Impulse Response of a Linear System **new** 2004/06/08 12:23:16.316 GMT-5 2004/06/08 13:30:51.032 GMT-5 Richard G. Baraniuk richb@rice.edu Richard G. Baraniuk richb@rice.edu Harika Basana ilsai@rice.edu Jeffrey Silverman jsilv@rice.edu impulse response Introduction to the impulse response of a linear system. All linear systems are matrix multiplies ().

ℋ is an N N matrix.

Row Interpretation of ℋ y ℋ x y n row n of ℋ | x i.e.: y n is the inner product of row n of ℋ with vector x . y n k 0 N 1 h n r k x k where ℋ h 0 r h 1 r ⋮ h N - 1 r .

Column Interpretation of ℋ ().

y ℋ x h 0 c h 1 c … h N - 1 c x 0 x 1 ⋮ x N 1 y n k 0 N 1 x k h k c i.e.: y equals the linear combination of the columns of ℋ weigthed by x k .

Special Case x δ r 0 ⋮ 0 1 0 ⋮ 0 where the 1 is in position r. Then .

where y ℋ δ r ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 ⋮ 1 ⋮ 0 h r c and h r c is the r th column of ℋ ().

δ r is the impulse and h r c is the impulse reponse.

By inputting in turn δ r , 0 n N 1 , we tease out the columns of ℋ one at a time. ℋ for a 3 point smoother: y n x n 1 N 2 x n x n 1 N 4 in 8 . ℋ for an edge detector: y n x n x n 1 N in 8 .