One period of a discrete-time sinusoidal signal of integer period length samples is given by
(1)
Figure 1 illustrates such a sinusoidal signal for .
Figure 1:
One period of a discrete-time sinusoidal signal.
Clearly, the sum of the signal values of is zero:
(2)
This result can be generalized to any sinusoidal signal of non-zero frequency when summed over an integral number of periods.
Figure 2 illustrates the result of performing a point-wise multiplication of with itself (essentially squaring each signal value). Note that the signal period is halved (the frequency is doubled) and that all the values are .
Figure 2:
The result of a point-wise multiplication of with itself.
The sum of the signal values of will be greater than zero (which indicates a DC offset). The sum of these values is equal to , where is the length of the period in samples.