ee 314 signal and linear system analysis
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EE 314 Signal and Linear System Analysis
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis Slide 1 of 14
Summary of Last Lecture
Lecture 8 EE 314 Signal and Linear System Analysis
โข Applying a causal input (๐ฅ๐ฅ(๐ก๐ก)) to a causal LTI systems with impulse response โ(๐ก๐ก) gives rise to a causal output ๐ฆ๐ฆ ๐ก๐ก :
0
( ) ( ) ( )t
y t x h t dฯ ฯ ฯ= โโซ
( )h t
0
( ) ( )t
h x t dฯ ฯ ฯ= โโซ
Slide 2 of 14
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis
โข Revisiting the prior RC example Let ๐ ๐ ๐ ๐ = 1/2 = ๐๐๐๐, hence,
Input: ๐ฃ๐ฃ๐๐๐๐ ๐ก๐ก = ๐ข๐ข ๐ก๐ก โ ๐ข๐ข(๐ก๐ก โ 1)โข Analytical soln is: For, 0 < t < 1
2( ) 2 ( )th t e u tโ=
0
( ) ( ) ( )t
out inv t v h t dฯ ฯ ฯ= โโซ ( ) ( )2
0
1 2t
te dฯ ฯโ โ= โซ 2 2
0
2t
te e dฯ ฯโ= โซ
2 2
0
tte e ฯโ = ( )2 2 1t te eโ= โ 21 teโ= โ( )( )21 ( ) ( 1)te u t u tโ= โ โ โ
Slide 3 of 14
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis
For, t > 1
Final result
1
0 1
( ) ( ) ( ) ( ) ( )t
out in inv t v h t d v h t dฯ ฯ ฯ ฯ ฯ ฯ= โ + โโซ โซ1
0
( ) 0h t dฯ ฯ= โ +โซ12 2
0
te e ฯโ =
( )2 2 1te eโ= โ 2( 1) 2t te eโ โ โ= โ( )2( 1) 2 ( 1)t te e u tโ โ โ= โ โ
( )( ) ( )2 2( 1) 2( ) 1 ( ) ( 1) ( 1)t t toutv t e u t u t e e u tโ โ โ โ= โ โ โ + โ โ
( ) ( )2 2( 1)1 ( ) 1 ( 1)t te u t e u tโ โ โ= โ + โ โ
Slide 4 of 14
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis
( ) ( )2 2( 1)( ) 1 ( ) 1 ( 1)t toutv t e u t e u tโ โ โ= โ + โ โ
Slide 5 of 14
โ โ๐๐ + ๐ก๐ก => Shift then flipโ(โ(๐๐ โ ๐ก๐ก)) => Flip then shift
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis
โข Letโs evaluate the convolution graphically
Overlay a plot of ๐ฃ๐ฃ๐๐๐๐(๐๐) with a plot of โ(๐ก๐ก โ ๐๐), compute the area under the product (for 0 โค ๐๐ โค ๐ก๐ก), then vary ๐ก๐ก.
0
( ) ( ) ( )t
out inv t v h t dฯ ฯ ฯ= โโซ
( )inv t( )h t
We need ๐ฃ๐ฃ๐๐๐๐(๐๐)
We need โ(๐ก๐ก โ ๐๐)
Flip then shift by ๐ก๐ก
The area under the product!!
ORโ(๐ก๐ก โ ๐๐) = โ(โ(๐๐ โ ๐ก๐ก))
Shift by ๐ก๐ก then flip
Slide 6 of 14
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis
โข Start with ๐ก๐ก < 0
0
( ) ( ) ( )t
y t x h t dฯ ฯ ฯ= โโซ
( ( ))h tฯโ โ ( )x ฯ๐ก๐ก = โ0.5 ๐ ๐ ๐ ๐ ๐ ๐
0=
( ( 0.5))h ฯโ โ
The area under the product?
0.5
0
( 0.5) ( ) ( )y x h t dฯ ฯ ฯโ
โ = โโซ
Slide 7 of 14
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis
โข Now, consider 0 โค ๐ก๐ก < 1
0
( ) ( ) ( )t
y t x h t dฯ ฯ ฯ= โโซ
( ( ))h tฯโ โ ( )x ฯ๐ก๐ก = +0.6 ๐ ๐ ๐ ๐ ๐ ๐
2( )
0
2t
te dฯ ฯโ โ= โซ
2( ) 2 ( )th t e u tโ=
( ) ( ( ))x h tฯ ฯโ โ
Slide 8 of 14
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis
โข Now, consider ๐ก๐ก > 1
0
( ) ( ) ( )t
y t x h t dฯ ฯ ฯ= โโซ
( ( ))h tฯโ โ
( )x ฯ
๐ก๐ก = +1.5 ๐ ๐ ๐ ๐ ๐ ๐
12( )
0
2 te dฯ ฯโ โ= โซ
Slide 9 of 14
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis
MATLAB code
Slide 10 of 14
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis
โข More Convolution Examples
MATLAB code
Slide 11 of 14
Graphical Convolution
Lecture 8 EE 314 Signal and Linear System Analysis
โข Systems connected in series
โข Systems connected in parallel
( )z t1( ) ( )* ( )z t x t h t=
2( ) ( )* ( )y t z t h t=
( )1 2( )* ( ) * ( )x t h t h t=
( )1 2( )** )( ()x t h t h t=
1( ) ( )* ( ) ( )* ( )Ny t x t h t x t h t= + +
[ ]1( )* ( ) ( )Nh t tx t h+ +=
Slide 12 of 14
Causal LTI System
Lecture 8 EE 314 Signal and Linear System Analysis
โข Causality A LTI system is causal if it does NOT rely on future inputs in
order to determine the current output.o All real/physical systems are causal โ They can not anticipate
future inputs!!
o i.e., A causal system has an impulse response that is a causal function.
( ) 0 for all 0h t tโ = <
( ) ( ) ( )y t x h t dฯ ฯ ฯโ
โโ
= โโซ Consider ๐๐ > ๐ก๐ก โผ
Would use future values of ๐ฅ๐ฅ ๐ก๐กto determine ๐ฆ๐ฆ(๐ก๐ก)!!
Does this system have memory?
Give an example of a system that does NOT have memory?
( )h t
0
( ) ( )t
x h t dฯ ฯ ฯ= โโซ
Slide 13 of 14
Next Lecture
Lecture 8 EE 314 Signal and Linear System Analysis
โข LTI Sinusoidal Response
โข Reading Assignment: Chap. 2.7
Slide 14 of 14