EECE 301 Signals & Systems
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1 EECE 01 Sgals & Systems Prof. Mark Fowler Note Set #9 Computg D-T Covoluto Readg Assgmet: Secto. of Kame ad Heck 1/
2 Course Flow Dagram The arrows here show coceptual flow betwee deas. Note the parallel structure betwee the pk blocks (C-T Freq. Aalyss) ad the blue blocks (D-T Freq. Aalyss). New Sgal Models Ch. 1 Itro C-T Sgal Model Fuctos o Real Le System Propertes LTI Causal Etc Ch. : CT Fourer Sgal Models Fourer Seres Perodc Sgals Fourer Trasform (CTFT) No-Perodc Sgals Ch. Dff Eqs C-T System Model Dfferetal Equatos D-T Sgal Model Dfferece Equatos Zero-State Respose Ch. 5: CT Fourer System Models Frequecy Respose Based o Fourer Trasform New System Model Ch. Covoluto C-T System Model Covoluto Itegral Ch. 6 & 8: Laplace Models for CT Sgals & Systems Trasfer Fucto New System Model New System Model D-T Sgal Model Fuctos o Itegers New Sgal Model Powerful Aalyss Tool Zero-Iput Respose Characterstc Eq. Ch. 4: DT Fourer Sgal Models DTFT (for Had Aalyss) DFT & FFT (for Computer Aalyss) D-T System Model Covoluto Sum Ch. 5: DT Fourer System Models Freq. Respose for DT Based o DTFT New System Model Ch. 7: Z Tras. Models for DT Sgals & Systems Trasfer Fucto New System Model /
3 . Computg D-T covoluto -We kow about the mpulse respose h -We foud out that h teracts wth x through covoluto to gve the zero-state respose: y x h How do we work ths? Ths s eeded for uderstadg how: (1) To aalyze systems () To mplemet systems Two cases, depedg o form of x: 1. x s kow aalytcally. x s kow umercally or graphcally Do t forget The desg process cludes aalyss Aalytcal Covoluto (used for by-had aalyss): Pretty straghtforward coceptually: - put fuctos to covoluto summato - explot math propertes to evaluate/smplfy /
4 4/ Example: u a x u b h Recall ths form from 1 st -order dfferece equato example b u u a? y h x y u b u a ) ( a fucto of gets summed away < 0 0, 0 1, u 0 ) ( u b a Now use: > u 0, 1, b a b b a 0 0 ) ( Now use: You should be able to go here drectly
5 5/ + + b a b a b a b a y, 1 1 1, 1 Geometrc Sum b a b y 0 If a b you are addg ( + 1) 1 s ad that gves + 1 So ow we smplfy ths summato If a b, the a stadard math relatoshp gves: 1, r r r r N N Kow Ths!!!
6 6/ Asde: Commutatvty Property of Covoluto A smple chage of varables shows that * * x h h x x h h x y So we ca use whch ever of these s easer
7 Graphcal Covoluto Steps Ca do covoluto ths way whe sgals are kow umercally or by equato - Covoluto volves the sum of a product of two sgals: xh - At each output dex, the product chages Commutatvty says we Step 1: Wrte both as fuctos of : x & h ca flp ether x or h ad get the same aswer Repeat for each Step : Flp h to get h- (The book calls ths fold ) Step : For each output dex value of terest, shft by to get h - (Note: postve gves rght shft!!!!) Step 4: Form product xh ad sum ts elemets to get the umber y 7/
8 Example of Graphcal Covoluto x h Fd yx*h for all teger values of Soluto For ths problem I choose to flp x My persoal preferece s to flp the shorter sgal although I sometmes do t follow that rule oly through lots of practce ca you lear how to best choose whch oe to flp. 8/
9 Step 1: Wrte both as fuctos of : x & h h 1 x Step : Flp x to get x- h Commutatvty says we ca flp ether x or h ad get the same aswer Here I flpped x x- 1 9/
10 We wat a soluto for -, -1, 0, 1,, so must do Steps &4 for all. But let s frst do: Steps &4 for 0 ad the proceed from there. Step : For 0, shft by to get x- h For 0 case there s o shft! x0 - x- x- x0-1 Step 4: For 0, Form the product xh ad sum ts elemets to gve y hx0 - Sum over y0 6-10/
11 Steps &4 for all < 0 Step : For < 0, shft by to get x- h x- x-1-1 Negatve gves a left-shft Show here for -1 Step 4: For < 0, Form the product xh ad sum ts elemets to gve y hx-1-0 Sum over y 0 < 0-11/
12 So what we kow so far s that: y 0, 6, < y y??? So ow we have to do Steps &4 for > 0 1/
13 Steps &4 for 1 Step : For 1, shft by to get x- h Postve gves a Rght-shft 1 x- x1 - shfted to the rght by oe Step 4: For 1, Form the product xh ad sum ts elemets to gve y x1 - h Sum over y /
14 Steps &4 for Step : For, shft by to get x- h Postve gves a Rght-shft 1 x- x - shfted to the rght by two Step 4: For, Form the product xh ad sum ts elemets to gve y x1 - h 1 Sum over y /
15 Steps &4 for Step : For, shft by to get x- h 1 x- x - Postve gves a Rght-shft shfted to the rght by three Step 4: For, Form the product xh ad sum ts elemets to gve y x1 - h 1 Sum over y /
16 Steps &4 for 4 Step : For 4, shft by to get x- h 1 x- x - Postve gves a Rght-shft shfted to the.. rght. by four Step 4: For 4, Form the product xh ad sum ts elemets to gve y x1 - h 1 Sum over y /
17 Steps &4 for 5 Step : For 5, shft by to get x- Postve gves a Rght-shft h 1 x- x - shfted to the rght by fve Step 4: For 5, Form the product xh ad sum ts elemets to gve y x1 - h 1 Sum over y /
18 Steps &4 for 6 Step : For 6, shft by to get x- Postve gves a Rght-shft h 1 x- x - shfted to the rght by sx Step 4: For 6, Form the product xh ad sum ts elemets to gve y x1 - h 1 Sum over y6 18/
19 Steps &4 for all > 6 Step : For > 6, shft by to get x- h Postve gves a Rght-shft x- x - 1 shfted to the rght by seve Step 4: For > 6, Form the product xh ad sum ts elemets to gve y 1 x1 - h 0 Sum over y 0 > 6 19/
20 So ow we kow the values of y for all values of We just eed to put t all together as a fucto Here t s easest to just plot t you could also lst t as a table. y Note that covolvg these kds of sgals gves a ramp-up at the begg ad a ramp-dow at the ed. Varous kds of trasets at the begg ad ed of a covoluto are commo. 0/
21 BIG PICTURE: So what we have just doe s foud the zero-state output of a system havg a mpulse respose gve by ths h whe the put s gve by ths x: x y x * h h x y h Lk: Web Demos of Graphcal D-T Covoluto Ths s a good ste that provdes sght to how to vsualze D-T covoluto However, be sure you ca do graphcal covoluto by had wthout the ad of ths ste!! 1/
22 Implemetato Issues Cosder a D-T system wth mpulse respose h that has fte durato Could Buld a dgtal hardware system or a software program for D-T covoluto lke ths: x(t) Clock x1 x0 0 Shft Regsters or Memory Locatos 0 clock ADC h0 h1 h h-1 Storage Regsters Note that the ormal order of sampled sgals comg to the shft regsters follows the flpped verso of the sgal. y1, y0, etc. /
23 Covoluto Propertes These are thgs you ca explot to make t easer to solve problems 1.Commutatvty x h h x You ca choose whch sgal to flp. Assocatvty x ( v w ) ( x v ) w Ca chage order sometmes oe order s easer tha aother. Dstrbutvty x ( h + h ) x h + x h 1 1 may be easer to splt complcated system h to sum of smple oes OR.. we ca splt complcated put to sum of smple oes (othg more tha learty ) δ 4. Covoluto wth mpulses x q x q Ths oe s VERY easy to see usg the graphcal covoluto steps. TRY IT!! /
24 4/
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