One period of a discrete-time sinusoidal signal of integer period length N samples is given by
Figure 1 illustrates such a sinusoidal signal for N = 32.
One period of a discrete-time sinusoidal signal.
Clearly, the sum of the signal values of w[n] is zero:
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 w[n] with itself (essentially squaring each signal value). Note that the signal period is halved (the frequency is doubled) and that all the values are .
The result of a point-wise multiplication of x[n] with itself.
The sum of the signal values of w2[n] will be greater than zero (which indicates a DC offset). The sum of these values is equal to N/2, where N is the length of the period in samples.