MDM Reference • Time & frequency
Time and Frequency Domains
A series may be described in two ways. In the time domain one describes what occurs at each instant; in the frequency domain one describes how much variation sits at each rate. The two are joined by a reversible transformation, and neither view loses anything the other holds.
The time view
The time view keeps the instants in their natural order and reports the value at each. It is the natural view for questions about onset, duration and shape: when a change began, how long a feature lasted, how sharply a level fell. Its summaries are the level, the dispersion, and the profile of correlation across lags.
The frequency view
The frequency view abandons the instants and reports the share of variation carried at each rate. It is the natural view for questions about periodicity and band: whether a repeating component is present, at what rate, and how broad the band of variation is. Its summaries are the spectrum and, between two series, the coherence.
The bridge
The two views are joined by a transformation that may be reversed without loss. Every statement in one view has a counterpart in the other, and the choice between them is a choice of convenience rather than of substance. The bridge is exact for a complete record and is an estimate for a finite slice, the shortfall being the same window effect met in the spectral cell.
Choosing a view
- Ask about timing and shape, and the time view is the clearer.
- Ask about repetition and band, and the frequency view is the clearer.
- Ask about the relation of two series, and often both are wanted.
A practised reader moves between the two without ceremony, using whichever exposes the feature under inquiry.
Localised views
Some features are both brief and periodic, and neither global view serves them well. A localised account, which reports the spectrum of a short window and slides that window along the record, recovers the missing detail. This local account restores the time information that the global spectrum discards, and it is the natural view when a series changes its character partway through.
The lesson of the two domains
Because the two views are equivalent, a claim made in one must survive translation into the other. A reader who finds a claimed feature in the time view should be able to say what it implies for the spectrum, and the reverse. Claims that fail this test are usually claims about the particular record rather than about the process, and they should be reported with that caution.
A worked translation
A record that rises and falls in a slow, even sweep looks simple in the time view and equally simple in the frequency view: a single narrow peak at the rate of the sweep. A record that is steady and flat in the time view, save for a sharp jolt, looks simple there and broad in the frequency view, since a jolt spreads its variation across many rates. The same record wears two faces, and the face that shows the feature is the one to consult.
Choosing the window
When a localised account is wanted, the length of the window is the chief decision. A short window follows swift changes but blurs the rates, while a long window separates the rates but smears the timing. The choice is made on the question at hand, and a reader should say which was made and why, so that the account may be read with that choice in mind.