A Discrete-Time Sinusoidal Signal

• One period of a discrete-time sinusoidal signal of integer period length $N$ samples is given by
    $$w[n] = \cos( 2 \pi n / N ) \hspace{0.2in} \mbox{for} \hspace{0.2in} n = 0, 1, 2, \ldots, N-1.$$ (1)

• Figure 1 illustrates such a sinusoidal signal for $N=32$.

  Figure 1: One period of a discrete-time sinusoidal signal.

  

• Clearly, the sum of the signal values of $w[n]$ is zero:
    $$\sum_{n=0}^{N-1} \cos( 2 \pi n / N ) = 0.$$ (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 $w[n]$ with itself (essentially squaring each signal value). Note that the signal period is halved (the frequency is doubled) and that all the values are $\ge 0$.

  Figure 2: The result of a point-wise multiplication of $x[n]$ with itself.

  

• The sum of the signal values of $w^2[n]$ 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.