Estimating the spectral density of a discrete stationary stochastic process is studied for the case when the observations consist of repeated groups of $\alpha$ equally spaced observations followed by $\beta$ missed observations, $(\alpha > \beta)$. The asymptotic variance of the estimate is derived for normally distributed variables. It is found that this variance depends not only on the value of the spectral density being estimated, but also on the spectral density at the harmonic frequencies brought in by the periodic method of sampling. Curves are presented for $\beta = 1$ showing the increase in the standard deviation and effective decrease in sample size as a function of $\alpha$.