Algorithms Fast Fourier Transform suggest change

Radix 2 FFT

The simplest and perhaps best-known method for computing the FFT is the Radix-2 Decimation in Time algorithm. The Radix-2 FFT works by decomposing an N point time domain signal into N time domain signals each composed of a single point.

Signal decomposition, or ‘decimation in time’ is achieved by bit reversing the indices for the array of time domain data. Thus, for a sixteen-point signal, sample 1 (Binary 0001) is swapped with sample 8 (1000), sample 2 (0010) is swapped with 4 (0100) and so on. Sample swapping using the bit reverse technique can be achieved simply in software, but limits the use of the Radix 2 FFT to signals of length N = 2^M.

The value of a 1-point signal in the time domain is equal to its value in the frequency domain, thus this array of decomposed single time-domain points requires no transformation to become an array of frequency domain points. The N single points; however, need to be reconstructed into one N-point frequency spectra. Optimal reconstruction of the complete frequency spectrum is performed using butterfly calculations. Each reconstruction stage in the Radix-2 FFT performs a number of two point butterflies, using a similar set of exponential weighting functions, Wn^R.

The FFT removes redundant calculations in the Discrete Fourier Transform by exploiting the periodicity of Wn^R. Spectral reconstruction is completed in log2(N) stages of butterfly calculations giving X[K]; the real and imaginary frequency domain data in rectangular form. To convert to magnitude and phase (polar coordinates) requires finding the absolute value, √(Re2 + Im2), and argument, tan-1(Im/Re).

The complete butterfly flow diagram for an eight point Radix 2 FFT is shown below. Note the input signals have previously been reordered according to the decimation in time procedure outlined previously.

The FFT typically operates on complex inputs and produces a complex output. For real signals, the imaginary part may be set to zero and real part set to the input signal, x[n], however many optimisations are possible involving the transformation of real-only data. Values of Wn^R used throughout the reconstruction can be determined using the exponential weighting equation.

The value of R (the exponential weighting power) is determined the current stage in the spectral reconstruction and the current calculation within a particular butterfly.

Code Example (C/C++)

A C/C++ code sample for computing the Radix 2 FFT can be found below. This is a simple implementation which works for any size N where N is a power of 2. It is approx 3x slower than the fastest FFTw implementation, but still a very good basis for future optimisation or for learning about how this algorithm works.

#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;
}

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