EEO 40 Digital Sigal Processig Prof. Mark Fowler Note Set #3 Covolutio & Impulse Respose Review Readig Assigmet: Sect. 2.3 of Proakis & Maolakis /
Covolutio for LTI D-T systems We are tryig to fid y(t) whe all i.e. o stored eergy x[] LTI D-T system y[] Before we ca fid the outptut we eed somethig first: Impulse Respose The impulse respose is what comes out whe [] goes i w/ ICs=0 [] [] LTI D-T system Note: If system is causal, the = 0 for < 0 2/
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[] 3/
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 : Usig time-ivariace we kow: [-i] (w/ ) h[-i] Shifted iput gives shifted output Step 2: Use homogeeity part of liearity: x[i][-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 4/
Let s see step 2 for a specific iput: x[i] 3 2 i x[i][-i] (w/ ) x[i]h[-i] - 2 3 4 5 6 x[0][] This I This Out 2 x[][-] 2h[-] x[2][-2] h[-2] 2.5 x[3][-3] 2.5h[-3] 5/
Step 3: Use additivity part of liearity I Step 2 we looked at iputs like this: x[i][-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: i x [ i] [ i] Output: x [ i] h[ i] i But what is this?? O the ext slide we show that it is the desired iput sigal x[]! 6/
Let s see step 3 for a specific iput: 3 2 x[i] i i x [ i] [ i] - 2 3 4 5 6 x[0][] Note: The Sum of these x-weighted impulses gives x[]!! 2 x[][-] x[2][-2] 2h[-] h[-2] 2.5 x[3][-3] 2.5h[-3] 7/
So what we ve see is this: Iput: i x [ i] [ i] Output: x [ i] h[ i] 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[] i 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: i y ] x[ i] h[ i] [ CONVOLUTION! Note: I your Sigals & Systems course you should have leared how to *do* covolutio. You should review that! 8/
Big Picture For a LTI D-T system i zero state characterized by impulse respose, we ca aalytically fid the output whe the iput is x[] by performig the covolutio betwee x[] ad. y [ ] xih [] [ i] hix [] [ i] i i What if the LTI system is causal? h [ ] 00 y [ ] xih [] [ i] hix [] [ i] i What if the iput starts @ = 0? x [ ] 00 i0 i0 y [ ] xih [ ] [ i] hix [ ] [ i] i What if the LTI system is causal ad iput starts @ = 0? y [ ] xih [ ] [ i] hix [ ] [ i] i0 i0 9/
Covolutio Properties These are thigs you ca exploit to make it easier to solve problems.commutativity x[ ] h[ ] h[ ] x[ ] You ca choose which sigal to flip 2. Associativity x[ ] ( v[ ] w[ ]) ( x[ ] v[ ]) w[ ] Ca chage order sometimes oe order is easier tha aother 3. Distributivity x ] ( h [ ] h [ ]) x[ ] h [ ] x[ ] h [ ] [ 2 2 may be easier to split complicated system ito sum of simple oes OR.. we ca split complicated iput ito sum of simple oes (othig more tha liearity ) 4. Covolutio with impulses x[ ] [ q] x[ q] This oe is VERY easy to see usig the graphical covolutio steps. TRY IT!! 0/
Checkig for Stability via the Impulse Respose A LTI system is BIBO stable if ad oly if its impulse respose is absolutely summable : h [ ] Systems w/ Ifiite-Duratio & Fiite-Duratio Impulse Resp. For simplicity of otatio we focus o causal systems here. y [ ] hix [ ] [ i] i0 I geeral, the impulse respose has ifiite duratio A system for which has ifiitely may o-zero values is said to be a ifiite-duratio impulse respose (IIR) system. A system for which has fiitely may o-zero values is said to be a fiite-duratio impulse respose (FIR) system. Suppose = 0 for < 0 ad for M the the covolutio sum becomes M This is said to be a order M y [ ] hix [ ] [ i] FIR system i0 /