CMPUT101 Introduction to Computing - Summer 2002

Transcription

1 CMPUT Introdution to Computing - Summer 22 %XLOGLQJ&RPSXWHU&LUFXLWV Chpter 4.4 3XUSRVH We hve looked t so fr how to uild logi gtes from trnsistors. Next we will look t how to uild iruits from logi gtes, for exmple: A iruit to hek if two numers re equl. A iruit to dd two numers. Gtes will eome our new uilding loks: Humn ody: ells orgns ody Computers: gtes iruits omputer CMPUT Introdution to Computing () Yngvi Bjornsson 2 &LUFXLW A iruit is olletion of interonneted logi gtes: tht trnsforms set of inry inputs into set of inry outputs, nd where the vlues of the outputs depend only on the urrent vlues of the inputs These kind of iruits re more urtely lled omintoril iruits. &LUFXLWH[WHUQDOYLHZ A iruit n hve ny numer of inputs nd outputs: Numer of inputs nd outputs n differ. The inputs nd outputs re either or. CMPUT Introdution to Computing () Yngvi Bjornsson 3 CMPUT Introdution to Computing () Yngvi Bjornsson 4 &LUFXLWH[WHUQDOYLHZFRQW Output depends only on urrent input vlues Eh set of input lwys genertes the sme output. Different sets of input n generte identil output. &LUFXLWLQWHUQDOYLHZ Ciruits re uild from interonneted AND, OR nd NOT gtes, in wy suh tht eh input omintion produes the desired output. CIRCUIT INPUT OUTPUT CMPUT Introdution to Computing () Yngvi Bjornsson 5 CMPUT Introdution to Computing () Yngvi Bjornsson 6 Chpter

2 CMPUT Introdution to Computing - Summer 22 ([DPSOH Wht re the output vlues nd d given input vlues,? CMPUT Introdution to Computing () Yngvi Bjornsson 7 d &LUFXLW'LDJUDPVDQG%RROHDQ([SUHVVLRQV The digrms we were looking t re lled iruit digrms. Reltionship etween iruit digrms nd Boolen expr.: Every Boolen expression n e represented pitorilly s iruit. Every output in iruit digrm n e written s Boolen expression. Exmple (output vlues nd d from previous digrm): ( OR ) d NOT ( ( OR ) AND (NOT ) ) CMPUT Introdution to Computing () Yngvi Bjornsson 8 &LUFXLWV'LDJUDPDQG%RROHDQ([SUHVVLRQV Deriving Boolen expressions for the output. &LUFXLWV'LDJUDPDQG%RROHDQ([SUHVVLRQV Rememer, when writing Boolen expressions for iruit digrms, we use different nottion! ( OR ) ( OR ) ( + ) ( + ) ( OR ) AND (NOT ) ( + ) NOT (( OR ) AND (NOT )) ( + ) (NOT ) CMPUT Introdution to Computing () Yngvi Bjornsson 9 CMPUT Introdution to Computing () Yngvi Bjornsson ([DPSOH Wht Boolen expression desries the output? + &RQVWUXFWLQJ&LUFXLWV How do we design nd onstrut iruits? We first hve to know wht we wnt the iruit to do! This implies, tht for ll possile input omintions we must deide wht the output should e. One we know tht, there exists methods we n use to design the lyout of the iruit. We will look t one suh method lled, sum-ofproduts lgorithm. CMPUT Introdution to Computing () Yngvi Bjornsson CMPUT Introdution to Computing () Yngvi Bjornsson 2 Chpter

3 CMPUT Introdution to Computing - Summer 22 6XPRI3URGXFWV$OJRULWKP Step : Truth Tle Constrution Repet steps 2, 3 nd 4 for eh output olumn Step 2: Su-expression onstrution using AND nd NOT gtes Step 3: Su-expression omintion using OR gtes Step 4: Ciruit Digrm Prodution Step 5: Comine Ciruit Digrms Step 6: Optimize Ciruit (optionl) Step 7: Stop 6WHS7UXWK7DEOH&RQVWUXFWLRQ Deide wht the iruit is supposed to do: tret the iruit itself s lk ox only interested in input/output signls Ciruit CMPUT Introdution to Computing () Yngvi Bjornsson 3 CMPUT Introdution to Computing () Yngvi Bjornsson 4 3 inputs possiilities 6WHSFRQW Write the desired output for ll possile input omintions: Outputs 2 CMPUT Introdution to Computing () Yngvi Bjornsson 5 6WHS6XEH[SUHVVLRQ&RQVWUXFWLRQ For eh output (seprtely): Use AND nd NOT gtes to onstrut suexpression for rows where the output is Outputs 2 Cse Cse 2 CMPUT Introdution to Computing () Yngvi Bjornsson 6 6WHSFRQW Look t the inputs, if the vlue is then use input s is in su-expression, ( e.g. ) then use input vlue omplemented ( e.g. ) CMPUT Introdution to Computing () Yngvi Bjornsson 7 Why do it this wy? Eh expression will evlute to for given input omintion (row), ut for ll other inputs! 6WHS6XEH[SUHVVLRQ&RPELQDWLRQ Use OR gtes to omine the su-expressions from previous step into one expression ( ) + ( ) This expression will evlute to for ll input omintions tht hve s output, ut for ll the other input omintions (rows)! CMPUT Introdution to Computing () Yngvi Bjornsson 8 Chpter

4 CMPUT Introdution to Computing - Summer 22 6WHS&LUFXLW'LDJUDP3URGXFWLRQ Construt iruit digrm from the expression generted in previous step: ( ) + ( ) CMPUT Introdution to Computing () Yngvi Bjornsson 9 5HSHDWVWHSVDQGIRUHDFKRXWSXW We need to repet steps 2, 3, 4 for eh output. In our exmple, there is one more output: Step2: Four su-expressions, one for eh row: Step 3: Comine su-expressions using + (OR): ( ) + ( ) + ( ) + ( ) Step 4: Drw iruit digrm (see p. 694 in text-ook) CMPUT Introdution to Computing () Yngvi Bjornsson 2 &RPELQH,QGLYLGXDO&LUFXLWV Comine the iruits for eh individul output into n one lrger iruit. Ciruit for Output Output 2SWLPL]HWKH&LUFXLW A iruit uild using this lgorithm will generte the orret output, ut it uses unneessrily mny gtes Why is tht importnt? Typilly we need to optimize the iruit, y minimize the numer of gtes used. An optimized iruit for the exmple would look like: Ciruit for Output 2 Output 2 CMPUT Introdution to Computing () Yngvi Bjornsson 2 CMPUT Introdution to Computing () Yngvi Bjornsson 22 ([DPSOH&RPSDUHIRU(TXDOLW\&LUFXLW&( We wnt to uild iruit tht heks if two numers re the sme? CMPUT Introdution to Computing () Yngvi Bjornsson 23 The sme numer if nd only if ll orresponding its re the identil. First step is to uild iruit tht ompres two its (n then use 6 of those to ompre two 6-it numers!) ([6WHS7UXWKWDEOHFRQVWUXFWLRQ The iruit to ompre two its hs: two inputs (the vlue of the two its) one output ( if the its re different, if the its re sme) -CE How does the truth-tle look like? Output CMPUT Introdution to Computing () Yngvi Bjornsson 24 Chpter

5 CMPUT Introdution to Computing - Summer 22 ([DPSOH 6WHS&RQVWUXFWVXEH[SUHVVLRQV Construt Boolen expression for eh row in the tle where the output is one: Output ([DPSOH6WHSDQG Comine into one su-expression using OR (+) ( ) + ( ) Drw iruit digrm CMPUT Introdution to Computing () Yngvi Bjornsson 25 CMPUT Introdution to Computing () Yngvi Bjornsson 26 5HSHDWIRUHDFKRXWSXW Need to repet step 2, 3, 4 for ll outputs: There is only one output, so we re done! So our -it ompre iruit ( -CE ) looks like: ELWFRPSDUH But we wnt to ompre N-it sized numers? CMPUT Introdution to Computing () Yngvi Bjornsson 27.. CMPUT Introdution to Computing () Yngvi Bjornsson 28 ([DPSOH$Q$GGLWLRQ&LUFXLWDGG We wnt to uild iruit tht dds two integers. How do we dd two inry numers the sme wy s deiml numers (ut different se) + CMPUT Introdution to Computing () Yngvi Bjornsson 29 s ([DPSOH$'' Let s strt y uilding iruit tht dds three its (two its + rry) We n then use N of these -ADD iruits to dd ny two N-it integers. rry -ADD rry s CMPUT Introdution to Computing () Yngvi Bjornsson 3 Chpter

6 CMPUT Introdution to Computing - Summer 22 ([DPSOH 6WHS RXWSXW ([ 6WHS 7UXWK WDEOH FRQVWUXFWLRQ Outputs s CMPUT Introdution to Computing Construt Boolen expression for eh -row Outputs s Comine into one Boolen expression s ( ) + ( ) + ( ) + ( ) () Yngvi Bjornsson 3 ([DPSOH 6WHS &LUFXLW 'LDJUDP RXWSXW CMPUT Introdution to Computing () Yngvi Bjornsson 33 ([DPSOH &RPELQLQJ RXWSXW DQG FLUFXLWV () Yngvi Bjornsson 32 ([DPSOH 6WHS RXWSXW Step2 : Construt Boolen expression for eh -row CMPUT Introdution to Computing rry Step 3: Comine into one Boolen expression s ( ) + ( ) + ( ) + ( ) Step 4: Drw iruit digrm (not shown) CMPUT Introdution to Computing () Yngvi Bjornsson 34 ([DPSOH $'' s rry CMPUT Introdution to Computing Chpter () Yngvi Bjornsson 35 CMPUT Introdution to Computing () Yngvi Bjornsson 36 6

7 CMPUT Introdution to Computing - Summer 22 ([DPSOH2SWLPL]HWKHFLUFXLW Eh -ADD iruit hs 25 gtes (47 trnsistors) 6 AND gtes ( x 2 trnsistors) 6 OR gmes ( x 2 trnsistors) 3 NOT gtes ( x trnsistors) To dd two 32-its its integers we need 32 -ADD iruits 32 * 25 8 gtes 54 trnsistors Optimized 32-its ddition iruit in modern omputers uses: 5-6 trnsistors We will not lern how to optimize iruits in this ourse &RQWURO&LUFXLWV Chpter 4.5 CMPUT Introdution to Computing () Yngvi Bjornsson 37 &RQWURO&LUFXLWV So fr we hve seen two types of iruits: Logil ( is?) Arithmeti ( + ) Computers use mny different logil (>, <, >. <,!, ), nd rithmeti (+,-,*,/) iruits. There re lso different kind of iruits tht re essentil for omputers Í ontrol iruits We will look t two different kind of ontrol iruits, multiplexors nd deoders. XOWLSOH[RU A multiplexor iruit hs: 2 N input lines (numered,, 2 N -) output line N seletor lines The seletor lines re used to hoose whih of the input signls eomes the output signl: Seletor lines interpreted s n N-it integer The signl on the input line with the orresponding numer eomes the output signl. CMPUT Introdution to Computing () Yngvi Bjornsson 39 CMPUT Introdution to Computing () Yngvi Bjornsson 4 XOWLSOH[RUFRQW XOWLSOH[RUFRQW 2 Multiplexor 3 CMPUT Introdution to Computing () Yngvi Bjornsson 4 CMPUT Introdution to Computing () Yngvi Bjornsson 42 Chpter

8 CMPUT Introdution to Computing - Summer 22 'HFRGHU A deoder iruit hs: N input lines (numered,,., N-) 2 N output line (numered,, 2 N -) Works s follows: The N input lines re interpreted s N-it integer vlue. The output line orresponding to the integer vlue is set to, ll other to 'HFRGHUFRQW CMPUT Introdution to Computing () Yngvi Bjornsson 43 CMPUT Introdution to Computing () Yngvi Bjornsson 44 'HFRGHUFRQW Deoder CMPUT Introdution to Computing () Yngvi Bjornsson 45 6XPPDU\ We looked t how omputers represent dt: Internl vs Externl Representtion Bsi storge unit is inry digit it Dt is represented internlly s inry dt. Use the inry numer system. We lerned why omputers use inry dt: Min reson is reliility Eletroni devies work est in i-stle environment. CMPUT Introdution to Computing () Yngvi Bjornsson 46 6XPPDU\FRQW 6XPPDU\FRQW We looked t the si uilding loks used in omputers: Binry Storge Devie Trnsistor We sw how to uild logi gtes (AND, OR, NOT): Trnsistors Gtes Boolen logi We sw how to uild iruits: Gtes Ciruits Looked t logil, rithmeti, nd ontrol iruits. CMPUT Introdution to Computing () Yngvi Bjornsson 47 Now tht we hve seen the si uilding loks (low-level view), in the next hpter we will look t the ig piture (high-level view). We will look t the si rhiteture underlying design of ll omputers: Look t higher level omputer omponents, suh s proessors nd memory. Understnd etter how omputers exeute progrms. CMPUT Introdution to Computing () Yngvi Bjornsson 48 Chpter