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We study the increase in per-sample differential entropy rate of randomsequences and processes after being passed through a non minimum-phase (NMP)discrete-time, linear time-invariant (LTI) filter G. For such filters andrandom processes, it has long been established that this entropy gain, Gain(G),equals the integral of log|G(exp(jw))|. It is also known that, if the firstsample of the impulse response of G has unit-magnitude, then this integralequals the sum of the logarithm of the magnitudes of the non-minimum phasezeros of G, say B(G). In this note, we begin by showing that existingtime-domain proofs of these results, which consider finite length-n sequencesand then let n tend to infinity, have neglected significant mathematical termsand, therefore, are inaccurate. We discuss some of the implications of thisoversight when considering random processes. We then present a rigoroustime-domain analysis of the entropy gain of LTI filters for random processes.In particular, we show that the entropy gain between equal-length input andoutput sequences is upper bounded by B(G) and arises if and only if thereexists an output additive disturbance with finite differential entropy (nomatter how small) or a random initial state. Instead, when comparing the inputdifferential entropy to that of the entire (longer) output of G, the entropygain equals B(G) without the need for additional exogenous random signals. Weillustrate some of the consequences of these results by presenting theirimplications in three different problems. Specifically: a simple derivation ofthe rate-distortion function for Gaussian non-stationary sources, conditionsfor equality in an information inequality of importance in networked controlproblems, and an observation on the capacity of auto-regressive Gaussianchannels with feedback.

The Entropy Gain of Linear Time-Invariant Filters and Some of its Implications