leo lam © 2010-2012 signals and systems ee235. leo lam © 2010-2012 today’s menu homework 2 due...
TRANSCRIPT
Leo Lam © 2010-2012
Signals and Systems
EE235
Leo Lam © 2010-2012
Today’s menu
• Homework 2 due now• Convolution!
Leo Lam © 2010-2011
y(t) at all t
3
• At all t
• t<0
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
The product of these two signals is zero where they don’t overlap
Shift Multiply Integrate
Leo Lam © 2010-2011
y(t) at all t
4
• At all t
• 0≤t<1
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
Shift Multiply Integrate
Leo Lam © 2010-2011
y(t) at all t
5
• At all t
• 1≤t<2
• y(t)=2-t for 1≤t<2
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
Shift Multiply Integrate
Leo Lam © 2010-2011
y(t) at all t
6
• At all t
• t≥2
• y(t)=0 for t≥2 (same as t<0, no overlap)
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
Shift Multiply Integrate
Leo Lam © 2010-2011
y(t) at all t
7
• Combine it all– y(t)=0 for t<0 and t>2– y(t)=t for 0≤t<1– y(t)=2-t for 1≤t<2
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
Leo Lam © 2010-2011
Another example
8
• At all t
• t<0
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
The product of these two signals is zero where they don’t overlap
Shift Multiply Integrate
Leo Lam © 2010-2011
Another example
9
• At all t
• 0≤t<0.5
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
Shift Multiply Integrate
h(t) moving right
Leo Lam © 2010-2011
Another example
10
• At all t
• 0.5≤t<1
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
h(t) moving right
Shift Multiply Integrate
5.0
0 2
11 d
Leo Lam © 2010-2011
Another example
11
• At all t
• 1≤t<1.5
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
Shift Multiply Integrate
h(t) moving right
Leo Lam © 2010-2011
Another example
12
• At all t
• 1.5≤t?
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
Shift Multiply Integrate
y(t)=0 because there is no more overlapping
Leo Lam © 2010-2011
Another example
13
• At all t
• Combining
• Can you plot and formulate it?
( ) ( )* ( ) ( ) ( )y t x t h t x h t d
0 0
0 0.5
( ) 0.5 0.5 1
0.5 ( 1) 1 1.5
0 1.5
t
t t
y t t
t t
t
Leo Lam © 2010-2011
Another example
14
• At all t ( ) ( )* ( ) ( ) ( )y t x t h t x h t d
0 0
0 0.5
( ) 0.5 0.5 1
0.5 ( 1) 1 1.5
0 1.5
t
t t
y t t
t t
t
( 0.5) ( 0.5) (( ) ( ) ( 1.5) ( 1.5)1) ( 1)t u t t uy t tu t t ut t
Leo Lam © 2010-2011
Few things to note
15
• Three things:– Width of y(t) = Width of x(t)+Width of h(t)– Start time adds– End time adds– y(t) is smoother than x(t) and h(t) (mostly)
• Stretching the thinking– What if one signal has infinite width?
( )y t x t h t
Leo Lam © 2010-2011
From yesterday
16
• Stretching the thinking– What if one signal has infinite width?
• Width = infinite (infinite overlapping)• Start time and end time all infinite
( )y t x t h t
Leo Lam © 2010-2011
One more example
17
• For all t: ( ) ( )* ( ) ( ) ( )y t x t h t x h t d
x(t)
2
1 t-1
Flip Shift
Can you guess the “width” of y(t)?
Leo Lam © 2010-2011
One more example
18
• For all t: ( ) ( )* ( ) ( ) ( )y t x t h t x h t d
x(t)
2
1 t-1
Multiply & integrate