How to find the phase difference of two sampled sine waves?

Question

I'm trying to measure a phase difference between two Sine functions I've acquired with a computer. I'm uploading one of the .txt files with the data I'm working with here: txt file. To remove the units of every row I'm using the function that @R.M posted here.

The first two columns of each file make a Sin function, the other columns make a different one. Here's a plot of those functions togheter:

I'm looking for a way to find all rows that have a maximum value on the second coordinate (that is, second and fourth column) and then be able to manipulate those lists in order to get the difference between the other two coordinates. For example, if this are the two lists with the maximums:

max1={{1.1,6},{2.2,6},{3.3,6},{4.4,6}{5.5,6}}
max2={{1.3,10},{2.4,10}{3.5,},{4.6,10},{5.7,10}}

Then I could easily get the difference between the first coordinates of each pair (1.3-1.1, 2.4-2.2, etc.), which is what I need.

I have tried doing the proposed methods in this question but none of them worked for me.

Furthermore, I got a lot of files like this to analyze, so I'm importing all of them with a For loop and putting them all on the same list with Table. It would be nice if I could get the maximums of all my files at the same time.

I'll apreciate any ideas, thank you.

Ps.: By the way, by asking and reading the contents of this page I realized Mathematica is a much more powerful tool than I thought, and I would like to learn more about how to properly use it. I know the Mathematica documentation is really good, but I would like to get a good book about Mathematica. Do you know one to reccommend?

Answer 1

If what you really want to do is to find the phase difference between two digitized sinusoids of the same frequency, then there is probably a better way to proceed than by counting the peaks. You can take the Fourier transform of the two signals, and then look at the phase difference between them. For example, say the sine waves are:

s1 = Table[Sin[2 Pi 10 t], {t, -1, 2, 1/1000}];
s2 = Table[0.2 Sin[2 Pi 10 t + 0.8], {t, -1, 2, 1/1000}];
ListLinePlot[{s1, s2}]

So you can see this is qualitatively like your situation. I've arbitrarily assigned the second (smaller) sine wave to be 0.8 radians out of phase with the first. Let's take the FFTs and recover this from the data.

ffts1 = Fourier[s1, FourierParameters -> {-1, 1}];
ffts2 = Fourier[s2, FourierParameters -> {-1, 1}];
max = Max[Abs[ffts1]];
pos = First[First[Position[Abs[ffts1], max]]];
Arg[ffts1[[pos]]] - Arg[ffts2[[pos]]]

which gives the answer

0.800167

Answer 2

Sometimes Correlation is useful too:

l1 = Table[Sin[x] + RandomReal[1/10], {x, 0, 10 Pi, 1/20}];
l2 = 1/4 Table[Sin[x + 1/2] + RandomReal[1/15], {x, 0, 10 Pi, 1/20}];
ListPlot[{l1, l2}]

And now:

h = IntegerPart[Length@l1/4]; 
lc = N@ListCorrelate[l1[[h - # ;; -h - #]], l2[[h ;; -h]]][[1]] & /@ Range[h - 1];
ListPlot[{RotateRight[l1, First@Ordering[lc] - 1], l2}]

Edit

Running with your data:

And the offset is

First@Ordering[lc] - 1
(*137 *)

or 2.74 μS

Answer 3

A simple and fast method for finding the phase difference would be to do it manually. Let's plot the two data sets in a Manipulate and let one have a manually determined offset. Since the time bases in both lists are the same I discard them for the moment so that the x-axis depicts sample number:

Manipulate[
    ListLinePlot[{24.7 lists[[1, All, 2]], lists[[2, offset ;; -1, 2]]}, ImageSize -> 600], 
    {offset, 1, 400, 1}
]

Note that I scaled one of the plots by 24.7 for reasons that will be made clear later on.

If you play with that you find that a 38 sample shift is about optimal.

How to do this with fourier analysis?

The amplitude spectra:

af1 = Abs@Fourier[lists[[1, All, 2]]];
af2 = Abs@Fourier[lists[[2, All, 2]]];

The phases:

pf1 = Arg@Fourier[lists[[1, All, 2]]];
pf2 = Arg@Fourier[lists[[2, All, 2]]];

A plot of the amplitude spectrum of the first sample:

ListLinePlot[af1, PlotRange -> All]

The two peaks are at:

Ordering[af1, -2]

{14, 2488}

The same for the second sample. At position 14 the ratio of the amplitudes is:

af2[[14]]/af1[[14]]

24.70345393

So, here we have our amplitude scaling factor.

The phase difference for this peak is:

pf2[[14]] - pf1[[14]]

1.240647232

or in terms of sample positions:

(pf2[[14]] - pf1[[14]])/(2 \[Pi]) 1/((14.-1)/Length[af1])

37.97214223

Yes, the same value we found manually.

Answer 4

Correlation[s1, s2] // ArcCos

Borrowing @bills definitions

s1 = N@Table[Sin[2 Pi 10 t], {t, -1, 2, 1/1000}];
s2 = N@Table[0.2 Sin[2 Pi 10 t + 0.8], {t, -1, 2, 1/1000}];

You get

0.800166

Borrowing @belisarius 's

s1 = Table[Sin[x] + RandomReal[1/10], {x, 0, 10 Pi, 1/20}];
s2 = 1/4 Table[Sin[x + 1/2] + RandomReal[1/15], {x, 0, 10 Pi, 1/20}];

Correlation[s1, s2] // ArcCos

0.501371

Answer 5

Using my function from here and the file you've provided, first import the data and group the first two columns and the last two columns as:

data = convertList /@ Import["/path/to/RLC1.txt", "Data"] /. {} -> Sequence[];
lists = Transpose /@ Partition[Transpose@data, 2];

You can plot them simply as ListPlot[lists]. To find the rows that have the maximum element in the second position, you can choose from Cases or Pick or Select. I like Select, and we can write a function that takes a list and returns those rows that have the maximum value in the second column:

maxList[list_List] := With[{max = Max@list[[;; , 2]]}, Select[list, #[[2]] == max &]];

maxList@First@lists // Short
(* {{-22.8,0.3},{-22.78,0.3},{-22.76,0.3},{-22.74,0.3},{-22.72,0.3},<<128>>,
    {24.24,0.3},{24.26,0.3},{24.28,0.3},{24.34,0.3},{24.36,0.3}} *)

maxList@Last@l // Short
(* {{-22.1,7.4},{-22.08,7.4},{-22.06,7.4},{-22.04,7.4},{-22.02,7.4},<<165>>,
    {24.9,7.4},{24.92,7.4},{24.94,7.4},{24.96,7.4},{24.98,7.4}} *)

You can get the corresponding rows for both lists as maxList /@ lists. With these in hand, you can easily manipulate it however you want. They're both of different lengths, so you'll have to decide how exactly you want to compare the two.

For several files, you can use FileNames to filter the data files, wrap the above commands in a Module and map them across your file list.