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基礎數學 淡江大學 資訊管理系 侯 永 昌 http: //mail. im. tku. edu. tw/~ychou ftp: //mail. im. tku. edu. tw/Prof_Hou
教科書:Discrete Mathematics, 5 th Edition,by J. A. Dossey, A. D. Otto, L. E. Spence and C. V. Eynden, Pearson Education出版, 滄海書局總經銷 淡江大學資訊管理系侯永昌 1
第十章 組合電路與有限狀態機 (Combinatorial Circuits and Finite State Machines)
邏輯閘門(logical gates) x y 0 0 1 1 0 1 x y x’ (AND) (OR) (NOT) 0 0 1 0 1 1 0 淡江大學資訊管理系侯永昌 3
電路設計(Circuit Design) § 例: x y 0 0 1 1 0 1 f 1(x, y) 0 1 0 0 f 1(x, y) = x’ y x y z 0 0 1 1 0 1 0 1 f 2(x, y, z) 0 0 0 1 0 0 f 2(x, y) = x’ y z 淡江大學資訊管理系侯永昌 20
卡諾圖(Karnaugh Map) § 例: f (x, y, z) = {0, 1, 3, 4, 7} y’ y x’ 1 10 3 2 x 1 04 1 50 7 6 z’ z z’ f = (y’ z’) (x’ y’) (y z) 0 1 2 3 4 5 6 7 x 0 0 1 1 y 0 0 1 1 z f (x, y, z) 0 1 1 1 0 0 0 1 1 淡江大學資訊管理系侯永昌 30
卡諾圖(Karnaugh Map) § 例: f (x, y, z) = {0, 2, 4, 6, 7} y’ y x’ 10 01 03 12 x 14 05 17 16 z’ z f = z’ (x y) z’ 0 1 2 3 4 5 6 7 x 0 0 1 1 y 0 0 1 1 z f (x, y, z) 0 1 1 0 0 1 1 1 淡江大學資訊管理系侯永昌 31
卡諾圖(Karnaugh Map) § 以四個輸入變數(x, y, z, w)為例: z’ z x’ x 00000 01004 1 1 0 012 10008 w’ 00011 01015 1 1 0 113 10019 w 00113 01117 1 115 1 0 1 111 00102 01106 1 1 1 014 1 010 y’ y y’ w’ 淡江大學資訊管理系侯永昌 33
卡諾圖(Karnaugh Map) § 例:f (x, y, z, w) = {0, 2, 5, 7, 8, 10, 13, 15} = (w y) ( w’ y’) z’ z x’ x 1 0 0 1 w’ 0 4 12 8 0 1 1 0 1 5 13 9 w 0 1 1 0 3 7 15 11 1 0 0 1 2 6 y’ y 14 10 y’ w’ 淡江大學資訊管理系侯永昌 34
卡諾圖(Karnaugh Map) § f (x, y, z, w) = {0, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15} = w’ (x’ y) (y z) (x y’ z’) z’ z x’ x 1 1 w’ 0 4 12 8 0 1 1 5 13 9 w 0 1 1 0 3 7 15 11 1 1 2 6 y’ y 14 10 y’ w’ 淡江大學資訊管理系侯永昌 37
卡諾圖(Karnaugh Map) § f (x, y, z, w) = {3, 4, 5, 6, 7, 11} = (x’ y) (y’ z w) z’ z x’ x 0 1 0 0 w’ 0 4 12 8 0 1 0 0 1 5 13 9 w 1 1 0 1 3 7 15 11 0 0 2 6 y’ y 14 10 y’ w’ 淡江大學資訊管理系侯永昌 38
具輸出的有限狀態機 § 具輸出的有限狀態機(finite state machine with output)包含: l l l 有限的狀態集合 S (a finite set of states S) 有限的輸入集合 I (a finite set of inputs I) 有限的輸出集合 O (a finite set of inputs O) 狀態轉換函數 f : I S S (the state the machine moves to when it is in state s and is giving input i) 起始狀態(initial state) s 0 淡江大學資訊管理系侯永昌 44
具輸出的有限狀態機 § 例: 輸入/輸出 Present State Next State L U Output L U q U U q Q d d D L L Input j j J L L s S 淡江大學資訊管理系侯永昌 45
具輸出的有限狀態機 § 例:計算輸入資料變成y的次數(但下圖最多只 算到 3次) § 輸入:x x y x x x y y y x x,輸出: 00111122222 § 輸入:y x x x y y y x x,輸出: 11222233333 State Next State A B C D E F x A C C E E F Input y B B D D F F Output x Input y State A B C D E F 0 1 1 2 2 3 3 淡江大學資訊管理系侯永昌 48
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