Add Inverse Discrete Fourier Transform.

Author: trekhleb

README.md

@@ -73,7 +73,7 @@ a set of rules that precisely define a sequence of operations.
   * `B` [Radian & Degree](src/algorithms/math/radian) - radians to degree and backwards conversion
   * `A` [Integer Partition](src/algorithms/math/integer-partition)
   * `A` [Liu Hui π Algorithm](src/algorithms/math/liu-hui) - approximate π calculations based on N-gons
-  * `A` [Fourier Transform (DFT, FFT)](src/algorithms/math/fourier-transform) - decompose a function of time (a signal) into the frequencies that make it up 
+  * `A` [Fourier Transform (DFT, IDFT, FFT)](src/algorithms/math/fourier-transform) - decompose a function of time (a signal) into the frequencies that make it up 
 * **Sets**
   * `B` [Cartesian Product](src/algorithms/sets/cartesian-product) - product of multiple sets
   * `B` [Fisher–Yates Shuffle](src/algorithms/sets/fisher-yates) - random permutation of a finite sequence

src/algorithms/math/fourier-transform/__test__/FourierTester.js

@@ -1,6 +1,6 @@
 import ComplexNumber from '../../complex-number/ComplexNumber';
 
-export const fourierDirectTestCases = [
+export const fourierTestCases = [
   {
     input: [
       { amplitude: 1 },
@@ -47,13 +47,13 @@ export const fourierDirectTestCases = [
     ],
     output: [
       {
-        frequency: 0, amplitude: 0.3333, phase: 0, re: 0.3333, im: 0,
+        frequency: 0, amplitude: 0.33333, phase: 0, re: 0.33333, im: 0,
       },
       {
-        frequency: 1, amplitude: 0.3333, phase: 0, re: 0.3333, im: 0,
+        frequency: 1, amplitude: 0.33333, phase: 0, re: 0.33333, im: 0,
       },
       {
-        frequency: 2, amplitude: 0.3333, phase: 0, re: 0.3333, im: 0,
+        frequency: 2, amplitude: 0.33333, phase: 0, re: 0.33333, im: 0,
       },
     ],
   },
@@ -179,13 +179,13 @@ export const fourierDirectTestCases = [
         frequency: 0, amplitude: 1.75, phase: 0, re: 1.75, im: 0,
       },
       {
-        frequency: 1, amplitude: 1.03077, phase: 14.0362, re: 0.99999, im: 0.25,
+        frequency: 1, amplitude: 1.03077, phase: 14.03624, re: 0.99999, im: 0.25,
       },
       {
         frequency: 2, amplitude: 0.25, phase: 0, re: 0.25, im: 0,
       },
       {
-        frequency: 3, amplitude: 1.03077, phase: -14.0362, re: 1, im: -0.25,
+        frequency: 3, amplitude: 1.03077, phase: -14.03624, re: 1, im: -0.25,
       },
     ],
   },
@@ -201,7 +201,7 @@ export const fourierDirectTestCases = [
         frequency: 0, amplitude: 1, phase: 0, re: 1, im: 0,
       },
       {
-        frequency: 1, amplitude: 1.76776, phase: 8.1301, re: 1.75, im: 0.25,
+        frequency: 1, amplitude: 1.76776, phase: 8.13010, re: 1.75, im: 0.25,
       },
       {
         frequency: 2, amplitude: 0.5, phase: 180, re: -0.5, im: 0,
@@ -240,17 +240,15 @@ export default class FourierTester {
    * @param {function} fourierTransform
    */
   static testDirectFourierTransform(fourierTransform) {
-    fourierDirectTestCases.forEach((testCase) => {
+    fourierTestCases.forEach((testCase) => {
       const { input, output: expectedOutput } = testCase;
 
-      // Convert input into complex numbers.
-      const complexInput = input.map(sample => new ComplexNumber({ re: sample.amplitude }));
-
       // Try to split input signal into sequence of pure sinusoids.
-      const currentOutput = fourierTransform(complexInput);
+      const formattedInput = input.map(sample => sample.amplitude);
+      const currentOutput = fourierTransform(formattedInput);
 
       // Check the signal has been split into proper amount of sub-signals.
-      expect(currentOutput.length).toBeGreaterThanOrEqual(complexInput.length);
+      expect(currentOutput.length).toBeGreaterThanOrEqual(formattedInput.length);
 
       // Now go through all the signals and check their frequency, amplitude and phase.
       expectedOutput.forEach((expectedSignal, frequency) => {
@@ -267,4 +265,31 @@ export default class FourierTester {
       });
     });
   }
+
+  /**
+   * @param {function} inverseFourierTransform
+   */
+  static testInverseFourierTransform(inverseFourierTransform) {
+    fourierTestCases.forEach((testCase) => {
+      const { input: expectedOutput, output: inputFrequencies } = testCase;
+
+      // Try to join frequencies into time signal.
+      const formattedInput = inputFrequencies.map((frequency) => {
+        return new ComplexNumber({ re: frequency.re, im: frequency.im });
+      });
+      const currentOutput = inverseFourierTransform(formattedInput);
+
+      // Check the signal has been combined of proper amount of time samples.
+      expect(currentOutput.length).toBeLessThanOrEqual(formattedInput.length);
+
+      // Now go through all the amplitudes and check their values.
+      expectedOutput.forEach((expectedAmplitudes, timer) => {
+        // Get template data we want to test against.
+        const currentAmplitude = currentOutput[timer];
+
+        // Check if current amplitude is close enough to the calculated one.
+        expect(currentAmplitude).toBeCloseTo(expectedAmplitudes.amplitude, 4);
+      });
+    });
+  }
 }

src/algorithms/math/fourier-transform/__test__/inverseDiscreteFourierTransform.test.js

@@ -0,0 +1,8 @@
+import inverseDiscreteFourierTransform from '../inverseDiscreteFourierTransform';
+import FourierTester from './FourierTester';
+
+describe('inverseDiscreteFourierTransform', () => {
+  it('should calculate output signal out of input frequencies', () => {
+    FourierTester.testInverseFourierTransform(inverseDiscreteFourierTransform);
+  });
+});

src/algorithms/math/fourier-transform/discreteFourierTransform.js

@@ -3,12 +3,14 @@ import ComplexNumber from '../complex-number/ComplexNumber';
 const CLOSE_TO_ZERO_THRESHOLD = 1e-10;
 
 /**
- * Discrete Fourier Transform.
+ * Discrete Fourier Transform (DFT): time to frequencies.
  *
  * Time complexity: O(N^2)
  *
- * @param {ComplexNumber[]} complexInputAmplitudes - Input signal amplitudes over time (complex
+ * @param {number[]} inputAmplitudes - Input signal amplitudes over time (complex
  * numbers with real parts only).
+ * @param {number} zeroThreshold - Threshold that is used to convert real and imaginary numbers
+ * to zero in case if they are smaller then this.
  *
  * @return {ComplexNumber[]} - Array of complex number. Each of the number represents the frequency
  * or signal. All signals together will form input signal over discrete time periods. Each signal's
@@ -17,10 +19,7 @@ const CLOSE_TO_ZERO_THRESHOLD = 1e-10;
  * @see https://gist.github.com/anonymous/129d477ddb1c8025c9ac
  * @see https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/
  */
-export default function dft(complexInputAmplitudes) {
-  // Convert complex amplitudes into real ones.
-  const inputAmplitudes = complexInputAmplitudes.map(complexAmplitude => complexAmplitude.re);
-
+export default function dft(inputAmplitudes, zeroThreshold = CLOSE_TO_ZERO_THRESHOLD) {
   const N = inputAmplitudes.length;
   const signals = [];
 
@@ -48,11 +47,11 @@ export default function dft(complexInputAmplitudes) {
     }
 
     // Close to zero? You're zero.
-    if (Math.abs(frequencySignal.re) < CLOSE_TO_ZERO_THRESHOLD) {
+    if (Math.abs(frequencySignal.re) < zeroThreshold) {
       frequencySignal.re = 0;
     }
 
-    if (Math.abs(frequencySignal.im) < CLOSE_TO_ZERO_THRESHOLD) {
+    if (Math.abs(frequencySignal.im) < zeroThreshold) {
       frequencySignal.im = 0;
     }
 

src/algorithms/math/fourier-transform/inverseDiscreteFourierTransform.js

@@ -0,0 +1,58 @@
+import ComplexNumber from '../complex-number/ComplexNumber';
+
+const CLOSE_TO_ZERO_THRESHOLD = 1e-10;
+
+/**
+ * Inverse Discrete Fourier Transform (IDFT): frequencies to time.
+ *
+ * Time complexity: O(N^2)
+ *
+ * @param {ComplexNumber[]} frequencies - Frequencies summands of the final signal.
+ * @param {number} zeroThreshold - Threshold that is used to convert real and imaginary numbers
+ * to zero in case if they are smaller then this.
+ *
+ * @return {number[]} - Discrete amplitudes distributed in time.
+ */
+export default function inverseDiscreteFourierTransform(
+  frequencies,
+  zeroThreshold = CLOSE_TO_ZERO_THRESHOLD,
+) {
+  const N = frequencies.length;
+  const amplitudes = [];
+
+  // Go through every discrete point of time.
+  for (let timer = 0; timer < N; timer += 1) {
+    // Compound amplitude at current time.
+    let amplitude = new ComplexNumber();
+
+    // Go through all discrete frequencies.
+    for (let frequency = 0; frequency < N; frequency += 1) {
+      const currentFrequency = frequencies[frequency];
+
+      // Calculate rotation angle.
+      const rotationAngle = (2 * Math.PI) * frequency * (timer / N);
+
+      // Remember that e^ix = cos(x) + i * sin(x);
+      const frequencyContribution = new ComplexNumber({
+        re: Math.cos(rotationAngle),
+        im: Math.sin(rotationAngle),
+      }).multiply(currentFrequency);
+
+      amplitude = amplitude.add(frequencyContribution);
+    }
+
+    // Close to zero? You're zero.
+    if (Math.abs(amplitude.re) < zeroThreshold) {
+      amplitude.re = 0;
+    }
+
+    if (Math.abs(amplitude.im) < zeroThreshold) {
+      amplitude.im = 0;
+    }
+
+    // Add current frequency signal to the list of compound signals.
+    amplitudes[timer] = amplitude.re;
+  }
+
+  return amplitudes;
+}