# Autocorrelation of noise - negative correlation

I am investigating autocorrelation of electrical noise as part of an undergraduate experiment (as detailed in http://physlab.lums.edu.pk/images/a/ab/Correlation.pdf). I sampled noise voltages using an 8-bit AtoD from a noise generator (whose description was not provided). I first sampled the noise directly, then I sampled it after passing it through a low-pass filter. I imported the data into MATLAB and used the autocorr function to autocorrelate it. The no-filter data autocorrelated as expected, being just 1 at t=0 (as in Fig 1 in paper); however, the filtered data decays initially as expected but then dips below 0 before coming back up and staying around 0 as in the image below. My three issues are:

1. I am not really sure what this means physically (is the correlation 'direction' changed so-to-speak?),

2. Why might this be occurring? I know that a description of the noise generator may be important here but I do not have that information myself.

3. Is there any way to correct this issue, and if not, what would be the best compromise?

EDIT:

Here are snapshots of the noise without and with the filter (RC=0.5ms) respectively:  • This question might be better suited for the signal processing SE. – Chris Mueller Apr 2 '14 at 22:30
• A negative autocorrelation implies that if a particular value is above average the next value (or for that matter the previous value) is more likely to be below average. If a particular value is below average, the next value is likely to be above average. You should plot your dataset spectrum before and after the filter perhaps you manipulated the data in an unexpected way. – user6972 Apr 2 '14 at 22:51

Letting $\mathbf{F}$ and $\mathbf{F}^{-1}$ be the forward and inverse discrete Fourier transform, the cyclic autocorrelation of a signal $A$ is given by $$S(A)=\mathbf{F}^{-1}\left[\mathbf{F}(A)\mathbf{F}(A)^*\right].$$ Let the low-passed signal $A_L$ be $$A_L=\mathbf{F}^{-1}\left[\mathbf{F}(A)\mathbf{F}(L)\right]$$ where $L$ is the low-pass filter in the time domain. Then $$S(A_L)=\mathbf{F}^{-1}\left[\mathbf{F}(A)\mathbf{F}(L)\mathbf{F}(A)^*\mathbf{F}(L)^*\right]=\mathbf{F}^{-1}\left[\mathbf{F}(A)\mathbf{F}(A)^*\mathbf{F}(L)\mathbf{F}(L)^*\right]\\=S(A)*S(L)$$ where the convolution theorem has been applied.
Since $S(A)\approx(A\cdot A,0,0,...)$ for uncorrelated noise, you should roughly be getting back a picture of $S(L)$ when you compute the autocorrelation of the low-pass signal.
So the question becomes: what sort of low-pass filter $L$ did you employ?