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LSI Systems and Convolution

Module by: Richard Baraniuk

Summary: In this section, you will learn about LSI Systems and Convolution.

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.

C^N Case

Figure 1
Figure 1 (figure1.png)

Where h h is the impulse response =0 0 th column of H H

yn=∑k=0N−1hn−kmodNxk y n k 0 N 1 h n k N x k (1)

circular convolution wheels

Figure 2
Figure 2 (figure2.png)

Let N→∞ N and we get

Figure 3
Figure 3 (figure3.png)

yn=∑n=-∞∞hn−kxk y n n h n k x k (2)

note:

hn−kmod∞ h n k has no effect.

Convolution / "Linear Convolution"

Figure 4
Figure 4 (fig1.png)

y=h*x y h x

  • Linear convolution computation is just like circular convolution but on a line but not a circle
    1. Flip h h
    2. Slide flipped h h
    3. Multiply by x x and add up
    4. Repeat

note:

circular shifts hn−kmodN h n k N are replaced by linear shifts hn−k h n k

Notation

yn=xn*hn y n x n h n (3)
or
(x*h)n ( x h ) n (4)
LTI CONVOLUTION

4 Easy Steps To Convolution

yn=∑k=-∞∞xkhn−k y n k x k h n k (5)

  1. Take hk h k and flip it around to h-k h k
    Figure 5
    Figure 5 (fig2.png)
  2. Slide h-k h k to point h-(k−n)=hn−k h k n h n k
    Figure 6
    Figure 6 (fig3.png)
  3. Weight by xk x k and sum.
    Figure 7
    Figure 7 (figure10.png)
  4. Repeat for all n n.

Example 1

Convolve δn*δn−5 δ n δ n 5 .

Method 1: Pictures

Method 2: Plug & Chug

yn=∑k=-∞∞xkhn−k y n k x k h n k

note:

Replace xk x k with δk δ k and hn−k h n k with δn−k−5 δ n k 5

=∑k=-∞∞δkδn−k−5 k δ k δ n k 5

note:

δk δ k is 0 0 unless k=0 k 0

=δn−5 δ n 5 .

Example 2

Convolve δn*xn δ n x n

note:

x x is anything you want it to be.

yn=∑k=-∞∞xkδn−k y n k x k δ n k

note:

n−k n k is 0 0 unless n=k n k

=xn x n .

  • Interpret δn δ n as the impulse response of an LTI system.

Example 3

Convolve xn*xn x n x n with

Figure 8
Figure 8 (fig4.png)

Remember the 4 steps!

Example 4

Convolve

Figure 9
Figure 9 (fig5.png)

Example 5

Convolve xn x n with itself

Figure 10
Figure 10 (fig6.png)

Example 6

Convolve xn=un−un−N x n u n u n N

=1if0≤n≤N−10ifotherwise 1 0 n N 1 0 otherwise

with hn=anun h n a n u n .

Using ∑k= N 1 N 2 αk=α N 1 −α N 2 +11−α k N 1 N 2 α k α N 1 α N 2 1 1 α N 2 ≥ N 1 N 2 N 1

SUPER DUPER USEFUL

Figure 11
Figure 11 (fig7.png)
( |a|<1 a 1 )

note:

x x and h h reversed in the text.

Length of Convolution

Figure 12
Figure 12 (figure9.png)

Length of support yn= y n

Bottom Line

Just as in ℂN N case, all LSI systems with infinite-length inputs and outputs can be characterized by convolution.

Figure 13
Figure 13 (figure4.png)
Figure 14
Figure 14 (figure5.png)
Figure 15
Figure 15 (figure6.png)
Figure 16
Figure 16 (figure7.png)
then
Figure 17
Figure 17 (figure8.png)

yn=∑k=-∞∞hn−kxk y n k h n k x k , -∞<n<∞ n

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