Talk:Discrete-time Fourier transform

message: The table uses X(omega), not the form defined in the lead.  And some of the entries have periodic DTFTs while some do not.  That is, the ones with exp(i omega) in them are obviously periodic, but the the ones like delta(omega+a) are not; they seem to just be copied from regular Fourier transforms.  These need to be fixed.  Dicklyon 04:09, 6 December 2006 (UTC)
replies: text: no problem :-) I added that line in the table because I made an exam about Z-transform and I had to use such a transform several times, but it was not in any table. So I calculated it in general, using generic parameters, and I published it on wikipedia, hoping it could be useful for somebody else. If you think the table is growing too big, we might consider moving it to its own page, keeping on this page just the main transforms such as sine and cosine, sinc, and a few more. Alessio Damato 12:41, 19 December 2006 (UTC)
thread_title: And the table...

message: The new table entries have errors.  For instance this equality:
:X_R(e^{i \omega}) = Re\{x[n]\}\!
One side is time domain and the other is frequency domain.
And this:
:\angle X(e^{i \omega}) = \angle X(e^{-i \omega}) \!
For instance, consider x[n] = \delta [n - k] \! and X(e^{i \omega}) = e^{-i k \omega} \!  :
:\angle X(e^{i \omega}) = -k\omega \!
:\angle X(e^{-i \omega}) = k\omega \!
So:   -k\omega \ \stackrel{\mathrm{???}}{=}\ \ k\omega\!
I haven't checked everything.  Those just leaped out at me.
--Bob K 04:17, 22 February 2007 (UTC)
thread_title:  Symmetry Properties 

message: The alternate notation (z-transform like) is used in the properties table, but it doesn't seem to say so, so the results are confusing.  We should clarify.  Dicklyon (talk) 19:41, 12 January 2013 (UTC)
thread_title: Alternate notation

message: The following paragraph was inserted as a "Definition" on Aug 21.  Besides the obvious flaws, it is not a definition, and it seems more concerned with the sampling theorem than with the subject of this article.
Discrete-Time Fourier Transform is related to Nyquist–Shannon sampling theorem. Continuous Fourier Transform (CFT) decomposite a non-periodic signal and gives its spectrum ratios/relations (due to integral over infinite period, unlike Fourier Series directly giving coeffiecients. The Inverse Continuous Fourier Transform is the Fourier coefficent expansion form for CFT). For digital processing such as using computer, discrete data is required. Shannon's original proof in Nyquist–Shannon sampling theorem logically showed that under proper sampling condition the original Continuous Fourier Transform X(ω) could be recovered by calculating its Discrete-time Fourier transform (the maximum unique frequency component DTFT represents cannot exceed the sampling frequency, and due to aliasing the maximum unique frequency is limited under half of the sampling frequency (Nyquist frequency), therefore the spectrum becomes periodic), and therefore its Fourier coefficient expansion form can be recovered, and the original signal. As long as the sampling frequency Fs >= 2B which is the Nyquist frequency (B is the bandwidth limit of the signal), using discrete samples to unambiguiously recover the original signal is achievable. Discrete Fourier transform is the counterpart of DTFT dealing with discrete sampling and reconstruction of periodic signal.
--Bob K (talk) 12:47, 24 August 2017 (UTC)
thread_title:  Rambling, sub-standard, misplaced paragraph