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

$$\begin{eqnarray*} \sum_{n=0}^{N-1} f(n) &\isdef & f(0) + f(1) + \dots + f(N-1)\\ x(t_n) &\isdef & \mbox{input signal \emph{amplitude} (real or complex) at time $t_n$\ (sec)} \\ t_n &\isdef & nT = \mbox{$n$th sampling instant (sec), $n$\ an integer $\ge 0$}\\ T &\isdef & \mbox{sampling interval (sec)} \\ X(\omega_k ) &\isdef & \mbox{\emph{spectrum}\index{spectrum\vert textbf} of $x$\ (complex valued), at frequency $\omega_k $}\\ \omega_k &\isdef & k\Omega = \mbox{$k$th frequency sample (radians per second)} \\ \Omega &\isdef & \frac{2\pi}{NT} = \mbox{radian-frequency sampling interval (rad/sec)} \\ f_s &\isdef & 1/T = \mbox{\emph{sampling rate}\index{sampling rate\vert textbf} (samples/sec, or Hertz (Hz))}\index{Hertz\vert textbf}\index{Hz\vert textbf}\\ N &=& \mbox{number of time samples = no.\ frequency samples (integer).} \end{eqnarray*}$$

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 $x\in\mathbb{R}^N$ . If they may be complex, we write $x\in\mathbb{C}^N$ . Finally, $n\in\mathbb{Z}$ means $n$ is any integer.