Also the time modulation implementation ofseveral optimization methods such as differential evolution (DE) [4, 5], particle swarm optimization (PSO) , or simulated annealing (SA)  may be considered amongst the early studies.
Sidebands occurring due to the periodic nature of time modulation are generally evaluated as a loss in many standard applications, but in the work of Tennant and Chambers, it is shown that these sidebands can be used on purpose .
As it can be understood from these expressions, the time modulation inevitably shifts some of the radiated power into sidebands.
Generally, in time modulation problems, a high directivity accompanied with a low power loss and a low sideband level is required.
All these distinct advantages in 4D arrays are based on the exploitation and utilization of the sideband signals caused by time modulation.
The time modulation period is set as [T.sub.p], with a time modulation frequency [f.sub.p] = 1/[T.sub.p] ([f.sub.p] [much less than] [f.sub.0], the radiation pattern is independent of [f.sub.p]).
The time modulation in 4D arrays can bring on equivalent amplitude excitations at the center frequency, and both amplitude and phase excitations at the sideband frequencies in a time-average sense.
It is known that the SLL of a conventional uniformly excited planar array without time modulation is about -16 dB.
As a result of time modulation in 4D arrays, sidebands are generated at multiples of the time modulation frequency.
In this paper, the time modulation scheme of pulse shifting proposed in  is adopted.
As [U.sub.k](t) is a periodic function with the time modulation period [T.sub.p], [U.sub.k](t) can be decomposed into a Fourier series, given by 
where [a.sub.mk] represents the complex excitation weighting for the kth element at the mth sideband and [f.sub.p] = 1/[T.sub.p] is the time modulation frequency.