EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive differece equatios; that is, differece equatios where the output is oly depedet o time-delayed values of the iput sigals x [ ], x [ ],..., x [ M+ ] : M y [ ] b k x [ k] k This class of LTI systems have the desirable property that as log as the iput sigal ad filter coefficiets are bouded, x [ ] <,, () () b k <, k, (3) the output sigal is bouded as well, such that, y [ ] <,. (4) This property is kow as BIBO (bouded-iput, bouded-output) stability. Recursive differece equatios, N M y [ ] a l y l + b k x [ k] l k do ot ecessarily have this property, as illustrated with the followig simple differece equatio, () y [ ] y[ ] + x [ ], (6) iitial coditio y[ ], ad iput sigal, x [ ] (7) Figure below plots the output y [ ] for {,,, }. Note that despite bouded filter coefficiets ad a sigle ozero iput at time idex, the output y [ ] grows without boud.. 8 x [ ] y [ ] 6 4. 4 6 8 Figure 4 6 8 The output of differece equatio (6) feeds back o itself to cause this ubouded result. While feedback ca be a very desirable property of systems, especially i the field of cotrol, the wrog kid of feedback ca lead to ubouded or ustable system outputs. Oe example of bad feedback that we are all familiar with occurs whe a microphoe is placed to close to its ow amplifier. Low-volume souds ear the microphoe are boosted by the amplifier, whose output the feeds back ito the microphoe, which the gets further amplified util this vicious cycle results i the very loud familiar screechig soud that we have all heard before. - -
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Because some recursive differece equatios ca result i udesirable system resposes, does ot mea that all recursive systems are udesirable. For example, just as we were able to derive o-recursive filters with desirable properties (such as low-pass filterig), we ca do the same for recursive filters. Cosider, for example, the simple recursive differece equatio below: y [ ] a y [ ] + b x [ ] (8) I equatio (6), we have already see how a bad choice of filter coefficiets ( a, b ) ca result i a ustable system respose; suppose, however, that we choose the followig filter coefficiets istead: a α, b ( α), < α < (9) such that equatio (8) becomes: y [ ] αy [ ] + ( α)x [ ]. () For values of α close to, the system equatio () becomes a simple recursive low-pass filter. I Figure, we plot the filtered output sigal y [ ], the magitude frequecy spectrums Xf () ad Yf (), ad the magitude frequecy respose Hf () for a sample iput sigal x [ ] ad α., α. ad α.9. (See Mathematica otebook, sectios Liear, recursive differece equatio example for these examples.) As i previous examples, the iput sigal x [ ] is a discrete-time sigal sampled at f s Hz from the cotiuous-time sigal, xt () si( π 4t) + oise where, for each sample, the oise compoet of the sigal cosists of a uiformly distributed radom umber i the iterval [, ]. Note that we are able to achieve low-pass filterig with may fewer coefficiets i the recursive case compared to the o-recursive case. I fact, it is true i geeral that for similar performace characteristics, recursive filters ca be desiged with fewer coefficiets tha o-recursive filters. Either way, filter desig largely boils dow to pickig appropriate filter coefficiets for both the recursive ad o-recursive case. As we have see, however, a importat additioal cosideratio i desigig recursive systems is that the output be BIBO-stable. Note that for a recursive LTI differece equatiao, N M y [ ] a l y l + b k x [ k] l k BIBO (bouded-iput, bouded-output) stability meas that whe the iput sigal ad filter coefficiets are bouded, x [ ] <,, (3) () () a l b k <, l, (4) <, k, () the output sigal is bouded as well, such that, y [ ] <,. (6) I this class, we will develop a aalysis tool, kow as the z -trasform, that will allow us to determie the stability of a recursive system without havig to test the system respose for every possible iput sigal. I fact, the z -trasform will allow us to make predictios about the system respose beyod stability; for example, we will be able to predict the exact time-domai system output for certai specific iput sigals.to illustrate some basic ideas, we ow look at three differet cases: () a ustable system, () a margially stable system ad (3) a stable system. Differece equatios for these three systems are give i equatios (7), (8) ad (9), respectively: y [ ] ( 4)y [ ] + ( 3)y [ ] ( )y [ 3] + x [ ] (ustable) (7) y [ ] ( )y [ ] + ( )y [ ] + x [ ] y 3 [ ] ( 3)y 3 [ ] + ( )y 3 [ ] + x [ ] (margially stable) (8) (stable) (9) - -
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary. iput sigal (sampled oisy sie wave) 4 magitude frequecy spectrum x [ ]. Xf () 3 3 -. - -. 3 4 - - f y [ ] Yf () Hf (). 4 α. α.. -. - -. 3 4.. -. - 3 3 4 3 3 - -.8.6.4..8.6.4. - - -. 3 4 - - - -. 4 α.9. -. - 3 3.8.6.4. -. 3 4 - - - - For each of these systems, we let the iput sequece be, Figure : Recursive, low-pass filterig examples x [ ] () which is kow as a discrete-time uit impulse, ad assume that, y p [ ], <, p { 3,, }. () Figure 3 illustrates the system output for iput () for the three differece equatios above; these system resposes are kow as the impulse respose of the system (i.e., the respose to a impulse iput). Note that for the ustable system, the output grows without boud; for the margially stable system, the output ever goes back to zero; while for the stable system, the output decays to zero over time. - 3 -
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary.. x [ ] system iput - - y [ ] ustable system 4 6 8-4 6 8.7.. -. -. margially stable system stable system y [ ] y 3 [ ] 4 6 8 -.4 4 6 8 Figure 3 Now, just lookig at the differece equatios above, it is ot obvious why the three systems should behave so radically differet. However, if we trasform these systems usig the z -trasform, to be studied later i this course, we will be able to make predictios about a system s stability without explicitly simulatig the system respose for a umber of differet iputs. For these three systems, the z -trasforms are give by :.8.6.4. -. H ( z) z --------------------------------------------------------------------------- 3 z 3 + ( 4)z ( 3)z +( ) (ustable) () H ( z) H 3 ( z) z ------------------------------------------------- z + ( )z ( ) z ------------------------------------------------- z + ( 3)z ( ) (margially stable) (3) (stable system) (4) As it turs out, the roots of the deomiators of the z -trasforms above gover the stability of each system. For causal LTI systems, if the roots lie iside the uit circle of the complex plae (i.e. horizotal real axis, imagiary vertical axis), the system will be stable; if oe or more roots lies o the uit circle, the system will be margially stable; ad, fially, if oe or more roots lies outside the uit circle, the system will be ustable. I Figure 4, we plot the roots of the deomiators i equatios () through (4) i the complex plae. We ca ow readily observe the stability (or lack thereof) of each system. System # has oe root outside the uit circle ad is therefore ustable; system # has oe root o the uit circle ad is therefore margially stable; ad system #3 has all its roots iside the uit circle ad. The observat reader will otice a very close relatioship betwee the coefficiets of the differece equatios ad the coefficiets of the umerator ad deomiator polyomials i the z-trasforms.. Causal meas that the output is depedet oly o previous system outputs, ad curret ad/or previous system iputs. - 4 -
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary ustable system margially stable system stable system... -. -. -. - - -.. - - -.. - - -.. Figure 4: root locatios of z-trasform deomiators is therefore stable. As such, the z -trasform is a very powerful aalysis tool i our study of recursive differece equatios. I fact, the z -trasform will allow us to fid closed-form solutios for the impulse respose of a system. For example, the system respose of the margially stable system is give by, y [ ] -- [ ( ),. () 3 + ] We cotiued our lecture discussio by lookig at a example of a simple oliear recursive differece equatio. Noliear differece equatios caot be expressed i the form or equatio () ad ca ofte behave i strage ad complicated ways; as such, they are much more difficult to aalyze tha LTI differece equatios. The example we cosidered was give by, y [ ] Ry[ ] ( y [ ] ), y[ ], R >. (6) Figure plots the output for differet values of R ; ote how eve this simple oliear system ca produce remarkably complex output patters, for example, whe R 3.7. (See Mathematica otebook, sectio Noliear, recursive differece equatio for this example.) Noliear systems simply do ot led themselves to the same kid of (relatively) straightforward aalysis as LTI systems. This lecture cocluded our qualitative overview of the material i this class. We eded the lecture with a slide presetatio of where we will go from here (see accompayig lecture slides). - -
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary. R. y [ ]. R. y [ ].4.4.3.3....8.6.4..7.7.6.6. 4 6 8 R. 4 6 8 R3.. 4 6 8 y [ ] y [ ]. 4 6 8.7.7.6.6.. 4 6 8.9.8.7.6..4.3 R3. R3.7 y [ ] y [ ] 4 6 8 Figure : output for differet values of R - 6 -