EECE 301 Sigals & Systems Prof. Mark Fowler Note Set #8 D-T Covolutio: The Tool for Fidig the Zero-State Respose Readig Assigmet: Sectio 2.1-2.2 of Kame ad Heck 1/14
Course Flow Diagram The arrows here show coceptual flow betwee ideas. Note the parallel structure betwee the pik blocks (C-T Freq. Aalysis) ad the blue blocks (D-T Freq. Aalysis). New Sigal Models Ch. 1 Itro C-T Sigal Model Fuctios o Real Lie System Properties LTI Causal Etc Ch. 3: CT Fourier Sigal Models Fourier Series Periodic Sigals Fourier Trasform (CTFT) No-Periodic Sigals Ch. 2 Diff Eqs C-T System Model Differetial Equatios D-T Sigal Model Differece Equatios Zero-State Respose Ch. 5: CT Fourier System Models Frequecy Respose Based o Fourier Trasform New System Model Ch. 2 Covolutio C-T System Model Covolutio Itegral Ch. 6 & 8: Laplace Models for CT Sigals & Systems Trasfer Fuctio New System Model New System Model D-T Sigal Model Fuctios o Itegers New Sigal Model Powerful Aalysis Tool Zero-Iput Respose Characteristic Eq. Ch. 4: DT Fourier Sigal Models DTFT (for Had Aalysis) DFT & FFT (for Computer Aalysis) D-T System Model Covolutio Sum Ch. 5: DT Fourier System Models Freq. Respose for DT Based o DTFT New System Model Ch. 7: Z Tras. Models for DT Sigals & Systems Trasfer Fuctio New System Model 2/14
Differetial Equatio zero zero + iput state solutio solutio Covolutio Our Iterest: Fidig the output of LTI systems (D-T & C-T cases) C-T (solve) LTI System D-T Differece Equatio (solve) zero zero + iput state solutio solutio Notice the parallel structure betwee C-T ad D-T systems! We ll see that they are solved usig similar but slightly differet tools!!! Use char. poly. roots HOW?? Use char. poly. roots HOW?? Our focus i this chapter will be o fidig the zero-state solutio (we already kow how to fid the zero-iput solutio for C-T differetial equatios ad later we ll lear how to do that for D-T differece equatios) 3/14
How do we fid the Zero-State Respose? (Remember that is the respose (i.e., output) of the system to a specific iput whe the system has zero iitial coditios) Recall that i the examples for differetial equatios we always saw: y ZS ( t) h( t λ) x( λ) dλ = t t 0 C-T covolutio Where does this come from? How do we deal with it? Recall that i the examples for differece equatios we saw: y ZS [ ] = 1 h[ i] x[ i] D-T covolutio Where does this come from? How do we deal with it? We ll hadle D-T systems first because they are easier to uderstad! 4/14
Covolutio for LTI D-T systems We are tryig to fid y ZS (t) so the i.e. o stored eergy We ll drop the zs subscript to make the otatio easier! x[] LTI D-T system y[] Before we ca fid the Z-S outptut we eed somethig first: Impulse Respose (Warig: book calls it uit-pulse respose ) The impulse respose is what comes out whe δ[] goes i w/ ICs=0 δ[] δ[] LTI D-T system Note: If system is causal, the = 0 for < 0 5/14
The impulse respose uiquely describes the system so we ca idetify the system by specifyig its impulse respose. Thus, we ofte show the system usig a block diagram with the system s impulse respose iside the box represetig the system: x[] LTI D-T system with y[] Because impulse respose is oly defied for LTI systems, if you see a box with the symbol iside it you ca assume that the system is a LTI system. x[] y[] 6/14
How do we kow/get the impulse respose? May possible ways: 1. Give by the desiger of D-T systems 2. Measured experimetally -Put i sequece... 0 0 1 0 0 0... -See what comes out 3. Mathematically aalyze the D-T system -Give differece equatio -Derive I what form will we kow? 1. kow aalytically as a fuctio Ex: 1 h[ ] = u[ ] 2 2. kow umerically as a fiite-duratio sequece Ex: 0 1 2 3 4 5 0.5 1 2.1 1.3.6 0... There are may ways to do this, as we will see! We assume that = 0 for < 0 7/14
Example of aalytically fidig Give a system described by a 1 st order differece equatio: y [ ] = ay[ 1] + bx[ ] Recall that is what comes out whe δ [] goes i (with zero ICs). So we ca re-write the above differece equatio as follows: h[ ] = ah[ 1] + bδ[ ] Here we solve for recursively ad the examie the results to deduce a closed-form solutio (ote: we ca t always use this deductive approach): δ [] -1 0 a 0 + b 0 = 0 0 1 a 0 + b 1 = b 1 0 ab + b 0 = ab 2 2 0 a ( ab) = ( a) b 3 0 2 3 a ( a) b = ( a) b (a ad b are arbitrary umbers) By examiig these results we see = b( a) u[ ] h[ ] = b( a) u[ ] So we ow have the impulse respose for this system!!! Next we ll lear how to use it to solve for the zero-state respose!!! 8/14
Q: How do we use to fid the Zero-State Respose? A: Covolutio We ll go through three aalysis steps that will derive The Geeral Aswer that covolutio is what we eed to do to fid the zerostate respose After that we wo t eed to re-do these steps we ll just Do Covolutio Step 1: Usig time-ivariace we kow: δ[-i] (w/ ) h[-i] Shifted iput gives shifted output Step 2: Use homogeeity part of liearity: x[i]δ[-i] (w/ ) x[i]h[-i] The iput is a fuctio of so we view x[i] as a fixed umber for a give i So we scale the output by the same fixed umber 9/14
Let s see step 2 for a specific iput: x[i] 3 2 1 i x[i]δ[-i] (w/ ) x[i]h[-i] -1 1 2 3 4 5 6 1 x[0]δ[] This I This Out 1 2 x[1]δ[-1] 2h[-1] 1 x[2]δ[-2] 1h[-2] 2.5 x[3]δ[-3] 2.5h[-3] 10/14
Step 3: Use additivity part of liearity I Step 2 we looked at iputs like this: x[i]δ[-i] x[i]h[-i] For each i, a differet iput For each i, a differet respose Now we use the additivity part of liearity: Put the Sum of Those Iputs I Get the Sum of Their Resposes Out Iput: x [ i] δ[ i] Output: x [ i] h[ i] But what is this?? O the ext slide we show that it is the desired iput sigal x[]! 11/14
Let s see step 3 for a specific iput: 3 2 1 x[i] i x [ i] δ[ i] -1 1 2 3 4 5 6 1 x[0]δ[] 1 Note: The Sum of these x-weighted impulses gives x[]!! 2 1 x[1]δ[-1] x[2]δ[-2] 2h[-1] 1h[-2] 2.5 x[3]δ[-3] 2.5h[-3] 12/14
So what we ve see is this: Iput: x [ i] δ[ i] Output: x [ i] h[ i] = x[] Or i other words we ve derived a expressio that tells what comes out of a D-T LTI system with iput x[]: x[] y ] = x[ i] h[ i] [ CONVOLUTION! y[ ] = x[ ] h[ ] Notatio for Covolutio So ow that we have derived this result we do t have to do these three steps we just use this equatio to fid the zero-state output: y ] = x[ i] h[ i] [ CONVOLUTION! 13/14
Big Picture For a LTI D-T system i zero state we o loger eed the differece equatio model -Istead we eed the impulse respose & covolutio New alterative model! Differece Equatio Covolutio & Impulse resp. Equivalet Models (for zero state) 14/14