N-point FFT and 2-radix FFT

Question

I am wondering what is the difference between a N-point FFT (output has same length as the input) and a 2-radix FFT (output is always of length $2^n$)

For example a is a sequence:

1     2     3     5     2     1     1

N-point FFT of a:

 15.0000 + 0.0000i    
 -4.3264 - 4.0333i    
  1.0930 + 2.2383i    
 -0.7666 - 1.7950i     
 -0.7666 + 1.7950i      
  1.0930 - 2.2383i    
 -4.3264 + 4.0333i

2-radix FFT (8-point in this case)

 15.0000 + 0.0000i    
 -3.8284 - 6.2426i    
 -1.0000 + 2.0000i      
  1.8284 - 2.2426i     
 -1.0000 + 0.0000i      
  1.8284 + 2.2426i    
 -1.0000 - 2.0000i    
 -3.8284 + 6.2426i

Also, is the relation between N-point FFT and 2-radix FFT, and how it they are? Can 2-radix FFT be converted to N-point FFT if possible?

Thanks a lot!

Answer 1

There is no relation, the 8 point FFT is of the modified sequence that has a zero added. So it is a result for a different input.

See Bluestein's chirp-z algorithm or Raders algorithm for a transformation of general N-point DFT into a form that a dyadic FFT may be applied.