Talk:Engineering Tables/DTFT Transform Table

Table all wrong
After staring at this table for quite a while it occured to me, that it doesn't actually list DTFT transforms, but ordinary Fourier transforms for angular frequency. Some books use $$\omega$$ for both normalized frequency and angular frequency. Wikipedia also follows this retarded convention, which got somebody really confused. I would suggest, that $$\Omega$$ is used for normalized frequency instead, which would be much clearer.

92.229.110.117 (talk)


 * I am not sure why you say this table lists Fourier transforms for angular frequencies. Without speaking on the convention wikipedia uses (I have had plenty of arguments over there.)  The use of $$\omega$$ for frequency variable is not unusual in this context.  For example this is the convention used by G. Strang & T. Nguyen "Wavelets and Filter Banks".  Could you please explain a bit more as to why you feel this table is all wrong? Thenub314 (talk) 14:32, 9 November 2009 (UTC)


 * The table is wrong because in the discrete-time Fourier domain every function $$F(\omega)$$ must give the same result as $$F(\omega + 2\pi)$$, which is not the case for the correspondencies listed in this table. Some books use the notation $$F(e^{\omega})$$ to emphasize that fact (obviously $$e^{\omega}$$ and $$e^{\omega + 2\pi}$$ are identical).  Other books use $$F(e^{\Omega})$$, which I find even better, as in this case there is no ambiguity: $$f$$ is ordinary frequency, $$\omega$$ is angular frequency and $$\Omega$$ is normalized frequency (normalized to the sampling rate).  In Wikipedia $$f$$ is ordinary frequency, and $$\omega$$ can refer to both angular frequency or normalized frequency, which is very confusing. 85.179.124.162 (talk) 09:42, 12 November 2009 (UTC)


 * After reading your comment again, I realized what maybe got you confused: the discrete-time Fourier domain is _not_ the same thing as the Fourier domain. The Fourier domain is infinite, while the discrete-time Fourier domain "repeats" after two $$\pi$$. To give you an idea, about the connection: Every discrete signal can be seen as a sampled continuous signal.  If the sampes are taken with a sampling interval of $$T$$, in the Fourier domain the baseband spectrum of the original continuous signal is repeated around multiples of $$f_{\text{sampling}} = \frac{1}{T}$$ (or $$\omega_{\text{sampling}} = 2\pi\frac{1}{T}$$ if you prefer angular frequencies for some reason).  The samples of the discrete signal contain no information about the sampling interval or the sampling frequency, therefore the discrete-time Fourier transform maps the discrete samples into the interval $$(-\pi:\pi)$$, where $$-\pi$$ would correspond (it would, but the information is lost) to the half of the negative sampling frequency ($$-\frac{1}{2}f_{\text{sampling}}$$), and $$\pi$$ to half of the positive one ($$\frac{1}{2}f_{\text{sampling}}$$) in the ordinary frequency domain, i.e. $$\Omega = 2\pi\frac{1}{f_{\text{sampling}}}$$. As mentioned earlier, Wikipedia writes $$\omega$$ not $$\Omega$$ as I just did, to refer to the normalized frequency.  This is confusing because the angular frequency is $$\omega = 2\pi f$$, which of course is something completely unrelated.85.179.124.162 (talk) 10:19, 12 November 2009 (UTC)