Spectral Density as the Fourier Transform of Autocovariance

From Autocovariance to Frequency

A covariance-stationary process can be described through how its observations covary across time. The autocovariance at lag \(h\) records dependence between observations \(h\) periods apart. The spectral density expresses the same second-order information in the frequency domain: instead of organizing variation by time separation, it organizes it by frequency.

This connection is a Fourier transform. Its usefulness depends on being precise about the convention, including the factor \(1/(2\pi)\). Throughout, let \(\{X_t\}\) be a real-valued covariance-stationary process with mean \(\mu\), and define its autocovariance by \(\gamma(h)=\operatorname{Cov}(X_t,X_{t-h})\). Assume that its autocovariances are absolutely summable:

This condition ensures that the Fourier series below converges absolutely and uniformly. Frequencies are measured in radians per observation and are considered on \([-\pi,\pi]\), with endpoints equivalent by periodicity.

The normalization is part of the definition used here. In particular, integrating the density over the full frequency interval gives the variance:

Thus the spectral density is scaled so that its total area, not its average height, equals the process variance. Since radians are dimensionless mathematically, its units can be described as squared units of \(X_t\) per unit frequency.

Fourier Inversion and the Real-Valued Case

The inversion formula follows from the orthogonality of complex exponentials. Absolute summability permits integrating the defining series term by term. Multiplying by \(e^{ih\omega}\) gives

Here the integral of \(e^{ik\omega}\) over \([-\pi,\pi]\) is \(2\pi\) if the integer \(k=0\), and zero otherwise. Setting \(h=0\) yields the variance identity.

For a real-valued covariance-stationary process, \(\gamma(-h)=\gamma(h)\). Consequently, paired positive and negative lags turn the complex Fourier series into a real cosine series:

It follows directly that \(f(-\omega)=f(\omega)\): the spectral density is even. The exponential form is often convenient for derivations, while the cosine form makes the real-valued symmetry explicit.

Why the Spectral Density Cannot Be Negative

The formula alone does not make nonnegativity obvious: individual autocovariances can be positive or negative, and cosine terms can have either sign. The key fact is that autocovariances come from variances of linear combinations of observations, and variances cannot be negative. The argument below connects that fact to the Fourier transform.

Let \(Y_t=X_t-\mu\), and for a fixed frequency \(\omega\), form the complex-valued finite sum \(S_N(\omega)=\sum_{t=1}^{N}Y_t e^{-it\omega}\). Its squared magnitude has nonnegative expectation. Grouping terms by the lag between observations gives

For each lag \(h\), there are \(N-|h|\) pairs of observations in the sum whose separation is \(h\), which explains the weight \(1-|h|/N\). As \(N\) increases, this weight tends to 1 for each fixed lag. Absolute summability allows the limit to pass through the series: the terms are bounded in absolute value by \(|\gamma(h)|\), whose sum is finite. Therefore,

This proves nonnegativity at every frequency. The complex sum in the argument is only a convenient mathematical device; for a real-valued process, the resulting density is real and even.

Working with a Summable Geometric Sequence

Absolute summability also makes familiar infinite-series calculations available. The following hypothetical autocovariance sequence illustrates how a Fourier series can be simplified before evaluating it at particular frequencies.

The Spectral Density of a Stationary AR(1)

Consider the stationary AR(1) process

where the innovations \(\varepsilon_t\) have mean zero, variance \(\sigma_\varepsilon^2\), and are uncorrelated across time. Under \(|\phi|<1\), the stationary linear-process representation gives the autocovariance sequence

Because \(|\phi|<1\), this sequence is absolutely summable. Substituting it into the Fourier formula and summing the two geometric series yields

The denominator is \(|1-\phi e^{-i\omega}|^2\), and is strictly positive for \(|\phi|<1\). Thus the formula is consistent with the general nonnegativity result.

Interpretation and a Common Pitfall

A spectral density is another representation of second-order dependence, not a separate probability distribution. Its values describe how the covariance structure is allocated across frequencies, while its area recovers the variance. It is therefore important to keep the definition’s scale and frequency interval in view when calculating or comparing spectral densities.

• Omitting the normalization. Using \(\sum_h\gamma(h)e^{-ih\omega}\) as the density without dividing by \(2\pi\) changes the convention and makes the integral over \([-\pi,\pi]\) equal \(2\pi\gamma(0)\), not \(\gamma(0)\).
• Assuming negative covariance terms imply a negative density. Individual terms in the Fourier series may be negative, but the full sum is nonnegative because it is the limit of scaled expected squared magnitudes.
• Forgetting symmetry. For a real-valued covariance-stationary process, positive and negative lags pair to produce cosines, so \(f(-\omega)=f(\omega)\).
• Reading density height as variance. A value \(f(\omega)\) is not the variance by itself. The variance is the area under the density over the full interval.

A reliable calculation begins by stating the Fourier convention, checking that the autocovariances are absolutely summable, and then checking symmetry, nonnegativity, and the variance integral. These checks catch many algebra and normalization errors.