This study has been concerned specifically with the problem of estimating the power spectrum associated with a random process. It has shown how the power spectral density function φ(ω) can be used to specify completely a stationary Gaussian process. Estimation of this function is therefore one of the fundamental problems in random time-series. The power spectral density function is given by

φ(ω) = [2/π] ∫₀^+∞ ρ(τ)cos ωτ dτ

And must be estimated from a partial realization of the process. To accomplish this, the usual procedure is to use estimated auto-covariance functions ρ̂(τ), computed from a set of observations X(t_i) from which φ(ω) is approximated by numerical integration. This gives

φ̂(ω) = 1/W [ρ̂(0)+2∑ _j=1^m-1 ρ̂[jπ/W]cos(ωjπ/W) + ρ̂[mπ/W]cos(ωmπ/W)]

where the ρ[jπ/W]’s are estimated from

ρ̂[jπ/W] = (1/n)∑ _i=1ⁿX(t_i)X(t_i + jπ/W) j = 0, 1, …, m.

φ̂(ω_α) is widely used in power spectral analysis and although it can be shown to be biased, the side lobes of its spectral window can be soothed in such a way that the bias is greatly reduced. The difficulty with the estimator is not so much with its bias, but rather with the considerable numerical task it creates even when digital computing equipment is available.

The primary objective of this research was to devise an estimator which would simulate the bias of the classical estimator φ̂(ω_α) but which would require much less work to compute. To this end the randomizes estimator

φ*(ω_α) = (1/n) _i=1ⁿ X(t_i)X(t_i + k_iΔt)G_α(k_i)

was considered. Unlike φ̂(ω_α) which was constructed by systematically forming all possible lagged products X(t_i)X(t_i + k_iΔt), i=1, 2, …, n and k=0, 1, …, m , the new estimator utilizes a random subsample of lagged products. This is made possible by choosing the k_i at random. The weighting function G_α(k_i) is determined in such a way that the bias of φ*(ω_α) is the same as the bias of φ̂(ω_α).

As would be expected, the sampling variance of φ*(ω_α) is larger than the variance of φ̂(ω_α), since φ*(ω_a) is based on considerably fewer points. It was discovered, however, that the variance of φ*(ω_α) was affected by the probabilities used in the selection of the k_i. Thus, the difference between the variances of the two estimators can be minimized by an appropriate choice of the probabilities P(j). It was shown also that by selecting the integers j = 0, 1, …, m with probabilities

P(j) = (√(f(j))) / ((∑ _j=1^m)(√(f(j)),

where

f(j) = (1/n)[ρ²(0) + ρ²(jΔt)]ρ²(j)G²_α²(j),

that the variance of φ*(ω_α) is minimized. For the special case, φ(ω) = λ and the point W/2, it was shown that sampling with equal probabilities is about half as efficient as with probabilities.

Finally, some of the areas in which research has been carried out using power spectral analysis were considered. In particular, a problem from the field of aeronautical engineering research was used to demonstrate how the randomized estimator φ*(ω_α) would be calculated from real data. Using 900 observations on the pitching velocity of an aircraft, the power spectrum was estimated at ten points. The new estimator proved very tractable and it is felt that the loss of precision due to sampling will be more than offset by the economy and ease with which it produces estimates. This will be especially true when the need is for quick pilot estimates of spectra to be used in preliminary studies, as guidance for future research.