This paper furnishes a new system-theoretic interpretation to the product of unstable poles of discrete-time linear systems. This index has been known to represent limitations of feedback control in various situations. Our result shows that the index gives the limitation of the signal-to-noise ratio (SNR) with respect to amplitude when the feedback loop is stabilized through a multidimensional channel with bounded noise. (C) 2018 Elsevier Ltd. All rights reserved.