Re: DFT or DFS: Are they the same thing?

On 7 Aug, 23:14, robert bristow-johnson <r...@xxxxxxxxxxxxxxxxxxxx>
wrote:
On Aug 7, 2:56 am, Rune Allnor <all...@xxxxxxxxxxxx> wrote:

The different variants take
either discrete or continuous data as input, and the data are
of either finite or infinite extent.

the simple rule to remember: discrete time means periodic in
frequency.  and vise versa (because of duality in the FT).

DFT (or DFS):   discrete    periodic t; discrete    periodic f
DTFT:           discrete   aperiodic t; continuous  periodic f
Fourier series: continuous  periodic t; discrete   aperiodic f
continuous FT:  continuous aperiodic t; continuous aperiodic f

did i miss a combination?

No, but you got the variants' defining characteristica wrong:

DFT (or DFS): discrete finite t; discrete finite f
DTFT: discrete infinite t; continuous finite f
Fourier series: continuous finite t; discrete infinite f
continuous FT: continuous infinite t; continuous infinite f

Since the digital computer can only work with finite amounts
of discrete data, the DFT is the only variation of the FT that
can be analyzed by means of numerics.

The question then becomes how to map the other varaints onto
the DFT. That's the main purpose of the sampling theorem.
It lets people understand what conversions and trade-offs are
involved in mapping from a continuous t domain to the
discrete t domain.

This talk about extensions of the data, are similar attempts
to try and explain what modifications and tradeoffs are
involved when one does computations on finite amounts
of data where one ought to do computations on infinite
amounts of data.

It gives people a hook on how to understand the difference
between CT data and the DT representations: Nyquist said
that 'the DT representation is exact if the bandwidth of
the CT signal meets certain criteria.'

Both you and I know that the CT signals never satisfy the
Nyquist criterion exactly, so there are errors involved
when sampling the CT data.

This idea about periodic extendions of the data serve
the same purpose: It is the special case where the
spectrum of the infinite sequence or function is practically
indistinguishable from the spectrum of the finite-length
sequence or function.

The data never are periodic, but this periodic extension
idea gives a hook on how to understand the errors involved.

These matters are trivial when presented this way from the
outset. Unfortunately, I have seen no DSP textbooks that
emphasize this POV.

Rune
.

Relevant Pages

  • Re: Zero Padding In The Frequency Domain
    ... the DFT is the version of the FT that processes *finite* ... amounts of *discrete* data. ... lengths of discrete or continuous data. ... Infinite amounts of data ...
    (comp.soft-sys.matlab)
  • Re: DFT or DFS: Are they the same thing?
    ... The DFT is a special case of the Discrete-Time ... discrete, infinite, and periodic sequence of period N in one domain ...
    (comp.dsp)
  • Re: Zero Padding in radix 2 FFT
    ... Their resulting Fourier transform is continuous and consists of delta functions, but not at rationally related frequencies. ... continuous, infinite time continuous, infinite frequency ... DISCRETE, infinite time continuous, infinite frequency and PERIODIC ... When a function is PERIODIC, then sums can be taken over a single period. ...
    (comp.dsp)
  • Re: Zero Padding in radix 2 FFT
    ... continuous, infinite time continuous, infinite frequency ... DISCRETE, infinite time <Fourier Transform> continuous, infinite frequency ... When a function is PERIODIC, then sums can be taken over a single period. ...
    (comp.dsp)
  • Re: Periodicity of the DFT - was Re: Phase of FFT compared to phase of Sinusoid
    ... The DFT output represents a periodic, ... "Discrete Fourier transform, the Fourier transform of a discrete periodic sequence. ... distinction between the DFT and the DTFT. ...
    (comp.dsp)