/*
July 15, 2015
Iowa Hills Software LLC
http://www.iowahills.com
*/
#include <math.h>
#include <new>
#include <iostream>
#include "wfir.h"
#undef M_PI
#define M_PI 3.14159265358979323846
#define M_2PI 6.28318530717958647692
// This first calculates the impulse response for a rectangular window.
// It then applies the windowing function of choice to the impulse response.
void WFIR::BasicFIR(double *FirCoeff, int NumTaps, TPassTypeName PassType,
double OmegaC, double BW, TWindowType WindowType, double WinBeta)
{
int j;
double Arg, OmegaLow, OmegaHigh;
switch (PassType)
{
case LPF:
for (j = 0; j < NumTaps; j++)
{
Arg = (double) j - (double) (NumTaps - 1) / 2.0;
FirCoeff[j] = OmegaC * Sinc(OmegaC * Arg * M_PI);
}
break;
case HPF:
if (NumTaps % 2 == 1) // Odd tap counts
{
for (j = 0; j < NumTaps; j++)
{
Arg = (double) j - (double) (NumTaps - 1) / 2.0;
FirCoeff[j] = Sinc(Arg * M_PI)
- OmegaC * Sinc(OmegaC * Arg * M_PI);
}
}
else // Even tap counts
{
for (j = 0; j < NumTaps; j++)
{
Arg = (double) j - (double) (NumTaps - 1) / 2.0;
if (Arg == 0.0)
FirCoeff[j] = 0.0;
else
FirCoeff[j] = cos(OmegaC * Arg * M_PI) / M_PI / Arg
+ cos(Arg * M_PI);
}
}
break;
case BPF:
OmegaLow = OmegaC - BW / 2.0;
OmegaHigh = OmegaC + BW / 2.0;
for (j = 0; j < NumTaps; j++)
{
Arg = (double) j - (double) (NumTaps - 1) / 2.0;
if (Arg == 0.0)
FirCoeff[j] = 0.0;
else
FirCoeff[j] = (cos(OmegaLow * Arg * M_PI)
- cos(OmegaHigh * Arg * M_PI)) / M_PI / Arg;
}
break;
case NOTCH: // If NumTaps is even for Notch filters, the response at Pi is attenuated.
OmegaLow = OmegaC - BW / 2.0;
OmegaHigh = OmegaC + BW / 2.0;
for (j = 0; j < NumTaps; j++)
{
Arg = (double) j - (double) (NumTaps - 1) / 2.0;
FirCoeff[j] = Sinc(Arg * M_PI)
- OmegaHigh * Sinc(OmegaHigh * Arg * M_PI)
- OmegaLow * Sinc(OmegaLow * Arg * M_PI);
}
break;
}
// WindowData can be used to window data before an FFT. When used for FIR filters we set
// Alpha = 0.0 to prevent a flat top on the window and
// set UnityGain = false to prevent the window gain from getting set to unity.
WindowData(FirCoeff, NumTaps, WindowType, 0.0, WinBeta, false);
}
//---------------------------------------------------------------------------
// This gets used with the Kaiser window.
double WFIR::Bessel(double x)
{
double Sum = 0.0, XtoIpower;
int i, j, Factorial;
for (i = 1; i < 10; i++)
{
XtoIpower = pow(x / 2.0, (double) i);
Factorial = 1;
for (j = 1; j <= i; j++)
Factorial *= j;
Sum += pow(XtoIpower / (double) Factorial, 2.0);
}
return (1.0 + Sum);
}
//-----------------------------------------------------------------------------
// This gets used with the Sinc window and various places in the BasicFIR function.
double WFIR::Sinc(double x)
{
if (x > -1.0E-5 && x < 1.0E-5)
return (1.0);
return (sin(x) / x);
}
//---------------------------------------------------------------------------
// These are the various windows definitions. These windows can be used for either
// FIR filter design or with an FFT for spectral analysis.
// Sourced verbatim from: ~MyDocs\Code\Common\FFTFunctions.cpp
// For definitions, see this article: http://en.wikipedia.org/wiki/Window_function
// This function has 6 inputs
// Data is the array, of length N, containing the data to to be windowed.
// This data is either a FIR filter sinc pulse, or the data to be analyzed by an fft.
// WindowType is an enum defined in the header file, which is at the bottom of this file.
// e.g. wtKAISER, wtSINC, wtHANNING, wtHAMMING, wtBLACKMAN, ...
// Alpha sets the width of the flat top.
// Windows such as the Tukey and Trapezoid are defined to have a variably wide flat top.
// As can be seen by its definition, the Tukey is just a Hanning window with a flat top.
// Alpha can be used to give any of these windows a partial flat top, except the Flattop and Kaiser.
// Alpha = 0 gives the original window. (i.e. no flat top)
// To generate a Tukey window, use a Hanning with 0 < Alpha < 1
// To generate a Bartlett window (triangular), use a Trapezoid window with Alpha = 0.
// Alpha = 1 generates a rectangular window in all cases. (except the Flattop and Kaiser)
// Beta is used with the Kaiser, Sinc, and Sine windows only.
// These three windows are primarily used for FIR filter design, not spectral analysis.
// In FIR filter design, Beta controls the filter's transition bandwidth and the sidelobe levels.
// The code ignores Beta except in the Kaiser, Sinc, and Sine window cases.
// UnityGain controls whether the gain of these windows is set to unity.
// Only the Flattop window has unity gain by design. The Hanning window, for example, has a gain
// of 1/2. UnityGain = true will set the gain of all these windows to 1.
// Then, when the window is applied to a signal, the signal's energy content is preserved.
// Don't use this with FIR filter design however. Since most of the enegy in an FIR sinc pulse
// is in the middle of the window, the window needs a peak amplitude of one, not unity gain.
// Setting UnityGain = true will simply cause the resulting FIR filter to have excess gain.
// If using these windows for FIR filters, start with the Kaiser, Sinc, or Sine windows and
// adjust Beta for the desired transition BW and sidelobe levels (set Alpha = 0).
// While the FlatTop is an excellent window for spectral analysis, don't use it for FIR filter design.
// It has a peak amplitude of ~ 4.7 which causes the resulting FIR filter to have about this much gain.
// It works poorly for FIR filters even if you adjust its peak amplitude.
// The Trapezoid also works poorly for FIR filter design.
// If using these windows with an fft for spectral analysis, start with the Hanning, Gauss, or Flattop.
// When choosing a window for spectral analysis, you must trade off between resolution and amplitude accuracy.
// The Hanning has the best resolution while the Flatop has the best amplitude accuracy.
// The Gauss is midway between these two for both accuracy and resolution.
// These three were the only windows available in the HP 89410A Vector Signal Analyzer. Which is to say,
// unless you have specific windowing requirements, use one of these 3 for general purpose signal analysis.
// Set UnityGain = true when using any of these windows for spectral analysis to preserve the signal's enegy level.
void WFIR::WindowData(double *Data, int N, TWindowType WindowType, double Alpha,
double Beta, bool UnityGain)
{
if (WindowType == wtNONE)
return;
int j, M, TopWidth;
double dM, *WinCoeff;
if (WindowType == wtKAISER || WindowType == wtFLATTOP)
Alpha = 0.0;
if (Alpha < 0.0)
Alpha = 0.0;
if (Alpha > 1.0)
Alpha = 1.0;
if (Beta < 0.0)
Beta = 0.0;
if (Beta > 10.0)
Beta = 10.0;
WinCoeff = new (std::nothrow) double[N + 2];
if (WinCoeff == 0)
{
std::cerr
<< "Failed to allocate memory in FFTFunctions::WindowFFTData() "
<< std::endl;
return;
}
TopWidth = (int) (Alpha * (double) N);
if (TopWidth % 2 != 0)
TopWidth++;
if (TopWidth > N)
TopWidth = N;
M = N - TopWidth;
dM = M + 1;
// Calculate the window for N/2 points, then fold the window over (at the bottom).
// TopWidth points will be set to 1.
if (WindowType == wtKAISER)
{
double Arg;
for (j = 0; j < M; j++)
{
Arg = Beta * sqrt(1.0 - pow(((double) (2 * j + 2) - dM) / dM, 2.0));
WinCoeff[j] = Bessel(Arg) / Bessel(Beta);
}
}
else if (WindowType == wtSINC) // Lanczos
{
for (j = 0; j < M; j++)
WinCoeff[j] = Sinc((double) (2 * j + 1 - M) / dM * M_PI);
for (j = 0; j < M; j++)
WinCoeff[j] = pow(WinCoeff[j], Beta);
}
else if (WindowType == wtSINE) // Hanning if Beta = 2
{
for (j = 0; j < M / 2; j++)
WinCoeff[j] = sin((double) (j + 1) * M_PI / dM);
for (j = 0; j < M / 2; j++)
WinCoeff[j] = pow(WinCoeff[j], Beta);
}
else if (WindowType == wtHANNING)
{
for (j = 0; j < M / 2; j++)
WinCoeff[j] = 0.5 - 0.5 * cos((double) (j + 1) * M_2PI / dM);
}
else if (WindowType == wtHAMMING)
{
for (j = 0; j < M / 2; j++)
WinCoeff[j] = 0.54 - 0.46 * cos((double) (j + 1) * M_2PI / dM);
}
else if (WindowType == wtBLACKMAN)
{
for (j = 0; j < M / 2; j++)
{
WinCoeff[j] = 0.42 - 0.50 * cos((double) (j + 1) * M_2PI / dM)
+ 0.08 * cos((double) (j + 1) * M_2PI * 2.0 / dM);
}
}
// See: http://www.bth.se/fou/forskinfo.nsf/0/130c0940c5e7ffcdc1256f7f0065ac60/$file/ICOTA_2004_ttr_icl_mdh.pdf
else if (WindowType == wtFLATTOP)
{
for (j = 0; j <= M / 2; j++)
{
WinCoeff[j] = 1.0
- 1.93293488969227 * cos((double) (j + 1) * M_2PI / dM)
+ 1.28349769674027
* cos((double) (j + 1) * M_2PI * 2.0 / dM)
- 0.38130801681619
* cos((double) (j + 1) * M_2PI * 3.0 / dM)
+ 0.02929730258511
* cos((double) (j + 1) * M_2PI * 4.0 / dM);
}
}
else if (WindowType == wtBLACKMAN_HARRIS)
{
for (j = 0; j < M / 2; j++)
{
WinCoeff[j] = 0.35875 - 0.48829 * cos((double) (j + 1) * M_2PI / dM)
+ 0.14128 * cos((double) (j + 1) * M_2PI * 2.0 / dM)
- 0.01168 * cos((double) (j + 1) * M_2PI * 3.0 / dM);
}
}
else if (WindowType == wtBLACKMAN_NUTTALL)
{
for (j = 0; j < M / 2; j++)
{
WinCoeff[j] = 0.3535819
- 0.4891775 * cos((double) (j + 1) * M_2PI / dM)
+ 0.1365995 * cos((double) (j + 1) * M_2PI * 2.0 / dM)
- 0.0106411 * cos((double) (j + 1) * M_2PI * 3.0 / dM);
}
}
else if (WindowType == wtNUTTALL)
{
for (j = 0; j < M / 2; j++)
{
WinCoeff[j] = 0.355768
- 0.487396 * cos((double) (j + 1) * M_2PI / dM)
+ 0.144232 * cos((double) (j + 1) * M_2PI * 2.0 / dM)
- 0.012604 * cos((double) (j + 1) * M_2PI * 3.0 / dM);
}
}
else if (WindowType == wtKAISER_BESSEL)
{
for (j = 0; j <= M / 2; j++)
{
WinCoeff[j] = 0.402 - 0.498 * cos(M_2PI * (double) (j + 1) / dM)
+ 0.098 * cos(2.0 * M_2PI * (double) (j + 1) / dM)
+ 0.001 * cos(3.0 * M_2PI * (double) (j + 1) / dM);
}
}
else if (WindowType == wtTRAPEZOID) // Rectangle for Alpha = 1 Triangle for Alpha = 0
{
int K = M / 2;
if (M % 2)
K++;
for (j = 0; j < K; j++)
WinCoeff[j] = (double) (j + 1) / (double) K;
}
// This definition is from http://en.wikipedia.org/wiki/Window_function (Gauss Generalized normal window)
// We set their p = 2, and use Alpha in the numerator, instead of Sigma in the denominator, as most others do.
// Alpha = 2.718 puts the Gauss window response midway between the Hanning and the Flattop (basically what we want).
// It also gives the same BW as the Gauss window used in the HP 89410A Vector Signal Analyzer.
// Alpha = 1.8 puts it quite close to the Hanning.
else if (WindowType == wtGAUSS)
{
for (j = 0; j < M / 2; j++)
{
WinCoeff[j] = ((double) (j + 1) - dM / 2.0) / (dM / 2.0) * 2.7183;
WinCoeff[j] *= WinCoeff[j];
WinCoeff[j] = exp(-WinCoeff[j]);
}
}
else // Error.
{
std::cerr << "Incorrect window type in WindowFFTData" << std::endl;
delete[] WinCoeff;
return;
}
// Fold the coefficients over.
for (j = 0; j < M / 2; j++)
WinCoeff[N - j - 1] = WinCoeff[j];
// This is the flat top if Alpha > 0. Cannot be applied to a Kaiser or Flat Top.
if (WindowType != wtKAISER && WindowType != wtFLATTOP)
{
for (j = M / 2; j < N - M / 2; j++)
WinCoeff[j] = 1.0;
}
// This will set the gain of the window to 1. Only the Flattop window has unity gain by design.
if (UnityGain)
{
double Sum = 0.0;
for (j = 0; j < N; j++)
Sum += WinCoeff[j];
Sum /= (double) N;
if (Sum != 0.0)
for (j = 0; j < N; j++)
WinCoeff[j] /= Sum;
}
// Apply the window to the data.
for (j = 0; j < N; j++)
Data[j] *= WinCoeff[j];
delete[] WinCoeff;
}