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:

• is the triangle function for arbitrary real-valued t:

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
		it works as a differentiator filter
		real numbers W, a
		Hilbert transform
		real numbers A, B complex C