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The labels in PDF Figure 2 match as follows:Series $1$ has a wandering level and an overall upward drift. Its empirical autocorrelation function stays close to $1$ and declines very slowly, matching panel $c$; this is characteristic of a trend or a record from a unit-root autoregressive process. Such an empirical ACF is not a population stationary ACF if the underlying model is a random walk.

Series $2$ shows relatively smooth runs of adjacent observations on the same side of its level. Positive short-lag dependence emphasizes low frequencies, matching the decreasing spectrum $d$.

Series $3$ is rougher, with much more rapid fluctuation. Negative short-lag dependence emphasizes high frequencies, matching the increasing spectrum $a$. A negative-coefficient AR(1), for example, has this type of spectrum; the figure does not uniquely identify its exact order or parameters.

Series $4$ has conspicuous seasonality. Its autocorrelation function should alternate between positive peaks at full seasonal periods and negative troughs between them, matching $b$. Here “characteristic function” is used descriptively for an ACF or spectrum, rather than for the probability-theoretic characteristic function of a random variable.