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. 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：
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. 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;
}
}
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其它相关文章！
