Digital filters are used to apply a mathematical operation to a sampled signal. One of the common formulations is the "direct form II transposed" which can represent both infinite impulse response (IIR) and finite impulse response (FIR) filters, as well as being more numerically stable than other forms. [1]

Task

Filter a signal using an order 3 low-pass Butterworth filter. The coefficients for the filter are a=[1.00000000, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17] and b = [0.16666667, 0.5, 0.5, 0.16666667]

The signal that needs filtering is the following vector: [-0.917843918645, 0.141984778794, 1.20536903482, 0.190286794412, -0.662370894973, -1.00700480494, -0.404707073677 ,0.800482325044, 0.743500089861, 1.01090520172, 0.741527555207, 0.277841675195, 0.400833448236, -0.2085993586, -0.172842103641, -0.134316096293, 0.0259303398477, 0.490105989562, 0.549391221511, 0.9047198589]

object ButterworthFilter extends App {
  private def filter(a: Vector[Double],
                     b: Vector[Double],
                     signal: Vector[Double]): Vector[Double] = {
 
    @scala.annotation.tailrec
    def outer(i: Int, acc: Vector[Double]): Vector[Double] = {
      if (i >= signal.length) acc
      else {
        @scala.annotation.tailrec
        def inner0(j: Int, tmp: Double): Double = if (j >= b.length) tmp
        else if ((i - j) >= 0) inner0(j + 1, tmp + b(j) * signal(i - j)) else inner0(j + 1, tmp)
 
        @scala.annotation.tailrec
        def inner1(j: Int, tmp: Double): Double = if (j >= a.length) tmp
        else if (i - j >= 0) inner1(j + 1, tmp - a(j) * acc(i - j)) else inner1(j + 1, tmp)
 
        outer(i + 1, acc :+ inner1(1, inner0(0, 0D)) / a(0))
      }
    }
 
    outer(0, Vector())
  }
 
  filter(Vector[Double](1, -2.77555756e-16, 3.33333333e-01, -1.85037171e-17),
    Vector[Double](0.16666667, 0.5, 0.5, 0.16666667),
    Vector[Double](
      -0.917843918645, 0.141984778794, 1.20536903482, 0.190286794412, -0.662370894973,
      -1.00700480494, -0.404707073677, 0.800482325044, 0.743500089861, 1.01090520172,
      0.741527555207, 0.277841675195, 0.400833448236, -0.2085993586, -0.172842103641,
      -0.134316096293, 0.0259303398477, 0.490105989562, 0.549391221511, 0.9047198589)
  ).grouped(5)
    .map(_.map(x => f"$x% .8f"))
    .foreach(line => println(line.mkString(" ")))
 
}