juce_FilterDesign.cpp

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/*
  ==============================================================================
 
   This file is part of the JUCE library.
   Copyright (c) 2022 - Raw Material Software Limited
 
   JUCE is an open source library subject to commercial or open-source
   licensing.
 
   By using JUCE, you agree to the terms of both the JUCE 7 End-User License
   Agreement and JUCE Privacy Policy.
 
   End User License Agreement: www.juce.com/juce-7-licence
   Privacy Policy: www.juce.com/juce-privacy-policy
 
   Or: You may also use this code under the terms of the GPL v3 (see
   www.gnu.org/licenses).
 
   JUCE IS PROVIDED "AS IS" WITHOUT ANY WARRANTY, AND ALL WARRANTIES, WHETHER
   EXPRESSED OR IMPLIED, INCLUDING MERCHANTABILITY AND FITNESS FOR PURPOSE, ARE
   DISCLAIMED.
 
  ==============================================================================
*/
 
namespace juce
{
namespace dsp
{
 
template <typename FloatType>
typename FIR::Coefficients<FloatType>::Ptr
    FilterDesign<FloatType>::designFIRLowpassWindowMethod (FloatType frequency,
                                                           double sampleRate, size_t order,
                                                           WindowingMethod type,
                                                           FloatType beta)
{
    jassert (sampleRate > 0);
    jassert (frequency > 0 && frequency <= sampleRate * 0.5);
 
    auto* result = new typename FIR::Coefficients<FloatType> (order + 1u);
 
    auto* c = result->getRawCoefficients();
    auto normalisedFrequency = frequency / sampleRate;
 
    for (size_t i = 0; i <= order; ++i)
    {
        if (i == order / 2)
        {
            c[i] = static_cast<FloatType> (normalisedFrequency * 2);
        }
        else
        {
            auto indice = MathConstants<double>::pi * (static_cast<double> (i) - 0.5 * static_cast<double> (order));
            c[i] = static_cast<FloatType> (std::sin (2.0 * indice * normalisedFrequency) / indice);
        }
    }
 
    WindowingFunction<FloatType> theWindow (order + 1, type, false, beta);
    theWindow.multiplyWithWindowingTable (c, order + 1);
 
    return *result;
}
 
template <typename FloatType>
typename FIR::Coefficients<FloatType>::Ptr
    FilterDesign<FloatType>::designFIRLowpassKaiserMethod (FloatType frequency, double sampleRate,
                                                           FloatType normalisedTransitionWidth,
                                                           FloatType amplitudedB)
{
    jassert (sampleRate > 0);
    jassert (frequency > 0 && frequency <= sampleRate * 0.5);
    jassert (normalisedTransitionWidth > 0 && normalisedTransitionWidth <= 0.5);
    jassert (amplitudedB >= -100 && amplitudedB <= 0);
 
    FloatType beta = 0;
 
    if (amplitudedB < -50)
        beta = static_cast<FloatType> (0.1102 * (-amplitudedB - 8.7));
    else if (amplitudedB <= -21)
        beta = static_cast<FloatType> (0.5842 * std::pow (-amplitudedB - 21, 0.4) + 0.07886 * (-amplitudedB - 21));
 
    int order = amplitudedB < -21 ? roundToInt (std::ceil ((-amplitudedB - 7.95) / (2.285 * normalisedTransitionWidth * MathConstants<double>::twoPi)))
                                    : roundToInt (std::ceil (5.79 / (normalisedTransitionWidth * MathConstants<double>::twoPi)));
 
    jassert (order >= 0);
 
    return designFIRLowpassWindowMethod (frequency, sampleRate, static_cast<size_t> (order),
                                         WindowingFunction<FloatType>::kaiser, beta);
}
 
 
template <typename FloatType>
typename FIR::Coefficients<FloatType>::Ptr
    FilterDesign<FloatType>::designFIRLowpassTransitionMethod (FloatType frequency, double sampleRate, size_t order,
                                                               FloatType normalisedTransitionWidth, FloatType spline)
{
    jassert (sampleRate > 0);
    jassert (frequency > 0 && frequency <= sampleRate * 0.5);
    jassert (normalisedTransitionWidth > 0 && normalisedTransitionWidth <= 0.5);
    jassert (spline >= 1.0 && spline <= 4.0);
 
    auto normalisedFrequency = frequency / static_cast<FloatType> (sampleRate);
 
    auto* result = new typename FIR::Coefficients<FloatType> (order + 1u);
    auto* c = result->getRawCoefficients();
 
    for (size_t i = 0; i <= order; ++i)
    {
        if (i == order / 2 && order % 2 == 0)
        {
            c[i] = static_cast<FloatType> (2 * normalisedFrequency);
        }
        else
        {
            auto indice  = MathConstants<double>::pi * ((double) i - 0.5 * (double) order);
            auto indice2 = MathConstants<double>::pi * normalisedTransitionWidth * ((double) i - 0.5 * (double) order) / spline;
            c[i] = static_cast<FloatType> (std::sin (2 * indice * normalisedFrequency)
                                            / indice * std::pow (std::sin (indice2) / indice2, spline));
        }
    }
 
    return *result;
}
 
template <typename FloatType>
typename FIR::Coefficients<FloatType>::Ptr
    FilterDesign<FloatType>::designFIRLowpassLeastSquaresMethod (FloatType frequency,
                                                                 double sampleRate, size_t order,
                                                                 FloatType normalisedTransitionWidth,
                                                                 FloatType stopBandWeight)
{
    jassert (sampleRate > 0);
    jassert (frequency > 0 && frequency <= sampleRate * 0.5);
    jassert (normalisedTransitionWidth > 0 && normalisedTransitionWidth <= 0.5);
    jassert (stopBandWeight >= 1.0 && stopBandWeight <= 100.0);
 
    auto normalisedFrequency = static_cast<double> (frequency) / sampleRate;
 
    auto wp = MathConstants<double>::twoPi * (static_cast<double> (normalisedFrequency - normalisedTransitionWidth / 2.0));
    auto ws = MathConstants<double>::twoPi * (static_cast<double> (normalisedFrequency + normalisedTransitionWidth / 2.0));
 
    auto N = order + 1;
 
    auto* result = new typename FIR::Coefficients<FloatType> (static_cast<size_t> (N));
    auto* c = result->getRawCoefficients();
 
    if (N % 2 == 1)
    {
        // Type I
        auto M = (N - 1) / 2;
 
        Matrix<double> b (M + 1, 1),
                       q (2 * M + 1, 1);
 
        auto sinc = [] (double x) { return x == 0 ? 1 : std::sin (x * MathConstants<double>::pi)
                                                          / (MathConstants<double>::pi * x); };
 
        auto factorp = wp / MathConstants<double>::pi;
        auto factors = ws / MathConstants<double>::pi;
 
        for (size_t i = 0; i <= M; ++i)
            b (i, 0) = factorp * sinc (factorp * (double) i);
 
        q (0, 0) = factorp + stopBandWeight * (1.0 - factors);
 
        for (size_t i = 1; i <= 2 * M; ++i)
            q (i, 0) = factorp * sinc (factorp * (double) i) - stopBandWeight * factors * sinc (factors * (double) i);
 
        auto Q1 = Matrix<double>::toeplitz (q, M + 1);
        auto Q2 = Matrix<double>::hankel (q, M + 1, 0);
 
        Q1 += Q2; Q1 *= 0.5;
 
        Q1.solve (b);
 
        c[M] = static_cast<FloatType> (b (0, 0));
 
        for (size_t i = 1; i <= M; ++i)
        {
            c[M - i] = static_cast<FloatType> (b (i, 0) * 0.5);
            c[M + i] = static_cast<FloatType> (b (i, 0) * 0.5);
        }
    }
    else
    {
        // Type II
        auto M = N / 2;
 
        Matrix<double> b (M, 1);
        Matrix<double> qp (2 * M, 1);
        Matrix<double> qs (2 * M, 1);
 
        auto sinc = [] (double x) { return x == 0 ? 1 : std::sin (x * MathConstants<double>::pi)
                                                          / (MathConstants<double>::pi * x); };
 
        auto factorp = wp / MathConstants<double>::pi;
        auto factors = ws / MathConstants<double>::pi;
 
        for (size_t i = 0; i < M; ++i)
            b (i, 0) = factorp * sinc (factorp * ((double) i + 0.5));
 
        for (size_t i = 0; i < 2 * M; ++i)
        {
            qp (i, 0) = 0.25 * factorp * sinc (factorp * (double) i);
            qs (i, 0) = -0.25 * stopBandWeight * factors * sinc (factors * (double) i);
        }
 
        auto Q1p = Matrix<double>::toeplitz (qp, M);
        auto Q2p = Matrix<double>::hankel (qp, M, 1);
        auto Q1s = Matrix<double>::toeplitz (qs, M);
        auto Q2s = Matrix<double>::hankel (qs, M, 1);
 
        auto Id = Matrix<double>::identity (M);
        Id *= (0.25 * stopBandWeight);
 
        Q1p += Q2p;
        Q1s += Q2s;
        Q1s += Id;
 
        auto& Q = Q1s;
        Q += Q1p;
 
        Q.solve (b);
 
        for (size_t i = 0; i < M; ++i)
        {
            c[M - i - 1] = static_cast<FloatType> (b (i, 0) * 0.25);
            c[M + i]     = static_cast<FloatType> (b (i, 0) * 0.25);
        }
    }
 
    return *result;
}
 
template <typename FloatType>
typename FIR::Coefficients<FloatType>::Ptr
    FilterDesign<FloatType>::designFIRLowpassHalfBandEquirippleMethod (FloatType normalisedTransitionWidth,
                                                                       FloatType amplitudedB)
{
    jassert (normalisedTransitionWidth > 0 && normalisedTransitionWidth <= 0.5);
    jassert (amplitudedB >= -300 && amplitudedB <= -10);
 
    auto wpT = (0.5 - normalisedTransitionWidth) * MathConstants<double>::pi;
 
    auto n = roundToInt (std::ceil ((amplitudedB - 18.18840664 * wpT + 33.64775300) / (18.54155181 * wpT - 29.13196871)));
    auto kp = (n * wpT - 1.57111377 * n + 0.00665857) / (-1.01927560 * n + 0.37221484);
    auto A = (0.01525753 * n + 0.03682344 + 9.24760314 / (double) n) * kp + 1.01701407 + 0.73512298 / (double) n;
    auto B = (0.00233667 * n - 1.35418408 + 5.75145813 / (double) n) * kp + 1.02999650 - 0.72759508 / (double) n;
 
    auto hn  = FilterDesign<FloatType>::getPartialImpulseResponseHn (n, kp);
    auto hnm = FilterDesign<FloatType>::getPartialImpulseResponseHn (n - 1, kp);
 
    auto diff = (hn.size() - hnm.size()) / 2;
 
    for (int i = 0; i < diff; ++i)
    {
        hnm.add (0.0);
        hnm.insert (0, 0.0);
    }
 
    auto hh = hn;
 
    for (int i = 0; i < hn.size(); ++i)
        hh.setUnchecked (i, A * hh[i] + B * hnm[i]);
 
    auto* result = new typename FIR::Coefficients<FloatType> (static_cast<size_t> (hh.size()));
    auto* c = result->getRawCoefficients();
 
    for (int i = 0; i < hh.size(); ++i)
        c[i] = (float) hh[i];
 
    auto NN = [&]
    {
        if (n % 2 == 0)
            return 2.0 * result->getMagnitudeForFrequency (0.5, 1.0);
 
        auto w01 = std::sqrt (kp * kp + (1 - kp * kp) * std::pow (std::cos (MathConstants<double>::pi / (2.0 * n + 1.0)), 2.0));
 
        if (std::abs (w01) > 1.0)
            return 2.0 * result->getMagnitudeForFrequency (0.5, 1.0);
 
        auto om01 = std::acos (-w01);
        return -2.0 * result->getMagnitudeForFrequency (om01 / MathConstants<double>::twoPi, 1.0);
    }();
 
    for (int i = 0; i < hh.size(); ++i)
        c[i] = static_cast<FloatType> ((A * hn[i] + B * hnm[i]) / NN);
 
    c[2 * n + 1] = static_cast<FloatType> (0.5);
 
    return *result;
}
 
template <typename FloatType>
Array<double> FilterDesign<FloatType>::getPartialImpulseResponseHn (int n, double kp)
{
    Array<double> alpha;
    alpha.resize (2 * n + 1);
 
    alpha.setUnchecked (2 * n, 1.0 / std::pow (1.0 - kp * kp, n));
 
    if (n > 0)
        alpha.setUnchecked (2 * n - 2, -(2 * n * kp * kp + 1) * alpha[2 * n]);
 
    if (n > 1)
        alpha.setUnchecked (2 * n - 4, -(4 * n + 1 + (n - 1) * (2 * n - 1) * kp * kp) / (2.0 * n) * alpha[2 * n - 2]
                             - (2 * n + 1) * ((n + 1) * kp * kp + 1) / (2.0 * n) * alpha[2 * n]);
 
    for (int k = n; k >= 3; --k)
    {
        auto c1 = (3 * (n*(n + 2) - k * (k - 2)) + 2 * k - 3 + 2 * (k - 2)*(2 * k - 3) * kp * kp) * alpha[2 * k - 4];
        auto c2 = (3 * (n*(n + 2) - (k - 1) * (k + 1)) + 2 * (2 * k - 1) + 2 * k*(2 * k - 1) * kp * kp) * alpha[2 * k - 2];
        auto c3 = (n * (n + 2) - (k - 1) * (k + 1)) * alpha[2 * k];
        auto c4 = (n * (n + 2) - (k - 3) * (k - 1));
 
        alpha.setUnchecked (2 * k - 6, -(c1 + c2 + c3) / c4);
    }
 
    Array<double> ai;
    ai.resize (2 * n + 1 + 1);
 
    for (int k = 0; k <= n; ++k)
        ai.setUnchecked (2 * k + 1, alpha[2 * k] / (2.0 * k + 1.0));
 
    Array<double> hn;
    hn.resize (2 * n + 1 + 2 * n + 1 + 1);
 
    for (int k = 0; k <= n; ++k)
    {
        hn.setUnchecked (2 * n + 1 + (2 * k + 1), 0.5 * ai[2 * k + 1]);
        hn.setUnchecked (2 * n + 1 - (2 * k + 1), 0.5 * ai[2 * k + 1]);
    }
 
    return hn;
}
 
template <typename FloatType>
ReferenceCountedArray<IIR::Coefficients<FloatType>>
    FilterDesign<FloatType>::designIIRLowpassHighOrderButterworthMethod (FloatType frequency, double sampleRate,
                                                                         FloatType normalisedTransitionWidth,
                                                                         FloatType passbandAmplitudedB,
                                                                         FloatType stopbandAmplitudedB)
{
    return designIIRLowpassHighOrderGeneralMethod (0, frequency, sampleRate, normalisedTransitionWidth,
                                                   passbandAmplitudedB, stopbandAmplitudedB);
}
 
template <typename FloatType>
ReferenceCountedArray<IIR::Coefficients<FloatType>>
    FilterDesign<FloatType>::designIIRLowpassHighOrderChebyshev1Method (FloatType frequency, double sampleRate,
                                                                        FloatType normalisedTransitionWidth,
                                                                        FloatType passbandAmplitudedB,
                                                                        FloatType stopbandAmplitudedB)
{
    return designIIRLowpassHighOrderGeneralMethod (1, frequency, sampleRate, normalisedTransitionWidth,
                                                   passbandAmplitudedB, stopbandAmplitudedB);
}
 
template <typename FloatType>
ReferenceCountedArray<IIR::Coefficients<FloatType>>
    FilterDesign<FloatType>::designIIRLowpassHighOrderChebyshev2Method (FloatType frequency, double sampleRate,
                                                                        FloatType normalisedTransitionWidth,
                                                                        FloatType passbandAmplitudedB,
                                                                        FloatType stopbandAmplitudedB)
{
    return designIIRLowpassHighOrderGeneralMethod (2, frequency, sampleRate, normalisedTransitionWidth,
                                                   passbandAmplitudedB, stopbandAmplitudedB);
}
 
template <typename FloatType>
ReferenceCountedArray<IIR::Coefficients<FloatType>>
    FilterDesign<FloatType>::designIIRLowpassHighOrderEllipticMethod (FloatType frequency, double sampleRate,
                                                                      FloatType normalisedTransitionWidth,
                                                                      FloatType passbandAmplitudedB,
                                                                      FloatType stopbandAmplitudedB)
{
    return designIIRLowpassHighOrderGeneralMethod (3, frequency, sampleRate, normalisedTransitionWidth,
                                                   passbandAmplitudedB, stopbandAmplitudedB);
}
 
template <typename FloatType>
ReferenceCountedArray<IIR::Coefficients<FloatType>>
    FilterDesign<FloatType>::designIIRLowpassHighOrderGeneralMethod (int type, FloatType frequency, double sampleRate,
                                                                     FloatType normalisedTransitionWidth,
                                                                     FloatType passbandAmplitudedB,
                                                                     FloatType stopbandAmplitudedB)
{
    jassert (0 < sampleRate);
    jassert (0 < frequency && frequency <= sampleRate * 0.5);
    jassert (0 < normalisedTransitionWidth && normalisedTransitionWidth <= 0.5);
    jassert (-20 < passbandAmplitudedB && passbandAmplitudedB < 0);
    jassert (-300 < stopbandAmplitudedB && stopbandAmplitudedB < -20);
 
    auto normalisedFrequency = frequency / sampleRate;
 
    auto fp = normalisedFrequency - normalisedTransitionWidth / 2;
    jassert (0.0 < fp && fp < 0.5);
 
    auto fs = normalisedFrequency + normalisedTransitionWidth / 2;
    jassert (0.0 < fs && fs < 0.5);
 
    double Ap = passbandAmplitudedB;
    double As = stopbandAmplitudedB;
    auto Gp = Decibels::decibelsToGain (Ap, -300.0);
    auto Gs = Decibels::decibelsToGain (As, -300.0);
    auto epsp = std::sqrt (1.0 / (Gp * Gp) - 1.0);
    auto epss = std::sqrt (1.0 / (Gs * Gs) - 1.0);
 
    auto omegap = std::tan (MathConstants<double>::pi * fp);
    auto omegas = std::tan (MathConstants<double>::pi * fs);
    constexpr auto halfPi = MathConstants<double>::halfPi;
 
    auto k = omegap / omegas;
    auto k1 = epsp / epss;
 
    int N;
 
    if (type == 0)
    {
        N = roundToInt (std::ceil (std::log (1.0 / k1) / std::log (1.0 / k)));
    }
    else if (type == 1 || type == 2)
    {
        N = roundToInt (std::ceil (std::acosh (1.0 / k1) / std::acosh (1.0 / k)));
    }
    else
    {
        double K, Kp, K1, K1p;
 
        SpecialFunctions::ellipticIntegralK (k, K, Kp);
        SpecialFunctions::ellipticIntegralK (k1, K1, K1p);
 
        N = roundToInt (std::ceil ((K1p * K) / (K1 * Kp)));
    }
 
    const int r = N % 2;
    const int L = (N - r) / 2;
    const double H0 = (type == 1 || type == 3) ? std::pow (Gp, 1.0 - r) : 1.0;
 
    Array<Complex<double>> pa, za;
    Complex<double> j (0, 1);
 
    if (type == 0)
    {
        if (r == 1)
            pa.add (-omegap * std::pow (epsp, -1.0 / (double) N));
 
        for (int i = 1; i <= L; ++i)
        {
            auto ui = (2 * i - 1.0) / (double) N;
            pa.add (omegap * std::pow (epsp, -1.0 / (double) N) * j * exp (ui * halfPi * j));
        }
    }
    else if (type == 1)
    {
        auto v0 = std::asinh (1.0 / epsp) / (N * halfPi);
 
        if (r == 1)
            pa.add (-omegap * std::sinh (v0 * halfPi));
 
        for (int i = 1; i <= L; ++i)
        {
            auto ui = (2 * i - 1.0) / (double) N;
            pa.add (omegap * j * std::cos ((ui - j * v0) * halfPi));
        }
    }
    else if (type == 2)
    {
        auto v0 = std::asinh (epss) / (N * halfPi);
 
        if (r == 1)
            pa.add(-1.0 / (k / omegap * std::sinh (v0 * halfPi)));
 
        for (int i = 1; i <= L; ++i)
        {
            auto ui = (2 * i - 1.0) / (double) N;
 
            pa.add (1.0 / (k / omegap * j * std::cos ((ui - j * v0) * halfPi)));
            za.add (1.0 / (k / omegap * j * std::cos (ui * halfPi)));
        }
    }
    else
    {
        auto v0 = -j * (SpecialFunctions::asne (j / epsp, k1) / (double) N);
 
        if (r == 1)
            pa.add (omegap * j * SpecialFunctions::sne (j * v0, k));
 
        for (int i = 1; i <= L; ++i)
        {
            auto ui = (2 * i - 1.0) / (double) N;
            auto zetai = SpecialFunctions::cde (ui, k);
 
            pa.add (omegap * j * SpecialFunctions::cde (ui - j * v0, k));
            za.add (omegap * j / (k * zetai));
        }
    }
 
    Array<Complex<double>> p, z, g;
 
    if (r == 1)
    {
        p.add ((1.0 + pa[0]) / (1.0 - pa[0]));
        g.add (0.5 * (1.0 - p[0]));
    }
 
    for (int i = 0; i < L; ++i)
    {
        p.add ((1.0 + pa[i + r]) / (1.0 - pa[i + r]));
        z.add (za.size() == 0 ? -1.0 : (1.0 + za[i]) / (1.0 - za[i]));
        g.add ((1.0 - p[i + r]) / (1.0 - z[i]));
    }
 
    ReferenceCountedArray<IIR::Coefficients<FloatType>> cascadedCoefficients;
 
    if (r == 1)
    {
        auto b0 = static_cast<FloatType> (H0 * std::real (g[0]));
        auto b1 = b0;
        auto a1 = static_cast<FloatType> (-std::real (p[0]));
 
        cascadedCoefficients.add (new IIR::Coefficients<FloatType> (b0, b1, 1.0f, a1));
    }
 
    for (int i = 0; i < L; ++i)
    {
        auto gain = std::pow (std::abs (g[i + r]), 2.0);
 
        auto b0 = static_cast<FloatType> (gain);
        auto b1 = static_cast<FloatType> (std::real (-z[i] - std::conj (z[i])) * gain);
        auto b2 = static_cast<FloatType> (std::real ( z[i] * std::conj (z[i])) * gain);
 
        auto a1 = static_cast<FloatType> (std::real (-p[i+r] - std::conj (p[i + r])));
        auto a2 = static_cast<FloatType> (std::real ( p[i+r] * std::conj (p[i + r])));
 
        cascadedCoefficients.add (new IIR::Coefficients<FloatType> (b0, b1, b2, 1, a1, a2));
    }
 
    return cascadedCoefficients;
}
 
template <typename FloatType>
ReferenceCountedArray<IIR::Coefficients<FloatType>>
    FilterDesign<FloatType>::designIIRLowpassHighOrderButterworthMethod (FloatType frequency,
                                                                         double sampleRate, int order)
{
    jassert (sampleRate > 0);
    jassert (frequency > 0 && frequency <= sampleRate * 0.5);
    jassert (order > 0);
 
    ReferenceCountedArray<IIR::Coefficients<FloatType>> arrayFilters;
 
    if (order % 2 == 1)
    {
        arrayFilters.add (*IIR::Coefficients<FloatType>::makeFirstOrderLowPass (sampleRate, frequency));
 
        for (int i = 0; i < order / 2; ++i)
        {
            auto Q = 1.0 / (2.0 * std::cos ((i + 1.0) * MathConstants<double>::pi / order));
            arrayFilters.add (*IIR::Coefficients<FloatType>::makeLowPass (sampleRate, frequency,
                                                                          static_cast<FloatType> (Q)));
        }
    }
    else
    {
        for (int i = 0; i < order / 2; ++i)
        {
            auto Q = 1.0 / (2.0 * std::cos ((2.0 * i + 1.0) * MathConstants<double>::pi / (order * 2.0)));
            arrayFilters.add (*IIR::Coefficients<FloatType>::makeLowPass (sampleRate, frequency,
                                                                          static_cast<FloatType> (Q)));
        }
    }
 
    return arrayFilters;
}
 
template <typename FloatType>
ReferenceCountedArray<IIR::Coefficients<FloatType>>
    FilterDesign<FloatType>::designIIRHighpassHighOrderButterworthMethod (FloatType frequency,
                                                                          double sampleRate, int order)
{
    jassert (sampleRate > 0);
    jassert (frequency > 0 && frequency <= sampleRate * 0.5);
    jassert (order > 0);
 
    ReferenceCountedArray<IIR::Coefficients<FloatType>> arrayFilters;
 
    if (order % 2 == 1)
    {
        arrayFilters.add (*IIR::Coefficients<FloatType>::makeFirstOrderHighPass (sampleRate, frequency));
 
        for (int i = 0; i < order / 2; ++i)
        {
            auto Q = 1.0 / (2.0 * std::cos ((i + 1.0) * MathConstants<double>::pi / order));
            arrayFilters.add (*IIR::Coefficients<FloatType>::makeHighPass (sampleRate, frequency,
                                                                           static_cast<FloatType> (Q)));
        }
    }
    else
    {
        for (int i = 0; i < order / 2; ++i)
        {
            auto Q = 1.0 / (2.0 * std::cos ((2.0 * i + 1.0) * MathConstants<double>::pi / (order * 2.0)));
            arrayFilters.add (*IIR::Coefficients<FloatType>::makeHighPass (sampleRate, frequency,
                                                                           static_cast<FloatType> (Q)));
        }
    }
 
    return arrayFilters;
}
 
template <typename FloatType>
typename FilterDesign<FloatType>::IIRPolyphaseAllpassStructure
    FilterDesign<FloatType>::designIIRLowpassHalfBandPolyphaseAllpassMethod (FloatType normalisedTransitionWidth,
                                                                             FloatType stopbandAmplitudedB)
{
    jassert (normalisedTransitionWidth > 0 && normalisedTransitionWidth <= 0.5);
    jassert (stopbandAmplitudedB > -300 && stopbandAmplitudedB < -10);
 
    const double wt = MathConstants<double>::twoPi * normalisedTransitionWidth;
    const double ds = Decibels::decibelsToGain (stopbandAmplitudedB, static_cast<FloatType> (-300.0));
 
    auto k = std::pow (std::tan ((MathConstants<double>::pi - wt) / 4), 2.0);
    auto kp = std::sqrt (1.0 - k * k);
    auto e = (1 - std::sqrt (kp)) / (1 + std::sqrt (kp)) * 0.5;
    auto q = e + 2 * std::pow (e, 5.0) + 15 * std::pow (e, 9.0) + 150 * std::pow (e, 13.0);
 
    auto k1 = ds * ds / (1 - ds * ds);
    int n = roundToInt (std::ceil (std::log (k1 * k1 / 16) / std::log (q)));
 
    if (n % 2 == 0)
        ++n;
 
    if (n == 1)
        n = 3;
 
    auto q1 = std::pow (q, (double) n);
    k1 = 4 * std::sqrt (q1);
 
    const int N = (n - 1) / 2;
    Array<double> ai;
 
    for (int i = 1; i <= N; ++i)
    {
        double num = 0.0;
        double delta = 1.0;
        int m = 0;
 
        while (std::abs (delta) > 1e-100)
        {
            delta = std::pow (-1, m) * std::pow (q, m * (m + 1))
                     * std::sin ((2 * m + 1) * MathConstants<double>::pi * i / (double) n);
            num += delta;
            m++;
        }
 
        num *= 2 * std::pow (q, 0.25);
 
        double den = 0.0;
        delta = 1.0;
        m = 1;
 
        while (std::abs (delta) > 1e-100)
        {
            delta = std::pow (-1, m) * std::pow (q, m * m)
                     * std::cos (m * MathConstants<double>::twoPi * i / (double) n);
            den += delta;
            ++m;
        }
 
        den = 1 + 2 * den;
 
        auto wi = num / den;
        auto api = std::sqrt ((1 - wi * wi * k) * (1 - wi * wi / k)) / (1 + wi * wi);
 
        ai.add ((1 - api) / (1 + api));
    }
 
    IIRPolyphaseAllpassStructure structure;
 
    for (int i = 0; i < N; i += 2)
        structure.directPath.add (new IIR::Coefficients<FloatType> (static_cast<FloatType> (ai[i]),
                                                                    0, 1, 1, 0, static_cast<FloatType> (ai[i])));
 
    structure.delayedPath.add (new IIR::Coefficients<FloatType> (0, 1, 1, 0));
 
    for (int i = 1; i < N; i += 2)
        structure.delayedPath.add (new IIR::Coefficients<FloatType> (static_cast<FloatType> (ai[i]),
                                                                     0, 1, 1, 0, static_cast<FloatType> (ai[i])));
 
    structure.alpha.addArray (ai);
 
    return structure;
}
 
 
template struct FilterDesign<float>;
template struct FilterDesign<double>;
 
} // namespace dsp
} // namespace juce