目录
1. C#实现复数类2. 递归法实现FFT3. 补充：窗函数1.>
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eix+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.>
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：
x(2r)和x(2r+1)都是长度为N/2−1的数据序列，不妨令
则原来的DFT就变成了：
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X1(k)和X2(k)周期都是N/2，所以有
从而得到：
综上，我们就可以得到递归法实现FFT的流程：
1.对于每组数据，按奇偶分解成两组数据
2.两组数据分别进行傅里叶变换，得到X1(k)和X2(k)
3.总体数据的X(k)由下式确定：
4.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.>
顺便提供几个常用的窗函数：
- RectangleBartlettHammingHanningBlackman
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;
}
}
以上就是C#实现FFT(递归法)的示例代码的详细内容，更多关于C# FFT递归法的资料请关注易采站长站其它相关文章！
