ListFourierSequenceTransform

✖
`ListFourierSequenceTransform`

✖

ListFourierSequenceTransform[list,ω]

gives the discrete-time Fourier transform (DTFT) of a list as a function of the parameter ω.

✖

ListFourierSequenceTransform[list,ω,k]

places the first element of list at integer time k on the infinite time axis.

✖

ListFourierSequenceTransform[list,{ω₁,ω₂,…},{k₁,k₂,…}]

gives the multidimensional discrete-time Fourier transform

Details and Options

By default, the one-dimensional discrete-time Fourier transform of a list of length is computed as .
ListFourierSequenceTransform[list,ω] is equivalent to ListFourierSequenceTransform[list,ω,0].
ListFourierSequenceTransform takes FourierParameters option. Common settings for FourierParameters include:
{1,1} default settings

{1,-2Pi} period 1

{a,b} general setting
ListFourierSequenceTransform[list,ω,k] effectively computes FourierSequenceTransform[f[r],r,ω] for a sequence f with f[r-1+k]=list[[r]] for 1<=r<=Length[list] and f[r]=0 otherwise.

Examples

open allclose all

Basic Examples (2)Summary of the most common use cases

Discrete-time Fourier transform of a constant vector:

In[1]:=1

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-hxcway

Out[1]=1

Two-dimensional DTFT:

In[1]:=1

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-d1f7mr

In[2]:=2

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-j1c4z9

Out[2]=2

Applications (1)Sample problems that can be solved with this function

Create a lowpass filter:

In[1]:=1

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-7yymgu

Visualize the frequency response:

In[2]:=2

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-iuvv1c

Out[2]=2

Apply to a noisy signal:

In[3]:=3

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-q8049x

Out[4]=4

Properties & Relations (4)Properties of the function, and connections to other functions

Discrete-time Fourier transform of a numeric list is equal to the Fourier sequence transform of a sum of shifted unit samples:

In[1]:=1

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-8b6of

In[2]:=2

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-dekd7k

Out[2]=2

Inverse of a discrete-time Fourier transform of a list:

In[1]:=1

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-5b8f8c

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-83e38

Out[2]=2

Fourier of a length- list returns samples of the ListFourierSequenceTransform at frequencies that are multiples of :

In[1]:=1

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-eodpg2

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-jhe1q1

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-eb46n

Out[3]=3

ListFourierSequenceTransform is equivalent to computing ListZTransform on the unit circle:

In[1]:=1

✖

https://wolfram.com/xid/0rkp3d44dzfe4a6-w7kd09

Out[1]=1

Top