So, instead of appearing as an impulse, target points in the scene will appear as since waves, which lend to
sidelobe effects and poor resolution. Increasing the bandwidth serves to narrow the main lobe of the since
function, improving resolution. As explained, one should seek to recreate a coherent image using as little
bandwidth as possible.
SPECTRAL ESTIMATION ALGORITHMS
The following provides a brief explanation of the fast Fourier transform (FFT) and CLEAN algorithms for a 1-D
discrete time data sequence x(n).
A. FFT
The radix-2 FFT algorithm, first described by Cooley and Tukey [4], efficiently computes the discrete
Fourier transform (DFT) using a divide and conquer approach. The inverse DFT of an array of complex spectral data
X[k] is given by
N-1
1 .2n-
x[n] = N E X k] " n. = 0,1,2 ...., - 1
k=O
where x[n] is a finite signal and N (constrained to be a power of two) represents its length. The formula is
then decomposed to model the signal over its even and odd indexed values:
s[n] = IDFT{X[k]}
N-1
= Z x[k]c n
= [x[I(o) + X[2]e'2" + + X[N - 21, J "-22) +
- [x . J' + .] '-3" + ... + X[N - I]ei N -1)]
N -1 -1
= X[2kflI (2) + + S X[2k + 1]' ,r1 1) (1
k=0 k=0
Upon rearranging, the final result is obtained
-1 -
[n] = - X[2k]e "n(2k) + - X[2k + l]eJ en(2k)
k=0 k=0
= IDFT.{X[2k]} + ' '"-7)IDFT{X[2k+ 1]}. (1.
2 2
The complex summations can thus be recursively broken down until the length of each IDFT is two. The
calculations then become trivial: