Cooley-Tukey in Scala

contents/cooley_tukey/code/scala/fft.scala

@@ -0,0 +1,133 @@
+
+case class Complex(val real: Double, val imag: Double) {
+  def +(other: Complex): Complex = new Complex(this.real + other.real, this.imag + other.imag)
+  def -(other: Complex): Complex = new Complex(this.real - other.real, this.imag - other.imag)
+
+  def *(other: Complex): Complex = new Complex(
+    this.real * other.real - this.imag * other.imag,
+    this.real * other.imag + this.imag * other.real)
+
+  def *(other: Double): Complex = new Complex(this.real * other, this.imag * other)
+}
+
+object Complex {
+  def fromPolar(mag: Double, phase: Double): Complex = 
+    new Complex(mag * Math.cos(phase), mag * Math.sin(phase))
+}
+
+object FT {
+
+  /** Calculates a single DFT coefficient. */
+  private def coefficient(n: Int, k: Int, ftLength: Int): Complex = 
+    Complex.fromPolar(1.0, -2.0 * Math.PI * k * n / ftLength)
+
+  /** Calculates one value of the DFT of signal */
+  private def dftValue(signal: IndexedSeq[Double], k: Int): Complex = {
+    // Multiply the signal with the coefficients vector
+    val terms = for (i <- signal.indices) yield coefficient(i, k, signal.length) * signal(i)
+
+    terms reduce { _ + _ }
+  }
+
+  def dft(signal: IndexedSeq[Double]): IndexedSeq[Complex] = 
+    signal.indices map { dftValue(signal, _) }
+
+  /** Combines the transforms of the even and odd indices */
+  private def mergeTransforms(evens: IndexedSeq[Complex], odds: IndexedSeq[Complex]): IndexedSeq[Complex] = {
+    val oddTerms = for (i <- odds.indices) 
+      yield coefficient(1, i, 2 * odds.length) * odds(i)
+
+    val pairs = evens.zip(oddTerms)
+
+    (pairs map { case (e, o) => e + o }) ++ (pairs map { case (e, o) => e - o })
+  }
+
+  def cooleyTukey(signal: IndexedSeq[Double]): IndexedSeq[Complex] = signal.length match {
+    case 2 => mergeTransforms(
+      Vector(new Complex(signal(0), 0)),
+      Vector(new Complex(signal(1), 0)))
+    case _ => {
+      // Split signal into even and odd indices and call cooleyTukey recursively on each.
+      val evens = cooleyTukey(for (i <- 0 until signal.length by 2) yield signal(i))
+      val odds = cooleyTukey(for (i <- 1 until signal.length by 2) yield signal(i))
+
+      mergeTransforms(evens, odds)
+    }
+  }
+
+  /** Reverses the bits in value. */
+  private def reverseBits(value: Int, length: Int): Int = length match {
+    case 1 => value
+    case _ => {
+      // Split bits in the middle.
+      val lowerHalf = length / 2
+
+      // The upper half will be longer if the number of bits is odd.
+      val upperHalf = length - lowerHalf
+      val mask = (1 << lowerHalf) - 1
+
+      // Reverse each half recursively and then swap them.
+      (reverseBits(value & mask, lowerHalf) << upperHalf) + 
+        reverseBits(value >> lowerHalf, upperHalf)
+    }
+  }
+
+
+  private def log2(x: Double): Double = Math.log(x) / Math.log(2.0)
+
+  def bitReverseIndices(signal: IndexedSeq[Double]): IndexedSeq[Double] = {
+    // Find the maximum number of bits needed.
+    val bitLength = log2(signal.length).ceil.toInt
+
+    for (i <- signal.indices)
+      yield signal(reverseBits(i, bitLength))
+  }
+
+  private def butterfly(x1: Complex, x2: Complex, coeff: Complex): (Complex, Complex) = 
+    (x1 + coeff * x2, x1 - coeff * x2)
+
+  @scala.annotation.tailrec
+  private def iterativeCooleyTukeyLoop(signal: IndexedSeq[Complex], dist: Int): IndexedSeq[Complex] = {
+    // Distance between subsequent groups of butterflies
+    val stride = 2 * dist
+    val result = new Array[Complex](signal.length)
+
+    for (groupStart <- 0 until signal.length by stride; i <- 0 until dist) {
+      val index = groupStart + i
+      val (r1, r2) = butterfly(
+        signal(index),
+        signal(index + dist),
+        coefficient(1, i, stride))
+
+      result(index) = r1
+      result(index + dist) = r2
+    }
+
+    if (stride >= signal.length)
+      result.toVector
+    else
+      iterativeCooleyTukeyLoop(result.toVector, dist * 2)
+  }
+
+  def iterativeCooleyTukey(signal: IndexedSeq[Double]): IndexedSeq[Complex] =
+    iterativeCooleyTukeyLoop(bitReverseIndices(signal) map { new Complex(_, 0.0) }, 1)
+}
+
+object Main {
+
+  private def approxEqual(a: IndexedSeq[Complex], b: IndexedSeq[Complex]): Boolean = {
+    val diffs = a.zip(b) map { case (x, y) => Math.abs(x.real - y.real + x.imag - y.imag) }
+    diffs map { _ < 1e-12 } reduce { _ && _ }
+  }
+
+  def main(args: Array[String]): Unit = {
+    val signal = for (i <- 0 until 16) yield Math.random() * 2 - 1
+    val x = FT.dft(signal)
+    val y = FT.cooleyTukey(signal)
+    val z = FT.iterativeCooleyTukey(signal)
+
+    println("DFT and Cooley-Tukey approx. equal: " + approxEqual(x, y))
+    println("DFT and iterative Cooley-Tukey approx. equal: " + approxEqual(x, z))
+    println("Cooley-Tukey and iterative Cooley-Tukey approx. equal: " + approxEqual(y, z))
+  }
+}

contents/cooley_tukey/cooley_tukey.md

@@ -89,6 +89,8 @@ For some reason, though, putting code to this transformation really helped me fi
 [import:3-15, lang:"javascript"](code/javascript/fft.js)
 {% sample lang="rs" %}
 [import:24-37, lang:"rust"](code/rust/fft.rs)
+{% sample lang="scala" %}
+[import:20-33, lang:"scala"](code/scala/fft.scala)
 {% endmethod %}
 
 In this function, we define `n` to be a set of integers from $$0 \rightarrow N-1$$ and arrange them to be a column.
@@ -142,6 +144,8 @@ In the end, the code looks like:
 [import:17-39, lang="javascript"](code/javascript/fft.js)
 {% sample lang="rs" %}
 [import:39-55, lang:"rust"](code/rust/fft.rs)
+{% sample lang="scala" %}
+[import:35-56, lang:"scala"](code/scala/fft.scala)
 {% endmethod %}
 
 As a side note, we are enforcing that the array must be a power of 2 for the operation to work.
@@ -255,6 +259,8 @@ Some rather impressive scratch code was submitted by Jie and can be found here:
 [import, lang:"javascript"](code/javascript/fft.js)
 {% sample lang="rs" %}
 [import, lang:"rust"](code/rust/fft.rs)
+{% sample lang="scala" %}
+[import, lang:"scala"](code/scala/fft.scala)
 {% endmethod %}
 
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