# Does averaging across multiple time series render higher Nyquist frequency?

Suppose I have three time series with the same length and the same temporal interval. If I upsample each of the three time series through interpolation, that will cause aliasing. However, if I'm interested in the information embedded in the average of the three time series, is such an upsampling justified? In other words, does the above upsamping and then averaging among three time series increase the Nyquist frequency by three times? Thanks!

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Please consider adding some references to this question (e.g. for the Nyquist frequency). –  Shane Oct 6 '10 at 19:57
What do you mean by I have 3 time series'? Do you have 1 time series that has been divided it into 3 pieces? Or do you have 3 devices all independently measuring the same signal' at the `same' time? –  Robby McKilliam Oct 7 '10 at 1:55
Shane, by references do you mean tagging? If so, it seems I'm not allowed to add new tags as I'm a newbie. –  bluepole Oct 7 '10 at 10:08
No, the three time series are not divided from one. Although independently sampled, they are strongly correlated with each other. –  bluepole Oct 7 '10 at 10:15

Simple Answer: No. (At least I don't think so -- It's been 10 years since my EE classes).

In general, multiple samples with independent white noise will, when averaged, raise the signal to noise ratio. But they will not increase the Nyquist frequency.

Short Example:

Construct three cell towers equidistant from a point source. Make a cell call in the center and record it independently on each of the three towers at a common sampling rate, say the 450 MHz GSM band. Then average each sample across the three towers and viola, triple the S/N (assuming that local noise characteristics of each tower are independent and all received the same signal at exactly the same strength).

However, even though you upped the S/N by a factor of three, you didn't change the Nyquist frequency. Reference included for those interested.

Note: In my example, I'm assuming that the samples collected at the three towers are "in phase" -- equidistant -- if so, then what I've stated above is correct. If the samples you're collecting are out of phase, then this might in fact raise the Nyquist frequency. That said, you might want to ask this question somewhere else -- Unfortunately the corresponding Signal, Image, & Video Processing and Electronics & Electrical Engineering boards have yet to really take off. You might post your question there, but I'm not sure if it will ever be answered.

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Really appreciate your help! Using your cell tower as an example, my scenario is more like the following: three cell phones at three equally-spaced locations receive signal from the same cell tower. Would this be considered "in phase"? Thanks again! –  bluepole Oct 7 '10 at 10:33
I'm not sure. If you look at your data, does the first sample collected in each of the three time series correspond to the same point in time in the underlying process? If so, then they're "in phase". Or is there a lag between when one series is collected and the other two are collected? If there is a lag, then they would be out of phase. There's a cute picture on this Wikipedia page which might provide some insight. –  M. Tibbits Oct 7 '10 at 11:28