基數 2 FFT

用於計算 FFT 的最簡單且可能是最著名的方法是 Radix-2 抽取時間演算法。Radix-2 FFT 通過將 N 點時域訊號分解為 N 個時域訊號來工作，每個時域訊號由單個點組成 。

通過位元反轉時域資料陣列的索引來實現訊號分解或時間抽取。因此，對於 16 點訊號，樣品 1（二元 0001）與樣品 8(1000) 交換，樣品 2(0010) 與 4（0100）交換，依此類推。使用位反轉技術的樣本交換可以簡單地在軟體中實現，但是將 Radix 2 FFT 的使用限制為長度為 N = 2 ^ M 的訊號。

時域中的 1 點訊號的值等於其在頻域中的值，因此該分解的單個時域點陣列不需要變換成為頻域點陣列。N 單點; 但是，需要重建為一個 N 點頻譜。使用蝶形計算執行完整頻譜的最佳重建。Radix-2 FFT 中的每個重建階段使用一組類似的指數加權函式 Wn ^ R 執行多個兩點蝴蝶。

FFT 利用 Wn ^ R 的週期性去除離散傅立葉變換中的冗餘計算。在蝴蝶計算的 log2(N) 階段完成光譜重建，給出 X [K]; 矩形的實數和虛數頻域資料。要轉換為幅度和相位（極座標），需要找到絕對值√（Re2 + Im2）和引數 tan-1（Im / Re）。

八點基數 2 FFT 的完整蝶形流程圖如下所示。注意，輸入訊號先前已根據先前概述的時間過程中的抽取重新排序。

FFT 通常在複雜輸入上執行併產生複雜輸出。對於實訊號，虛部可以設定為零並且實部設定為輸入訊號 x [n]，然而，涉及純實資料的變換的許多優化是可能的。可以使用指數加權方程確定在整個重建期間使用的 Wn ^ R 的值。

R 的值（指數加權功率）確定光譜重建中的當前階段和特定蝶形內的當前計算。

程式碼示例（C / C++）

用於計算 Radix 2 FFT 的 AC / C++程式碼示例可以在下面找到。這是一個簡單的實現，適用於任何大小 N，其中 N 是 2 的冪。它比最快的 FFTw 實現慢大約 3 倍，但仍然是未來優化或瞭解該演算法如何工作的非常好的基礎。

#include <math.h>

#define PI       3.1415926535897932384626433832795    // PI for sine/cos calculations
#define TWOPI    6.283185307179586476925286766559     // 2*PI for sine/cos calculations
#define Deg2Rad  0.017453292519943295769236907684886  // Degrees to Radians factor
#define Rad2Deg  57.295779513082320876798154814105    // Radians to Degrees factor
#define log10_2  0.30102999566398119521373889472449   // Log10 of 2 
#define log10_2_INV 3.3219280948873623478703194294948 // 1/Log10(2)

// complex variable structure (double precision)
struct complex
{
public:
    double  Re, Im;        // Not so complicated after all
};

// Returns true if N is a power of 2
bool isPwrTwo(int N, int *M)
{
    *M = (int)ceil(log10((double)N) * log10_2_INV);// M is number of stages to perform. 2^M = N
    int NN = (int)pow(2.0, *M);
    if ((NN != N) || (NN == 0)) // Check N is a power of 2. 
        return false;

    return true;
}

void rad2FFT(int N, complex *x, complex *DFT)
{
    int M = 0;

    // Check if power of two. If not, exit        
    if (!isPwrTwo(N, &M))
        throw "Rad2FFT(): N must be a power of 2 for Radix FFT";

    // Integer Variables

    int BSep;                  // BSep is memory spacing between butterflies
    int BWidth;                // BWidth is memory spacing of opposite ends of the butterfly
    int P;                     // P is number of similar Wn's to be used in that stage
    int j;                     // j is used in a loop to perform all calculations in each stage
    int stage = 1;             // stage is the stage number of the FFT. There are M stages in total (1 to M).
    int HiIndex;               // HiIndex is the index of the DFT array for the top value of each butterfly calc 
    unsigned int iaddr;        // bitmask for bit reversal 
    int ii;                    // Integer bitfield for bit reversal (Decimation in Time)
    int MM1 = M - 1;

    unsigned int i;
    int l;
    unsigned int nMax = (unsigned int)N;

    // Double Precision Variables
    double TwoPi_N = TWOPI / (double)N;    // constant to save computational time.  = 2*PI / N
    double TwoPi_NP;

    // complex Variables (See 'struct complex')
    complex WN;               // Wn is the exponential weighting function in the form a + jb
    complex TEMP;             // TEMP is used to save computation in the butterfly calc
    complex *pDFT = DFT;      // Pointer to first elements in DFT array
    complex *pLo;             // Pointer for lo / hi value of butterfly calcs
    complex *pHi;
    complex *pX;              // Pointer to x[n]

    // Decimation In Time - x[n] sample sorting
    for (i = 0; i < nMax; i++, DFT++)
    {
        pX = x + i;             // Calculate current x[n] from base address *x and index i.
        ii = 0;                 // Reset new address for DFT[n]
        iaddr = i;              // Copy i for manipulations
        for (l = 0; l < M; l++) // Bit reverse i and store in ii...
        {
            if (iaddr & 0x01)     // Detemine least significant bit
                ii += (1 << (MM1 - l));    // Increment ii by 2^(M-1-l) if lsb was 1
            iaddr >>= 1;                // right shift iaddr to test next bit. Use logical operations for speed increase
            if (!iaddr)
                break;
        }
        DFT = pDFT + ii;        // Calculate current DFT[n] from base address *pDFT and bit reversed index ii    
        DFT->Re = pX->Re;       // Update the complex array with address sorted time domain signal x[n]
        DFT->Im = pX->Im;       // NB: Imaginary is always zero
    }

    // FFT Computation by butterfly calculation
    for (stage = 1; stage <= M; stage++) // Loop for M stages, where 2^M = N
    {
        BSep = (int)(pow(2, stage)); // Separation between butterflies = 2^stage
        P = N / BSep;             // Similar Wn's in this stage = N/Bsep
        BWidth = BSep / 2;     // Butterfly width (spacing between opposite points) = Separation / 2.

        TwoPi_NP = TwoPi_N*P;

        for (j = 0; j < BWidth; j++) // Loop for j calculations per butterfly
        {
            if (j != 0)              // Save on calculation if R = 0, as WN^0 = (1 + j0)
            {
                //WN.Re = cos(TwoPi_NP*j)
                WN.Re = cos(TwoPi_N*P*j);     // Calculate Wn (Real and Imaginary)
                WN.Im = -sin(TwoPi_N*P*j);
            }

            for (HiIndex = j; HiIndex < N; HiIndex += BSep) // Loop for HiIndex Step BSep butterflies per stage
            {
                pHi = pDFT + HiIndex;                  // Point to higher value
                pLo = pHi + BWidth;                    // Point to lower value (Note VC++ adjusts for spacing between elements)

                if (j != 0)                            // If exponential power is not zero...
                {
                    //CMult(pLo, &WN, &TEMP);          // Perform complex multiplication of Lovalue with Wn
                    TEMP.Re = (pLo->Re * WN.Re) - (pLo->Im * WN.Im);
                    TEMP.Im = (pLo->Re * WN.Im) + (pLo->Im * WN.Re);

                    //CSub (pHi, &TEMP, pLo);
                    pLo->Re = pHi->Re - TEMP.Re;       // Find new Lovalue (complex subtraction)
                    pLo->Im = pHi->Im - TEMP.Im;

                    //CAdd (pHi, &TEMP, pHi);          // Find new Hivalue (complex addition)
                    pHi->Re = (pHi->Re + TEMP.Re);
                    pHi->Im = (pHi->Im + TEMP.Im);
                }
                else
                {
                    TEMP.Re = pLo->Re;
                    TEMP.Im = pLo->Im;

                    //CSub (pHi, &TEMP, pLo);
                    pLo->Re = pHi->Re - TEMP.Re;       // Find new Lovalue (complex subtraction)
                    pLo->Im = pHi->Im - TEMP.Im;

                    //CAdd (pHi, &TEMP, pHi);          // Find new Hivalue (complex addition)
                    pHi->Re = (pHi->Re + TEMP.Re);
                    pHi->Im = (pHi->Im + TEMP.Im);
                }
            }
        }
    }

    pLo = 0;    // Null all pointers
    pHi = 0;
    pDFT = 0;
    DFT = 0;
    pX = 0;
}