Discrete-time Fourier Transform - Table of Discrete-time Fourier Transforms

Table of Discrete-time Fourier Transforms

Some common transform pairs are shown below. The following notation applies:

  • is an integer representing the discrete-time domain (in samples)
  • is a real number in, representing continuous angular frequency (in radians per sample).
    • The remainder of the transform is defined by:
  • is the discrete-time unit step function
  • is the normalized sinc function
  • is the Dirac delta function
  • is the Kronecker delta
  • is the rectangle function for arbitrary real-valued t:
\mathrm{rect}(t) = \sqcap(t) = \begin{cases} 0 & \mbox{if } |t| > \frac{1}{2} \\ \frac{1}{2} & \mbox{if } |t| = \frac{1}{2} \\ 1 & \mbox{if } |t| < \frac{1}{2} \end{cases}
  • is the triangle function for arbitrary real-valued t:
\operatorname{tri}(t) = \and (t) = \begin{cases} 1 + t; & - 1 \leq t \leq 0 \\ 1 - t; & 0 < t \leq 1 \\ 0 & \mbox{otherwise} \end{cases}
Time domain
 Frequency domain
 Remarks
integer M
integer M
The term must be interpreted as a distribution in the sense of a Cauchy principal value around its poles at .
real number a
real number a
real number a
integer M
real number a
real number W
real numbers W
\begin{cases} 0 & n=0 \\ \frac{(-1)^n}{n} & \mbox{elsewhere} \end{cases} it works as a differentiator filter
real numbers W, a
\begin{cases} 0; & n \mbox{ even} \\ \frac{2}{\pi n} ; & n \mbox{ odd} \end{cases} \begin{cases} j & \omega < 0 \\ 0 & \omega = 0 \\ -j & \omega > 0 \end{cases} Hilbert transform
real numbers A, B
complex C

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