Writing Exams: Writing Exams. Exam Review

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1 riting Exms: riting Exms Exm Review riting Exms synhronous iruits += Res, yles n Stte ssignment Synhronous iruits Stte-Grph onstrution n Smll Prolems lso Multiple Outputs, n Hrer omintionl Prolem

2 riting Exms: riting Exms RLETON UNIVERSITY pril 28 Finl pril, 6 28; 9: hrs DURTION: 3 HOURS No. of Stuents: 22 Deprtment Nme & ourse Numer: Eletronis ELE 267 n ourse Instrutor(s) T.G.Ry n J. Knight Nme: Numer: Signture: UTHORIED MEMORND TURN OFF ell phones n personl ommunitions equipment n LEVE THEM T THE FRONT. Notes, ooks, n non-ommuniting lultor re llowe. Stuents MUST ount the numer of pges in this exmintion question pper efore eginning to write, n report ny isrepny immeitely to protor. This question pper hs pges. This exmintion pper My Not e tken from the exmintion room. In ition to this question pper, stuents require: n exmintion ooklet yes no x my request Sntron sheet yes no x 2% Plese nswer on the exmintion pper. If your nswer oes not fit, hek you re not oing long hr wy. You my sk for ooklet if you nee it. oolen If you use mp, inite in the spe unerneth it, for whih funtion it is. If you use lger, inite the rules use t the right sie of eh line. Espeilly for sorption, D2, Swp, onesus n other less ovious ones Mrks will e eute if you on t! D D mp of 2

3 riting Exms: riting Exms riting Exms Timing yourself: 3 hours, %.5 min for % or 5 min for %. /2 hour for lenup. Give Up: If you n t get it move on. RTFQ: Re The Foolish Question; re it gin DMN IT ht ws ske for? Just the stte tle? Then on t mke K-mps. Di it wnt iruit. Then rw it! Look for simple methos: Don t o ll lger questions y multiplying out. (++)(++)(+) = First look for simplifitions. re there three m spe for the nswer? Do you nee thirty? re you sure. Simplify t eh step: Right elow: +nything = Left elow: +nything = + nything Nose: += NO NO NO! F oes not equl its ul F = (++)(++)(+) = ++(+) = +++ = + = (+)(+) Tke ul Dist +nythng = (use simp twie) 3 += Exm Rey + = + = +

4 riting Exms: riting Exms riting Exms Timing yourself: 3 hours, %.5 min for % or 5 min for %. /2 hour for lenup. Give Up: If you n t get it move on. RTFQ: Re The Foolish Question; re it gin DMN IT ht ws ske for? Just the stte tle? Then on t mke K-mps. Di it wnt iruit. Then rw it! Look for simple methos: Don t o ll lger questions y multiplying out. (++)(++)(+)= First look for simplifitions. re there three m spe for the nswer? Do you nee thirty? No! see simplifition elow. Simplify t eh step: Right elow: +nything = Left elow: +nything = + nything F n F ul Nose: += F = (++)(++)(+) F ul = ++(+) = +++ = + (F ul ) ul = F = (+)(+) Tke ul Dist +nythng = (use simp twie) + =

5 ommon Mistkes: ommon Mistkes ommon Mistkes. Sying is the sme s 2. Sying n expression is equl to its ul. 3. Not using + = to simplify expressions efore using omplex rules. Not reuing using + E = + E. Simplifying n reuing first sves lger. 4. Sying + = Everyoy knows etter thn this, ut they still o it. 5. hen you tke ul, or generl Demorgn, o not put in the rkets in your he. ( + ) + DE ==> (+)(D +E) ( + ) + DE ==> (() + ) + (DE) ==> ((+))(D +E) 6. Not knowing D2. + =( + )( +) 7. Krnugh mp my not give the simplest iruit, ut it oes give the simplest Σ of Π iruit. Unless you mess up the irles 5

6 oolen lger: oolen lger oolen lger. hnge to Σ of Π ( + ) 2. Simplify D + 3. Tke the ul of F=( + )( + Y) + 4. Ftor + 5. Ftor + 6. Fin the ul of G=( + )( + ) + D 7. Ftor + D 8. onstrut simplest iruit with MUs + D + D 6

7 oolen lger: oolen lger oolen lger. hnge to Σ of Π ( + ) 2. Simplify D + + D + = rule += rule DeMorg rule += 3. Tke the ul of F=( + )( + Y) + 4. Ftor + 5. Ftor + F={( + )( + Y)} + rket NDs F ul ={()+(Y)} ( + )( + ) rule D2 ( +)( + ) rule D2 =( + )( +)( + ) rule D2 6. Fin the ul of G=( + )( + ) + D {( + )( + )} + D rket NDs G ul ={()+()}D ul =D rules =; x+=x 7. Ftor + D ( + )( + D) rule D2 8. onstrut with MUs + D + D =( + )( + )( + D)( + D) rule D2 + D D + D = =D 7

8 oolen lger: 8 oolen lger K-Mps; ommon Errors Mp of F Mp of G hek Your Mp Entries One vrile in the wrong squre, you re tost! Don t Tret Multiple Output Prolems Like Unrelte iruits Mp of F Mp of G 2 gtes, none shre 9 gtes, 3 shre hek for rp roun, Poor it of ftoring might lower gte ount. SHRE GTES Goo Fin iggest loops Poor n ier rp roun You fin two etter loops Using lger fter mp simplifition. Usully not help: K-mp gives the simplest Σ of Π iruit. three gtes + = ( + ) two gtes

9 oolen lger: oolen lger K-Mps; ommon Errors Do not onfuse 5-vrile mps n ul-output mps. 5-vrile use the lrgest irles, Try to enlrge irle y using oth mps. Multiple output Shring is very importnt to sve gtes Often (esp exms) some smller irles will give fewer gtes. shring lowers gte ount. lne these smller irles inrese letter ount. Multiple Output Mpping Rules Do Hlf-Mps First (Exept for PLs) No frien rule ll friens gone to rk sie rule Lst two rules re heuristis They help, ut ut not lwys, They o not reple ll thinking. 9

10 Multiple Outputs Detils: Multiple Outputs Detils Multiple Outputs Detils Exmple.-33 Fin the equtions with minimum logi Minimiztion with shring () Hlf Mps Look for hlf-mp irles (one letter terms) These o not require n ND gte. n n e irle without loss of potentil gte shring. No Friens (Lone )Rule (2) irle squres tht re on only one mp thus nnot e shre. No Friens, They ent To The Drk Sie, Rule (2) Fin squres tht nnot e usefully shre euse tht squre on other mps is lrey irle. irle them hlf-mp qurter-mp Lone Mp Y f No frien Y Lone Y Mp Y g Y

11 Multiple Outputs Detils: Multiple Outputs Detils No est Friens Rule Lonesome rule) (2) Loop squres tht pper on only one mp There is no wy to shre them. Loop them with s mny rothers, or s, s possile. No est friens Just rothers irle him with his rothers est irle y (fmily) No frien here Mp of Mp of No frien here No frien here Mp of Mp of Y

12 Multiple Outputs Detils: Multiple Outputs Detils My est Friens re Gone Rule. (They went over to the rk sie) (2) ith no est friens left, we nnot usefully shre loops. Loop these new frienless s with fmily. h one est frien Frien went to the rk sie Mp of Loop him with fmily Mp of est frien s gone est frien s gone Mp of Loop him His est frien is gone. Mp of Y 2

13 Multiple Outputs Detils: Multiple Outputs Detils Exmp.-33Fin the minimum Σ of Π Minimum numer of ND terms Poor Metho F Y G Y H Y etter Metho () irle hlf mps (none) Lrgest irles 7 ND terms (2) No Friens, ( s on only one mp) F Y G Y H Y Loop Frienless s, they will never shre Unfortuntely you hve hoies; severl wys to loop some squres F Y G Y hoies H Y hoies F Y G Y H Y 3

14 Fin the minimum Σ of Π(ont): Fin the minimum Σ of Π(ont) Fin the minimum Σ of Π(ont) hoies hoie () hosen irles re she Shows unneessry reunny hoie (2) hoie (3) hoies John s Solution 6 ND terms Tom s Solution 5 ND terms nie s Solution 5 ND terms F F Y F Y Y G Y G Y G Y F Y G Y H H H Y Y H Y Y 4

15 Design of Stte Grphs: Design of Stte Grphs Design of Stte Grphs Exmple.-34 or Mhine min sequene G- G- = got - = re Detet or with overlp. Output shoul lst for full lok yle. First tret s iniviul mhines:. Do the grph vertilly. re got got Fill in extr rrows got = = re re re re re Do the grph horizontlly. Do the min sequene first G- = got - got got got got Fill in extr rrows G- go for go for go for go for Mens Moore = got for 5

16 Design of Stte Grphs: or Moore Mhine Detet or with overlp. Mke the prout grph. The two grphs long the xes etermine where the rrows go. got - got G- = R = Design of Stte Grphs got - got got got got G- G = = = = = = R G got = got = got = 6

17 Design of Stte Grphs: or Mhine Detet or with overlp. omplete the prout grph. e sure eh stte hs two exit rrows. LUE got - G- = got Design of Stte Grphs got - got got got got G- = = = = = got got got = = = = = = 7

18 Design of Stte Grphs: Design of Stte Grphs or Mhine, lternte Solution Detet or with overlp. The t omes in serilly with the leftmost it first Output lsts for one whole lok yle. rrnge the sequenes long two xes. The xes show how fr long eh sequene one is. G- got - got - got - got got - got got got got got got got got got got got G- got - got - R got - got R = got - got R got got R got got got got R R got got R = got got 8

19 Prolems from De 96.: Prolems from De 96. Sketh the Output veforms D L Prolems from De 96. D Lth (inverte lok) Trnsprent when? Lthe when? L D Q D Flip-Flop ege triggere Q Equilivlent to gte D G L L D Lth Trnsprent when Lthe when 9

20 Prolems from De 96.: Lthes et from De 96. Exmple.-35 Sketh the Output veforms D L T T T Prolems from De 96. D Lth (inverte lok) Trnsprent when low Lthe when high L D Q D Flip-Flop Rising ege triggere Q D G L L T T T NOR gte D Lth Trnsprent when = Lthe when = 2

21 Prolems from De 96.: Prolems from De 96. Lthes et from De 96.Lthes et from De 96. Exmple.-36 Sketh the Output veforms D L T T T D Lth (inverte lok) Trnsprent when lok low Lthe when high L D Q D Flip-Flop Rising ege triggere Q D G L L T T T NOR gte D Lth Trnsprent when = Lthe when= 2

22 Prolems from De 96.: Lthes et. from De 96. Exmple.-37 Sketh the Output veforms R S E L R S L Prolems from De 96. Reset(L) - set(l) Lth Set output to when R= Set output = when RS= Store output when SR= F Synhronous Stte Grph x= x= x= z= x= z= LK Stte= Synhronous Stte Grph G x= x= z= x= LK x=/z= x=/z= Stte Stte Stte Show stte s well s 22

23 Prolems from De 96.: Lthes et. from De 96. Exmple.-38 Sketh the Output veforms R S E L R(L) S(L) L Prolems from De 96. Reset(L)-set(L) Lth Set output = when RS= Set output to when R= Store output when SR= Set ominnt F Synhronous Stte Grph x= x= x= z= x= z= Sketh the iruit. LK Stte Stte Stte Stte Stte= Stte only hnges on the lok ege Synhronous Stte Grph G x= x= = x= LK = Stte Stte Stte Show stte s well s Stte Stte Stte 23

24 Prolems from De 96.: Prolems from De 96. Summry So Fr T flip flop Q En LK R Re the question! ll of it! lwys hek for x + = Use this to simplify nywhere. lwys reue + x = + x Use nywhere exept for hzrs Too mny people sy x + = x Do not onfuse synhronous n synhronous. Synhronous Not othere y res. ssign sttes to reue logi. Not othere y hzrs. Don t try to ure them. 2. synhronous ssign sttes to e re free. Must not hve hzrs. To tell Moore from Mely in wor prolems LK. Moore Outputs Outputs will pper fter the next tive lok ege. 2. Mely Outputs Outputs will pper fter the input hnges Flip flops Smple D input just efore the lok.. Trnsfer this to the output just fter the lok. Q never hnges exept t lok eges T-flip flops Toggle Q fter every lok ege if - enle (provie it hs n enle) D LK D R D LK Q Input using output Moore Mely 24

25 Prolems from De 96.: Prolems from De 96. Must Do ith Finite-Stte Mhines. Rerrnge the stte tle into Krnugh mp orer.. hoose Moore or Mely outputs from esription. Mely output prolems hve wors like: The output eomes true s soon s the finl it in the sequene rrives. The output oes not hve to wit for the lok. hoose n output type most likely to reue the size of the iruit. Moore output prolems hve wors like: The output eomes true uring the next stte fter the finl input it is reeive. The output must not trnsfer ny glithes tht re on the input.. hnge sttes only t n tive lok ege. Often one when on timing igrms. NoNo. Rememer tht FSMs re multiple output mhines. th for shre gtes when you loop mps. Next-Stte Logi Output Logi Q 2 Q 2 Q 2 Q 2 Q 2 Q Q Q Q Q Q Q Q Q Q D 2 D D Y Q D Q 25

26 Fin Res In synhronous Stte Fin Res In synhronous Stte Tles Exmple.4-.Fin Res ) Stte Tles for espile mhines Present Stte FG Next Stte F + G + For inputs: = = = = FG= FG= FG= FG= Fin Res In synhronous Stte ) irle stle sttes Present Stte FG Next Stte F + G + For inputs: = = = = FG= FG= FG= FG= ) Strting t stle sttes hnge one input, either or hek for oth F + n G + hnging. Present Stte FG Next Stte F + G + For inputs: = = = = FG= FG= 2 FG= FG= ) hek re Present Stte FG Next Stte F + G + For inputs: = = FG= FG= FG= FG= Re Strts t FG=, = hnges -> My en in FG=, = FG=, = Doule input hnges never hppen. Res tht on t strt in stle stte re who res unless prt of lrger pth. 26

27 Fin Res In synhronous Stte Fin Res In synhronous Stte Fin Res In synhronous Stte Tles (ont) Exmple.4-2.Fin Res ontinue ) Strting t stle sttes hnge one input, either or hek for oth F + n G + hnging. Present Stte FG Next Stte F + G + For inputs: = = = = FG= FG= FG= FG= 2 e) hek Re 2 Present Stte FG Next Stte F + G + For inputs: = = FG= FG= FG= Fixe thnks to Priti Re 2 Strts t FG=, = hnges -> ill en in FG=, = FG= Nonritil re f) If the first step is not re. hek for oth F + n G + hnging s trnsitions go own the olumn. Present Stte FG Next Stte F + G + For inputs: = = = = FG= 3 FG= FG= FG= g) hek Re 3 Present Stte FG Next Stte F + G + For inputs: = = FG= FG= FG= FG= Re 3 Strts t FG=, = hnges -> Re invoves FG=, = FG=, = FG=, = ill en in FG=, = Nonritil re etween trnsient sttes 27

28 Fin Res In synhronous Stte Fin Res In synhronous Stte Fining Nstys In synhronous Stte Tles Exmple.4-3.Fin Res ) Fin res irle stle sttes x + y + for inputs : x + y + for inputs : ) Y hek st re x + y + for inputs : Y ) ) Strting from stle sttes look for oule stte hnges x + y + for inputs : Y ritil. n en in three ifferent ifferent sttes Strt stte (=) En: one of stte (=) stte (=) stte (=) hek 2 n re x + y + for inputs : Y 2 Y 2 Rememer oule input hnges re not llowe. Res tht on t strt in stle stte re who res unless prt of lrger pth. e) No re: ) Not llowe input hnge ) Doesn t strt in stle stte ritil. n en in two ifferent sttes Strt stte (=) En: one of stte (=) stte (=) 28

29 Fin Res In synhronous Stte Fin Res In synhronous Stte Fining More Nstys In synhronous Stte Tles Exmple.4-4 Fin Nonritil Res One entry is hnge from previous stte tle. ) Fin nonritil res x + y + for inputs : Y hnge ) irle stle sttes x + y + for inputs : Y ) hek new re e) hek seonry pths x + y + for inputs : Re. x + y + for inputs : Y My go Y to three sttes. Strt in =-> ) Strting from stle sttes look for res x + y + for inputs : Y ut from ny stte we soon he k to. e will eventully reh stte (=). Nonritil Re. 29

30 Fin Res In synhronous Stte Fin Res In synhronous Stte Fining Still More Nstys Fin True yle Exmple.4-5 One whih osilltes until n input hnges.. ) Fin true yle ) irle stle sttes x + y + for inputs : x + y + for inputs : Y Y ) Look own olumns for yles x + y + for inputs : Y ) hek for other yles x + y + for inputs : Y Not yle eventully goes to yle No stle stte Osilltes until input hnges. 3

31 Fin Res In synhronous Stte Milly Diffiult Stte ssignments Exmple.4-6 ) ) Fin Re Free Stte ssignment Simultneous input hnges never hppen. synhronous Stte Tle irle Stle Sttes Next Stte for inputs x y: FG - - on t res re - Look t using - Next Stte for inputs x y: FG Fill in the on t res Fin Res In synhronous Stte ) Show llowe trnsitions ) Mke grph of Next Stte trnsitions. for inputs x y: Sme grph FG rwn ifferently - - Rerw grph nnot fit tringles on K-mp e) Grph trnsitions through - - eome trnsient sttes New grph with - use f) ssign Sttes Possile ssignment g) omplete tle Next stte for inputs x y: FG = = = = 3

32 Fin Res In synhronous Stte Fin Res In synhronous Stte nother Stte ssignments Exmple.4-7 ): synhronous Stte Tle; irle Stle Stes Fin Re Free Stte ssignment Simultneous input hnges never hppen. Present Next Stte Q + Q + 2 Output Stte Q Q 2 xy= xy= xy= xy= = F D D D F D D F F F D ): Drw Trnsitions Trnsitions tht must e 2 step ith ssignment () () F ): Look t two-step trnsitions Next Stte for inputs x y: F D Q Q 2 F D? F D F D n t fix D? F D D (2) F F F D? D ill not work Present Next Stte Q + + Q 2 Output Stte Q Q 2 xy= xy= xy= xy= = F D D D F D D F F F D ith ssignment (2) ): Look t two-step trnsitions orks Next Stte for inputs x y: Q Q 2 F D D D F D D F F F D D F F D e) ssign Sttes F () D (2) 32