fft - fast Fourier transform.

ifft - fast Fourier transform.

Calling Sequence

x=fft(a ,-1) or x=fft(a)

x=fft(a,1) or x=ifft(a)

x=fft(a,-1,dim,incr)

x=fft(a,1,dim,incr)

Parameters

• x : real or complex vector. Real or complex matrix (2-dim fft)
• a : real or complex vector, matrix or multidimensionnal array.
• dim : integer
• incr : integer

Description

Short syntax

direct

x=fft(a,-1) or x=fft(a) gives a direct transform.

single variate

If a is a vector a single variate direct FFT is computed that is:

x(k)=sum over m from 1 to n of a(m)*exp(-2i*pi*(m-1)*(k-1)/n)

for k varying from 1 to n (n=size of vector a ).

(the -1 argument refers to the sign of the exponent..., NOT to "inverse"),

multivariate

If a is a matrix or or a multidimensionnal array a multivariate direct FFT is performed.

inverse

a=fft(x,1) or a=ifft(x) performs the inverse transform normalized by 1/n .

single variate

If a is a vector a single variate inverse FFT is computed

multivariate

If a is a matrix or or a multidimensionnal array a multivariate inverse FFT is performed.

Long syntax for multidimensional FFT

x=fft(a,-1,dim,incr) allows to perform an multidimensional fft.

If a is a real or complex vector implicitly indexed by j1,j2,..,jp i.e. a(j1,j2,..,jp) where j1 lies in 1:dim(1), j2 in 1:dim(2),... one gets a p-variate FFT by calling p times fft as follows

incrk=1; x=a; for k=1:p x=fft(x ,-1,dim(k),incrk)
        incrk=incrk*dim(k) end

where dimk is the dimension of the current variable w.r.t which one is integrating and incrk is the increment which separates two successive jk elements in a .

In particular,if a is an mxn matrix, x=fft(a,-1) is equivalent to the two instructions:

a1=fft(a,-1,m,1) and x=fft(a1,-1,n,m) .

Examples

  //Comparison with explicit formula
  //----------------------------------
  a=[1;2;3];n=size(a,'*');
  norm(1/n*exp(2*%i*%pi*(0:n-1)'.*.(0:n-1)/n)*a -fft(a,1))
  norm(exp(-2*%i*%pi*(0:n-1)'.*.(0:n-1)/n)*a -fft(a,-1)) 
 
  //Frequency components of a signal
  //----------------------------------
  // build a noides signal sampled at 1000hz  containing to pure frequencies 
  // at 50 and 70 Hz
  sample_rate=1000;
  t = 0:1/sample_rate:0.6;
  N=size(t,'*'); //number of samples
  s=sin(2*%pi*50*t)+sin(2*%pi*70*t+%pi/4)+grand(1,N,'nor',0,1);
  
  y=fft(s);
  //the fft response is symetric we retain only the first N/2 points
  f=sample_rate*(0:(N/2))/N; //associated frequency vector
  n=size(f,'*')
  clf()
  plot2d(f,abs(y(1:n)))
 
  

See Also

corr ,