姜书艳数字逻辑设计及应用8

1、1 1chapter 4 combinational logic chapter 4 combinational logic design principlesdesign principles( (组合逻辑设计原理组合逻辑设计原理) ) basic logic algebra (逻辑代数基础逻辑代数基础) combinational-circuit analysis (组合电路分析组合电路分析) combinational-circuit synthesis (组合电路综合组合电路综合)digital logic design and application ( (数字逻辑设计及应用数字逻辑

2、设计及应用) )2 2review of chapter 3review of chapter 3electronic behavior of cmos circuitslogic voltage levels (逻辑电压电平逻辑电压电平)dc noise margins (直流噪声容限直流噪声容限)fan-in（扇入）（扇入）fun-out (扇出扇出)digital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )3 3review of chapter 3review of chapter 3transmission gates

3、(传输门传输门)schmitt-trigger inputs （hysteresis）three-state outputs (tri-state output)open-drain outputs (open-collector gate)enen_labaenout逻辑符号逻辑符号abzdigital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )4 4review of chapter 3review of chapter 3logic levelscmos(0-1.5v, 3.5-5v)ttl(0-0.8v, 2-5v)ecl

4、(l=-1.8v, h=-0.9v)(l=3.6v, h=4.4v)high (高态高态)abnomal(不正常状态不正常状态)low (低态低态)volmaxvilmaxvihminvohmindigital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )5 5review of chapter 3review of chapter 3wired and (线与线与)open-drain outputs (open-collector gate)wired or (线或线或)emitter-coupled logic gate （e

5、cl, 发射极耦合逻辑门）发射极耦合逻辑门）abzdigital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )6 6digital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )review of chapter 3review of chapter 3positive logic and negative logic(正逻辑和负逻辑正逻辑和负逻辑)three basic logic functions: and, or, and not (三种基本逻辑：与、三种基本逻辑：

6、与、或、非或、非)voutvinvccr获得高、低电平的基本原理获得高、低电平的基本原理7 7review of chapter 3 (review of chapter 3 (第三章内容回顾第三章内容回顾) )digital logic design and application (数字逻辑设计及应用数字逻辑设计及应用)three kinds of description method (三种描述方法三种描述方法)： truth table (真值表真值表) logic expression (逻辑表达式逻辑表达式) logic circuit (逻辑符号逻辑符号)nand and nor

7、 (与非和或非与非和或非)8 88introductionintroductionlets learn to design digital circuits, starting with a simple form of circuit:combinational circuitoutputs depend solely on the present combination of the circuit inputs values2.1digitalsystemb=0f=0digitalsystemif b=0, then f=0if b=1, then f=1b=1f=1digitalsys

8、temb=0f=0(a)digitalsystemb=1f=1 vs. sequential circuit: has “memory” that impacts outputs toodigitalsystemb=0f=1cannot determine value of f solely from present input value(b)9 9digital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )basic concepts (basic concepts (基本概念基本概念) )two types of logic

9、circuits(逻辑电路分为两大类逻辑电路分为两大类):combinational logic circuit（组合逻辑电路组合逻辑电路）sequential logic circuit（时序逻辑电路时序逻辑电路）outputs depend only on its current inputs.(任何时刻的输出仅取决与当时的输入任何时刻的输出仅取决与当时的输入)outputs depends not only on the current inputs but also on the past sequence of inputs.(任一时刻的输出不仅取决与当时的输入，任一时刻的输出不仅取

10、决与当时的输入，还取决于过去的输入序列还取决于过去的输入序列)电路特点：无反馈回路、无记忆元件电路特点：无反馈回路、无记忆元件1010digital logic design and application (数字逻辑设计及应用数字逻辑设计及应用)4.1 switching algebra (4.1 switching algebra (开关代数开关代数) )4.1.1 axioms (公理公理)x = 0 , if x 1 x = 1, if x 0 0 = 1 1 = 0 00 = 0 1+1 = 1 11 = 1 0+0 = 0 01 = 10 = 0 1+0 = 0+1 = 1f =

11、0 + 1 ( 0 + 1 0 ) = 0 + 1 111 114.1.2 single-variable theorems4.1.2 single-variable theorems( (单变量开关代数定理单变量开关代数定理) )identities (自等律自等律)：x + 0 = x x 1 = xnull elements (0-1律律)：x + 1 = 1 x 0 = 0involution (还原律还原律)：( x ) = xidempotency(同一律同一律)：x + x = x x x = xcomplements(互补律互补律)：x + x = 1 x x = 0变量和变量

12、和常量的常量的关系关系变量和变量和其自身其自身的关系的关系digital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )1212digital logic design and application (数字逻辑设计及应用数字逻辑设计及应用)4.1.3 two-and three-variable theorems (4.1.3 two-and three-variable theorems (二变二变量或三变量开关代数定理量或三变量开关代数定理) ) similar relationship with general algebra

13、 (与普通代数相似的关系与普通代数相似的关系)commutativity (交换律交换律) a b = b a a + b = b + aassociativity (结合律结合律) a(bc) = (ab)c a+(b+c) = (a+b)+cdistributivity (分配律分配律) a(b+c) = ab+ac a+bc = (a+b)(a+c)可以利用真值表证明公式和定理可以利用真值表证明公式和定理1313perfect induction of the theoremuse the truth table to prove the functions on both side a

14、re same !zxyxzyzxyxto prove, just evaluate all possibilities141414example uses of the propertiesexample uses of the propertiesshow abc equivalent to cba.use commutative property:a*b*c = a*c*b = c*a*b = c*b*ashow abc + abc = ab.use first distributive propertyabc + abc = ab(c+c). complement property r

15、eplace c+c by 1: ab(c+c) = ab(1). identity property ab(1) = ab*1 = ab.a151515example uses of the propertiesexample uses of the propertiesshow x + xz equivalent to x + z.second distributive property replace x+xz by (x+x)*(x+z). complement property replace (x+x) by 1, identity property replace 1*(x+z)

16、 by x+z.a1616notes (notes (几点注意几点注意) )不存在变量的指数不存在变量的指数 aaa a3允许提取公因子允许提取公因子 ab+ac = a(b+c)没有定义除法没有定义除法 if ab=bc a=c ? 没有定义减法没有定义减法 if a+b=a+c b=c ?a=1, b=0, c=0ab=ac=0, a ca=1, b=0, c=1错！错！错！错！digital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )1717some special relationshipssome special rel

17、ationships( (一些特殊的关系一些特殊的关系) )covering (吸收律吸收律)x + xy = x x(x+y) = xcombining (组合律组合律)xy + xy = x (x+y)(x+y) = xconsensus 添加律（一致性定理）添加律（一致性定理）xy + xz + yz = xy + xz(x+y)(x+z)(y+z) = (x+y)(x+z)digital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )1818对上述的公式、定理要熟记，做到举一反三对上述的公式、定理要熟记，做到举一反三(x+y)

18、 + (x+y) = 1a + a = 1xy + xy = x(a+b)(a(b+c) + (a+b)(a(b+c) = (a+b)代入定理：代入定理： 在含有变量在含有变量 x x 的逻辑等式中，如果将式中的逻辑等式中，如果将式中所有出现所有出现 x x 的地方都用另一个函数的地方都用另一个函数 f f 来代替，来代替，则等式仍然成立。则等式仍然成立。digital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )1919prove (证明证明)： xy + xz + yz = xy + xzyz = 1yz = (x+x)yzxy

19、 + xz + (x+x)yz= xy + xz + xyz +xyz= xy(1+z) + xz(1+y)= xy + xzdigital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )20204.1.4 n-variable theorems 4.1.4 n-variable theorems (n(n变量定理变量定理) )generalized idempotency theorem ( 广义同一律广义同一律 )x + x + + x = x x x x = xshannons expansion theorems ( 香农展开

20、定理香农展开定理 ), 0(), 1 (),(f212121nnnxxfxxxfxxxx), 1 (), 0(),(f212121nnnxxfxxxfxxxxdigital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )2121prove (证明证明)： ad + ac + cd + abcd = ad + ac= a ( 1d + 1c + cd + 1bcd ) + a ( 0d + 0c + cd + 0bcd )= a ( d + cd + bcd ) + a ( c + cd )= ad( 1 + c + bc ) + ac

21、( 1 + d )= ad + acdigital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )22224.1.4 n-variable theorems 4.1.4 n-variable theorems ( n( n变量定理变量定理 ) )demorgans theorems (摩根定理摩根定理)2121)(nnxxxxxx 2121)(nnxxxxxx ),(),(2121 nnxxxfxxxf complement theorems ( (反演定理反演定理) )(a b) = a + b(a + b) = a bdigita

22、l logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )2323complement rules (反演规则反演规则)： +，0 1，complementing variables ( 变量取反变量取反 )follow the priority sequence as before ( 遵循原来的运算优先次序遵循原来的运算优先次序 )keep the complement symbol of non-single variables ( 不属于单个变量上的反号应保留不变不属于单个变量上的反号应保留不变 )digital logic desi

23、gn and application ( (数字逻辑设计及应用数字逻辑设计及应用) )2424example 1：write the complement function for each of the following logic functions. (写出下面函数的反函数写出下面函数的反函数 ) f1 = a (b + c) + c d f2 = (a b) + c d eexample 2：prove (ab + ac) = ab + ac 合理地运用反演定理能够将一些问题简化合理地运用反演定理能够将一些问题简化2525合理地运用反演定理能够将一些问题简化合理地运用反演定理能够将一

24、些问题简化prove：ab + ac = ab + acab + ac + bc = ab + ac(a+b)(a+c)aa +ac + ab + bcac + ab ac + ab + bcdigital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )26264.1.5 duality (4.1.5 duality (对偶性对偶性) )duality rule ( 对偶规则对偶规则 ) +；0 1变换时不能破坏原来的运算顺序（优先级）变换时不能破坏原来的运算顺序（优先级）principle of duality ( 对偶原理对偶原理

25、 )若两逻辑式相等，则它们的对偶式也相等若两逻辑式相等，则它们的对偶式也相等例：例： write the duality function for each of the following logic functions. (写出下面函数的对偶函数写出下面函数的对偶函数) f1 = a + b (c + d) f2 = ( a(b+c) + (c+d) )x + x y = xx x + y = xx + y = xx ( x + y ) = x fd(x1 , x2 , , xn , + , , ) = f(x1 , x2 , , xn , , + , ) digital logic de

26、sign and application ( (数字逻辑设计及应用数字逻辑设计及应用) )27274.1.5 duality (4.1.5 duality (对偶性对偶性) )证明公式：证明公式：a+bc = (a+b)(a+c)a(b+c)ab+acdigital logic design and application ( (数字逻辑设计及应用数字逻辑设计及应用) )duality rule ( 对偶规则对偶规则 ) +；0 1变换时不能破坏原来的运算顺序（优先级）变换时不能破坏原来的运算顺序（优先级）principle of duality ( 对偶原理对偶原理 )若两逻辑式相等，则它们的对偶式也相等若两逻辑式相等，则它们的对偶式也相等2828two kind of logic positive logic : 1 ( high level ) 0 (low level)negative logic: 0 ( high level ) 1 (low level) if a logic relation exist in positive logic, it must be exist in negativ