  This paper deals with the problem of finding a low-complexity estimate of the
impulse response of a linear time-invariant discrete-time dynamic system from
noise-corrupted input-output data. To this purpose, we introduce an
identification criterion formed by the average (over the input perturbations)
of a standard prediction error cost, plus a weighted l1 regularization term
which promotes sparse solutions. While it is well known that such criteria do
provide solutions with many zeros, a critical issue in our identification
context is where these zeros are located, since sensible low-order models
should be zero in the tail of the impulse response. The flavor of the key
results in this paper is that, under quite standard assumptions (such as i.i.d.
input and noise sequences and system stability), the estimate of the impulse
response resulting from the proposed criterion is indeed identically zero from
a certain time index (named the leading order) onwards, with arbitrarily high
probability, for a sufficiently large data cardinality. Numerical experiments
are reported that support the theoretical results, and comparisons are made
with some other state-of-the-art methodologies.
