Some fundamental properties of a new impulse response Gramian for linear, time-invariant, asymptotically stable, discrete single-input-single-output (SISO) systems are derived in this note. This Gramian is system invariant and can be found by solving a Lyapunov equation, The connection with standard controllability, observability, and cross Gramians is proven, The significance of these results in model-order reduction is highlighted with an efficient procedure.