15.8. Discrete Fourier Transform
It is observed that DTFT is defined for any infinite sequence. In case of finite-length sequence x(n), 0 ≤ n ≤ N − 1, there exists a relation between sequence and its DTFT X(ejw). Thus, for length N sequence, N values of X(ejw) at different frequency points are sufficient to determine x(n) and hence X(ejw). This evaluation of finite length sequence leads to Discrete Fourier Transform (DFT).
Programming Tip 15.2
DTFT is applicable to any arbitrary sequence whereas DFT is used for finite length sequence only.
Mathematically, DFT of discrete-time sequence x(n) is given as
15.8.
| or | WN = e−j2π/N |
we have,
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