Friday 31 January 2014

Fast Calculation of Autocorrelation Coefficients in Julia

Autocorrelation is one of those things that gets used a lot in signal processing which has a very simple formula, but which is easy to speed up. In essence, autocorrelation is a measure of how similar a signal is to itself. Periodic signals are exactly those signals that are self-similar, so autocorrelation is good for locationg periodicities. The basic formula for autocorrelation of a signal \(x(i)\) is:

\[ R_{xx}(j) = \sum_{i=0}^N x(i)x(i-j) \]

In the above equation, \( R_{xx}(j) \) is the notation for the \(j^{th}\) autocorrelation coefficient of signal \(x\), where \(x\) is of length \(N\) samples. Note that the above equation is \(O(N^2)\) in complexity if we want all the autocorrelation coefficients (of which there are around \( 2N - 1\) ). To speed it up, we are going to use the Wiener–Khinchin theorem, which states that the power spectrum of a signal and the autocorrelation sequence of a signal are related to each other via the Fourier Transform (FFT). This means:

\[ S_{xx} = FFT(R_{xx})\]


\[ R_{xx} = IFFT(S_{xx}) = IFFT( |FFT(x)|^2 ) \]

where \(S_{xx}\) is the power spectrum of our signal \(x\). Note that the complexity of the FFT is \(O(n\log n)\), so this is a significant speedup over the original equation. This means our code for fast autocorrelation is simply:

Alternatively, Julia provides a xcorr function for computing cross correlation, to produce the same output as the function above, you would need: xcorr(x,x)[length(x):], this is the preferred way of achieving it.

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