  We show that metric temporal logic can be viewed as linear time-invariant
filtering, by interpreting addition, multiplication, and their neutral
elements, over the (max,min,0,1) idempotent dioid. Moreover, by interpreting
these operators over the field of reals (+,*,0,1), one can associate various
quantitative semantics to a metric-temporal-logic formula, depending on the
filter's kernel used: square, rounded-square, Gaussian, low-pass, band-pass, or
high-pass. This remarkable connection between filtering and metric temporal
logic allows us to freely navigate between the two, and to regard
signal-feature detection as logical inference. To the best of our knowledge,
this connection has not been established before. We prove that our qualitative,
filtering semantics is identical to the classical MTL semantics. We also
provide a quantitative semantics for MTL, which measures the normalized,
maximum number of times a formula is satisfied within its associated kernel, by
a given signal. We show that this semantics is sound, in the sense that, if its
measure is 0, then the formula is not satisfied, and it is satisfied otherwise.
We have implemented both of our semantics in Matlab, and illustrate their
properties on various formulas and signals, by plotting their computed
measures.
