diff --git a/README.md b/README.md
new file mode 100644
index 0000000..ce165b9
--- /dev/null
+++ b/README.md
@@ -0,0 +1,3 @@
+## Fast Fourier Tranform 
+[FFT Code] (https://github.com/rorysroes/Fast_Algorithm_FFT/blob/Fast_Fourier_Transform/修正版mid.cpp)：
+This project I implemented Cooley–Tukey FFT algorithm with radix_2, radix_3 and radix_5
diff --git a/fft ver2.cpp b/fft ver2.cpp
new file mode 100644
index 0000000..75543da
--- /dev/null
+++ b/fft ver2.cpp	
@@ -0,0 +1,431 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <time.h>
+int SFT(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int FFT(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int Generate_N(int p, int q, int r);
+int Initial(double *x, double *y, int N);
+int Print_Complex_Vector(double *x, double *y, int N);
+int Bit_Increase(int *D,int b, int N);
+int Bit_Reserve(int *D, int b, int N);
+int Bit_Reserve_Integer(int N);
+int FFTver3(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int Group(int N);
+
+int main()
+{
+	int D[4];
+	//Bit_Reserve(D, 5, 4);
+	//Bit_Reserve_Integer(125);
+	//Group(8);
+	/*
+	Bit_Increase(D, 2, 4);
+	Bit_Reserve(D, 2, 4);                               // 2 ¶i¦ì 
+	*/
+	// y_k = sum(x_n * w^{-kn}, n=0..N-1)
+	// w = cos(2*pi/N)+isin(2*pi/N)
+	int k ,n ,N, p, q, r;
+	double *y_r, *y_i, *x_r, *x_i, w_r, w_i;
+	clock_t t1,t2;
+	//input N
+	printf("Please input p q r=");
+	scanf("%d %d %d", &p, &q, &r);    
+	N = Generate_N(p, q, r);
+	printf("N=2^%d 3^%d 5^%d = %d\n", p, q, r, N);                       
+	//N = 1 << p;                     //²¾¦ìºâ¤l 2^p ,¥ª²¾ p ­Ó¦ì¸m(ex:p=4 => 1= 0001 => 1000 = 16)  
+	//printf("N=%d\n",N);
+	//create memory for x and y
+	x_r = (double * ) malloc(N*sizeof(double));
+	x_i = (double * ) malloc(N*sizeof(double));
+	y_r = (double * ) malloc(N*sizeof(double));
+	y_i = (double * ) malloc(N*sizeof(double));
+	/*
+    Initial(x_r, x_i, N);
+	t1 = clock();
+    SFT(x_r, x_i, y_r, y_i, N);
+	t2 = clock();
+	printf("%f secs\n",1.0*(t2 -t1)/CLOCKS_PER_SEC);
+	//Print_Complex_Vector(y_r,y_i, N);
+	*/
+	Initial(x_r, x_i, N);
+	t1 = clock();
+	FFT(x_r, x_i, y_r, y_i, N);
+    t2 = clock();
+    printf("%f secs\n",1.0*(t2 -t1)/CLOCKS_PER_SEC);
+    //Print_Complex_Vector(y_r,y_i, N);
+    
+    Initial(x_r, x_i, N);
+	t1 = clock();
+    FFTver3(x_r, x_i, y_r, y_i, N);
+    t2 = clock();
+    printf("%f secs\n",1.0*(t2 -t1)/CLOCKS_PER_SEC);
+}
+int SFT(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+{
+	int k, n;
+	double w_r, w_i;
+	for(k=0;k<N;++k)
+	{
+		y_r[k] = 0.0;
+		y_i[k] = 0.0;
+		for(n=0;n<N;n++)
+		{
+			// w^{-kn}
+			// w^{-kn} = cos(-kn*2*pi/N)+ i sin(-kn*2*pi/N)
+			w_r = cos(-k*n*2*M_PI/N);
+			w_i = sin(-k*n*2*M_PI/N);
+			y_r[k] = y_r[k] + x_r[n]*w_r - x_i[n]*w_i;
+			y_i[k] = y_i[k] + x_r[n]*w_i + x_i[n]*w_r;
+		}
+	}
+}
+int FFT(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+{
+	if(N==1)
+	{
+		y_r[0] = x_r[0];
+		y_i[0] = x_i[0];
+	    return 0; 	
+	}
+	// input : x = x_r + i * x_i
+	// output : y = y_r + i * y_i
+	int k, n;
+	//double *u_r, *u_i, *v_r, *v_i, w_r, w_i;
+	double *even_r, *even_i, *odd_r, *odd_i, w_r, w_i;
+    double *even_FT_r, *even_FT_i, *odd_FT_r, *odd_FT_i;
+    
+    //¶}8­Ó°O¾ÐÅé 
+	even_r = (double *) malloc(N*sizeof(double));
+	even_i = (double *) malloc(N*sizeof(double));
+	//¤@¦¸¶}N­Ó 
+	odd_r = even_r + N/2; //(double *) malloc(N/2*sizeof(double)); 
+	odd_i = even_i + N/2; //(double *) malloc(N/2*sizeof(double));
+	even_FT_r = (double *) malloc(N*sizeof(double));
+	even_FT_i = (double *) malloc(N*sizeof(double));
+	odd_FT_r = even_FT_r + N/2;//(double *) malloc(N/2*sizeof(double));
+	odd_FT_i = even_FT_i + N/2;//(double *) malloc(N/2*sizeof(double));
+	for(n=0;n<N/2;n++)
+	{
+		even_r[n] = x_r[2*n];
+		even_i[n] = x_i[2*n];
+		odd_r[n] = x_r[2*n+1];
+		odd_i[n] = x_i[2*n+1];
+	}
+	FFT(even_r,even_i,even_FT_r,even_FT_i,N/2);
+	FFT(odd_r,odd_i,odd_FT_r,odd_FT_i,N/2);
+	
+	for(k=0;k<N/2;++k)
+	{
+		// w^{-k}
+		w_r = cos(-k*2*M_PI/N);
+		w_i = sin(-k*2*M_PI/N);
+		//printf("N=%d, w_r = %f, w_i = %f\n",N, w_r, w_i);
+		y_r[k] = even_FT_r[k] + (w_r*odd_FT_r[k] - w_i*odd_FT_i[k]);
+		y_i[k] = even_FT_i[k] + (w_r*odd_FT_i[k] + w_i*odd_FT_r[k]);
+		y_r[k+N/2] = even_FT_r[k] - (w_r*odd_FT_r[k] - w_i*odd_FT_i[k]);
+		y_i[k+N/2] = even_FT_i[k] - (w_r*odd_FT_i[k] + w_i*odd_FT_r[k]); 
+	    
+	}
+	//§â°O¾ÐÅéÁÙ§@·~¨t²Î 
+	free(even_r);
+	free(even_i);
+	free(even_FT_r);
+	free(even_FT_i);
+	/*
+	for(k=N/2;k<N-1;++k)
+	{
+		// w^{-k}
+		w_r = cos(-k*2*M_PI/N);
+		w_i = sin(-k*2*M_PI/N);
+		y_r[k] = even_r[k-N/2] + w_r*odd_r[k-N/2] - w_i*odd_i[K-N/2];
+		y_i[k] = even_i[k-N/2] + w_r*odd_i[k-N/2] + w_i*odd_r[k-N/2];
+	}	
+	*/
+	/*
+	u_r = (double * ) malloc(N*sizeof(double));
+	u_i = (double * ) malloc(N*sizeof(double));
+	v_r = (double * ) malloc(N*sizeof(double));
+	v_i = (double * ) malloc(N*sizeof(double));
+	
+	for(n=0;n<N/2;n++)
+	{
+		u_r[n] = x_r[2*n];
+		u_i[n] = x_i[2*n];
+		u_r[n+N/2] = x_r[2*n+1];
+		u_i[n+N/2] = x_i[2*n+1];
+	}
+	FFT(u_r, u_i, v_r, v_i, N/2);
+	FFT(u_r+N/2, u_i+N/2, v_r+N/2, v_i+N/2, N/2);
+	*/
+	return 0;
+}
+int Generate_N(int p, int q, int r)
+{
+	int N = 1,i;
+	// N = 2^p 3^q 5^r
+	for(i=0;i<p;i++) N = N * 2;
+	for(i=0;i<q;i++) N = N * 3;
+	for(i=0;i<r;i++) N = N * 5;
+	return N;
+}
+
+int Initial(double *x, double *y, int N)
+{
+	int n;
+	for(n=0;n<N;++n)
+	{
+		x[n] = n;
+		y[n] = 0;
+	}
+}
+
+int Print_Complex_Vector(double *x, double *y, int N)
+{
+	int n;
+	for(n=0;n<N;++n)
+	{
+	 	if (y[n] >=0) printf("%d : %f +%f i\n", n, x[n], y[n]);
+		else printf("%d : %f %f i\n", n, x[n], y[n]);
+	}
+}
+int Bit_Increase(int *D, int b, int N)
+{
+	int i;
+	//      D[3] D[2] D[1] D[0] 
+	//print 0000 0001 0010 0011 0100 0101 ...
+	for(i=0;i<N;++i) D[i] = 0;
+	
+	while(D[N-1] < b)
+	{
+		for(i=N-1;i>=0;i--) printf("%d", D[i]);
+		printf("\n");
+		D[0] = D[0] + 1;
+		//check every bit , if D[i] = b => D[i] = 0 , D[i+1]+1       0002 => 0010
+	    i = 0;
+	    while(D[i]==b & i < N-1)
+	    {
+	    	D[i] = 0;
+	    	i = i + 1;
+	    	D[i] = D[i] + 1; 
+		}
+		system("pause");
+    }
+    return 0;
+}
+
+int Bit_Reserve(int *D, int b, int N)
+{
+	int i;
+	//      D[3] D[2] D[1] D[0] 
+	//print 0000 1000 0100 1100 0010 1010 ...
+	for(i=0;i<N;++i) D[i] = 0;
+	
+	while(D[0] < b)
+	{
+		for(i=N-1;i>=0;i--) printf("%d", D[i]);
+		printf("\n");
+
+		//check every bit , if D[i] = b => D[i] = 0 , D[i+1]+1       2000 => 0100
+	    i = N-1;
+	    while(D[i]==b-1 & i > 0)
+	    {
+	    	D[i] = 0;
+	    	i = i - 1;
+	    	//D[i] = D[i] + 1; 
+		}
+		D[i] = D[i] + 1;
+		system("pause");
+    }
+    return 0;
+}
+int Bit_Reserve_Integer(int N)
+{
+	
+	//      N = 8       
+	//         D[2] D[1] D[0] 
+	// x print 000  100  010  110  001  101 ...
+	//   print  0    4    2    6    1    5
+	// N = 27
+    //          0 9 18 3 12 ..... 
+	// N = 125
+	//          0 25  ....              	
+	int i=0, j=0, M ;
+	
+	while(i < N)
+	{
+		printf("%d <-> %d", i, j);
+
+	    M = N/5 ;    //check 25     
+
+	    while( j >=  4 * M & M > 0)
+	    {
+	    	j = j - 4 * M ;
+	        M = M / 5 ;  
+		}	
+		j = j + M ;
+		i = i + 1 ;
+	
+		system("pause");
+    }
+
+	/*3ªºª©¥» 
+	while(i < N)
+	{
+		printf("%d <-> %d", i, j);
+
+	    M = N/3 ;    //check 9     
+
+	    while( j >= 2 * M & M > 0)
+	    {
+	    	j = j - 2*M ;
+	        M = M / 3 ;  
+		}	
+		j = j + M ;
+		i = i + 1 ;
+	
+		system("pause");
+    }
+    */
+	/*2ªºª©¥» 
+	while(i< N)
+	{
+		printf("%d <-> %d", i, j);
+
+		//check every bit , if D[i] = b => D[i] = 0 , D[i+1]+1       2000 => 0100
+	    //i = N-1;
+	    M = N/2 ;
+	    // while(D[i]==b-1 & i>0)
+	    while( j >= M & M > 0)
+	    {
+	    	//D[i] = 0;
+	    	j = j - M ;
+	    	//i = i - 1; 
+	        M = M / 2 ;  
+		}
+		//D[i] = D[i] + 1; 
+		j = j + M ;
+		i = i + 1 ;
+	
+		system("pause");
+    }
+    */
+    return 0;
+}
+
+int FFTver3(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+{
+
+	// input : x = x_r + i * x_i
+	// output : y = y_r + i * y_i
+	int n,k;
+	for(n=0;n<N;++n)
+	{
+		y_r[n] = x_r[n];
+		y_i[n] = x_r[n];
+	}
+	
+	int i=0, j=0, M ;
+	double t_r, t_i;
+	while(i < N)
+	{
+		if(i < j)
+		{
+			//swap y[i], y[j]
+			t_r = y_r[i];
+			t_i = y_i[i];
+			y_r[i] = y_r[j];
+			y_i[i] = y_i[j];
+			y_r[j] = t_r;
+			y_i[j] = t_i;
+		}
+	    M = N/2 ;        
+	    while( j >= M & M > 0)
+	    {
+	    	j = j - M ;
+	        M = M / 2 ;  
+		}	
+		j = j + M ;
+		i = i + 1 ;
+    }
+    n=1;
+    double u_r,u_i,w_r,w_i;
+	while(n < N)
+	{
+		u_r = cos(-2.0*M_PI/(2*n));
+		u_i = sin(-2.0*M_PI/(2*n));		
+		w_r = 1;
+		w_i = 0;
+		for(i=0;i<n;++i)
+		{	
+		    w_r = w_r * u_r - w_i * u_i;
+			w_i = w_r * u_i + w_i * u_r;  		
+			for(j=i;j<N;j=j+2*n)
+			{
+				k = j + n;
+				t_r = w_r*y_r[k]-w_i*y_i[k];
+				t_i = w_r*y_i[k]+w_i*y_r[k]; 
+				//jth jth + w^(-i)*kth
+				//k   jth - w^(-i)*kth
+				y_r[k] = y_r[k] - t_r;
+				y_i[k] = y_i[k] - t_r;
+				y_r[j] = y_r[j] + t_r;
+				y_i[j] = y_i[i] + t_i;
+		    } 
+		}
+		n = n * 2;
+    }
+	return 0;
+}
+
+int Group(int N)
+{
+	// N = 8
+	// ((0,1) (2,3)   (4,5) (6,7))                 Big Group number = 1
+	// ((0,2) (4,6))   ((1,3) (5,7))               Big Group number = 2
+	// ((0,4)) ((1,5)) ((2,6)) ((3,7))             Big Group number = 4
+	
+    int n = 1, i, j;
+	while(n < N)
+	{
+		printf("n=%d\n", n);
+		for(i=0;i<n;++i)
+		{
+  			printf("%d:", i); 
+			// 0
+			// 0 1
+			// 0 1 2
+			// 0 1 2 3
+			for(j=i;j<N;j=j+2*n)
+			{
+				//(0, )(2, )(4, )(6, )
+				//(0, (4,
+			    printf("(%d %d)\n",j,j+n);
+		    } 
+		}
+		n = n * 2;
+    }
+  
+}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/fft.cpp b/fft.cpp
new file mode 100644
index 0000000..e69de29
diff --git a/midexam.cpp b/midexam.cpp
new file mode 100644
index 0000000..9ad62b9
--- /dev/null
+++ b/midexam.cpp
@@ -0,0 +1,482 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <time.h>
+int SFT(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int FFT_radix_2(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int FFT_radix_3(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int FFT_radix_5(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int Initial(double *x, double *y, int N);
+int Print_Complex_Vector(double *x, double *y, int N);
+
+int main()
+{
+	// y_k = sum(x_n * w^{-kn}, n=0..N-1)
+	// w = cos(2*pi/N)+isin(2*pi/N)
+	int k ,n ,N;
+	double *y_r, *y_i, *x_r, *x_i;
+	clock_t t1,t2;
+	printf("Please input N=");
+	scanf("%d", &N);    
+	printf("N = %d\n", N);                       
+
+	x_r = (double * ) malloc(N*sizeof(double));
+	x_i = (double * ) malloc(N*sizeof(double));
+	y_r = (double * ) malloc(N*sizeof(double));
+	y_i = (double * ) malloc(N*sizeof(double));
+	/*
+    Initial(x_r, x_i, N);
+	t1 = clock();
+    SFT(x_r, x_i, y_r, y_i, N);
+	t2 = clock();
+	printf("%f secs\n",1.0*(t2 -t1)/CLOCKS_PER_SEC);
+	//Print_Complex_Vector(y_r,y_i, N);
+	*/
+    Initial(x_r, x_i, N);
+	if((N%2) == 0)
+	{
+		t1 = clock();
+		FFT_radix_2(x_r, x_i, y_r, y_i, N);
+		t2 = clock();
+		printf("%f secs\n", 1.0*(t2-t1)/CLOCKS_PER_SEC);	
+		system("pause");
+		Print_Complex_Vector(y_r, y_i, N);	
+	}
+	else if((N%3) == 0){
+		t1 = clock();
+		FFT_radix_3(x_r, x_i, y_r, y_i, N);
+		t2 = clock();
+		printf("%f secs\n", 1.0*(t2-t1)/CLOCKS_PER_SEC);	
+		system("pause");
+		Print_Complex_Vector(y_r, y_i, N);	
+	}
+	else if((N%5) == 0){
+		t1 = clock();
+		FFT_radix_5(x_r, x_i, y_r, y_i, N);
+		t2 = clock();
+		printf("%f secs\n", 1.0*(t2-t1)/CLOCKS_PER_SEC);	
+		system("pause");
+		Print_Complex_Vector(y_r, y_i, N);	
+	}
+	else {
+		printf("Error");
+		return 0;
+	}
+    return 0;
+}
+int SFT(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+{
+	int k, n;
+	double w_r, w_i;
+	for(k=0;k<N;++k)
+	{
+		y_r[k] = 0.0;
+		y_i[k] = 0.0;
+		for(n=0;n<N;n++)
+		{
+			// w^{-kn}
+			// w^{-kn} = cos(-kn*2*pi/N)+ i sin(-kn*2*pi/N)
+			w_r = cos(-k*n*2*M_PI/N);
+			w_i = sin(-k*n*2*M_PI/N);
+			y_r[k] = y_r[k] + x_r[n]*w_r - x_i[n]*w_i;
+			y_i[k] = y_i[k] + x_r[n]*w_i + x_i[n]*w_r;
+		}
+	}
+}
+
+int FFT_radix_2(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+{
+	if(N==1)
+	{
+		y_r[0] = x_r[0];
+		y_i[0] = x_i[0];
+	    return 0; 	
+	}
+	// input : x = x_r + i * x_i
+	// output : y = y_r + i * y_i
+	int k, n;
+
+	double *even_r, *even_i, *odd_r, *odd_i, w_r, w_i;
+    double *even_FT_r, *even_FT_i, *odd_FT_r, *odd_FT_i;
+
+	even_r = (double *) malloc(N*sizeof(double));
+	even_i = (double *) malloc(N*sizeof(double));
+	odd_r = even_r + N/2; //(double *) malloc(N/2*sizeof(double)); 
+	odd_i = even_i + N/2; //(double *) malloc(N/2*sizeof(double));
+	even_FT_r = (double *) malloc(N*sizeof(double));
+	even_FT_i = (double *) malloc(N*sizeof(double));
+	odd_FT_r = even_FT_r + N/2;//(double *) malloc(N/2*sizeof(double));
+	odd_FT_i = even_FT_i + N/2;//(double *) malloc(N/2*sizeof(double));
+	for(n=0;n<N/2;n++)
+	{
+		even_r[n] = x_r[2*n];
+		even_i[n] = x_i[2*n];
+		odd_r[n] = x_r[2*n+1];
+		odd_i[n] = x_i[2*n+1];
+	}
+	//FFT_radix_2(even_r,even_i,even_FT_r,even_FT_i,N/2);
+	//FFT_radix_2(odd_r,odd_i,odd_FT_r,odd_FT_i,N/2);
+	if((n%2) == 0){
+		FFT_radix_2(even_r,even_i,even_FT_r,even_FT_i,N/2);
+		FFT_radix_2(odd_r,odd_i,odd_FT_r,odd_FT_i,N/2);
+	}
+	else if((n%3) == 0){
+		FFT_radix_3(even_r,even_i,even_FT_r,even_FT_i,N/2);
+	    FFT_radix_3(odd_r,odd_i,odd_FT_r,odd_FT_i,N/2);
+	}
+	else{
+		FFT_radix_5(even_r,even_i,even_FT_r,even_FT_i,N/2);
+		FFT_radix_5(odd_r,odd_i,odd_FT_r,odd_FT_i,N/2);
+	}
+	
+    //w_r = cos(-2.0*M_PI/(2*N));
+	//w_i = sin(-2.0*M_PI/(2*N));
+	for(k=0;k<N/2;++k)
+	{
+		// w^{-k}
+		w_r = cos(-k*2*M_PI/N);
+		w_i = sin(-k*2*M_PI/N);
+		//printf("N=%d, w_r = %f, w_i = %f\n",N, w_r, w_i);
+		y_r[k] = even_FT_r[k] + (w_r*odd_FT_r[k] - w_i*odd_FT_i[k]);
+		y_i[k] = even_FT_i[k] + (w_r*odd_FT_i[k+N/2] + w_i*odd_FT_r[k]);
+		y_r[k+N/2] = even_FT_r[k] - (w_r*odd_FT_r[k] - w_i*odd_FT_i[k]);
+		y_i[k+N/2] = even_FT_i[k] - (w_r*odd_FT_i[k] + w_i*odd_FT_r[k]); 
+	    
+	}
+	free(even_r);
+	free(even_i);
+	free(even_FT_r);
+	free(even_FT_i);
+	return 0;
+}
+int FFT_radix_3(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+{
+	if(N==1)
+	{
+		y_r[0] = x_r[0];
+		y_i[0] = x_i[0];
+	    return 0; 	
+	}
+	// input : x = x_r + i * x_i
+	// output : y = y_r + i * y_i
+	int k, n;
+	double *three_1_r, *three_1_i, *three_2_r, *three_2_i, *three_3_r, *three_3_i,  w_r, w_i;
+    double *three_1_FT_r, *three_1_FT_i, *three_2_FT_r, *three_2_FT_i, *three_3_FT_r, *three_3_FT_i;
+    
+	three_1_r = (double *) malloc(N*sizeof(double));
+	three_1_i = (double *) malloc(N*sizeof(double));
+	three_2_r = three_1_r + N/3;
+	three_2_i = three_1_i + N/3; 
+	three_3_r = three_2_r + N/3;
+	three_3_i = three_2_i + N/3; 
+	three_1_FT_r = (double *) malloc(N*sizeof(double));
+	three_1_FT_i = (double *) malloc(N*sizeof(double));
+	three_2_FT_r = three_1_FT_r + N/3;
+	three_2_FT_i = three_1_FT_i + N/3;
+	three_3_FT_r = three_2_FT_r + N/3;
+	three_3_FT_i = three_2_FT_i + N/3;
+	
+	for(n=0;n<N/3;n++)
+	{
+		three_1_r[n] = x_r[3*n];
+		three_1_i[n] = x_i[3*n];
+		three_2_r[n] = x_r[3*n+1];
+		three_2_i[n] = x_i[3*n+1];
+		three_3_r[n] = x_r[3*n+2];
+	    three_3_i[n] = x_i[3*n+2];
+	}
+	
+	//FFT_radix_3(three_1_r,three_1_i,three_1_FT_r,three_1_FT_i,N/3);
+	//FFT_radix_3(three_2_r,three_2_i,three_2_FT_r,three_2_FT_i,N/3);
+	//FFT_radix_3(three_3_r,three_3_i,three_3_FT_r,three_3_FT_i,N/3);
+	
+	if((n%5) == 0){
+	  FFT_radix_5(three_1_r,three_1_i,three_1_FT_r,three_1_FT_i,N/3);
+	  FFT_radix_5(three_2_r,three_2_i,three_2_FT_r,three_2_FT_i,N/3);
+	  FFT_radix_5(three_3_r,three_3_i,three_3_FT_r,three_3_FT_i,N/3);
+	}
+	else if((n%3) == 0){
+	  FFT_radix_3(three_1_r,three_1_i,three_1_FT_r,three_1_FT_i,N/3);
+	  FFT_radix_3(three_2_r,three_2_i,three_2_FT_r,three_2_FT_i,N/3);
+	  FFT_radix_3(three_3_r,three_3_i,three_3_FT_r,three_3_FT_i,N/3);
+	}
+	else{
+	  FFT_radix_2(three_1_r,three_1_i,three_1_FT_r,three_1_FT_i,N/3);
+	  FFT_radix_2(three_2_r,three_2_i,three_2_FT_r,three_2_FT_i,N/3);
+	  FFT_radix_2(three_3_r,three_3_i,three_3_FT_r,three_3_FT_i,N/3);
+    }
+	for(k=0;k<N/3;++k)
+	{
+		//wN^{-3kn} = cos(-n*3*pi/N)+ i sin(-kn*3*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3kn})
+		//wN^{-3k(n+1)} = cos(-(n+1)*3*pi/N)+ i sin(-k(n+1)*3*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+1)})
+		//wN^{-3k(n+2)} = cos(-(n+2)*3*pi/N)+ i sin(-k(n+2)*3*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+1)})
+		
+		w_r = cos(-k*2*M_PI/N);
+		w_i = sin(-k*2*M_PI/N);
+		y_r[k] = three_1_FT_r[k] + w_r*three_1_FT_r[k+N/3] - w_i*three_1_FT_i[k+N/3];
+		y_i[k] = three_1_FT_i[k] + w_r*three_1_FT_i[k+N/3] + w_i*three_1_FT_r[k+N/3];
+		w_r = cos(-k*4*M_PI/N);
+		w_i = sin(-k*4*M_PI/N);
+		y_r[k] += w_r*three_1_FT_r[k+2*N/3] - w_i*three_1_FT_i[k+2*N/3];
+		y_i[k] += w_r*three_1_FT_i[k+2*N/3] + w_i*three_1_FT_r[k+2*N/3];
+		
+		//f(k + N/3)
+		w_r = cos(-(k+N/3)*2*M_PI/N);
+		w_i = sin(-(k+N/3)*2*M_PI/N);
+		y_r[k+N/3] = three_1_FT_r[k] + w_r*three_1_FT_r[k+N/3] - w_i*three_1_FT_i[k+N/3];
+		y_i[k+N/3] = three_1_FT_i[k] + w_r*three_1_FT_i[k+N/3] + w_i*three_1_FT_r[k+N/3];
+		w_r = cos(-(k+N/3)*4*M_PI/N);
+		w_i = sin(-(k+N/3)*4*M_PI/N);
+		y_r[k+N/3] += w_r*three_1_FT_r[k+2*N/3] - w_i*three_1_FT_i[k+2*N/3];
+		y_i[k+N/3] += w_r*three_1_FT_i[k+2*N/3] + w_i*three_1_FT_r[k+2*N/3];
+		
+		//f(k + N/3)
+		w_r = cos(-(k+2*N/3)*2*M_PI/N);
+		w_i = sin(-(k+2*N/3)*2*M_PI/N);
+		y_r[k+2*N/3] = three_1_FT_r[k] + w_r*three_1_FT_r[k+N/3] - w_i*three_1_FT_i[k+N/3];
+		y_i[k+2*N/3] = three_1_FT_i[k] + w_r*three_1_FT_i[k+N/3] + w_i*three_1_FT_r[k+N/3] ;
+		w_r = cos(-(k+2*N/3)*4*M_PI/N);
+		w_i = sin(-(k+2*N/3)*4*M_PI/N);
+		y_r[k+2*N/3] += w_r*three_1_FT_r[k+2*N/3] - w_i*three_1_FT_i[k+2*N/3];
+		y_i[k+2*N/3] += w_r*three_1_FT_i[k+2*N/3] + w_i*three_1_FT_r[k+2*N/3]; 
+	}
+	
+	free(three_1_r);
+	free(three_1_i);
+	free(three_1_FT_r);
+	free(three_1_FT_i);
+
+	return 0;
+}
+
+int FFT_radix_5(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+{
+	if(N==1)
+	{
+		y_r[0] = x_r[0];
+		y_i[0] = x_i[0];
+	    return 0; 	
+	}
+	// input : x = x_r + i * x_i
+	// output : y = y_r + i * y_i
+	int k, n;
+	double *five_1r, *five_1i, *five_2r, *five_2i, *five_3r, *five_3i, *five_4r, *five_4i, *five_5r, *five_5i, w_r, w_i;
+    double *five_1_FTr, *five_1_FTi, *five_2_FTr, *five_2_FTi, *five_3_FTr, *five_3_FTi, *five_4_FTr, *five_4_FTi, *five_5_FTr, *five_5_FTi;
+
+	five_1r = (double *) malloc(N*sizeof(double));
+	five_1i = (double *) malloc(N*sizeof(double));
+	five_2r = five_1r + N/5;
+	five_2i = five_1i + N/5; 
+	five_3r = five_2r + N/5;
+	five_3i = five_2i + N/5;
+	five_4r = five_3r + N/5;
+	five_4i = five_3i + N/5;
+	five_5r = five_4r + N/5;
+	five_5i = five_4i + N/5;
+	five_1_FTr = (double *) malloc(N*sizeof(double));
+	five_1_FTi = (double *) malloc(N*sizeof(double));
+	five_2_FTr = five_1_FTr + N/5;
+	five_2_FTi = five_1_FTi + N/5;
+	five_3_FTr = five_2_FTr + N/5;
+	five_3_FTi = five_2_FTi + N/5;
+	five_4_FTr = five_3_FTr + N/5;
+	five_4_FTi = five_3_FTi + N/5;
+	five_5_FTr = five_4_FTr + N/5;
+	five_5_FTi = five_4_FTi + N/5;	
+	
+	
+	for(n=0;n<N/5;n++)
+	{
+		five_1r[n] = x_r[5*n];
+		five_1i[n] = x_i[5*n];
+		five_2r[n] = x_r[5*n+1];
+		five_2i[n] = x_i[5*n+1];
+		five_3r[n] = x_r[5*n+2];
+	    five_3i[n] = x_i[5*n+2];
+		five_4r[n] = x_r[5*n+3];
+	    five_4i[n] = x_i[5*n+3];	    
+		five_5r[n] = x_r[5*n+4];
+	    five_5i[n] = x_i[5*n+4];	     
+	}
+	
+	//FFT_radix_5(five_1r, five_1i,five_1_FTr, five_1_FTi,N/5);
+	//FFT_radix_5(five_2r, five_2i,five_2_FTr, five_2_FTi,N/5);
+	//FFT_radix_5(five_3r, five_3i,five_3_FTr, five_3_FTi,N/5);
+	//FFT_radix_5(five_4r, five_4i,five_4_FTr, five_4_FTi,N/5);
+	//FFT_radix_5(five_5r, five_5i,five_5_FTr, five_5_FTi,N/5);
+	
+	if((n%5) == 0){
+		FFT_radix_5(five_1r, five_1i,five_1_FTr, five_1_FTi,N/5);
+		FFT_radix_5(five_2r, five_2i,five_2_FTr, five_2_FTi,N/5);
+		FFT_radix_5(five_3r, five_3i,five_3_FTr, five_3_FTi,N/5);
+		FFT_radix_5(five_4r, five_4i,five_4_FTr, five_4_FTi,N/5);
+		FFT_radix_5(five_5r, five_5i,five_5_FTr, five_5_FTi,N/5);
+	}
+	else if((n%3) == 0){
+		FFT_radix_3(five_1r, five_1i,five_1_FTr, five_1_FTi,N/5);
+		FFT_radix_3(five_2r, five_2i,five_2_FTr, five_2_FTi,N/5);
+		FFT_radix_3(five_3r, five_3i,five_3_FTr, five_3_FTi,N/5);
+		FFT_radix_3(five_4r, five_4i,five_4_FTr, five_4_FTi,N/5);
+		FFT_radix_3(five_5r, five_5i,five_5_FTr, five_5_FTi,N/5);
+	}
+	else{
+		FFT_radix_2(five_1r, five_1i,five_1_FTr, five_1_FTi,N/5);
+		FFT_radix_2(five_2r, five_2i,five_2_FTr, five_2_FTi,N/5);
+		FFT_radix_2(five_3r, five_3i,five_3_FTr, five_3_FTi,N/5);
+		FFT_radix_2(five_4r, five_4i,five_4_FTr, five_4_FTi,N/5);
+		FFT_radix_2(five_5r, five_5i,five_5_FTr, five_5_FTi,N/5);
+	}
+    for(k=0;k<N/5;++k)
+	{
+		//wN^{-5kn} = cos(-n*5*pi/N)+ i sin(-kn*5*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-5kn})
+		//wN^{-5k(n+1)} = cos(-(n+1)*5*pi/N)+ i sin(-k(n+1)*5*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+1)})
+		//wN^{-5k(n+2)} = cos(-(n+2)*5*pi/N)+ i sin(-k(n+2)*5*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+2)})
+		//wN^{-5k(n+3)} = cos(-(n+3)*5*pi/N)+ i sin(-k(n+3)*5*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+3)})
+		//wN^{-5k(n+3)} = cos(-(n+4)*5*pi/N)+ i sin(-k(n+4)*5*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+4)})
+				
+		w_r = cos(-k*2*M_PI/N);
+		w_i = sin(-k*2*M_PI/N);
+		y_r[k] = five_1_FTr[k] + w_r*five_1_FTr[k+N/5] - w_i*five_1_FTi[k+N/5];
+		y_i[k] = five_1_FTi[k] + w_r*five_1_FTi[k+N/5] + w_i*five_1_FTr[k+N/5];
+		w_r = cos(-k*4*M_PI/N);
+		w_i = sin(-k*4*M_PI/N);
+		y_r[k] += w_r*five_1_FTr[k+2*N/5] - w_i*five_1_FTi[k+2*N/5]; 
+		y_i[k] += w_r*five_1_FTi[k+2*N/5] + w_i*five_1_FTr[k+2*N/5];
+		w_r = cos(-k*6*M_PI/N);
+		w_i = sin(-k*6*M_PI/N);
+		y_r[k] += w_r*five_1_FTr[k+3*N/5] - w_i*five_1_FTi[k+3*N/5];
+		y_i[k] += w_r*five_1_FTi[k+3*N/5] + w_i*five_1_FTr[k+3*N/5];
+		w_r = cos(-k*8*M_PI/N);
+		w_i = sin(-k*8*M_PI/N);
+		y_r[k] += w_r*five_1_FTr[k+4*N/5] - w_i*five_1_FTi[k+4*N/5];
+		y_i[k] += w_r*five_1_FTi[k+4*N/5] + w_i*five_1_FTr[k+4*N/5];
+		
+		w_r = cos(-(k+N/5)*2*M_PI/N);
+		w_i = sin(-(k+N/5)*2*M_PI/N);
+		y_r[k+N/5] = five_1_FTr[k] + w_r*five_1_FTr[k+N/5] - w_i*five_1_FTi[k+N/5];
+		y_i[k+N/5] = five_1_FTi[k] + w_r*five_1_FTi[k+N/5] + w_i*five_1_FTr[k+N/5];
+		w_r = cos(-(k+N/5)*4*M_PI/N);
+		w_i = sin(-(k+N/5)*4*M_PI/N);
+		y_r[k+N/5] += w_r*five_1_FTr[k+2*N/5] - w_i*five_1_FTi[k+2*N/5]; 
+		y_i[k+N/5] += w_r*five_1_FTi[k+2*N/5] + w_i*five_1_FTr[k+2*N/5];
+		w_r = cos(-(k+N/5)*6*M_PI/N);
+		w_i = sin(-(k+N/5)*6*M_PI/N);
+		y_r[k+N/5] += w_r*five_1_FTr[k+3*N/5] - w_i*five_1_FTi[k+3*N/5];
+		y_i[k+N/5] += w_r*five_1_FTi[k+3*N/5] + w_i*five_1_FTr[k+3*N/5];
+		w_r = cos(-(k+N/5)*8*M_PI/N);
+		w_i = sin(-(k+N/5)*8*M_PI/N);
+		y_r[k+N/5] += w_r*five_1_FTr[k+4*N/5] - w_i*five_1_FTi[k+4*N/5];
+		y_i[k+N/5] += w_r*five_1_FTi[k+4*N/5] + w_i*five_1_FTr[k+4*N/5];
+		
+		w_r = cos(-(k+2*N/5)*2*M_PI/N);
+		w_i = sin(-(k+2*N/5)*2*M_PI/N);
+		y_r[k+2*N/5] = five_1_FTr[k] + w_r*five_1_FTr[k+N/5] - w_i*five_1_FTi[k+N/5];
+		y_i[k+2*N/5] = five_1_FTi[k] + w_r*five_1_FTi[k+N/5] + w_i*five_1_FTr[k+N/5];
+		w_r = cos(-(k+2*N/5)*4*M_PI/N);
+		w_i = sin(-(k+2*N/5)*4*M_PI/N);
+		y_r[k+2*N/5] += w_r*five_1_FTr[k+2*N/5] - w_i*five_1_FTi[k+2*N/5]; 
+		y_i[k+2*N/5] += w_r*five_1_FTi[k+2*N/5] + w_i*five_1_FTr[k+2*N/5];
+		w_r = cos(-(k+2*N/5)*6*M_PI/N);
+		w_i = sin(-(k+2*N/5)*6*M_PI/N);
+		y_r[k+2*N/5] += w_r*five_1_FTr[k+3*N/5] - w_i*five_1_FTi[k+3*N/5];
+		y_i[k+2*N/5] += w_r*five_1_FTi[k+3*N/5] + w_i*five_1_FTr[k+3*N/5];
+		w_r = cos(-(k+2*N/5)*8*M_PI/N);
+		w_i = sin(-(k+2*N/5)*8*M_PI/N);
+		y_r[k+2*N/5] += w_r*five_1_FTr[k+4*N/5] - w_i*five_1_FTi[k+4*N/5];
+		y_i[k+2*N/5] += w_r*five_1_FTi[k+4*N/5] + w_i*five_1_FTr[k+4*N/5];
+		
+		w_r = cos(-(k+3*N/5)*2*M_PI/N);
+		w_i = sin(-(k+3*N/5)*2*M_PI/N);
+		y_r[k+3*N/5] = five_1_FTr[k] + w_r*five_1_FTr[k+N/5] - w_i*five_1_FTi[k+N/5];
+		y_i[k+3*N/5] = five_1_FTi[k] + w_r*five_1_FTi[k+N/5] + w_i*five_1_FTr[k+N/5];
+		w_r = cos(-(k+3*N/5)*4*M_PI/N);
+		w_i = sin(-(k+3*N/5)*4*M_PI/N);
+		y_r[k+3*N/5] += w_r*five_1_FTr[k+2*N/5] - w_i*five_1_FTi[k+2*N/5]; 
+		y_i[k+3*N/5] += w_r*five_1_FTi[k+2*N/5] + w_i*five_1_FTr[k+2*N/5];
+		w_r = cos(-(k+3*N/5)*6*M_PI/N);
+		w_i = sin(-(k+3*N/5)*6*M_PI/N);
+		y_r[k+3*N/5] += w_r*five_1_FTr[k+3*N/5] - w_i*five_1_FTi[k+3*N/5];
+		y_i[k+3*N/5] += w_r*five_1_FTi[k+3*N/5] + w_i*five_1_FTr[k+3*N/5];
+		w_r = cos(-(k+3*N/5)*8*M_PI/N);
+		w_i = sin(-(k+3*N/5)*8*M_PI/N);
+		y_r[k+3*N/5] += w_r*five_1_FTr[k+4*N/5] - w_i*five_1_FTi[k+4*N/5];
+		y_i[k+3*N/5] += w_r*five_1_FTi[k+4*N/5] + w_i*five_1_FTr[k+4*N/5];
+		
+		w_r = cos(-(k+4*N/5)*2*M_PI/N);
+		w_i = sin(-(k+4*N/5)*2*M_PI/N);
+		y_r[k+4*N/5] = five_1_FTr[k] + w_r*five_1_FTr[k+N/5] - w_i*five_1_FTi[k+N/5];
+		y_i[k+4*N/5] = five_1_FTi[k] + w_r*five_1_FTi[k+N/5] + w_i*five_1_FTr[k+N/5];
+		w_r = cos(-(k+4*N/5)*4*M_PI/N);
+		w_i = sin(-(k+4*N/5)*4*M_PI/N);
+		y_r[k+4*N/5] += w_r*five_1_FTr[k+2*N/5] - w_i*five_1_FTi[k+2*N/5]; 
+		y_i[k+4*N/5] += w_r*five_1_FTi[k+2*N/5] + w_i*five_1_FTr[k+2*N/5];
+		w_r = cos(-(k+4*N/5)*6*M_PI/N);
+		w_i = sin(-(k+4*N/5)*6*M_PI/N);
+		y_r[k+4*N/5] += w_r*five_1_FTr[k+3*N/5] - w_i*five_1_FTi[k+3*N/5];
+		y_i[k+4*N/5] += w_r*five_1_FTi[k+3*N/5] + w_i*five_1_FTr[k+3*N/5];
+		w_r = cos(-(k+4*N/5)*8*M_PI/N);
+		w_i = sin(-(k+4*N/5)*8*M_PI/N);
+		y_r[k+4*N/5] += w_r*five_1_FTr[k+4*N/5] - w_i*five_1_FTi[k+4*N/5];
+		y_i[k+4*N/5] += w_r*five_1_FTi[k+4*N/5] + w_i*five_1_FTr[k+4*N/5];
+	}
+	
+	free(five_1r);
+	free(five_1i);
+	free(five_1_FTr);
+	free(five_1_FTi);
+	
+	return 0;
+	
+}
+
+int Initial(double *x, double *y, int N)
+{
+	int n;
+	for(n=0;n<N;++n)
+	{
+		x[n] = n;
+		y[n] = 0;
+	}
+}
+
+int Print_Complex_Vector(double *x, double *y, int N)
+{
+	int n;
+	for(n=0;n<N;++n)
+	{
+	 	if (y[n] >=0) printf("%d : %f +%f i\n", n, x[n], y[n]);
+		else printf("%d : %f %f i\n", n, x[n], y[n]);
+	}
+}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git "a/\344\277\256\346\255\243\347\211\210mid.cpp" "b/\344\277\256\346\255\243\347\211\210mid.cpp"
new file mode 100644
index 0000000..3c0257e
--- /dev/null
+++ "b/\344\277\256\346\255\243\347\211\210mid.cpp"
@@ -0,0 +1,477 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <time.h>
+int SFT(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int FFT_radix_2(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int FFT_radix_3(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int FFT_radix_5(double *x_r, double *x_i, double *y_r, double *y_i, int N);
+int Initial(double *x, double *y, int N);
+int Print_Complex_Vector(double *x, double *y, int N);
+
+int main()
+{
+	// y_k = sum(x_n * w^{-kn}, n=0..N-1)
+	// w = cos(2*pi/N)+isin(2*pi/N)
+	int k ,n ,N=1,q,p,r;
+	double *y_r, *y_i, *x_r, *x_i;
+	clock_t t1,t2;
+	printf("Please input p q r=");
+	scanf("%d %d %d", &p,&q,&r);
+	for(int i = 0;i<p ;i++)N*=2;    
+	for(int i = 0;i<q ;i++)N*=3;    
+	for(int i = 0;i<r ;i++)N*=5;    
+	printf("N = %d\n", N);                       
+
+	x_r = (double * ) malloc(N*sizeof(double));
+	x_i = (double * ) malloc(N*sizeof(double));
+	y_r = (double * ) malloc(N*sizeof(double));
+	y_i = (double * ) malloc(N*sizeof(double));
+	/*
+    Initial(x_r, x_i, N);
+	t1 = clock();
+    SFT(x_r, x_i, y_r, y_i, N);
+	t2 = clock();
+	printf("%f secs\n",1.0*(t2 -t1)/CLOCKS_PER_SEC);
+	Print_Complex_Vector(y_r,y_i, N);
+	*/
+    Initial(x_r, x_i, N);
+	if((N%2) == 0)
+	{
+		t1 = clock();
+		FFT_radix_2(x_r, x_i, y_r, y_i, N);
+		t2 = clock();
+		printf("%f secs\n", 1.0*(t2-t1)/CLOCKS_PER_SEC);	
+		system("pause");
+		Print_Complex_Vector(y_r, y_i, N);	
+	}
+	else if((N%3) == 0){
+		t1 = clock();
+		FFT_radix_3(x_r, x_i, y_r, y_i, N);
+		t2 = clock();
+		printf("%f secs\n", 1.0*(t2-t1)/CLOCKS_PER_SEC);	
+		system("pause");
+		Print_Complex_Vector(y_r, y_i, N);	
+	}
+	else if((N%5) == 0){
+		t1 = clock();
+		FFT_radix_5(x_r, x_i, y_r, y_i, N);
+		t2 = clock();
+		printf("%f secs\n", 1.0*(t2-t1)/CLOCKS_PER_SEC);	
+		system("pause");
+		Print_Complex_Vector(y_r, y_i, N);	
+	}
+	else {
+		printf("Error");
+		return 0;
+	}
+    return 0;
+}
+int SFT(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+{
+	int k, n;
+	double w_r, w_i;
+	for(k=0;k<N;++k)
+	{
+		y_r[k] = 0.0;
+		y_i[k] = 0.0;
+		for(n=0;n<N;n++)
+		{
+			// w^{-kn}
+			// w^{-kn} = cos(-kn*2*pi/N)+ i sin(-kn*2*pi/N)
+			w_r = cos(-k*n*2*M_PI/N);
+			w_i = sin(-k*n*2*M_PI/N);
+			y_r[k] = y_r[k] + x_r[n]*w_r - x_i[n]*w_i;
+			y_i[k] = y_i[k] + x_r[n]*w_i + x_i[n]*w_r;
+		}
+	}
+}
+
+int FFT_radix_2(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+{
+	if(N==1)
+	{
+		y_r[0] = x_r[0];
+		y_i[0] = x_i[0];
+	    return 0; 	
+	}
+	// input : x = x_r + i * x_i
+	// output : y = y_r + i * y_i
+	int k, n;
+
+	double *even_r, *even_i, *odd_r, *odd_i, w_r, w_i;
+    double *even_FT_r, *even_FT_i, *odd_FT_r, *odd_FT_i;
+
+	even_r = (double *) malloc(N*sizeof(double));
+	even_i = (double *) malloc(N*sizeof(double));
+	odd_r = even_r + N/2; //(double *) malloc(N/2*sizeof(double));
+	odd_i = even_i + N/2; //(double *) malloc(N/2*sizeof(double));
+	even_FT_r = (double *) malloc(N*sizeof(double));
+	even_FT_i = (double *) malloc(N*sizeof(double));
+	odd_FT_r = even_FT_r + N/2;//(double *) malloc(N/2*sizeof(double));
+	odd_FT_i = even_FT_i + N/2;//(double *) malloc(N/2*sizeof(double));
+	
+	for(n=0;n<N/2;++n)
+	{
+		even_r[n] = x_r[2*n];
+		even_i[n] = x_i[2*n];
+		odd_r[n] = x_r[2*n+1];
+		odd_i[n] = x_i[2*n+1];
+	}
+	//FFT_radix_2(even_r,even_i,even_FT_r,even_FT_i,N/2);
+	//FFT_radix_2(odd_r,odd_i,odd_FT_r,odd_FT_i,N/2);
+	if((n%2) == 0){
+		FFT_radix_2(even_r,even_i,even_FT_r,even_FT_i,N/2);
+		FFT_radix_2(odd_r,odd_i,odd_FT_r,odd_FT_i,N/2);
+	}
+	else if((n%3) == 0){
+		FFT_radix_3(even_r,even_i,even_FT_r,even_FT_i,N/2);
+	    FFT_radix_3(odd_r,odd_i,odd_FT_r,odd_FT_i,N/2);
+	}
+	else if(n%5==0){
+		FFT_radix_5(even_r,even_i,even_FT_r,even_FT_i,N/2);
+		FFT_radix_5(odd_r,odd_i,odd_FT_r,odd_FT_i,N/2);
+	}else{
+	    FFT_radix_2(even_r,even_i,even_FT_r,even_FT_i,N/2);
+		FFT_radix_2(odd_r,odd_i,odd_FT_r,odd_FT_i,N/2);	
+	}
+	
+	for(k=0;k<N/2;++k)
+	{
+		// w^{-k}
+		w_r = cos(-k*2*M_PI/N);
+		w_i = sin(-k*2*M_PI/N);
+		//printf("N=%d, w_r = %f, w_i = %f\n",N, w_r, w_i);
+		y_r[k] = even_FT_r[k] + (w_r*odd_FT_r[k] - w_i*odd_FT_i[k]);
+		y_i[k] = even_FT_i[k] + (w_r*odd_FT_i[k] + w_i*odd_FT_r[k]);
+		y_r[k+N/2] = even_FT_r[k] - (w_r*odd_FT_r[k] - w_i*odd_FT_i[k]);
+		y_i[k+N/2] = even_FT_i[k] - (w_r*odd_FT_i[k] + w_i*odd_FT_r[k]); 
+	    
+	}
+	free(even_r);
+	free(even_i);
+	free(even_FT_r);
+	free(even_FT_i);
+	return 0;
+}
+
+int FFT_radix_3(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+{
+	if(N==1)
+	{
+		y_r[0] = x_r[0];
+		y_i[0] = x_i[0];
+	    return 0; 	
+	}
+	// input : x = x_r + i * x_i
+	// output : y = y_r + i * y_i
+	int k, n;
+	double *three_1_r, *three_1_i, *three_2_r, *three_2_i, *three_3_r, *three_3_i,  w_r, w_i;
+    double *three_1_FT_r, *three_1_FT_i, *three_2_FT_r, *three_2_FT_i, *three_3_FT_r, *three_3_FT_i;
+    
+	three_1_r = (double *) malloc(N*sizeof(double));
+	three_1_i = (double *) malloc(N*sizeof(double));
+	three_2_r = three_1_r + N/3;
+	three_2_i = three_1_i + N/3; 
+	three_3_r = three_2_r + N/3;
+	three_3_i = three_2_i + N/3; 
+	three_1_FT_r = (double *) malloc(N*sizeof(double));
+	three_1_FT_i = (double *) malloc(N*sizeof(double));
+	three_2_FT_r = three_1_FT_r + N/3;
+	three_2_FT_i = three_1_FT_i + N/3;
+	three_3_FT_r = three_2_FT_r + N/3;
+	three_3_FT_i = three_2_FT_i + N/3;
+	
+	for(n=0;n<N/3;n++)
+	{
+		three_1_r[n] = x_r[3*n];
+		three_1_i[n] = x_i[3*n];
+		three_2_r[n] = x_r[3*n+1];
+		three_2_i[n] = x_i[3*n+1];
+		three_3_r[n] = x_r[3*n+2];
+	    three_3_i[n] = x_i[3*n+2];
+	}
+	
+	//FFT_radix_3(three_1_r,three_1_i,three_1_FT_r,three_1_FT_i,N/3);
+	//FFT_radix_3(three_2_r,three_2_i,three_2_FT_r,three_2_FT_i,N/3);
+	//FFT_radix_3(three_3_r,three_3_i,three_3_FT_r,three_3_FT_i,N/3);
+	
+	if((n%5) == 0){
+	  FFT_radix_5(three_1_r,three_1_i,three_1_FT_r,three_1_FT_i,N/3);
+	  FFT_radix_5(three_2_r,three_2_i,three_2_FT_r,three_2_FT_i,N/3);
+	  FFT_radix_5(three_3_r,three_3_i,three_3_FT_r,three_3_FT_i,N/3);
+	}
+	else if((n%3) == 0){
+	  FFT_radix_3(three_1_r,three_1_i,three_1_FT_r,three_1_FT_i,N/3);
+	  FFT_radix_3(three_2_r,three_2_i,three_2_FT_r,three_2_FT_i,N/3);
+	  FFT_radix_3(three_3_r,three_3_i,three_3_FT_r,three_3_FT_i,N/3);
+	}	
+	else if((n%2) == 0){
+	  FFT_radix_2(three_1_r,three_1_i,three_1_FT_r,three_1_FT_i,N/3);
+	  FFT_radix_2(three_2_r,three_2_i,three_2_FT_r,three_2_FT_i,N/3);
+	  FFT_radix_2(three_3_r,three_3_i,three_3_FT_r,three_3_FT_i,N/3);
+    }else{
+      FFT_radix_3(three_1_r,three_1_i,three_1_FT_r,three_1_FT_i,N/3);
+	  FFT_radix_3(three_2_r,three_2_i,three_2_FT_r,three_2_FT_i,N/3);
+	  FFT_radix_3(three_3_r,three_3_i,three_3_FT_r,three_3_FT_i,N/3);
+	}
+	
+	for(k=0;k<N/3;++k)
+	{
+		//wN^{-3kn} = cos(-n*3*pi/N)+ i sin(-kn*3*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3kn})
+		//wN^{-3k(n+1)} = cos(-(n+1)*3*pi/N)+ i sin(-k(n+1)*3*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+1)})
+		//wN^{-3k(n+2)} = cos(-(n+2)*3*pi/N)+ i sin(-k(n+2)*3*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+1)})
+		
+		w_r = cos(-k*2*M_PI/N);
+		w_i = sin(-k*2*M_PI/N);
+		y_r[k] = three_1_FT_r[k] + w_r*three_1_FT_r[k+N/3] - w_i*three_1_FT_i[k+N/3];
+		y_i[k] = three_1_FT_i[k] + w_r*three_1_FT_i[k+N/3] + w_i*three_1_FT_r[k+N/3];
+		w_r = cos(-k*4*M_PI/N);
+		w_i = sin(-k*4*M_PI/N);
+		y_r[k] += w_r*three_1_FT_r[k+2*N/3] - w_i*three_1_FT_i[k+2*N/3];
+		y_i[k] += w_r*three_1_FT_i[k+2*N/3] + w_i*three_1_FT_r[k+2*N/3];
+		
+		//f(k + N/3)
+		w_r = cos(-(k+N/3)*2*M_PI/N);
+		w_i = sin(-(k+N/3)*2*M_PI/N);
+		y_r[k+N/3] = three_1_FT_r[k] + w_r*three_1_FT_r[k+N/3] - w_i*three_1_FT_i[k+N/3];
+		y_i[k+N/3] = three_1_FT_i[k] + w_r*three_1_FT_i[k+N/3] + w_i*three_1_FT_r[k+N/3];
+		w_r = cos(-(k+N/3)*4*M_PI/N);
+		w_i = sin(-(k+N/3)*4*M_PI/N);
+		y_r[k+N/3] += w_r*three_1_FT_r[k+2*N/3] - w_i*three_1_FT_i[k+2*N/3];
+		y_i[k+N/3] += w_r*three_1_FT_i[k+2*N/3] + w_i*three_1_FT_r[k+2*N/3];
+		
+		//f(k + N/3)
+		w_r = cos(-(k+2*N/3)*2*M_PI/N);
+		w_i = sin(-(k+2*N/3)*2*M_PI/N);
+		y_r[k+2*N/3] = three_1_FT_r[k] + w_r*three_1_FT_r[k+N/3] - w_i*three_1_FT_i[k+N/3];
+		y_i[k+2*N/3] = three_1_FT_i[k] + w_r*three_1_FT_i[k+N/3] + w_i*three_1_FT_r[k+N/3] ;
+		w_r = cos(-(k+2*N/3)*4*M_PI/N);
+		w_i = sin(-(k+2*N/3)*4*M_PI/N);
+		y_r[k+2*N/3] += w_r*three_1_FT_r[k+2*N/3] - w_i*three_1_FT_i[k+2*N/3];
+		y_i[k+2*N/3] += w_r*three_1_FT_i[k+2*N/3] + w_i*three_1_FT_r[k+2*N/3]; 
+	}
+	
+	free(three_1_r);
+	free(three_1_i);
+	free(three_1_FT_r);
+	free(three_1_FT_i);
+
+	return 0;
+}
+int FFT_radix_5(double *x_r, double *x_i, double *y_r, double *y_i, int N)
+ {
+ 
+	if(N==1)
+	{
+		y_r[0] = x_r[0];
+		y_i[0] = x_i[0];
+	    return 0; 	
+	}
+	// input : x = x_r + i * x_i
+	// output : y = y_r + i * y_i
+	int k, n;
+	double *five_1r, *five_1i, *five_2r, *five_2i, *five_3r, *five_3i, *five_4r, *five_4i, *five_5r, *five_5i, w_r, w_i;
+    double *five_1_FTr, *five_1_FTi, *five_2_FTr, *five_2_FTi, *five_3_FTr, *five_3_FTi, *five_4_FTr, *five_4_FTi, *five_5_FTr, *five_5_FTi;
+
+	five_1r = (double *) malloc(N*sizeof(double));
+	five_1i = (double *) malloc(N*sizeof(double));
+	five_2r = five_1r + N/5;
+	five_2i = five_1i + N/5; 
+	five_3r = five_2r + N/5;
+	five_3i = five_2i + N/5;
+	five_4r = five_3r + N/5;
+	five_4i = five_3i + N/5;
+	five_5r = five_4r + N/5;
+	five_5i = five_4i + N/5;
+	five_1_FTr = (double *) malloc(N*sizeof(double));
+	five_1_FTi = (double *) malloc(N*sizeof(double));
+	five_2_FTr = five_1_FTr + N/5;
+	five_2_FTi = five_1_FTi + N/5;
+	five_3_FTr = five_2_FTr + N/5;
+	five_3_FTi = five_2_FTi + N/5;
+	five_4_FTr = five_3_FTr + N/5;
+	five_4_FTi = five_3_FTi + N/5;
+	five_5_FTr = five_4_FTr + N/5;
+	five_5_FTi = five_4_FTi + N/5;	
+	
+	
+	for(n=0;n<N/5;n++)
+	{
+		five_1r[n] = x_r[5*n];
+		five_1i[n] = x_i[5*n];
+		five_2r[n] = x_r[5*n+1];
+		five_2i[n] = x_i[5*n+1];
+		five_3r[n] = x_r[5*n+2];
+	    five_3i[n] = x_i[5*n+2];
+		five_4r[n] = x_r[5*n+3];
+	    five_4i[n] = x_i[5*n+3];	    
+		five_5r[n] = x_r[5*n+4];
+	    five_5i[n] = x_i[5*n+4];	     
+	}
+	
+	//FFT_radix_5(five_1r, five_1i,five_1_FTr, five_1_FTi,N/5);
+	//FFT_radix_5(five_2r, five_2i,five_2_FTr, five_2_FTi,N/5);
+	//FFT_radix_5(five_3r, five_3i,five_3_FTr, five_3_FTi,N/5);
+	//FFT_radix_5(five_4r, five_4i,five_4_FTr, five_4_FTi,N/5);
+	//FFT_radix_5(five_5r, five_5i,five_5_FTr, five_5_FTi,N/5);
+	
+	if((n%5) == 0){
+		FFT_radix_5(five_1r, five_1i,five_1_FTr, five_1_FTi,N/5);
+		FFT_radix_5(five_2r, five_2i,five_2_FTr, five_2_FTi,N/5);
+		FFT_radix_5(five_3r, five_3i,five_3_FTr, five_3_FTi,N/5);
+		FFT_radix_5(five_4r, five_4i,five_4_FTr, five_4_FTi,N/5);
+		FFT_radix_5(five_5r, five_5i,five_5_FTr, five_5_FTi,N/5);
+	}	
+	else if((n%3) == 0){
+		FFT_radix_3(five_1r, five_1i,five_1_FTr, five_1_FTi,N/5);
+		FFT_radix_3(five_2r, five_2i,five_2_FTr, five_2_FTi,N/5);
+		FFT_radix_3(five_3r, five_3i,five_3_FTr, five_3_FTi,N/5);
+		FFT_radix_3(five_4r, five_4i,five_4_FTr, five_4_FTi,N/5);
+		FFT_radix_3(five_5r, five_5i,five_5_FTr, five_5_FTi,N/5);
+	}
+	else if((n%2) == 0){
+		FFT_radix_2(five_1r, five_1i,five_1_FTr, five_1_FTi,N/5);
+		FFT_radix_2(five_2r, five_2i,five_2_FTr, five_2_FTi,N/5);
+		FFT_radix_2(five_3r, five_3i,five_3_FTr, five_3_FTi,N/5);
+		FFT_radix_2(five_4r, five_4i,five_4_FTr, five_4_FTi,N/5);
+		FFT_radix_2(five_5r, five_5i,five_5_FTr, five_5_FTi,N/5);
+	}else{
+		FFT_radix_5(five_1r, five_1i,five_1_FTr, five_1_FTi,N/5);
+		FFT_radix_5(five_2r, five_2i,five_2_FTr, five_2_FTi,N/5);
+		FFT_radix_5(five_3r, five_3i,five_3_FTr, five_3_FTi,N/5);
+		FFT_radix_5(five_4r, five_4i,five_4_FTr, five_4_FTi,N/5);
+		FFT_radix_5(five_5r, five_5i,five_5_FTr, five_5_FTi,N/5);
+		
+	}
+    for(k=0;k<N/5;++k)
+	{
+		//wN^{-5kn} = cos(-n*5*pi/N)+ i sin(-kn*5*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-5kn})
+		//wN^{-5k(n+1)} = cos(-(n+1)*5*pi/N)+ i sin(-k(n+1)*5*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+1)})
+		//wN^{-5k(n+2)} = cos(-(n+2)*5*pi/N)+ i sin(-k(n+2)*5*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+2)})
+		//wN^{-5k(n+3)} = cos(-(n+3)*5*pi/N)+ i sin(-k(n+3)*5*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+3)})
+		//wN^{-5k(n+3)} = cos(-(n+4)*5*pi/N)+ i sin(-k(n+4)*5*pi/N)
+		//f(k) = sum((x_r + i * x_i) * wN^{-3k(n+4)})
+				
+		w_r = cos(-k*2*M_PI/N);
+		w_i = sin(-k*2*M_PI/N);
+		y_r[k] = five_1_FTr[k] + w_r*five_1_FTr[k+N/5] - w_i*five_1_FTi[k+N/5];
+		y_i[k] = five_1_FTi[k] + w_r*five_1_FTi[k+N/5] + w_i*five_1_FTr[k+N/5];
+		w_r = cos(-k*4*M_PI/N);
+		w_i = sin(-k*4*M_PI/N);
+		y_r[k] += w_r*five_1_FTr[k+2*N/5] - w_i*five_1_FTi[k+2*N/5]; 
+		y_i[k] += w_r*five_1_FTi[k+2*N/5] + w_i*five_1_FTr[k+2*N/5];
+		w_r = cos(-k*6*M_PI/N);
+		w_i = sin(-k*6*M_PI/N);
+		y_r[k] += w_r*five_1_FTr[k+3*N/5] - w_i*five_1_FTi[k+3*N/5];
+		y_i[k] += w_r*five_1_FTi[k+3*N/5] + w_i*five_1_FTr[k+3*N/5];
+		w_r = cos(-k*8*M_PI/N);
+		w_i = sin(-k*8*M_PI/N);
+		y_r[k] += w_r*five_1_FTr[k+4*N/5] - w_i*five_1_FTi[k+4*N/5];
+		y_i[k] += w_r*five_1_FTi[k+4*N/5] + w_i*five_1_FTr[k+4*N/5];
+		
+		w_r = cos(-(k+N/5)*2*M_PI/N);
+		w_i = sin(-(k+N/5)*2*M_PI/N);
+		y_r[k+N/5] = five_1_FTr[k] + w_r*five_1_FTr[k+N/5] - w_i*five_1_FTi[k+N/5];
+		y_i[k+N/5] = five_1_FTi[k] + w_r*five_1_FTi[k+N/5] + w_i*five_1_FTr[k+N/5];
+		w_r = cos(-(k+N/5)*4*M_PI/N);
+		w_i = sin(-(k+N/5)*4*M_PI/N);
+		y_r[k+N/5] += w_r*five_1_FTr[k+2*N/5] - w_i*five_1_FTi[k+2*N/5]; 
+		y_i[k+N/5] += w_r*five_1_FTi[k+2*N/5] + w_i*five_1_FTr[k+2*N/5];
+		w_r = cos(-(k+N/5)*6*M_PI/N);
+		w_i = sin(-(k+N/5)*6*M_PI/N);
+		y_r[k+N/5] += w_r*five_1_FTr[k+3*N/5] - w_i*five_1_FTi[k+3*N/5];
+		y_i[k+N/5] += w_r*five_1_FTi[k+3*N/5] + w_i*five_1_FTr[k+3*N/5];
+		w_r = cos(-(k+N/5)*8*M_PI/N);
+		w_i = sin(-(k+N/5)*8*M_PI/N);
+		y_r[k+N/5] += w_r*five_1_FTr[k+4*N/5] - w_i*five_1_FTi[k+4*N/5];
+		y_i[k+N/5] += w_r*five_1_FTi[k+4*N/5] + w_i*five_1_FTr[k+4*N/5];
+		
+		w_r = cos(-(k+2*N/5)*2*M_PI/N);
+		w_i = sin(-(k+2*N/5)*2*M_PI/N);
+		y_r[k+2*N/5] = five_1_FTr[k] + w_r*five_1_FTr[k+N/5] - w_i*five_1_FTi[k+N/5];
+		y_i[k+2*N/5] = five_1_FTi[k] + w_r*five_1_FTi[k+N/5] + w_i*five_1_FTr[k+N/5];
+		w_r = cos(-(k+2*N/5)*4*M_PI/N);
+		w_i = sin(-(k+2*N/5)*4*M_PI/N);
+		y_r[k+2*N/5] += w_r*five_1_FTr[k+2*N/5] - w_i*five_1_FTi[k+2*N/5]; 
+		y_i[k+2*N/5] += w_r*five_1_FTi[k+2*N/5] + w_i*five_1_FTr[k+2*N/5];
+		w_r = cos(-(k+2*N/5)*6*M_PI/N);
+		w_i = sin(-(k+2*N/5)*6*M_PI/N);
+		y_r[k+2*N/5] += w_r*five_1_FTr[k+3*N/5] - w_i*five_1_FTi[k+3*N/5];
+		y_i[k+2*N/5] += w_r*five_1_FTi[k+3*N/5] + w_i*five_1_FTr[k+3*N/5];
+		w_r = cos(-(k+2*N/5)*8*M_PI/N);
+		w_i = sin(-(k+2*N/5)*8*M_PI/N);
+		y_r[k+2*N/5] += w_r*five_1_FTr[k+4*N/5] - w_i*five_1_FTi[k+4*N/5];
+		y_i[k+2*N/5] += w_r*five_1_FTi[k+4*N/5] + w_i*five_1_FTr[k+4*N/5];
+		
+		w_r = cos(-(k+3*N/5)*2*M_PI/N);
+		w_i = sin(-(k+3*N/5)*2*M_PI/N);
+		y_r[k+3*N/5] = five_1_FTr[k] + w_r*five_1_FTr[k+N/5] - w_i*five_1_FTi[k+N/5];
+		y_i[k+3*N/5] = five_1_FTi[k] + w_r*five_1_FTi[k+N/5] + w_i*five_1_FTr[k+N/5];
+		w_r = cos(-(k+3*N/5)*4*M_PI/N);
+		w_i = sin(-(k+3*N/5)*4*M_PI/N);
+		y_r[k+3*N/5] += w_r*five_1_FTr[k+2*N/5] - w_i*five_1_FTi[k+2*N/5]; 
+		y_i[k+3*N/5] += w_r*five_1_FTi[k+2*N/5] + w_i*five_1_FTr[k+2*N/5];
+		w_r = cos(-(k+3*N/5)*6*M_PI/N);
+		w_i = sin(-(k+3*N/5)*6*M_PI/N);
+		y_r[k+3*N/5] += w_r*five_1_FTr[k+3*N/5] - w_i*five_1_FTi[k+3*N/5];
+		y_i[k+3*N/5] += w_r*five_1_FTi[k+3*N/5] + w_i*five_1_FTr[k+3*N/5];
+		w_r = cos(-(k+3*N/5)*8*M_PI/N);
+		w_i = sin(-(k+3*N/5)*8*M_PI/N);
+		y_r[k+3*N/5] += w_r*five_1_FTr[k+4*N/5] - w_i*five_1_FTi[k+4*N/5];
+		y_i[k+3*N/5] += w_r*five_1_FTi[k+4*N/5] + w_i*five_1_FTr[k+4*N/5];
+		
+		w_r = cos(-(k+4*N/5)*2*M_PI/N);
+		w_i = sin(-(k+4*N/5)*2*M_PI/N);
+		y_r[k+4*N/5] = five_1_FTr[k] + w_r*five_1_FTr[k+N/5] - w_i*five_1_FTi[k+N/5];
+		y_i[k+4*N/5] = five_1_FTi[k] + w_r*five_1_FTi[k+N/5] + w_i*five_1_FTr[k+N/5];
+		w_r = cos(-(k+4*N/5)*4*M_PI/N);
+		w_i = sin(-(k+4*N/5)*4*M_PI/N);
+		y_r[k+4*N/5] += w_r*five_1_FTr[k+2*N/5] - w_i*five_1_FTi[k+2*N/5]; 
+		y_i[k+4*N/5] += w_r*five_1_FTi[k+2*N/5] + w_i*five_1_FTr[k+2*N/5];
+		w_r = cos(-(k+4*N/5)*6*M_PI/N);
+		w_i = sin(-(k+4*N/5)*6*M_PI/N);
+		y_r[k+4*N/5] += w_r*five_1_FTr[k+3*N/5] - w_i*five_1_FTi[k+3*N/5];
+		y_i[k+4*N/5] += w_r*five_1_FTi[k+3*N/5] + w_i*five_1_FTr[k+3*N/5];
+		w_r = cos(-(k+4*N/5)*8*M_PI/N);
+		w_i = sin(-(k+4*N/5)*8*M_PI/N);
+		y_r[k+4*N/5] += w_r*five_1_FTr[k+4*N/5] - w_i*five_1_FTi[k+4*N/5];
+		y_i[k+4*N/5] += w_r*five_1_FTi[k+4*N/5] + w_i*five_1_FTr[k+4*N/5];
+	}
+	
+	free(five_1r);
+	free(five_1i);
+	free(five_1_FTr);
+	free(five_1_FTi);
+	
+	return 0;
+	
+}
+
+int Initial(double *x, double *y, int N)
+{
+	int n;
+	for(n=0;n<N;++n)
+	{
+		x[n] = n;
+		y[n] = 0;
+	}
+}
+
+int Print_Complex_Vector(double *x, double *y, int N)
+{
+	int n;
+	for(n=0;n<N;++n)
+	{
+	 	if (y[n] >=0) printf("%d : %f +%f i\n", n, x[n], y[n]);
+		else printf("%d : %f %f i\n", n, x[n], y[n]);
+	}
+}
+