LSI Systems and Convolution **new** 2004/07/13 10:15:49.039 GMT-5 2004/07/29 15:26:47.969 GMT-5 Richard G. Baraniuk richb@rice.edu Malan Shiralkar mshiralkar05@hotmail.com Harika Basana ilsai@rice.edu Richard G. Baraniuk richb@rice.edu LSI Systems and Convolution LSI systems convolution elec 301 In this section, you will learn about LSI Systems and Convolution.

C^N Case

Where h is the impulse response 0 th column of H y n k 0 N 1 h n k N x k circular convolution wheels

Let N and we get

y n n h n k x k h n k has no effect.

Convolution / "Linear Convolution"

y h x Linear convolution computation is just like circular convolution but on a line but not a circle Flip h Slide flipped h Multiply by x and add up Repeat circular shifts h n k N are replaced by linear shifts h n k

Notation y n x n h n or ( x h ) n LTI CONVOLUTION

4 Easy Steps To Convolution y n k x k h n k Take h k and flip it around to h k

Slide h k to point h k n h n k

Weight by x k and sum.

Repeat for all n .

Convolve δ n δ n 5 .

Method 1: Pictures

Method 2: Plug & Chug y n k x k h n k Replace x k with δ k and h n k with δ n k 5 k δ k δ n k 5 δ k is 0 unless k 0 δ n 5 .

Convolve δ n x n x is anything you want it to be. y n k x k δ n k n k is 0 unless n k x n . Interpret δ n as the impulse response of an LTI system. Convolve x n x n with

Remember the 4 steps! Convolve

Convolve x n with itself

Convolve x n u n u n N 1 0 n N 1 0 otherwise with h n a n u n . Using k N 1 N 2 α k α N 1 α N 2 1 1 α N 2 N 1 SUPER DUPER USEFUL

( a 1 ) x and h reversed in the text.

Length of Convolution

Length of support y n

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

then

y n k h n k x k , n