# FourierSequenceTransform

FourierSequenceTransform[expr,n,ω]

gives the Fourier sequence transform of expr.

FourierSequenceTransform[expr,{n1,n2,},{ω1,ω2,}]

gives the multidimensional Fourier sequence transform.

# Details and Options • FourierSequenceTransform is also known as discrete-time Fourier transform (DTFT).
• FourierSequenceTransform[expr,n,ω] takes a sequence whose n term is given by expr, and yields a function of the continuous parameter ω.
• The Fourier sequence transform of is by default defined to be .
• The multidimensional transform of is defined to be .
• The following options can be given:
•  Assumptions \$Assumptions assumptions on parameters FourierParameters {1,1} parameters to definite discrete-time Fourier transform GenerateConditions False whether to generate results that involve conditions on parameters
• Common settings for FourierParameters include:
•  {1,1} default settings {1,-2Pi} period 1 {a,b} general setting

# Examples

open allclose all

## Basic Examples(2)

Find the discrete-time Fourier transform of a simple signal:

Find a bivariate discrete-time Fourier transform:

## Scope(4)

Compute the transform for each frequency ω:

Plot the spectrum:

The phase:

Plot both spectrum and phase using color:

Constant:

Periodic:

Impulse:

Exponential:

Exponential polynomial:

Rational sequence:

Rational-trigonometric:

Hypergeometric terms:

Multivariate sequences:

## Options(2)

### FourierParameters(1)

Use a non-default setting for FourierParameters:

### GenerateConditions(1)

Obtain conditions on parameters:

## Properties & Relations(5)

FourierSequenceTransform is defined by a doubly infinite sum:

FourierSequenceTransform and InverseFourierSequenceTransform are inverses:

FourierSequenceTransform is closely related to ZTransform:

A discrete analog of FourierTransform being closely related to LaplaceTransform:

FourierSequenceTransform provides a -analog generating function:

FourierSequenceTransform is closely related to BilateralZTransform: