# DFT VS DTFT

Started by July 21, 2003
```Hello,

I wanted to know the difference between discrete fourier transform and
discrete time fourier transform.

praveen
```
```"praveen" <praveenkumar1979@rediffmail.com> wrote in message
> Hello,
>
> I wanted to know the difference between discrete fourier transform and
> discrete time fourier transform.

You can consider four versions of the 1-dimensional Fourier Transform:

Time                       Frequency

Continuous                 Continuous

Discrete                   Continuous

Continuous                 Discrete

Discrete                   Discrete

DTFT generally means the Discrete/Continous version.
If you Google on DTFT, you will find the Discrete/Continuous version.
The time series is infinite discrete and the frequency function is
continuous, periodic.
So, a discrete sum is used in the transform to go from time to frequency and
a continous integral over one period is used in the transform to go from
frequency to time.

Fred

```
```praveen <praveenkumar1979@rediffmail.com> wrote:
> Hello,
>
> I wanted to know the difference between discrete fourier transform and
> discrete time fourier transform.
>
>
> waiting for reply
> praveen

The DTFT is aperiodic-discrete and the DFT is periodic-discrete.

Tom

--
```
```Tom <T.Otermans_REMOVE_THIS_@home.nl> wrote in message
news:<slrnbho1p5.aeh.T.Otermans_REMOVE_THIS@noritake.basement>...
> praveen <praveenkumar1979@rediffmail.com> wrote:
> > Hello,
> >
> > I wanted to know the difference between discrete fourier transform and
> > discrete time fourier transform.
> >
> >
> > waiting for reply
> > praveen
>
> The DTFT is aperiodic-discrete and the DFT is periodic-discrete.
>
>
> Tom
>
> --

Can DFT be viewed as sampled version of DTFT and hence it is periodic?
I am not sure about this but intuitively it is easier to think so.
-Nithin
```
```"Tom" <T.Otermans_REMOVE_THIS_@home.nl> wrote in message
news:slrnbho1p5.aeh.T.Otermans_REMOVE_THIS@noritake.basement...
> praveen <praveenkumar1979@rediffmail.com> wrote:
> > Hello,
> >
> > I wanted to know the difference between discrete fourier transform and
> > discrete time fourier transform.
> >
> >
> > waiting for reply
> > praveen
>
> The DTFT is aperiodic-discrete and the DFT is periodic-discrete.

Tom,

How can that be for the DTFT case?  That is, if I understand what domains
you're referring to....
I would have said: discrete <-> continous periodic for DTFT
where <-> denotes transform pair.
and
discrete periodic <-> discrete periodic for DFT.

Fred

```
```Fred Marshall wrote:
> "Tom" <T.Otermans_REMOVE_THIS_@home.nl> wrote in message
> news:slrnbho1p5.aeh.T.Otermans_REMOVE_THIS@noritake.basement...
>
>>praveen <praveenkumar1979@rediffmail.com> wrote:
>>
>>>Hello,
>>>
>>>I wanted to know the difference between discrete fourier transform and
>>>discrete time fourier transform.
>>>
>>>
>>>praveen
>>
>>The DTFT is aperiodic-discrete and the DFT is periodic-discrete.
>
>
> Tom,
>
> How can that be for the DTFT case?  That is, if I understand what domains
> you're referring to....
> I would have said: discrete <-> continous periodic for DTFT
> where <-> denotes transform pair.
> and
> discrete periodic <-> discrete periodic for DFT.
>
> Fred
>
>
Fred,
The Z transform is continuous for any finite length sequence.
There are formulas relating the DFT and the Z transform in Oppenheim.
The chirp Z transform is a valid DFT, isn't it?

Stan

```
```"Stan Pawlukiewicz" <stanp@nospam_mitre.org> wrote in message
news:bfjrv8\$4rm\$1@newslocal.mitre.org...
> Fred Marshall wrote:
> > "Tom" <T.Otermans_REMOVE_THIS_@home.nl> wrote in message
> > news:slrnbho1p5.aeh.T.Otermans_REMOVE_THIS@noritake.basement...
> >
> >>praveen <praveenkumar1979@rediffmail.com> wrote:
> >>
> >>>Hello,
> >>>
> >>>I wanted to know the difference between discrete fourier transform and
> >>>discrete time fourier transform.
> >>>
> >>>
> >>>waiting for reply
> >>>praveen
> >>
> >>The DTFT is aperiodic-discrete and the DFT is periodic-discrete.
> >
> >
> > Tom,
> >
> > How can that be for the DTFT case?  That is, if I understand what
domains
> > you're referring to....
> > I would have said: discrete <-> continous periodic for DTFT
> > where <-> denotes transform pair.
> > and
> > discrete periodic <-> discrete periodic for DFT.
> >
> > Fred
> >
> >
> Fred,
>       The Z transform is continuous for any finite length sequence.
> There are formulas relating the DFT and the Z transform in Oppenheim.
> The chirp Z transform is a valid DFT, isn't it?

Stan,

You probably know better than I.  It appears so from a quick look.  Not
something I've dealt with directly.

Fred

```
```All I know about CTFT, DTFT, and DFT

Given a continous signal x[t}
and take the fourier transfrom we got CTFT X(jw).
Notice: X(jw) is a continuous spectrum.

Given a discrete sequence f[n]
and take the fourier transform we got DTFT F(e^jw).
Notice: X(e^jw) is also a continuous spectrum.

If we take the discrete samples of the continous spectrum DTFT X(e^jw),
we got the discrete spectrum DFT X[k].

that is my 2 cents

good luck
```
```Shouldn't one of the headings be "Amplitude" ?

> You can consider four versions of the 1-dimensional Fourier Transform:
>
> Time                       Frequency
>
> Continuous                 Continuous

<Snip>

```
```In article <3f211e89\$0\$49116\$e4fe514c@news.xs4all.nl>,
"hans" <nospam@microsoft.com> wrote:

>Subject: Re: DFT VS DTFT
>From: "hans" <nospam@microsoft.com>
>Date: Fri, 25 Jul 2003 14:11:52 +0200
>Newsgroups: comp.dsp
>
>Shouldn't one of the headings be "Amplitude" ?

No.

Time can be either continuous or discrete and either unbounded or
periodic. There are four cases.

Frequency can be either continuous or discrete and either unbounded
or periodic. There are four cases.

The pairing of the cases is quite simple.

Unbounded time goes with continuous frequencies and periodic time
goes with discrete frequencies. Continuous time goes with unbounded
frequencies and discrete time goes with periodic frequencies. Notice
that if the time and requency labels are exchanged nothing is changed.

Time                 Frequency             Name
Continuous Unbounded    Unbounded Continuous    Fourier Integral
Continuous Periodic     Unbounded Discrete      Fourier Series
Discrete   Unbounded    Periodic  Continuous    Fourier Sequences
Discrete   Periodic     Periodic  Discrete      Discrete Fourier

The names are by no means standardized.

Fourier Integral  is real line    for time to real line    for frequency
Fourier Series    is circle       for time to integers     for frequency
Fourier Sequences is integers     for time to circle       for frequency
Discrete Fourier  is mod integers for time to mod integers for frequency

Fourier Integral has no sampling.
Fourier Series and Fourier Sequences have sampling of either time or
frequency but not both. You have to say which to separate these two
which are often confused as their names are not well separated.
Discrete Fourier has sampling of both time and frequency.

In each case time forms a group and the operation is the Fourier
Transform approriate to that group. So rather than say that one
Fourier Transform is an approximation to another one should say
that the group that it is defined over approximates the other group.

So the integers approximate the real line or the mod integers with a
large modulus approximate the integers.

>
>> You can consider four versions of the 1-dimensional Fourier Transform:
>>
>> Time                       Frequency
>>
>> Continuous                 Continuous
>
><Snip>
>
>

```