 is used for no interpolation.</p>
<h4 id="usage_8">Usage</h4>
<pre><code>interpolator_select(gen, idv, sel) : _,_... (equal to N = outputs(gen))
</code></pre>
<p>Where:</p>
<ul>
<li><code>gen</code>: a circuit with an 'idv' reader input that produces N outputs</li>
<li><code>idv</code>: a fractional read index expressed as a float value, or a (int,frac) pair</li>
<li><code>sel</code>: an interpolation algorithm selector in [0..3] (0 = linear, 1 = cosine, 2 = cubic, 3 = nointerp)</li>
</ul>
<h2 id="generic-piecewise-linear-interpolation">Generic piecewise linear interpolation</h2>
<hr />
<h3 id="itlerp"><code>(it.)lerp</code></h3>
<p>Linear interpolation between two points.</p>
<h4 id="usage_9">Usage</h4>
<pre><code>lerp(x0, x1, y0, y1, x) : si.bus(1);
</code></pre>
<p>Where:</p>
<ul>
<li><code>x0</code>: x-coordinate origin</li>
<li><code>x1</code>: x-coordinate destination</li>
<li><code>y0</code>: y-coordinate origin</li>
<li><code>y1</code>: y-coordinate destination</li>
<li><code>x</code>: x-coordinate input</li>
</ul>
<hr />
<h3 id="itpiecewise"><code>(it.)piecewise</code></h3>
<p>Linear piecewise interpolation between N points.</p>
<h4 id="usage_10">Usage</h4>
<pre><code>piecewise(xList, yList, x) : si.bus(1);
</code></pre>
<p>Where:</p>
<ul>
<li><code>xList</code>: x-coordinates list</li>
<li><code>yList</code>: y-coordinates list</li>
<li><code>x</code>: x-coordinate input</li>
</ul>
<h4 id="example-test-program">Example test program</h4>
<p>The code below will output the values of linear segments going through the
y coordinates as the input goes from -5 to 5:</p>
<pre><code>x = hslider("x", -5, -5.0, 5.0, .001);
process = it.piecewise((-5, -3, 0, 3, 5), (2, 0, 3, -3, -2), x);
</code></pre>
<h2 id="lagrange-based-interpolators">Lagrange based interpolators</h2>
<hr />
<h3 id="itlagrangecoeffs"><code>(it.)lagrangeCoeffs</code></h3>
<p>This is a function to generate N + 1 coefficients for an Nth-order Lagrange
basis polynomial with arbitrary spacing of the points.</p>
<h4 id="usage_11">Usage</h4>
<pre><code>lagrangeCoeffs(N, xCoordsList, x) : si.bus(N + 1)
</code></pre>
<p>Where:</p>
<ul>
<li><code>N</code>: order of the interpolation filter, known at compile-time</li>
<li><code>xCoordsList</code>: a list of N + 1 elements determining the x-axis coordinates of N + 1 values, known at compile-time</li>
<li><code>x</code>: a fractional position on the x-axis to obtain the interpolated y-value</li>
</ul>
<h4 id="reference_3">Reference</h4>
<ul>
<li><a href="https://ccrma.stanford.edu/~jos/pasp/Lagrange_Interpolation.html">https://ccrma.stanford.edu/~jos/pasp/Lagrange_Interpolation.html</a></li>
<li><a href="https://en.wikipedia.org/wiki/Lagrange_polynomial">https://en.wikipedia.org/wiki/Lagrange_polynomial</a></li>
</ul>
<hr />
<h3 id="itlagrangeinterpolation"><code>(it.)lagrangeInterpolation</code></h3>
<p>Nth-order Lagrange interpolator to interpolate between a set of arbitrarily spaced N + 1 points.</p>
<h4 id="usage_12">Usage</h4>
<pre><code>x , yCoords : lagrangeInterpolation(N, xCoordsList) : _
</code></pre>
<p>Where:</p>
<ul>
<li><code>N</code>: order of the interpolator, known at compile-time</li>
<li><code>xCoordsList</code>: a list of N + 1 elements determining the x-axis spacing of the points, known at compile-time</li>
<li><code>x</code>: an x-axis position to interpolate between the y-values</li>
<li><code>yCoords</code>: N + 1 elements determining the values of the interpolation points</li>
</ul>
<p>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]:</p>
<pre><code>N = 3;
xCoordsList = (0, 1, 2, 3);
x = N / 2.0;
yCoords = 2, 5, -1, 3;
process = x, yCoords : it.lagrangeInterpolation(N, xCoordsList);
</code></pre>
<p>which outputs ~1.938.</p>
<ul>
<li>Example: output the dashed curve showed on the Wikipedia page (top figure in <a href="https://en.wikipedia.org/wiki/Lagrange_polynomial">https://en.wikipedia.org/wiki/Lagrange_polynomial</a>):</li>
</ul>
<pre><code>N = 3;
xCoordsList = (-9, -4, -1, 7);
x = os.phasor(16, 1) - 9;
yCoords = 5, 2, -2, 9;
process = x, yCoords : it.lagrangeInterpolation(N, xCoordsList);
</code></pre>
<h4 id="reference_4">Reference</h4>
<ul>
<li><a href="https://ccrma.stanford.edu/~jos/pasp/Lagrange_Interpolation.html">https://ccrma.stanford.edu/~jos/pasp/Lagrange_Interpolation.html</a>
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.</li>
<li><a href="https://dafx2020.mdw.ac.at/proceedings/papers/DAFx20in21_paper_38.pdf">https://dafx2020.mdw.ac.at/proceedings/papers/DAFx20in21_paper_38.pdf</a></li>
</ul>
<hr />
<h3 id="itfrdtable"><code>(it.)frdtable</code></h3>
<p>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.</p>
<h4 id="usage_13">Usage</h4>
<pre><code>frdtable(N, S, init, idx) : _
</code></pre>
<p>Where:</p>
<ul>
<li><code>N</code>: Lagrange interpolation order, known at compile-time</li>
<li><code>S</code>: table size in samples, known at compile-time</li>
<li><code>init</code>: the initial table content, known at compile-time</li>
<li><code>idx</code>: fractional index wrapped-around 0 and S</li>
</ul>
<h4 id="example-test-program_1">Example test program</h4>
<p>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:</p>
<pre><code>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);
</code></pre>
<hr />
<h3 id="itfrwtable"><code>(it.)frwtable</code></h3>
<p>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.</p>
<h4 id="usage_14">Usage</h4>
<pre><code>frwtable(N, S, init, w_idx, x, r_idx) : _
</code></pre>
<p>Where:</p>
<ul>
<li><code>N</code>: Lagrange interpolation order, known at compile-time</li>
<li><code>S</code>: table size in samples, known at compile-time</li>
<li><code>init</code>: the initial table content, known at compile-time</li>
<li><code>w_idx</code>: it should be an INT between 0 and S - 1</li>
<li><code>x</code>: input signal written on the w_idx positions</li>
<li><code>r_idx</code>: fractional index wrapped-around 0 and S</li>
</ul>
<h4 id="example-test-program_2">Example test program</h4>
<p>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:</p>
<pre><code>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)) ,
it.frwtable(N, S, os.sinwaveform(S), wIdx, os.sinwaveform(S), rIdx);
</code></pre>
<h2 id="misc-functions">Misc functions</h2>
<hr />
<h3 id="itremap"><code>(it.)remap</code></h3>
<p>Linearly map from an input domain to an output range.</p>
<h4 id="usage_15">Usage</h4>
<pre><code>_ : remap(from1, from2, to1, to2) : _
</code></pre>
<p>Where:</p>
<ul>
<li><code>from1</code>: the domain's lower bound.</li>
<li><code>from2</code>: the domain's upper bound.</li>
<li><code>to1</code>: the range's lower bound.</li>
<li><code>to2</code>: the range's upper bound.</li>
</ul>
<p>Note that having <code>from1</code> == <code>from2</code> in the mapping will cause a division by zero that has to be taken in account.</p>
<h4 id="example-test-program_3">Example test program</h4>
<p>An oscillator remapped from [-1., 1.] to [100., 1000.]:</p>
<pre><code>os.osc(440) : it.remap(-1., 1., 100., 1000.)
</code></pre></div>
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