Cyclic Autocorrelation

For sequences of length $N$ , the cyclic autocorrelation operator is defined by

$$(v\star v)_l \isdefs \sum_{n=0}^{N-1} \overline{v(n)} v(n+l) \;\longleftrightarrow\;\left\vert V(\omega_k)\right\vert^2, \quad k=0,1,2,\ldots,N-1,$$ (7.19)

where $\omega_k\isdef 2\pi k/N$ and the index $n+l$ is interpreted modulo $N$ .

By using zero padding by a factor of 2 or more, cyclic autocorrelation also implements acyclic autocorrelation as defined in (6.16).

An unbiased cyclic autocorrelation is obtained, in the zero-mean case, by simply normalizing $v\star v$ by the number of terms in the sum:

$$\hat{r}_v(l) = \frac{1}{N}(v\star v)_l$$ (7.20)