Re: Discrete Fourier Transform
- To: mathgroup at smc.vnet.net
- Subject: [mg44893] Re: Discrete Fourier Transform
- From: AES/newspost <siegman at stanford.edu>
- Date: Sat, 6 Dec 2003 04:45:31 -0500 (EST)
- References: <bm0juq$prl$1@smc.vnet.net> <bqpn51$91v$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <bqpn51$91v$1 at smc.vnet.net>, David Wood <me at floyd.attbi.com> wrote: > > I am attempting to perform a discrete Fourier transform on a time series > > using Mathematica 4.0. (I'd like to determine the period of the signal.) > > Unfortunately, I noticed that I cannot control the frequency resolution of > > this transform. I also have no idea what units the frequency is in. > > Do you know of any way to get around the problem? > I'd suggest that you need to write down on paper whatever definition of the continuous Fourier transform (CFT) pair you like (f and t, or omega and t, or whatever), making sure you've scaled them so they really are a self-transforming pair; and then write down the formal definition of the discrete Fourier transform (DFT) as given in The Mathematica Book in terms of r, s and n (or change this to i, j and np (number of points), or n, m and np, or whatever indices you like. Then discretize the continuous transform by writing f = f_i deltaF, t = t_j deltaT, or whatever indices you like, and then make the adjustments you have to make to match up your discretized continuous Fourier transform (DCFT) to the Mathematica DFT for a given np value. Note in particular that both of the DFT series repeat periodically, such that the highest frequency in the DFT is not np deltaF, but np deltaF/2. You may find yourself still confused after doing this -- at least I still get myself messed up occasionally, after decades of working with transforms and using the DFT -- but at least you'll see where the conversion comes from. [And as a side point, note that despite the suggestion above, the DFT is really *not* simply a discretized approximation of the CFT. The DFT and CFT are separate free-standing totally independent exact transforms, in their separate discrete and continuous domains, though you can use the DFT to get a discrete approximation to the FFT if you use large enough np.]