From 0320d0b02530fb33eb15508bc3620221f814b11f Mon Sep 17 00:00:00 2001
From: Robin Haberkorn <robin.haberkorn@googlemail.com>
Date: Mon, 11 Dec 2023 03:46:54 +0300
Subject: added FFT Jupyter notebook

* This still refers to a wave file that shouldn't be checked in.
---
 examples/fft.ipynb | 2154 ++++++++++++++++++++++++++++++++++++++++++++++++++++
 examples/fft.lua   |    2 +-
 2 files changed, 2155 insertions(+), 1 deletion(-)
 create mode 100644 examples/fft.ipynb

diff --git a/examples/fft.ipynb b/examples/fft.ipynb
new file mode 100644
index 0000000..ff447f2
--- /dev/null
+++ b/examples/fft.ipynb
@@ -0,0 +1,2154 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "4a93eae1-5c0e-4cf1-846a-621d0fcdd29e",
+   "metadata": {},
+   "source": [
+    "# Fast Fourier Transforms (FFT)\n",
+    "\n",
+    "Let $x_0, \\ldots, x_{n-1}$ be complex numbers. The [Discrete Fourier Transform (DFT)](https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform) is defined by the formula:\n",
+    "\n",
+    "$$\n",
+    "  X_k = \\sum_{m=0}^{n-1} x_m e^{-i2\\pi k m/n} \\qquad k = 0,\\ldots,n-1,\n",
+    "$$\n",
+    "\n",
+    "Let's define a stream, consisting of exactly 20 sine waves in 1024 samples:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "554cb334-283a-40c1-89aa-5fe1fd41161c",
+   "metadata": {},
+   "outputs": [
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+       "\t\tL84.6,19.7 L85.1,24.5 L85.6,32.4 L86.0,43.2 L86.5,56.9 L87.0,73.3 L87.5,92.0 L87.9,112.7\n",
+       "\t\tL88.4,135.3 L88.9,159.3 L89.3,184.4 L89.8,210.2 L90.3,236.3 L90.8,262.3 L91.2,287.8 L91.7,312.5\n",
+       "\t\tL92.2,336.0 L92.7,357.9 L93.1,377.9 L93.6,395.7 L94.1,411.0 L94.5,423.6 L95.0,433.3 L95.5,439.9\n",
+       "\t\tL96.0,443.4 L96.4,443.7 L96.9,440.9 L97.4,434.8 L97.9,425.7 L98.3,413.7 L98.8,398.9 L99.3,381.6\n",
+       "\t\tL99.7,362.1 L100.2,340.5 L100.7,317.3 L101.2,292.9 L101.6,267.5 L102.1,241.5 L102.6,215.4 L103.1,189.5\n",
+       "\t\tL103.5,164.3 L104.0,140.0 L104.5,117.1 L104.9,96.0 L105.4,76.8 L105.9,60.0 L106.4,45.8 L106.8,34.3\n",
+       "\t\tL107.3,25.8 L107.8,20.4 L108.3,18.2 L108.7,19.1 L109.2,23.3 L109.7,30.5 L110.1,40.8 L110.6,54.0\n",
+       "\t\tL111.1,69.8 L111.6,88.0 L112.0,108.4 L112.5,130.7 L113.0,154.4 L113.5,179.3 L113.9,205.0 L114.4,231.0\n",
+       "\t\tL114.9,257.1 L115.3,282.8 L115.8,307.7 L116.3,331.4 L116.8,353.7 L117.2,374.1 L117.7,392.3 L118.2,408.1\n",
+       "\t\tL118.7,421.3 L119.1,431.6 L119.6,438.8 L120.1,443.0 L120.5,443.9 L121.0,441.7 L121.5,436.3 L122.0,427.8\n",
+       "\t\tL122.4,416.3 L122.9,402.1 L123.4,385.3 L123.9,366.1 L124.3,345.0 L124.8,322.1 L125.3,297.8 L125.7,272.6\n",
+       "\t\tL126.2,246.7 L126.7,220.6 L127.2,194.6 L127.6,169.2 L128.1,144.8 L128.6,121.6 L129.1,100.0 L129.5,80.5\n",
+       "\t\tL130.0,63.2 L130.5,48.4 L130.9,36.4 L131.4,27.3 L131.9,21.2 L132.4,18.4 L132.8,18.7 L133.3,22.2\n",
+       "\t\tL133.8,28.8 L134.3,38.5 L134.7,51.1 L135.2,66.4 L135.7,84.2 L136.1,104.2 L136.6,126.1 L137.1,149.6\n",
+       "\t\tL137.6,174.3 L138.0,199.8 L138.5,225.8 L139.0,251.9 L139.5,277.7 L139.9,302.8 L140.4,326.8 L140.9,349.4\n",
+       "\t\tL141.3,370.1 L141.8,388.8 L142.3,405.2 L142.8,418.9 L143.2,429.7 L143.7,437.6 L144.2,442.4 L144.6,444.0\n",
+       "\t\tL145.1,442.4 L145.6,437.6 L146.1,429.7 L146.5,418.9 L147.0,405.2 L147.5,388.8 L148.0,370.1 L148.4,349.4\n",
+       "\t\tL148.9,326.8 L149.4,302.8 L149.8,277.7 L150.3,251.9 L150.8,225.8 L151.3,199.8 L151.7,174.3 L152.2,149.6\n",
+       "\t\tL152.7,126.1 L153.2,104.2 L153.6,84.2 L154.1,66.4 L154.6,51.1 L155.0,38.5 L155.5,28.8 L156.0,22.2\n",
+       "\t\tL156.5,18.7 L156.9,18.4 L157.4,21.2 L157.9,27.3 L158.4,36.4 L158.8,48.4 L159.3,63.2 L159.8,80.5\n",
+       "\t\tL160.2,100.0 L160.7,121.6 L161.2,144.8 L161.7,169.2 L162.1,194.6 L162.6,220.6 L163.1,246.7 L163.6,272.6\n",
+       "\t\tL164.0,297.8 L164.5,322.1 L165.0,345.0 L165.4,366.1 L165.9,385.3 L166.4,402.1 L166.9,416.3 L167.3,427.8\n",
+       "\t\tL167.8,436.3 L168.3,441.7 L168.8,443.9 L169.2,443.0 L169.7,438.8 L170.2,431.6 L170.6,421.3 L171.1,408.1\n",
+       "\t\tL171.6,392.3 L172.1,374.1 L172.5,353.7 L173.0,331.4 L173.5,307.7 L174.0,282.8 L174.4,257.1 L174.9,231.0\n",
+       "\t\tL175.4,205.0 L175.8,179.3 L176.3,154.4 L176.8,130.7 L177.3,108.4 L177.7,88.0 L178.2,69.8 L178.7,54.0\n",
+       "\t\tL179.2,40.8 L179.6,30.5 L180.1,23.3 L180.6,19.1 L181.0,18.2 L181.5,20.4 L182.0,25.8 L182.5,34.3\n",
+       "\t\tL182.9,45.8 L183.4,60.0 L183.9,76.8 L184.4,96.0 L184.8,117.1 L185.3,140.0 L185.8,164.3 L186.2,189.5\n",
+       "\t\tL186.7,215.4 L187.2,241.5 L187.7,267.5 L188.1,292.9 L188.6,317.3 L189.1,340.5 L189.6,362.1 L190.0,381.6\n",
+       "\t\tL190.5,398.9 L191.0,413.7 L191.4,425.7 L191.9,434.8 L192.4,440.9 L192.9,443.7 L193.3,443.4 L193.8,439.9\n",
+       "\t\tL194.3,433.3 L194.8,423.6 L195.2,411.0 L195.7,395.7 L196.2,377.9 L196.6,357.9 L197.1,336.0 L197.6,312.5\n",
+       "\t\tL198.1,287.8 L198.5,262.3 L199.0,236.3 L199.5,210.2 L199.9,184.4 L200.4,159.3 L200.9,135.3 L201.4,112.7\n",
+       "\t\tL201.8,92.0 L202.3,73.3 L202.8,56.9 L203.3,43.2 L203.7,32.4 L204.2,24.5 L204.7,19.7 L205.1,18.1\n",
+       "\t\tL205.6,19.7 L206.1,24.5 L206.6,32.4 L207.0,43.2 L207.5,56.9 L208.0,73.3 L208.5,92.0 L208.9,112.7\n",
+       "\t\tL209.4,135.3 L209.9,159.3 L210.3,184.4 L210.8,210.2 L211.3,236.3 L211.8,262.3 L212.2,287.8 L212.7,312.5\n",
+       "\t\tL213.2,336.0 L213.7,357.9 L214.1,377.9 L214.6,395.7 L215.1,411.0 L215.5,423.6 L216.0,433.3 L216.5,439.9\n",
+       "\t\tL217.0,443.4 L217.4,443.7 L217.9,440.9 L218.4,434.8 L218.9,425.7 L219.3,413.7 L219.8,398.9 L220.3,381.6\n",
+       "\t\tL220.7,362.1 L221.2,340.5 L221.7,317.3 L222.2,292.9 L222.6,267.5 L223.1,241.5 L223.6,215.4 L224.1,189.5\n",
+       "\t\tL224.5,164.3 L225.0,140.0 L225.5,117.1 L225.9,96.0 L226.4,76.8 L226.9,60.0 L227.4,45.8 L227.8,34.3\n",
+       "\t\tL228.3,25.8 L228.8,20.4 L229.3,18.2 L229.7,19.1 L230.2,23.3 L230.7,30.5 L231.1,40.8 L231.6,54.0\n",
+       "\t\tL232.1,69.8 L232.6,88.0 L233.0,108.4 L233.5,130.7 L234.0,154.4 L234.5,179.3 L234.9,205.0 L235.4,231.0\n",
+       "\t\tL235.9,257.1 L236.3,282.8 L236.8,307.7 L237.3,331.4 L237.8,353.7 L238.2,374.1 L238.7,392.3 L239.2,408.1\n",
+       "\t\tL239.7,421.3 L240.1,431.6 L240.6,438.8 L241.1,443.0 L241.5,443.9 L242.0,441.7 L242.5,436.3 L243.0,427.8\n",
+       "\t\tL243.4,416.3 L243.9,402.1 L244.4,385.3 L244.9,366.1 L245.3,345.0 L245.8,322.1 L246.3,297.8 L246.7,272.6\n",
+       "\t\tL247.2,246.7 L247.7,220.6 L248.2,194.6 L248.6,169.2 L249.1,144.8 L249.6,121.6 L250.1,100.0 L250.5,80.5\n",
+       "\t\tL251.0,63.2 L251.5,48.4 L251.9,36.4 L252.4,27.3 L252.9,21.2 L253.4,18.4 L253.8,18.7 L254.3,22.2\n",
+       "\t\tL254.8,28.8 L255.3,38.5 L255.7,51.1 L256.2,66.4 L256.7,84.2 L257.1,104.2 L257.6,126.1 L258.1,149.6\n",
+       "\t\tL258.6,174.3 L259.0,199.8 L259.5,225.8 L260.0,251.9 L260.4,277.7 L260.9,302.8 L261.4,326.8 L261.9,349.4\n",
+       "\t\tL262.3,370.1 L262.8,388.8 L263.3,405.2 L263.8,418.9 L264.2,429.7 L264.7,437.6 L265.2,442.4 L265.6,444.0\n",
+       "\t\tL266.1,442.4 L266.6,437.6 L267.1,429.7 L267.5,418.9 L268.0,405.2 L268.5,388.8 L269.0,370.1 L269.4,349.4\n",
+       "\t\tL269.9,326.8 L270.4,302.8 L270.8,277.7 L271.3,251.9 L271.8,225.8 L272.3,199.8 L272.7,174.3 L273.2,149.6\n",
+       "\t\tL273.7,126.1 L274.2,104.2 L274.6,84.2 L275.1,66.4 L275.6,51.1 L276.0,38.5 L276.5,28.8 L277.0,22.2\n",
+       "\t\tL277.5,18.7 L277.9,18.4 L278.4,21.2 L278.9,27.3 L279.4,36.4 L279.8,48.4 L280.3,63.2 L280.8,80.5\n",
+       "\t\tL281.2,100.0 L281.7,121.6 L282.2,144.8 L282.7,169.2 L283.1,194.6 L283.6,220.6 L284.1,246.7 L284.6,272.6\n",
+       "\t\tL285.0,297.8 L285.5,322.1 L286.0,345.0 L286.4,366.1 L286.9,385.3 L287.4,402.1 L287.9,416.3 L288.3,427.8\n",
+       "\t\tL288.8,436.3 L289.3,441.7 L289.8,443.9 L290.2,443.0 L290.7,438.8 L291.2,431.6 L291.6,421.3 L292.1,408.1\n",
+       "\t\tL292.6,392.3 L293.1,374.1 L293.5,353.7 L294.0,331.4 L294.5,307.7 L295.0,282.8 L295.4,257.1 L295.9,231.0\n",
+       "\t\tL296.4,205.0 L296.8,179.3 L297.3,154.4 L297.8,130.7 L298.3,108.4 L298.7,88.0 L299.2,69.8 L299.7,54.0\n",
+       "\t\tL300.2,40.8 L300.6,30.5 L301.1,23.3 L301.6,19.1 L302.0,18.2 L302.5,20.4 L303.0,25.8 L303.5,34.3\n",
+       "\t\tL303.9,45.8 L304.4,60.0 L304.9,76.8 L305.4,96.0 L305.8,117.1 L306.3,140.0 L306.8,164.3 L307.2,189.5\n",
+       "\t\tL307.7,215.4 L308.2,241.5 L308.7,267.5 L309.1,292.9 L309.6,317.3 L310.1,340.5 L310.6,362.1 L311.0,381.6\n",
+       "\t\tL311.5,398.9 L312.0,413.7 L312.4,425.7 L312.9,434.8 L313.4,440.9 L313.9,443.7 L314.3,443.4 L314.8,439.9\n",
+       "\t\tL315.3,433.3 L315.7,423.6 L316.2,411.0 L316.7,395.7 L317.2,377.9 L317.6,357.9 L318.1,336.0 L318.6,312.5\n",
+       "\t\tL319.1,287.8 L319.5,262.3 L320.0,236.3 L320.5,210.2 L320.9,184.4 L321.4,159.3 L321.9,135.3 L322.4,112.7\n",
+       "\t\tL322.8,92.0 L323.3,73.3 L323.8,56.9 L324.3,43.2 L324.7,32.4 L325.2,24.5 L325.7,19.7 L326.1,18.1\n",
+       "\t\tL326.6,19.7 L327.1,24.5 L327.6,32.4 L328.0,43.2 L328.5,56.9 L329.0,73.3 L329.5,92.0 L329.9,112.7\n",
+       "\t\tL330.4,135.3 L330.9,159.3 L331.3,184.4 L331.8,210.2 L332.3,236.3 L332.8,262.3 L333.2,287.8 L333.7,312.5\n",
+       "\t\tL334.2,336.0 L334.7,357.9 L335.1,377.9 L335.6,395.7 L336.1,411.0 L336.5,423.6 L337.0,433.3 L337.5,439.9\n",
+       "\t\tL338.0,443.4 L338.4,443.7 L338.9,440.9 L339.4,434.8 L339.9,425.7 L340.3,413.7 L340.8,398.9 L341.3,381.6\n",
+       "\t\tL341.7,362.1 L342.2,340.5 L342.7,317.3 L343.2,292.9 L343.6,267.5 L344.1,241.5 L344.6,215.4 L345.1,189.5\n",
+       "\t\tL345.5,164.3 L346.0,140.0 L346.5,117.1 L346.9,96.0 L347.4,76.8 L347.9,60.0 L348.4,45.8 L348.8,34.3\n",
+       "\t\tL349.3,25.8 L349.8,20.4 L350.3,18.2 L350.7,19.1 L351.2,23.3 L351.7,30.5 L352.1,40.8 L352.6,54.0\n",
+       "\t\tL353.1,69.8 L353.6,88.0 L354.0,108.4 L354.5,130.7 L355.0,154.4 L355.5,179.3 L355.9,205.0 L356.4,231.0\n",
+       "\t\tL356.9,257.1 L357.3,282.8 L357.8,307.7 L358.3,331.4 L358.8,353.7 L359.2,374.1 L359.7,392.3 L360.2,408.1\n",
+       "\t\tL360.7,421.3 L361.1,431.6 L361.6,438.8 L362.1,443.0 L362.5,443.9 L363.0,441.7 L363.5,436.3 L364.0,427.8\n",
+       "\t\tL364.4,416.3 L364.9,402.1 L365.4,385.3 L365.9,366.1 L366.3,345.0 L366.8,322.1 L367.3,297.8 L367.7,272.6\n",
+       "\t\tL368.2,246.7 L368.7,220.6 L369.2,194.6 L369.6,169.2 L370.1,144.8 L370.6,121.6 L371.1,100.0 L371.5,80.5\n",
+       "\t\tL372.0,63.2 L372.5,48.4 L372.9,36.4 L373.4,27.3 L373.9,21.2 L374.4,18.4 L374.8,18.7 L375.3,22.2\n",
+       "\t\tL375.8,28.8 L376.2,38.5 L376.7,51.1 L377.2,66.4 L377.7,84.2 L378.1,104.2 L378.6,126.1 L379.1,149.6\n",
+       "\t\tL379.6,174.3 L380.0,199.8 L380.5,225.8 L381.0,251.9 L381.4,277.7 L381.9,302.8 L382.4,326.8 L382.9,349.4\n",
+       "\t\tL383.3,370.1 L383.8,388.8 L384.3,405.2 L384.8,418.9 L385.2,429.7 L385.7,437.6 L386.2,442.4 L386.6,444.0\n",
+       "\t\tL387.1,442.4 L387.6,437.6 L388.1,429.7 L388.5,418.9 L389.0,405.2 L389.5,388.8 L390.0,370.1 L390.4,349.4\n",
+       "\t\tL390.9,326.8 L391.4,302.8 L391.8,277.7 L392.3,251.9 L392.8,225.8 L393.3,199.8 L393.7,174.3 L394.2,149.6\n",
+       "\t\tL394.7,126.1 L395.2,104.2 L395.6,84.2 L396.1,66.4 L396.6,51.1 L397.0,38.5 L397.5,28.8 L398.0,22.2\n",
+       "\t\tL398.5,18.7 L398.9,18.4 L399.4,21.2 L399.9,27.3 L400.4,36.4 L400.8,48.4 L401.3,63.2 L401.8,80.5\n",
+       "\t\tL402.2,100.0 L402.7,121.6 L403.2,144.8 L403.7,169.2 L404.1,194.6 L404.6,220.6 L405.1,246.7 L405.6,272.6\n",
+       "\t\tL406.0,297.8 L406.5,322.1 L407.0,345.0 L407.4,366.1 L407.9,385.3 L408.4,402.1 L408.9,416.3 L409.3,427.8\n",
+       "\t\tL409.8,436.3 L410.3,441.7 L410.8,443.9 L411.2,443.0 L411.7,438.8 L412.2,431.6 L412.6,421.3 L413.1,408.1\n",
+       "\t\tL413.6,392.3 L414.1,374.1 L414.5,353.7 L415.0,331.4 L415.5,307.7 L416.0,282.8 L416.4,257.1 L416.9,231.0\n",
+       "\t\tL417.4,205.0 L417.8,179.3 L418.3,154.4 L418.8,130.7 L419.3,108.4 L419.7,88.0 L420.2,69.8 L420.7,54.0\n",
+       "\t\tL421.2,40.8 L421.6,30.5 L422.1,23.3 L422.6,19.1 L423.0,18.2 L423.5,20.4 L424.0,25.8 L424.5,34.3\n",
+       "\t\tL424.9,45.8 L425.4,60.0 L425.9,76.8 L426.4,96.0 L426.8,117.1 L427.3,140.0 L427.8,164.3 L428.2,189.5\n",
+       "\t\tL428.7,215.4 L429.2,241.5 L429.7,267.5 L430.1,292.9 L430.6,317.3 L431.1,340.5 L431.5,362.1 L432.0,381.6\n",
+       "\t\tL432.5,398.9 L433.0,413.7 L433.4,425.7 L433.9,434.8 L434.4,440.9 L434.9,443.7 L435.3,443.4 L435.8,439.9\n",
+       "\t\tL436.3,433.3 L436.7,423.6 L437.2,411.0 L437.7,395.7 L438.2,377.9 L438.6,357.9 L439.1,336.0 L439.6,312.5\n",
+       "\t\tL440.1,287.8 L440.5,262.3 L441.0,236.3 L441.5,210.2 L441.9,184.4 L442.4,159.3 L442.9,135.3 L443.4,112.7\n",
+       "\t\tL443.8,92.0 L444.3,73.3 L444.8,56.9 L445.3,43.2 L445.7,32.4 L446.2,24.5 L446.7,19.7 L447.1,18.1\n",
+       "\t\tL447.6,19.7 L448.1,24.5 L448.6,32.4 L449.0,43.2 L449.5,56.9 L450.0,73.3 L450.5,92.0 L450.9,112.7\n",
+       "\t\tL451.4,135.3 L451.9,159.3 L452.3,184.4 L452.8,210.2 L453.3,236.3 L453.8,262.3 L454.2,287.8 L454.7,312.5\n",
+       "\t\tL455.2,336.0 L455.7,357.9 L456.1,377.9 L456.6,395.7 L457.1,411.0 L457.5,423.6 L458.0,433.3 L458.5,439.9\n",
+       "\t\tL459.0,443.4 L459.4,443.7 L459.9,440.9 L460.4,434.8 L460.9,425.7 L461.3,413.7 L461.8,398.9 L462.3,381.6\n",
+       "\t\tL462.7,362.1 L463.2,340.5 L463.7,317.3 L464.2,292.9 L464.6,267.5 L465.1,241.5 L465.6,215.4 L466.1,189.5\n",
+       "\t\tL466.5,164.3 L467.0,140.0 L467.5,117.1 L467.9,96.0 L468.4,76.8 L468.9,60.0 L469.4,45.8 L469.8,34.3\n",
+       "\t\tL470.3,25.8 L470.8,20.4 L471.3,18.2 L471.7,19.1 L472.2,23.3 L472.7,30.5 L473.1,40.8 L473.6,54.0\n",
+       "\t\tL474.1,69.8 L474.6,88.0 L475.0,108.4 L475.5,130.7 L476.0,154.4 L476.5,179.3 L476.9,205.0 L477.4,231.0\n",
+       "\t\tL477.9,257.1 L478.3,282.8 L478.8,307.7 L479.3,331.4 L479.8,353.7 L480.2,374.1 L480.7,392.3 L481.2,408.1\n",
+       "\t\tL481.7,421.3 L482.1,431.6 L482.6,438.8 L483.1,443.0 L483.5,443.9 L484.0,441.7 L484.5,436.3 L485.0,427.8\n",
+       "\t\tL485.4,416.3 L485.9,402.1 L486.4,385.3 L486.9,366.1 L487.3,345.0 L487.8,322.1 L488.3,297.8 L488.7,272.6\n",
+       "\t\tL489.2,246.7 L489.7,220.6 L490.2,194.6 L490.6,169.2 L491.1,144.8 L491.6,121.6 L492.0,100.0 L492.5,80.5\n",
+       "\t\tL493.0,63.2 L493.5,48.4 L493.9,36.4 L494.4,27.3 L494.9,21.2 L495.4,18.4 L495.8,18.7 L496.3,22.2\n",
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+       "\n"
+      ]
+     },
+     "metadata": {
+      "": ""
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "sines = Stream.SinOsc(samplerate*20/1024):sub(1, 1024)\n",
+    "sines:gnuplot()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "580e13a3-f910-472c-b887-76929aab1d87",
+   "metadata": {},
+   "source": [
+    "The bins of the corresponding frequency spectrum correspond to frequencies according to the following formula:\n",
+    "\n",
+    "$$\n",
+    "  f_i = {i {f_{srate} \\over N}}\n",
+    "$$\n",
+    "\n",
+    "Plotting the frequency spectrum's magnitudes of `sines` should therefore give us a clear signal around bin 20 (Lua index 21).\n",
+    "\n",
+    "**NOTE:** This does not need windowing since the waves fit perfectly into 1024 sample buffer."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "02aec3d0-4ae9-4bcb-a034-764418b7827f",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "{0, 0, 0, 1.173742634399e-14, 0, 0, 0, 3.9584680075211e-15, 0, 0, 0, 512, 0, 0, 0, 2.2315199565268e-14, 0, 0, 0, 8.6716166150709e-15, 0}\n"
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+       "\t</g>\n",
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+       "</g>\n",
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+       "</g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "\t<path stroke='black'  d='M53.9,18.1 L53.9,444.0 L575.0,444.0 L575.0,18.1 L53.9,18.1 Z  '/></g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "</g>\n",
+       "</svg>\n",
+       "\n"
+      ]
+     },
+     "metadata": {
+      "": ""
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "spectrum = tostream(magnitude(FFT(sines)))\n",
+    "print(spectrum:sub(10, 30))\n",
+    "spectrum:gnuplot()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7ca7e0b3-1854-41b4-813f-016eb148387e",
+   "metadata": {},
+   "source": [
+    "Testing the inverse FFT - looks just like the original wave:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "e7fc0ea8-d510-41aa-815e-f29055f5b8e1",
+   "metadata": {},
+   "outputs": [
+    {
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+       "\t\t<text><tspan font-family=\"Arial\" > 0.025</tspan></text>\n",
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+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "\t<g id=\"gnuplot_plot_1\" ><title>gnuplot_plot_1</title>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "\t<path stroke='rgb(148,   0, 211)'  d='M54.4,205.0 L54.8,179.3 L55.3,154.4 L55.8,130.7 L56.3,108.4 L56.7,88.0 L57.2,69.8 L57.7,54.0\n",
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+       "\t\t '/></g>\n",
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+       "\t<path stroke='black'  d='M53.9,18.1 L53.9,444.0 L575.0,444.0 L575.0,18.1 L53.9,18.1 Z  '/></g>\n",
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+       "\n"
+      ]
+     },
+     "metadata": {
+      "": ""
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "IFFT(FFT(sines)):gnuplot()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6c306c6e-dddc-4d8c-bac8-04f009c406df",
+   "metadata": {},
+   "source": [
+    "With 20.5 sines in the fragment, the plot is much better when applying a windowing function:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "03d8fa36-aad6-4de9-9866-1551c5010d25",
+   "metadata": {},
+   "outputs": [
+    {
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+       "</g>\n",
+       "\t<g id=\"gnuplot_plot_1\" ><title>gnuplot_plot_1</title>\n",
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+      ]
+     },
+     "metadata": {
+      "": ""
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "tostream(magnitude(FFT(Hamming(Stream.SinOsc(samplerate*20.5/1024):sub(1, 1024))))):gnuplot()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c1e86db7-d3a6-4626-a71c-66bea1c04ae2",
+   "metadata": {},
+   "source": [
+    "Testing phase plotting. This is 20 sines but with phase 0.7 ($0.7 \\times 2\\pi$).\n",
+    "\n",
+    "**FIXME:** How exactly to interpret these numbers?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "id": "0d8e9009-255c-4840-b930-408eb3c419ed",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.9390625, 0, 0, 0, 0, 0, 0, 0, 0, 0}\n"
+     ]
+    },
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+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "\t<g id=\"gnuplot_plot_1\" ><title>gnuplot_plot_1</title>\n",
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+       "\t\tL543.3,231.0 L544.3,231.0 L545.3,231.0 L546.2,231.0 L547.2,231.0 L548.2,231.0 L549.2,231.0 L550.2,231.0\n",
+       "\t\tL551.2,231.0 L552.2,231.0 L553.1,231.0 L554.1,231.0 L555.1,231.0 L556.1,231.0 L557.1,231.0 L558.1,231.0\n",
+       "\t\tL559.0,231.0  '/></g>\n",
+       "\t</g>\n",
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+       "</g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"black\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "\t<path stroke='black'  d='M53.9,18.1 L53.9,444.0 L575.0,444.0 L575.0,18.1 L53.9,18.1 Z  '/></g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "</g>\n",
+       "</svg>\n",
+       "\n"
+      ]
+     },
+     "metadata": {
+      "": ""
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "spectrum = tostream(phase(FFT(Stream.SinOsc(samplerate*20/1024, 0.7):sub(1, 1024))))\n",
+    "print(spectrum:sub(10, 30))\n",
+    "spectrum:gnuplot()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "12ed863d-99a4-4372-bb11-e0ce4b5013d3",
+   "metadata": {},
+   "source": [
+    "Let's try some naive real-time pitch shifting.\n",
+    "This however is [not easy to get right](https://www.reddit.com/r/DSP/comments/k6t24c/pitch_shifting_algorithm_in_frequency_domain/)!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "id": "bd0e1d6d-ad05-46dc-9701-92846bc8a683",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "WARNING: Buffer underrun detected\n"
+     ]
+    }
+   ],
+   "source": [
+    "-- FIXME: Use a different sound file!\n",
+    "robin = SndfileStream(\"tracks/robin-mono.wav\"):sub(sec(10.284), sec(17.466)):eval()\n",
+    "robin:play()\n",
+    "robin:FFT(1024):map(function(spectrum)\n",
+    "    assert(#spectrum == 513)\n",
+    "    -- NOTE: We cannot use Stream:resample() as it won't work with complex samples.\n",
+    "    for i = 1, 512/2 do spectrum[i] = spectrum[i*2] end\n",
+    "    for i = 512/2+1, 512 do spectrum[i] = 0i end\n",
+    "    return spectrum\n",
+    "end):IFFT(1024):play()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cdf40dfb-b7ac-42ae-a6c3-178c7c8db1ff",
+   "metadata": {},
+   "source": [
+    "Let's generate and plot the spectrum of a sine mixed with some noise:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "58845a87-ed29-45e8-85c7-ac4b24013e5c",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
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+       "<svg \n",
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+       "</g>\n",
+       "<g fill=\"none\" color=\"gray\" stroke=\"currentColor\" stroke-width=\"0.50\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "\t<path stroke='gray' stroke-dasharray='2,4' class=\"gridline\"  d='M575.0,444.0 L575.0,18.1  '/></g>\n",
+       "<g fill=\"none\" color=\"gray\" stroke=\"gray\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "\t<path stroke='black'  d='M575.0,444.0 L575.0,435.0 M575.0,18.1 L575.0,27.1  '/>\t<g transform=\"translate(575.0,465.9)\" stroke=\"none\" fill=\"black\" font-family=\"Arial\" font-size=\"12.00\"  text-anchor=\"middle\">\n",
+       "\t\t<text><tspan font-family=\"Arial\" > 0.012</tspan></text>\n",
+       "\t</g>\n",
+       "</g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "\t<path stroke='black'  d='M53.9,18.1 L53.9,444.0 L575.0,444.0 L575.0,18.1 L53.9,18.1 Z  '/></g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "\t<g id=\"gnuplot_plot_1\" ><title>gnuplot_plot_1</title>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
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+       "\t\tL497.0,186.7 L498.0,206.5 L499.0,204.9 L500.0,183.5 L501.0,200.1 L501.9,205.3 L502.9,202.8 L503.9,164.7\n",
+       "\t\tL504.9,204.9 L505.9,182.1 L506.9,188.0 L507.8,197.1 L508.8,221.2 L509.8,198.1 L510.8,163.9 L511.8,172.4\n",
+       "\t\tL512.8,209.5 L513.8,172.3 L514.7,191.9 L515.7,201.8 L516.7,197.6 L517.7,203.3 L518.7,187.1 L519.7,206.7\n",
+       "\t\tL520.6,184.6 L521.6,180.6 L522.6,168.8 L523.6,206.3 L524.6,216.1 L525.6,173.1 L526.6,194.6 L527.5,156.2\n",
+       "\t\tL528.5,182.2 L529.5,198.5 L530.5,179.9 L531.5,167.0 L532.5,218.9 L533.4,188.5 L534.4,203.3 L535.4,197.7\n",
+       "\t\tL536.4,199.3 L537.4,188.3 L538.4,161.8 L539.4,175.4 L540.3,194.2 L541.3,156.1 L542.3,150.4 L543.3,224.2\n",
+       "\t\tL544.3,200.1 L545.3,214.8 L546.2,189.5 L547.2,220.7 L548.2,181.8 L549.2,171.8 L550.2,183.3 L551.2,204.1\n",
+       "\t\tL552.2,193.5 L553.1,226.0 L554.1,198.3 L555.1,228.4 L556.1,160.9 L557.1,154.2 L558.1,198.3 L559.0,228.5\n",
+       "\t\t '/></g>\n",
+       "\t</g>\n",
+       "<g fill=\"none\" color=\"white\" stroke=\"rgb(148,   0, 211)\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"2.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"black\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "\t<path stroke='black'  d='M53.9,18.1 L53.9,444.0 L575.0,444.0 L575.0,18.1 L53.9,18.1 Z  '/></g>\n",
+       "<g fill=\"none\" color=\"black\" stroke=\"currentColor\" stroke-width=\"1.00\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\">\n",
+       "</g>\n",
+       "</g>\n",
+       "</svg>\n",
+       "\n"
+      ]
+     },
+     "metadata": {
+      "": ""
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "noisy = Stream.SinOsc(440):mix(NoiseStream, 0.4)\n",
+    "tostream(magnitude(FFT(Hamming(noisy:sub(1, 1024))))):mul(0.05):gnuplot()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2141e582-948c-454e-834b-8148465e3869",
+   "metadata": {},
+   "source": [
+    "Try some very naive noise cancelling.\n",
+    "\n",
+    "**WARNING:** This plays indefinitely. Interrupt via Kernel &rarr; \"Interrupt Kernel\"."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "id": "b43ca02e-16b8-4738-a157-db6dc1bb5b22",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "WARNING: Buffer underrun detected\n"
+     ]
+    },
+    {
+     "ename": "n/a",
+     "evalue": "fft.lua:124: interrupted!",
+     "execution_count": 22,
+     "output_type": "error",
+     "traceback": [
+      "fft.lua:124: interrupted!",
+      "stack traceback:",
+      "\tfft.lua:124: in function 'FFT_internal'",
+      "\tfft.lua:119: in function 'FFT_internal'",
+      "\tfft.lua:119: in function 'FFT_internal'",
+      "\tfft.lua:136: in function 'fnc'",
+      "\tapplause.lua:1871: in function 'tick'",
+      "\tapplause.lua:1870: in function 'tick'",
+      "\tapplause.lua:1870: in function 'stream_tick'",
+      "\tapplause.lua:1634: in function 'tick'",
+      "\tapplause.lua:726: in function <applause.lua:717>",
+      "\t[C]: in function 'xpcall'",
+      "\tapplause.lua:683: in function <applause.lua:663>",
+      "\t[C]: in function 'xpcall'",
+      "\t...ies/ilua/env/lib/python3.8/site-packages/ilua/interp.lua:65: in function 'handle_execute'",
+      "\t...ies/ilua/env/lib/python3.8/site-packages/ilua/interp.lua:176: in main chunk",
+      "stack traceback:",
+      "\t[C]: in function 'error'",
+      "\tapplause.lua:707: in function <applause.lua:663>",
+      "\t[C]: in function 'xpcall'",
+      "\t...ies/ilua/env/lib/python3.8/site-packages/ilua/interp.lua:65: in function 'handle_execute'",
+      "\t...ies/ilua/env/lib/python3.8/site-packages/ilua/interp.lua:176: in main chunk"
+     ]
+    }
+   ],
+   "source": [
+    "noisy:FFT(1024, Hamming):map(function(spectrum)\n",
+    "    assert(#spectrum == 513)\n",
+    "    for i = 1, #spectrum do\n",
+    "        if (spectrum[i].re^2 + spectrum[i].im^2)^.5*0.05 < 0.8 then spectrum[i] = 0i end\n",
+    "    end\n",
+    "    return spectrum\n",
+    "end):IFFT(1024):play()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fc38550e-835c-47a8-a792-7164109dafee",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Lua",
+   "language": "lua",
+   "name": "lua"
+  },
+  "language_info": {
+   "file_extension": ".lua",
+   "mimetype": "text/x-lua",
+   "name": "lua",
+   "version": "n/a"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/examples/fft.lua b/examples/fft.lua
index 85d0bc2..417f3c6 100644
--- a/examples/fft.lua
+++ b/examples/fft.lua
@@ -28,7 +28,7 @@ end):IFFT(1024):play()
 noisy = Stream.SinOsc(440):mix(NoiseStream, 0.4)
 tostream(magnitude(FFT(Hamming(noisy:sub(1, 1024))))):mul(0.05):gnuplot()
 
--- Naive noise canceling
+-- Naive noise cancelling
 noisy:FFT(1024, Hamming):map(function(spectrum)
 	assert(#spectrum == 513)
 	for i = 1, #spectrum do
-- 
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