The periodic autocorrelation of a finite sequence 
of complex numbers is the 
-periodic sequence
 |
(1)
|
where 
denotes the complex conjugate and the final
subscript is reduced modulo 
(Zwillinger 1995, p. 223; Tang et al. 2018; Seberry
and Yamada 2020, p. 33). Some conventions replace 
by 
. Unlike aperiodic
autocorrelation, periodic autocorrelation wraps indices around the ends of the
original sequence. It satisfies
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
If
 |
(5)
|
is the discrete Fourier transform of the original sequence, then
 |
(6)
|
Thus the periodic autocorrelation is the inverse discrete Fourier transform of the absolute square of the transform (Tang et al. 2018).
Similarly, the periodic autocorrelation of a periodic array 
with 
and 
is the 
array
 |
(7)
|
where the final subscripts are reduced modulo 
and 
, respectively.
