autocorrelation is symmetry, , which is easy to prove from the definition. In the continuous case, the autocorrelation is an even function when is a real function, and the autocorrelation is a Hermitian function when is a complex function. The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay , . This is a consequence of the rearrangement inequality. The same result holds in the discrete case. The autocorrelation of a periodic function is, itself, periodic with the same period. The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all ) is the sum of the autocorrelations of each function separately. Since autocorrelation is a specific type of cross-correlation, it maintains all the properties of cross-correlation. By using the symbol to represent convolution and is a function which manipulates the function and is defined as , the definition for may be written as: Multi-dimensional autocorrelation Multi-dimensional autocorrelation is defined similarly. For example, in three dimensions the autocorrelation of a square-summable discrete signal would be When mean values are subtracted from signals before computing an autocorrelation function, the resulting function is usually called an auto-covariance function. Efficient computation For data expressed as a discrete sequence, it is frequently necessary to compute the autocorrelation with high computational efficiency. A brute force method based on the signal processing definition can be used when the signal size is small. For example, to calculate the autocorrelation of the real signal sequence (i.e. , and for all other values of ) by hand, we first recognize that the definition just given is the same as the "usual" multiplication, but with right shifts, where each vertical addition gives the autocorrelation for particular lag values: Thus the required autocorrelation sequence is , where and the autocorrelation for other lag values being zero. In this calculation we do not perform the carry-over operation during addition as is usual in normal multiplication. Note that we can halve the number of operations required by exploiting the inherent symmetry of the autocorrelation. If the signal happens to be periodic, i.e. then we get a circular autocorrelation (similar to circular convolution) where the left and right tails of the previous autocorrelation sequence will overlap and give which has the same period as the signal sequence The procedure can be regarded as an application of the convolution property