Re: Attach frequency list to a list of fourier tranformed numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg14696] Re: Attach frequency list to a list of fourier tranformed numbers*From*: "Peltio" <peltioNOS at PAMyahoo.com>*Date*: Sun, 8 Nov 1998 21:15:53 -0500*Organization*: Peltio Inc.*References*: <720rsl$1sn@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi, Raymond Your question: How do I attach the actual frequency values to a list of numbers transformed using the Fourier[list] command? My answer: Here's what I do. Actually the answer depends on what the initial data mean. Let's suppose we're in time domain, and that you have sampled a waveform over an interval T with sampling time Ts, that is you have the values at t= 0,Ts,2Ts,...,kTs,..(n-1)Ts. (henceTs=T/n) The frequencies associated are f=1/T and fmax=1/Ts. Your Fourier transformed data will span an interval fmax wide with step f. But, beware!, your spectrum is folded, so you will go with the "real" values from frequency 0 to frequency fnyq=fmax/2 and from then on (fnyq to fmax) you'll have a folded copy of the first part. To associate the frequencies to the spectrum we need to know either the sampling time Ts or the interval duration T, but let's start with the continuous signal of frquency 2 Hz f[t_]:=Sin[ 2Pi 2t ] data=Table[ { t , f[t]+0.Random[Real,{-.5,.5}]}, {t,0,5,5/255} ]; Length[data] Out[1]=256 (incidentally, powers of two are what Fourier crunching motors do expect) Let's scorporate time values from signal values trange=#[[1]]&/@data; signal=#[[2]]&/@data; the interval duration and the sampling interval are T=(Max[trange]-Min[trange])// N ; Ts=T/Length[trange//N ; frequency will vary from 0 to 1/Ts in 256 steps 1/T wide frange=Range[0,1/Ts-1/T,1/T]; Length[frange] Out[48]=256 Now we calculate the Fourier transform of the 256 point data transform=Fourier[signal]/Sqrt[Length[signal]]//Chop; Length[transform] Out[49]=256 and finally we put the frequency data together modulo=Transpose[{frange,Abs[transform]}]; Try this to see that indeed your spectrum shows two peaks at 2Hz and at 49.2 (that is 51.2-2) Hx. The value at 51.2 is 1/Ts, while 2 is the frequency of the sinusoid we used to build our data set ListPlot[modulo,PlotRange->All,PlotJoined->True] Show[%,PlotRange->{{0,10},All}] In case you prefer it you can even unfold your spectrum so to show the components at negative frequencies, spanning from -fnyq to +fnyq. But at this point it's quite straightforward. What worries me is that to do this in general, *I think*, you'll have to discern between the case with an even number of points and an odd number, and worst still you need to know wether frequency 0 is comprised in the data set or not (-1,0,1,... or -1.5,-0.5,0.5,1.5,..). So I, too, have a question for the newsgroup: is there a simple and straigthforward method to unfold the spectrum regardless of this matter? Hope that helps. Regards, Peltio