DFT Definition

The Discrete Fourier Transform (DFT) of a signal $x$ may be defined by

$$X(\omega_k ) \isdef \sum_{n=0}^{N-1}x(t_n)e^{-j\omega_k t_n}, \qquad k=0,1,2,\ldots,N-1,$$

where ` $\isdeftext$' means ``is defined as'' or ``equals by definition'', and

$$\begin{eqnarray*} \sum_{n=0}^{N-1} f(n) &\isdef & f(0) + f(1) + \dots + f(N-1)\\... ...mbox{number of time samples = no.\ frequency samples (integer).} \end{eqnarray*}$$

The sampling interval $T$ is also called the sampling period. For a tutorial on sampling continuous-time signals to obtain non-aliased discrete-time signals, see Appendix D.

When all $N$ signal samples $x(t_n)$ are real, we say $x\in{\bf R}^N$. If they may be complex, we write $x\in{\bf C}^N$. Finally, $n\in{\bf Z}$ means $n$ is any integer.