third_party/blink/renderer/platform/audio/biquad.cc

/*
 * Copyright (C) 2010 Google Inc. All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1.  Redistributions of source code must retain the above copyright
 *     notice, this list of conditions and the following disclaimer.
 * 2.  Redistributions in binary form must reproduce the above copyright
 *     notice, this list of conditions and the following disclaimer in the
 *     documentation and/or other materials provided with the distribution.
 * 3.  Neither the name of Apple Computer, Inc. ("Apple") nor the names of
 *     its contributors may be used to endorse or promote products derived
 *     from this software without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY APPLE AND ITS CONTRIBUTORS "AS IS" AND ANY
 * EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
 * WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
 * DISCLAIMED. IN NO EVENT SHALL APPLE OR ITS CONTRIBUTORS BE LIABLE FOR ANY
 * DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
 * (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
 * ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */

#include "third_party/blink/renderer/platform/audio/biquad.h"

#include "build/build_config.h"
#include "third_party/blink/renderer/platform/audio/audio_utilities.h"
#include "third_party/blink/renderer/platform/audio/denormal_disabler.h"
#include "third_party/blink/renderer/platform/wtf/math_extras.h"

#include <algorithm>
#include <complex>
#include <stdio.h>
#if defined(OS_MACOSX)
#include <Accelerate/Accelerate.h>
#endif

namespace blink {

#if defined(OS_MACOSX)
const int kBiquadBufferSize = 1024;
#endif

// Compute 10^x = exp(x*log(10))
static double pow10(double x) {
  return expf(x * 2.30258509299404568402);
}

Biquad::Biquad() : has_sample_accurate_values_(false) {
#if defined(OS_MACOSX)
  // Allocate two samples more for filter history
  input_buffer_.Allocate(kBiquadBufferSize + 2);
  output_buffer_.Allocate(kBiquadBufferSize + 2);
#endif

  // Allocate enough space for the a-rate filter coefficients to handle a
  // rendering quantum of 128 frames.
  b0_.Allocate(audio_utilities::kRenderQuantumFrames);
  b1_.Allocate(audio_utilities::kRenderQuantumFrames);
  b2_.Allocate(audio_utilities::kRenderQuantumFrames);
  a1_.Allocate(audio_utilities::kRenderQuantumFrames);
  a2_.Allocate(audio_utilities::kRenderQuantumFrames);

  // Initialize as pass-thru (straight-wire, no filter effect)
  SetNormalizedCoefficients(0, 1, 0, 0, 1, 0, 0);

  Reset();  // clear filter memory
}

Biquad::~Biquad() = default;

void Biquad::Process(const float* source_p,
                     float* dest_p,
                     size_t frames_to_process) {
  // WARNING: sourceP and destP may be pointing to the same area of memory!
  // Be sure to read from sourceP before writing to destP!
  if (HasSampleAccurateValues()) {
    int n = frames_to_process;

    // Create local copies of member variables
    double x1 = x1_;
    double x2 = x2_;
    double y1 = y1_;
    double y2 = y2_;

    const double* b0 = b0_.Data();
    const double* b1 = b1_.Data();
    const double* b2 = b2_.Data();
    const double* a1 = a1_.Data();
    const double* a2 = a2_.Data();

    for (int k = 0; k < n; ++k) {
      // FIXME: this can be optimized by pipelining the multiply adds...
      float x = *source_p++;
      float y = b0[k] * x + b1[k] * x1 + b2[k] * x2 - a1[k] * y1 - a2[k] * y2;

      *dest_p++ = y;

      // Update state variables
      x2 = x1;
      x1 = x;
      y2 = y1;
      y1 = y;
    }

    // Local variables back to member. Flush denormals here so we
    // don't slow down the inner loop above.
    x1_ = DenormalDisabler::FlushDenormalFloatToZero(x1);
    x2_ = DenormalDisabler::FlushDenormalFloatToZero(x2);
    y1_ = DenormalDisabler::FlushDenormalFloatToZero(y1);
    y2_ = DenormalDisabler::FlushDenormalFloatToZero(y2);

    // There is an assumption here that once we have sample accurate values we
    // can never go back to not having sample accurate values.  This is
    // currently true in the way AudioParamTimline is implemented: once an
    // event is inserted, sample accurate processing is always enabled.
    //
    // If so, then we never have to update the state variables for the MACOSX
    // path.  The structure of the state variable in these cases aren't well
    // documented so it's not clear how to update them anyway.
  } else {
#if defined(OS_MACOSX)
    double* input_p = input_buffer_.Data();
    double* output_p = output_buffer_.Data();

    // Set up filter state.  This is needed in case we're switching from
    // filtering with variable coefficients (i.e., with automations) to
    // fixed coefficients (without automations).
    input_p[0] = x2_;
    input_p[1] = x1_;
    output_p[0] = y2_;
    output_p[1] = y1_;

    // Use vecLib if available
    ProcessFast(source_p, dest_p, frames_to_process);

    // Copy the last inputs and outputs to the filter memory variables.
    // This is needed because the next rendering quantum might be an
    // automation which needs the history to continue correctly.  Because
    // sourceP and destP can be the same block of memory, we can't read from
    // sourceP to get the last inputs.  Fortunately, processFast has put the
    // last inputs in input[0] and input[1].
    x1_ = input_p[1];
    x2_ = input_p[0];
    y1_ = dest_p[frames_to_process - 1];
    y2_ = dest_p[frames_to_process - 2];

#else
    int n = frames_to_process;

    // Create local copies of member variables
    double x1 = x1_;
    double x2 = x2_;
    double y1 = y1_;
    double y2 = y2_;

    double b0 = b0_[0];
    double b1 = b1_[0];
    double b2 = b2_[0];
    double a1 = a1_[0];
    double a2 = a2_[0];

    while (n--) {
      // FIXME: this can be optimized by pipelining the multiply adds...
      float x = *source_p++;
      float y = b0 * x + b1 * x1 + b2 * x2 - a1 * y1 - a2 * y2;

      *dest_p++ = y;

      // Update state variables
      x2 = x1;
      x1 = x;
      y2 = y1;
      y1 = y;
    }

    // Local variables back to member. Flush denormals here so we
    // don't slow down the inner loop above.
    x1_ = DenormalDisabler::FlushDenormalFloatToZero(x1);
    x2_ = DenormalDisabler::FlushDenormalFloatToZero(x2);
    y1_ = DenormalDisabler::FlushDenormalFloatToZero(y1);
    y2_ = DenormalDisabler::FlushDenormalFloatToZero(y2);
#endif
  }
}

#if defined(OS_MACOSX)

// Here we have optimized version using Accelerate.framework

void Biquad::ProcessFast(const float* source_p,
                         float* dest_p,
                         size_t frames_to_process) {
  double filter_coefficients[5];
  filter_coefficients[0] = b0_[0];
  filter_coefficients[1] = b1_[0];
  filter_coefficients[2] = b2_[0];
  filter_coefficients[3] = a1_[0];
  filter_coefficients[4] = a2_[0];

  double* input_p = input_buffer_.Data();
  double* output_p = output_buffer_.Data();

  double* input2p = input_p + 2;
  double* output2p = output_p + 2;

  // Break up processing into smaller slices (kBiquadBufferSize) if necessary.

  int n = frames_to_process;

  while (n > 0) {
    int frames_this_time = n < kBiquadBufferSize ? n : kBiquadBufferSize;

    // Copy input to input buffer
    for (int i = 0; i < frames_this_time; ++i)
      input2p[i] = *source_p++;

    ProcessSliceFast(input_p, output_p, filter_coefficients, frames_this_time);

    // Copy output buffer to output (converts float -> double).
    for (int i = 0; i < frames_this_time; ++i)
      *dest_p++ = static_cast<float>(output2p[i]);

    n -= frames_this_time;
  }
}

void Biquad::ProcessSliceFast(double* source_p,
                              double* dest_p,
                              double* coefficients_p,
                              size_t frames_to_process) {
  // Use double-precision for filter stability
  vDSP_deq22D(source_p, 1, coefficients_p, dest_p, 1, frames_to_process);

  // Save history.  Note that sourceP and destP reference m_inputBuffer and
  // m_outputBuffer respectively.  These buffers are allocated (in the
  // constructor) with space for two extra samples so it's OK to access array
  // values two beyond framesToProcess.
  source_p[0] = source_p[frames_to_process - 2 + 2];
  source_p[1] = source_p[frames_to_process - 1 + 2];
  dest_p[0] = dest_p[frames_to_process - 2 + 2];
  dest_p[1] = dest_p[frames_to_process - 1 + 2];
}

#endif  // defined(OS_MACOSX)

void Biquad::Reset() {
#if defined(OS_MACOSX)
  // Two extra samples for filter history
  double* input_p = input_buffer_.Data();
  input_p[0] = 0;
  input_p[1] = 0;

  double* output_p = output_buffer_.Data();
  output_p[0] = 0;
  output_p[1] = 0;

#endif
  x1_ = x2_ = y1_ = y2_ = 0;
}

void Biquad::SetLowpassParams(int index, double cutoff, double resonance) {
  // Limit cutoff to 0 to 1.
  cutoff = clampTo(cutoff, 0.0, 1.0);

  if (cutoff == 1) {
    // When cutoff is 1, the z-transform is 1.
    SetNormalizedCoefficients(index, 1, 0, 0, 1, 0, 0);
  } else if (cutoff > 0) {
    // Compute biquad coefficients for lowpass filter

    resonance = pow10(resonance / 20);

    double theta = kPiDouble * cutoff;
    double alpha = sin(theta) / (2 * resonance);
    double cosw = cos(theta);
    double beta = (1 - cosw) / 2;

    double b0 = beta;
    double b1 = 2 * beta;
    double b2 = beta;

    double a0 = 1 + alpha;
    double a1 = -2 * cosw;
    double a2 = 1 - alpha;

    SetNormalizedCoefficients(index, b0, b1, b2, a0, a1, a2);
  } else {
    // When cutoff is zero, nothing gets through the filter, so set
    // coefficients up correctly.
    SetNormalizedCoefficients(index, 0, 0, 0, 1, 0, 0);
  }
}

void Biquad::SetHighpassParams(int index, double cutoff, double resonance) {
  // Limit cutoff to 0 to 1.
  cutoff = clampTo(cutoff, 0.0, 1.0);

  if (cutoff == 1) {
    // The z-transform is 0.
    SetNormalizedCoefficients(index, 0, 0, 0, 1, 0, 0);
  } else if (cutoff > 0) {
    // Compute biquad coefficients for highpass filter

    resonance = pow10(resonance / 20);
    double theta = kPiDouble * cutoff;
    double alpha = sin(theta) / (2 * resonance);
    double cosw = cos(theta);
    double beta = (1 + cosw) / 2;

    double b0 = beta;
    double b1 = -2 * beta;
    double b2 = beta;

    double a0 = 1 + alpha;
    double a1 = -2 * cosw;
    double a2 = 1 - alpha;

    SetNormalizedCoefficients(index, b0, b1, b2, a0, a1, a2);
  } else {
    // When cutoff is zero, we need to be careful because the above
    // gives a quadratic divided by the same quadratic, with poles
    // and zeros on the unit circle in the same place. When cutoff
    // is zero, the z-transform is 1.
    SetNormalizedCoefficients(index, 1, 0, 0, 1, 0, 0);
  }
}

void Biquad::SetNormalizedCoefficients(int index,
                                       double b0,
                                       double b1,
                                       double b2,
                                       double a0,
                                       double a1,
                                       double a2) {
  double a0_inverse = 1 / a0;

  b0_[index] = b0 * a0_inverse;
  b1_[index] = b1 * a0_inverse;
  b2_[index] = b2 * a0_inverse;
  a1_[index] = a1 * a0_inverse;
  a2_[index] = a2 * a0_inverse;
}

void Biquad::SetLowShelfParams(int index, double frequency, double db_gain) {
  // Clip frequencies to between 0 and 1, inclusive.
  frequency = clampTo(frequency, 0.0, 1.0);

  double a = pow10(db_gain / 40);

  if (frequency == 1) {
    // The z-transform is a constant gain.
    SetNormalizedCoefficients(index, a * a, 0, 0, 1, 0, 0);
  } else if (frequency > 0) {
    double w0 = kPiDouble * frequency;
    double s = 1;  // filter slope (1 is max value)
    double alpha = 0.5 * sin(w0) * sqrt((a + 1 / a) * (1 / s - 1) + 2);
    double k = cos(w0);
    double k2 = 2 * sqrt(a) * alpha;
    double a_plus_one = a + 1;
    double a_minus_one = a - 1;

    double b0 = a * (a_plus_one - a_minus_one * k + k2);
    double b1 = 2 * a * (a_minus_one - a_plus_one * k);
    double b2 = a * (a_plus_one - a_minus_one * k - k2);
    double a0 = a_plus_one + a_minus_one * k + k2;
    double a1 = -2 * (a_minus_one + a_plus_one * k);
    double a2 = a_plus_one + a_minus_one * k - k2;

    SetNormalizedCoefficients(index, b0, b1, b2, a0, a1, a2);
  } else {
    // When frequency is 0, the z-transform is 1.
    SetNormalizedCoefficients(index, 1, 0, 0, 1, 0, 0);
  }
}

void Biquad::SetHighShelfParams(int index, double frequency, double db_gain) {
  // Clip frequencies to between 0 and 1, inclusive.
  frequency = clampTo(frequency, 0.0, 1.0);

  double a = pow10(db_gain / 40);

  if (frequency == 1) {
    // The z-transform is 1.
    SetNormalizedCoefficients(index, 1, 0, 0, 1, 0, 0);
  } else if (frequency > 0) {
    double w0 = kPiDouble * frequency;
    double s = 1;  // filter slope (1 is max value)
    double alpha = 0.5 * sin(w0) * sqrt((a + 1 / a) * (1 / s - 1) + 2);
    double k = cos(w0);
    double k2 = 2 * sqrt(a) * alpha;
    double a_plus_one = a + 1;
    double a_minus_one = a - 1;

    double b0 = a * (a_plus_one + a_minus_one * k + k2);
    double b1 = -2 * a * (a_minus_one + a_plus_one * k);
    double b2 = a * (a_plus_one + a_minus_one * k - k2);
    double a0 = a_plus_one - a_minus_one * k + k2;
    double a1 = 2 * (a_minus_one - a_plus_one * k);
    double a2 = a_plus_one - a_minus_one * k - k2;

    SetNormalizedCoefficients(index, b0, b1, b2, a0, a1, a2);
  } else {
    // When frequency = 0, the filter is just a gain, A^2.
    SetNormalizedCoefficients(index, a * a, 0, 0, 1, 0, 0);
  }
}

void Biquad::SetPeakingParams(int index,
                              double frequency,
                              double q,
                              double db_gain) {
  // Clip frequencies to between 0 and 1, inclusive.
  frequency = clampTo(frequency, 0.0, 1.0);

  // Don't let Q go negative, which causes an unstable filter.
  q = std::max(0.0, q);

  double a = pow10(db_gain / 40);

  if (frequency > 0 && frequency < 1) {
    if (q > 0) {
      double w0 = kPiDouble * frequency;
      double alpha = sin(w0) / (2 * q);
      double k = cos(w0);

      double b0 = 1 + alpha * a;
      double b1 = -2 * k;
      double b2 = 1 - alpha * a;
      double a0 = 1 + alpha / a;
      double a1 = -2 * k;
      double a2 = 1 - alpha / a;

      SetNormalizedCoefficients(index, b0, b1, b2, a0, a1, a2);
    } else {
      // When Q = 0, the above formulas have problems. If we look at
      // the z-transform, we can see that the limit as Q->0 is A^2, so
      // set the filter that way.
      SetNormalizedCoefficients(index, a * a, 0, 0, 1, 0, 0);
    }
  } else {
    // When frequency is 0 or 1, the z-transform is 1.
    SetNormalizedCoefficients(index, 1, 0, 0, 1, 0, 0);
  }
}

void Biquad::SetAllpassParams(int index, double frequency, double q) {
  // Clip frequencies to between 0 and 1, inclusive.
  frequency = clampTo(frequency, 0.0, 1.0);

  // Don't let Q go negative, which causes an unstable filter.
  q = std::max(0.0, q);

  if (frequency > 0 && frequency < 1) {
    if (q > 0) {
      double w0 = kPiDouble * frequency;
      double alpha = sin(w0) / (2 * q);
      double k = cos(w0);

      double b0 = 1 - alpha;
      double b1 = -2 * k;
      double b2 = 1 + alpha;
      double a0 = 1 + alpha;
      double a1 = -2 * k;
      double a2 = 1 - alpha;

      SetNormalizedCoefficients(index, b0, b1, b2, a0, a1, a2);
    } else {
      // When Q = 0, the above formulas have problems. If we look at
      // the z-transform, we can see that the limit as Q->0 is -1, so
      // set the filter that way.
      SetNormalizedCoefficients(index, -1, 0, 0, 1, 0, 0);
    }
  } else {
    // When frequency is 0 or 1, the z-transform is 1.
    SetNormalizedCoefficients(index, 1, 0, 0, 1, 0, 0);
  }
}

void Biquad::SetNotchParams(int index, double frequency, double q) {
  // Clip frequencies to between 0 and 1, inclusive.
  frequency = clampTo(frequency, 0.0, 1.0);

  // Don't let Q go negative, which causes an unstable filter.
  q = std::max(0.0, q);

  if (frequency > 0 && frequency < 1) {
    if (q > 0) {
      double w0 = kPiDouble * frequency;
      double alpha = sin(w0) / (2 * q);
      double k = cos(w0);

      double b0 = 1;
      double b1 = -2 * k;
      double b2 = 1;
      double a0 = 1 + alpha;
      double a1 = -2 * k;
      double a2 = 1 - alpha;

      SetNormalizedCoefficients(index, b0, b1, b2, a0, a1, a2);
    } else {
      // When Q = 0, the above formulas have problems. If we look at
      // the z-transform, we can see that the limit as Q->0 is 0, so
      // set the filter that way.
      SetNormalizedCoefficients(index, 0, 0, 0, 1, 0, 0);
    }
  } else {
    // When frequency is 0 or 1, the z-transform is 1.
    SetNormalizedCoefficients(index, 1, 0, 0, 1, 0, 0);
  }
}

void Biquad::SetBandpassParams(int index, double frequency, double q) {
  // No negative frequencies allowed.
  frequency = std::max(0.0, frequency);

  // Don't let Q go negative, which causes an unstable filter.
  q = std::max(0.0, q);

  if (frequency > 0 && frequency < 1) {
    double w0 = kPiDouble * frequency;
    if (q > 0) {
      double alpha = sin(w0) / (2 * q);
      double k = cos(w0);

      double b0 = alpha;
      double b1 = 0;
      double b2 = -alpha;
      double a0 = 1 + alpha;
      double a1 = -2 * k;
      double a2 = 1 - alpha;

      SetNormalizedCoefficients(index, b0, b1, b2, a0, a1, a2);
    } else {
      // When Q = 0, the above formulas have problems. If we look at
      // the z-transform, we can see that the limit as Q->0 is 1, so
      // set the filter that way.
      SetNormalizedCoefficients(index, 1, 0, 0, 1, 0, 0);
    }
  } else {
    // When the cutoff is zero, the z-transform approaches 0, if Q
    // > 0. When both Q and cutoff are zero, the z-transform is
    // pretty much undefined. What should we do in this case?
    // For now, just make the filter 0. When the cutoff is 1, the
    // z-transform also approaches 0.
    SetNormalizedCoefficients(index, 0, 0, 0, 1, 0, 0);
  }
}

void Biquad::GetFrequencyResponse(int n_frequencies,
                                  const float* frequency,
                                  float* mag_response,
                                  float* phase_response) {
  // Evaluate the Z-transform of the filter at given normalized
  // frequency from 0 to 1.  (1 corresponds to the Nyquist
  // frequency.)
  //
  // The z-transform of the filter is
  //
  // H(z) = (b0 + b1*z^(-1) + b2*z^(-2))/(1 + a1*z^(-1) + a2*z^(-2))
  //
  // Evaluate as
  //
  // b0 + (b1 + b2*z1)*z1
  // --------------------
  // 1 + (a1 + a2*z1)*z1
  //
  // with z1 = 1/z and z = exp(j*pi*frequency). Hence z1 = exp(-j*pi*frequency)

  // Make local copies of the coefficients as a micro-optimization.
  double b0 = b0_[0];
  double b1 = b1_[0];
  double b2 = b2_[0];
  double a1 = a1_[0];
  double a2 = a2_[0];

  for (int k = 0; k < n_frequencies; ++k) {
    if (frequency[k] < 0 || frequency[k] > 1) {
      // Out-of-bounds frequencies should return NaN.
      mag_response[k] = std::nanf("");
      phase_response[k] = std::nanf("");
    } else {
      double omega = -kPiDouble * frequency[k];
      std::complex<double> z = std::complex<double>(cos(omega), sin(omega));
      std::complex<double> numerator = b0 + (b1 + b2 * z) * z;
      std::complex<double> denominator =
          std::complex<double>(1, 0) + (a1 + a2 * z) * z;
      std::complex<double> response = numerator / denominator;
      mag_response[k] = static_cast<float>(abs(response));
      phase_response[k] =
          static_cast<float>(atan2(imag(response), real(response)));
    }
  }
}

static double RepeatedRootResponse(double n,
                                   double c1,
                                   double c2,
                                   double r,
                                   double log_eps) {
  // The response is h(n) = r^(n-2)*[c1*(n+1)*r^2+c2]. We're looking
  // for n such that |h(n)| = eps.  Equivalently, we want a root
  // of the equation log(|h(n)|) - log(eps) = 0 or
  //
  //   (n-2)*log(r) + log(|c1*(n+1)*r^2+c2|) - log(eps)
  //
  // This helps with finding a nuemrical solution because this
  // approximately linearizes the response for large n.

  return (n - 2) * log(r) + log(fabs(c1 * (n + 1) * r * r + c2)) - log_eps;
}

// Regula Falsi root finder, Illinois variant
// (https://en.wikipedia.org/wiki/False_position_method#The_Illinois_algorithm).
//
// This finds a root of the repeated root response where the root is
// assumed to lie between |low| and |high|.  The response is given by
// |c1|, |c2|, and |r| as determined by |RepeatedRootResponse|.
// |log_eps| is the log the the maximum allowed amplitude in the
// response.
static double RootFinder(double low,
                         double high,
                         double log_eps,
                         double c1,
                         double c2,
                         double r) {
  // Desired accuray of the root (in frames).  This doesn't need to be
  // super-accurate, so half frame is good enough, and should be less
  // than 1 because the algorithm may prematurely terminate.
  const double kAccuracyThreshold = 0.5;
  // Max number of iterations to do.  If we haven't converged by now,
  // just return whatever we've found.
  const int kMaxIterations = 10;

  int side = 0;
  double root = 0;
  double f_low = RepeatedRootResponse(low, c1, c2, r, log_eps);
  double f_high = RepeatedRootResponse(high, c1, c2, r, log_eps);

  // The function values must be finite and have opposite signs!
  DCHECK(std::isfinite(f_low));
  DCHECK(std::isfinite(f_high));
  DCHECK_LE(f_low * f_high, 0);

  int iteration;
  for (iteration = 0; iteration < kMaxIterations; ++iteration) {
    root = (f_low * high - f_high * low) / (f_low - f_high);
    if (fabs(high - low) < kAccuracyThreshold * fabs(high + low))
      break;
    double fr = RepeatedRootResponse(root, c1, c2, r, log_eps);

    DCHECK(std::isfinite(fr));

    if (fr * f_high > 0) {
      // fr and f_high have same sign.  Copy root to f_high
      high = root;
      f_high = fr;
      side = -1;
    } else if (f_low * fr > 0) {
      // fr and f_low have same sign. Copy root to f_low
      low = root;
      f_low = fr;
      if (side == 1)
        f_high /= 2;
      side = 1;
    } else {
      // f_low * fr looks like zero, so assume we've converged.
      break;
    }
  }

  // Want to know if the max number of iterations is ever exceeded so
  // we can understand why that happened.
  DCHECK_LT(iteration, kMaxIterations);

  return root;
}

double Biquad::TailFrame(int coef_index, double max_frame) {
  // The Biquad filter is given by
  //
  //   H(z) = (b0 + b1/z + b2/z^2)/(1 + a1/z + a2/z^2).
  //
  // To compute the tail time, compute the impulse response, h(n), of
  // H(z), which we can do analytically.  From this impulse response,
  // find the value n0 where |h(n)| <= eps for n >= n0.
  //
  // Assume first that the two poles of H(z) are not repeated, say r1
  // and r2.  Then, we can compute a partial fraction expansion of
  // H(z):
  //
  //   H(z) = (b0+b1/z+b2/z^2)/[(1-r1/z)*(1-r2/z)]
  //        = b0 + C2/(z-r2) - C1/(z-r1)
  //
  //  where
  //    C2 = (b0*r2^2+b1*r2+b2)/(r2-r1)
  //    C1 = (b0*r1^2+b1*r1+b2)/(r2-r1)
  //
  // Expand H(z) then this in powers of 1/z gives:
  //
  //   H(z) = b0 -(C2/r2+C1/r1) + sum(C2*r2^(i-1)/z^i + C1*r1^(i-1)/z^i)
  //
  // Thus, for n > 1 (we don't care about small n),
  //
  //   h(n) = C2*r2^(n-1) + C1*r1^(n-1)
  //
  // We need to find n0 such that |h(n)| < eps for n > n0.
  //
  // Case 1: r1 and r2 are real and distinct, with |r1|>=|r2|.
  //
  // Then
  //
  //   h(n) = C1*r1^(n-1)*(1 + C2/C1*(r2/r1)^(n-1))
  //
  // so
  //
  //   |h(n)| = |C1|*|r1|^(n-1)*|1+C2/C1*(r2/r1)^(n-1)|
  //          <= |C1|*|r1|^(n-1)*[1 + |C2/C1|*|r2/r1|^(n-1)]
  //          <= |C1|*|r1|^(n-1)*[1 + |C2/C1|]
  //
  // by using the triangle inequality and the fact that |r2|<=|r1|.
  // And we want |h(n)|<=eps which is true if
  //
  //   |C1|*|r1|^(n-1)*[1 + |C2/C1|] <= eps
  //
  // or
  //
  //   n >= 1 + log(eps/C)/log(|r1|)
  //
  // where C = |C1|*[1+|C2/C1|] = |C1| + |C2|.
  //
  // Case 2: r1 and r2 are complex
  //
  // Thne we can write r1=r*exp(i*p) and r2=r*exp(-i*p).  So,
  //
  //   |h(n)| = |C2*r^(n-1)*exp(-i*p*(n-1)) + C1*r^(n-1)*exp(i*p*(n-1))|
  //          = |C1|*r^(n-1)*|1 + C2/C1*exp(-i*p*(n-1))/exp(i*n*(n-1))|
  //          <= |C1|*r^(n-1)*[1 + |C2/C1|]
  //
  // Again, this is easily solved to give
  //
  //   n >= 1 + log(eps/C)/log(r)
  //
  // where C = |C1|*[1+|C2/C1|] = |C1| + |C2|.
  //
  // Case 3: Repeated roots, r1=r2=r.
  //
  // In this case,
  //
  //   H(z) = (b0+b1/z+b2/z^2)/[(1-r/z)^2
  //
  // Expanding this in powers of 1/z gives:
  //
  //   H(z) = C1*sum((i+1)*r^i/z^i) - C2 * sum(r^(i-2)/z^i) + b2/r^2
  //        = b2/r^2 + sum([C1*(i+1)*r^i + C2*r^(i-2)]/z^i)
  // where
  //   C1 = (b0*r^2+b1*r+b2)/r^2
  //   C2 = b1*r+2*b2
  //
  // Thus, the impulse response is
  //
  //   h(n) = C1*(n+1)*r^n + C2*r^(n-2)
  //        = r^(n-2)*[C1*(n+1)*r^2+C2]
  //
  // So
  //
  //   |h(n)| = |r|^(n-2)*|C1*(n+1)*r^2+C2|
  //
  // To find n such that |h(n)| < eps, we need a numerical method in
  // general, so there's no real reason to simplify this or use other
  // approximations.  Just solve |h(n)|=eps directly.
  //
  // Thus, for an set of filter coefficients, we can compute the tail
  // time.
  //

  // If the maximum amplitude of the impulse response is less than
  // this, we assume that we've reached the tail of the response.
  // Currently, this means that the impulse is less than 1 bit of a
  // 16-bit PCM value.
  const double kMaxTailAmplitude = 1 / 32768.0;

  // Find the roots of 1+a1/z+a2/z^2 = 0.  Or equivalently,
  // z^2+a1*z+a2 = 0.  From the quadratic formula the roots are
  // (-a1+/-sqrt(a1^2-4*a2))/2.

  double a1 = a1_[coef_index];
  double a2 = a2_[coef_index];
  double b0 = b0_[coef_index];
  double b1 = b1_[coef_index];
  double b2 = b2_[coef_index];

  double tail_frame = 0;
  double discrim = a1 * a1 - 4 * a2;

  if (discrim > 0) {
    // Compute the real roots so that r1 has the largest magnitude.
    double rplus = (-a1 + sqrt(discrim)) / 2;
    double rminus = (-a1 - sqrt(discrim)) / 2;
    double r1 = fabs(rplus) >= fabs(rminus) ? rplus : rminus;
    // Use the fact that a2 = r1*r2
    double r2 = a2 / r1;

    double c1 = (b0 * r1 * r1 + b1 * r1 + b2) / (r2 - r1);
    double c2 = (b0 * r2 * r2 + b1 * r2 + b2) / (r2 - r1);

    DCHECK(std::isfinite(r1));
    DCHECK(std::isfinite(r2));
    DCHECK(std::isfinite(c1));
    DCHECK(std::isfinite(c2));

    // It's possible for kMaxTailAmplitude to be greater than c1 + c2.
    // This may produce a negative tail frame.  Just clamp the tail
    // frame to 0.
    tail_frame = clampTo(
        1 + log(kMaxTailAmplitude / (fabs(c1) + fabs(c2))) / log(fabs(r1)), 0);

    DCHECK(std::isfinite(tail_frame));
  } else if (discrim < 0) {
    // Two complex roots.
    // One root is -a1/2 + i*sqrt(-discrim)/2.
    double x = -a1 / 2;
    double y = sqrt(-discrim) / 2;
    std::complex<double> r1(x, y);
    std::complex<double> r2(x, -y);
    double r = hypot(x, y);

    DCHECK(std::isfinite(r));

    // It's possible for r to be 1. (LPF with Q very large can cause this.)
    if (r == 1) {
      tail_frame = max_frame;
    } else {
      double c1 = abs((b0 * r1 * r1 + b1 * r1 + b2) / (r2 - r1));
      double c2 = abs((b0 * r2 * r2 + b1 * r2 + b2) / (r2 - r1));

      DCHECK(std::isfinite(c1));
      DCHECK(std::isfinite(c2));

      tail_frame = 1 + log(kMaxTailAmplitude / (c1 + c2)) / log(r);
      if (c1 == 0 && c2 == 0) {
        // If c1 = c2 = 0, then H(z) = b0.  Hence, there's no tail
        // because this is just a wire from input to output.
        tail_frame = 0;
      } else {
        // Otherwise, check that the tail has finite length.  Not
        // strictly necessary, but we want to know if this ever
        // happens.
        DCHECK(std::isfinite(tail_frame));
      }
    }
  } else {
    // Repeated roots.  This should be pretty rare because all the
    // coefficients need to be just the right values to get a
    // discriminant of exactly zero.
    double r = -a1 / 2;

    if (r == 0) {
      // Double pole at 0.  This just delays the signal by 2 frames,
      // so set the tail frame to 2.
      tail_frame = 2;
    } else {
      double c1 = (b0 * r * r + b1 * r + b2) / (r * r);
      double c2 = b1 * r + 2 * b2;

      DCHECK(std::isfinite(c1));
      DCHECK(std::isfinite(c2));

      // It can happen that c1=c2=0.  This basically means that H(z) =
      // constant, which is the limiting case for several of the
      // biquad filters.
      if (c1 == 0 && c2 == 0) {
        tail_frame = 0;
      } else {
        // The function c*(n+1)*r^n is not monotonic, but it's easy to
        // find the max point since the derivative is
        // c*r^n*(1+(n+1)*log(r)).  This has a root at
        // -(1+log(r))/log(r). so we can start our search from that
        // point to max_frames.

        double low = clampTo(-(1 + log(r)) / log(r), 1.0,
                             static_cast<double>(max_frame - 1));
        double high = max_frame;

        DCHECK(std::isfinite(low));
        DCHECK(std::isfinite(high));

        tail_frame = RootFinder(low, high, log(kMaxTailAmplitude), c1, c2, r);
      }
    }
  }

  return tail_frame;
}

}  // namespace blink