Language Reference

IFFT Function

computes the inverse finite Fourier transform

IFFT( f)

where is an numeric matrix.

The IFFT function expands a set of sine and cosine coefficients into a sequence equal to the sum of the coefficients times the sine and cosine functions. The argument is an matrix; the value returned is an vector.

Note: If the element in the last row and second column of is exactly 0, then is ; otherwise, is .

The inverse finite Fourier transform of a two column matrix , denoted by the vector , is

for , where if is even, or if is odd.

Note: For most efficient use of the IFFT function, should be a power of 2. If is a power of 2, a fast Fourier transform is used (Singleton 1969); otherwise, a Chirp-Z algorithm is used (Monro and Branch 1976).

IFFT(FFT(X)) returns times , where is the dimension of . If is not the Fourier transform of a real sequence, then the vector generated by the IFFT function is not a true inverse Fourier transform. However, applications exist where the FFT and IFFT functions can be used for operations on multidimensional or complex data (Gentleman and Sande 1966; Nussbaumer 1982).

As an example, the convolution of two vectors () and () can be accomplished by using the following module:

  
    start conv(u,v); 
    /* w = conv(u,v) convolves vectors u and v. 
       Algebraically, convolution is the same operation as 
       multiplying the polynomials whose coefficients are the 
       elements of u and v. Straight convolution is too slow, 
       so use the FFT. 
  
       Both of u and v are column vectors. 
    */ 
       m = nrow(u); 
       n = nrow(v); 
  
       wn = m + n - 1; 
       /* find p so that 2##(p-1) < wn <= 2##p */ 
       p = ceil( log(wn)/ log(2) ); 
       nice = 2##p; 
  
       a = fft( u // j(nice-m,1,0) ); 
       b = fft( v // j(nice-n,1,0) ); 
       /* complex multiplication of a and b */ 
       wReal = a[,1]#b[,1] - a[,2]#b[,2]; 
       wImag = a[,1]#b[,2] + a[,2]#b[,1]; 
       w = wReal || wImag; 
       z=ifft(w); 
       z = z[1:wn,1] / nice;  /* take real part and first wn elements */ 
       return (z); 
    finish; 
  
    /* example of convolution of two waveforms */ 
    TimeStep = 0.01; 
    t = T( do(0,8,TimeStep) ); 
  
    Signal = j(nrow(t),1,5); 
    Signal[ loc(t>4) ] = -5; 
  
    ImpulseResponse = j(nrow(t),1,0); 
    ImpulseResponse[ loc(t<=2) ] = 3; 
  
    /* The time domain for this convolution is [0,16] 
       with the same time step. 
       For waveforms, rescale amplitude by the time step. */ 
    y = conv(Signal,ImpulseResponse) * TimeStep;
 

Other applications of the FFT and IFFT functions include windowed spectral estimates and the inverse autocorrelation function.