C#实现FFT（递归法）

1. C#实现复数类

我们在进行信号分析的时候，难免会使用到复数。但是遗憾的是，C#没有自带的复数类，以下提供了一种复数类的构建方法。

复数相比于实数，可以理解为一个二维数，构建复数类，我们需要实现以下这些内容：

复数实部与虚部的属性
复数与复数的加减乘除运算
复数与实数的加减乘除运算
复数取模
复数取相位角
欧拉公式（即$e^{ix+y}$）

C#实现的代码如下：

 public class Complex

    {

        double real;

        double imag;

        public Complex(double x, double y)   //构造函数

        {

            this.real = x;

            this.imag = y;

        }

        //通过属性实现对复数实部与虚部的单独查看和设置

        public double Real

        {

            set { this.real = value; }

            get { return this.real; }

        }

        public double Imag

        {

            set { this.imag = value; }

            get { return this.imag; }

        }

        //重载加法

        public static Complex operator +(Complex c1, Complex c2)

        {

            return new Complex(c1.real + c2.real, c1.imag + c2.imag);

        }

        public static Complex operator +(double c1, Complex c2)

        {

            return new Complex(c1 + c2.real, c2.imag);

        }

        public static Complex operator +(Complex c1, double c2)

        {

            return new Complex(c1.Real + c2, c1.imag);

        }

        //重载减法

        public static Complex operator -(Complex c1, Complex c2)

        {

            return new Complex(c1.real - c2.real, c1.imag - c2.imag);

        }

        public static Complex operator -(double c1, Complex c2)

        {

            return new Complex(c1 - c2.real, -c2.imag);

        }

        public static Complex operator -(Complex c1, double c2)

        {

            return new Complex(c1.real - c2, c1.imag);

        }

        //重载乘法

        public static Complex operator *(Complex c1, Complex c2)

        {

            double cr = c1.real * c2.real - c1.imag * c2.imag;

            double ci = c1.imag * c2.real + c2.imag * c1.real;

            return new Complex(Math.Round(cr, 4), Math.Round(ci, 4));

        }

        public static Complex operator *(double c1, Complex c2)

        {

            double cr = c1 * c2.real;

            double ci = c1 * c2.imag;

            return new Complex(Math.Round(cr, 4), Math.Round(ci, 4));

        }

        public static Complex operator *(Complex c1, double c2)

        {

            double cr = c1.Real * c2;

            double ci = c1.Imag * c2;

            return new Complex(Math.Round(cr, 4), Math.Round(ci, 4));

        }

        //重载除法

        public static Complex operator /(Complex c1, Complex c2)

        {

            if (c2.real == 0 && c2.imag == 0)

            {

                return new Complex(double.NaN, double.NaN);

            }

            else

            {

                double cr = (c1.imag * c2.imag + c2.real * c1.real) / (c2.imag * c2.imag + c2.real * c2.real);

                double ci = (c1.imag * c2.real - c2.imag * c1.real) / (c2.imag * c2.imag + c2.real * c2.real);

                return new Complex(Math.Round(cr, 4), Math.Round(ci, 4));           //保留四位小数后输出

            }

        }

        public static Complex operator /(double c1, Complex c2)

        {

            if (c2.real == 0 && c2.imag == 0)

            {

                return new Complex(double.NaN, double.NaN);

            }

            else

            {

                double cr = c1 * c2.Real / (c2.imag * c2.imag + c2.real * c2.real);

                double ci = -c1 * c2.imag / (c2.imag * c2.imag + c2.real * c2.real);

                return new Complex(Math.Round(cr, 4), Math.Round(ci, 4));           //保留四位小数后输出

            }

        }

        public static Complex operator /(Complex c1, double c2)

        {

            if (c2 == 0)

            {

                return new Complex(double.NaN, double.NaN);

            }

            else

            {

                double cr = c1.Real / c2;

                double ci = c1.imag / c2;

                return new Complex(Math.Round(cr, 4), Math.Round(ci, 4));           //保留四位小数后输出

            }

        }

        //创建一个取模的方法

        public static double Abs(Complex c)

        {

            return Math.Sqrt(c.imag * c.imag + c.real * c.real);

        }

        //创建一个取相位角的方法

        public static double Angle(Complex c)

        {

            return Math.Round(Math.Atan2(c.real, c.imag), 6);//保留6位小数输出

        }

        //重载字符串转换方法,便于显示复数

        public override string ToString()

        {

            if (imag >= 0)

                return string.Format("{0}+i{1}", real, imag);

            else

                return string.Format("{0}-i{1}", real, -imag);

        }

        //欧拉公式

        public static Complex Exp(Complex c)

        {

            double amplitude = Math.Exp(c.real);

            double cr = amplitude * Math.Cos(c.imag);

            double ci = amplitude * Math.Sin(c.imag);

            return new Complex(Math.Round(cr, 4), Math.Round(ci, 4));//保留四位小数输出

        }

    }

2. 递归法实现FFT

以下的递归法是基于奇偶分解实现的。

奇偶分解的原理推导如下：

\[\begin{split}
X(k)=DFT[x(n)]&=\sum_{n=0}^{N-1}x(n)W_N^{nk}\\
&=\sum_{r=0}^{N/2-1}x(2r)W_N^{2rk}+\sum_{r=0}^{N/2-1}x(2r+1)W_N^{(2r+1)k}，将x(n)按奇偶分解\\
&=\sum_{r=0}^{N/2-1}x(2r)(W_N^{2})^{rk}+W_N^k\sum_{r=0}^{N/2-1}x(2r+1)(W_N^2)^{rk}
\end{split}
\]

$x(2r)$和$x(2r+1)$都是长度为$N/2-1$的数据序列，不妨令

\[x_1(n)=x(2r)\\
x_2(n)=x(2r+1)
\]

则原来的DFT就变成了：

\[\begin{split}
X(k)&=\sum_{n=0}^{N/2-1}x_1(n)(W_N^{2})^{nk}+W_N^k\sum_{n=0}^{N/2-1}x_2(n)(W_N^2)^{nk}\\
&=F(x_1(n))+W_N^kF(x_2(n))\\
&=X_1(k)+W_N^kX_2(k)
\end{split}
\]

于是，将原来的N点傅里叶变换变成了两个N/2点傅里叶变换的线性组合。

但是，N/2点傅里叶变换只能确定N/2个频域数据，另外N/2个数据怎么确定呢？

因为$X_1(k)$和$X_2(k)$周期都是$N/2$，所以有

\[X_1(k+N/2)=X_1(k),X_2(k+N/2)=X_2(k)\\
W_N^{k+N/2}=-W_N^k\\
\]

从而得到：

\[\begin{split}
X(k+N/2)&=X_1(k+N/2)+W_N^{k+N/2}X_2(k+N/2)\\
&=X_1(k)-W_n^kX_2(k)
\end{split}
\]

综上，我们就可以得到递归法实现FFT的流程：

对于每组数据，按奇偶分解成两组数据
两组数据分别进行傅里叶变换，得到$X_1(k)$和$X_2(k)$
总体数据的$X(k)$由下式确定：

\[X(k)==X_1(k)+W_N^kX_2(k)\\
X(k+N/2)=X_1(k)-W_n^kX_2(k)\\
0\le k \le N/2 -1
\]
对上述过程进行递归

具体代码实现如下：

public Complex[] FFTre(Complex[] c)

{

    int n = c.Length;

    Complex[] cout = new Complex[n];

    if (n == 1)

    {

        cout[0] = c[0];

        return cout;

    }

    else

    {

        double n_2_f = n / 2;

        int n_2 = (int)Math.Floor(n_2_f);

        Complex[] c1 = new Complex[n / 2];

        Complex[] c2 = new Complex[n / 2];

        for (int i = 0; i < n_2; i++)

        {

            c1[i] = c[2 * i];

            c2[i] = c[2 * i + 1];

        }

        Complex[] c1out = FFTre(c1);

        Complex[] c2out = FFTre(c2);

        Complex[] c3 = new Complex[n / 2];

        for (int i = 0; i < n / 2; i++)

        {

            c3[i] = new Complex(0, -2 * Math.PI * i / n);

        }

        for (int i = 0; i < n / 2; i++)

        {

            c2out[i] = c2out[i] * Complex.Exp(c3[i]);

        }

        for (int i = 0; i < n / 2; i++)

        {

            cout[i] = c1out[i] + c2out[i];

            cout[i + n / 2] = c1out[i] - c2out[i];

        }

        return cout;

    }

}

3. 补充：窗函数

顺便提供几个常用的窗函数：

Rectangle
Bartlett
Hamming
Hanning
Blackman

    public class WDSLib

    {

        //以下窗函数均为periodic

        public double[] Rectangle(int len)

        {

            double[] win = new double[len];

            for (int i = 0; i < len; i++)

            {

                win[i] = 1;

            }

            return win;

        }

        public double[] Bartlett(int len)

        {

            double length = (double)len - 1;

            double[] win = new double[len];

            for (int i = 0; i < len; i++)

            {

                if (i < len / 2) { win[i] = 2 * i / length; }

                else { win[i] = 2 - 2 * i / length; }

            }

            return win;

        }

        public double[] Hamming(int len)

        {

            double[] win = new double[len];

            for (int i = 0; i < len; i++)

            {

                win[i] = 0.54 - 0.46 * Math.Cos(Math.PI * 2 * i / len);

            }

            return win;

        }

        public double[] Hanning(int len)

        {

            double[] win = new double[len];

            for (int i = 0; i < len; i++)

            {

                win[i] = 0.5 * (1 - Math.Cos(2 * Math.PI * i / len));

            }

            return win;

        }

        public double[] Blackman(int len)

        {

            double[] win = new double[len];

            for (int i = 0; i < len; i++)

            {

                win[i] = 0.42 - 0.5 * Math.Cos(Math.PI * 2 * (double)i / len) + 0.08 * Math.Cos(Math.PI * 4 * (double)i / len);

            }

            return win;

        }

    }