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Impulse Response of a Linear System

Module by: Richard Baraniuk

Summary: Introduction to the impulse response of a linear system.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

All linear systems are matrix multiplies (Figure 1).

Figure 1: ℋ ℋ is an N×N N N matrix.
Figure 1 (matmult.png)

Row Interpretation of ℋ

y=ℋx y ℋ x yn= row n of ℋ | x y n row n of ℋ | x i.e.: yn y n is the inner product of row n n of ℋ ℋ with vector x x.

yn=∑k=0N−1 h n r kxk y n k 0 N 1 h n r k x k (1)
where ℋ=h0rh1r⋮h N - 1 r ℋ h 0 r h 1 r ⋮ h N - 1 r .

Column Interpretation of ℋ

(Figure 2).

Figure 2
Figure 2 (col.png)
y=ℋx=h0ch1c…h N - 1 cx0x1⋮xN−1 y ℋ x h 0 c h 1 c … h N - 1 c x 0 x 1 ⋮ x N 1 (2)
yn=∑k=0N−1xkhkc y n k 0 N 1 x k h k c (3)
i.e.: y y equals the linear combination of the columns of ℋ ℋ weigthed by xk x k .

Special Case

x=δr=0⋮010⋮0 x δ r 0 ⋮ 0 1 0 ⋮ 0 (4)
where the 1 is in position rr. Then Figure 3.
Figure 3
Figure 3 (special1.png)
where
y=ℋδr=⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮0⋮1⋮0=hrc y ℋ δ r ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 ⋮ 1 ⋮ 0 h r c (5)
and hrc h r c is the r th r th column of ℋ ℋ (Figure 4).
Figure 4: δr δ r is the impulse and hrc h r c is the impulse reponse.
Figure 4 (special2.png)
By inputting in turn δr δ r , 0≤n≤N−1 0 n N 1 , we tease out the columns of ℋℋ one at a time.

Example 1

ℋ ℋ for a 3 point smoother:

yn=xn+1modN+2xn+xn−1modN4 y n x n 1 N 2 x n x n 1 N 4 (6)
in ℂ8 8 .

Example 2

ℋ ℋ for an edge detector:

yn=xn−xn−1modN y n x n x n 1 N (7)
in ℂ8 8 .

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