CSE 140 Lecture 11 Standard Combinational Modules CK Cheng and Diba Mirza CSE Dept. UC San Diego 1
Part III - Standard Combinational Modules (Harris: 2.8, 5) Signal Transport • Decoder: Decode address • Encoder: Encode address • Multiplexer (Mux): Select data by address • Demultiplexier (DeMux): Direct data by address • Shifter: Shift bit location Data Operator • Adder: Add two binary numbers • Multiplier: Multiply two binary numbers 2
1. Decoder • Definition • Logic Diagram • Application (Universal Set) • Tree of Decoders 3
iClicker: Decoder Definition A. A device that decodes B. An electronic device that converts signals from one form to another C. A machine that converts a coded text into ordinary language D. A device or program that translates encoded data into its original format E. All of the above 4
Decoder Definition: A digital module that converts a binary address to the assertion of the addressed device E (enable) y 0 0 I 0 0 y 1 1 . 2 1 I 1 3 . 4 5 I 2 2 6 y 7 n to 2 n decoder 7 2 n outputs n inputs function: 2 3 = 8 n= 3 y i = 1 if E= 1 & (I 2, I 1, I 0 ) = i y i = 0 otherwise 5
Interconnect: Decoder, Encoder, Mux, DeMux Processors Arbiter Data 1 Memory Bank Mux P1 Address 1 Data P2 Demux n-m Address 2 Mux Address m n 2 m Address k Decoder Data k Decoder: Decode the address to assert the addressed device Pk Mux: Select the inputs according to the index addressed by the control signals 6
1. Decoder: Definition PI Q: What is the output Y 3:0 of the 2:4 decoder for (A 1 , A 0 ) = (1,0)? 2:4 Decoder 11 Y 3 A. (1, 1, 0, 0 ) A 1 10 Y 2 A 0 01 Y 1 B. (1, 0, 1, 1) 00 Y 0 C. (0, 0, 1, 0) A 1 A 0 Y 3 Y 2 Y 1 Y 0 D. (0, 1, 0, 0) 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 1 0 0 0 7
1. Decoder: Definition • N inputs, 2 N outputs • One-hot outputs: only one output HIGH at once E 2:4 Decoder 11 Y 3 A 1 10 Y 2 A 0 01 Y 1 00 Y 0 E= 1 A 1 A 0 Y 3 Y 2 Y 1 Y 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 1 0 0 0 8
Decoder: Logic Diagram E Output Expression: y i = E·m i I 0 ’ I 1 ’ y 0 I 2 ’ y 0 =1 if E=1 & (I 2, I 1, I 0 )=(0,0,0) I 0 ’ I 1 ’ y 1 I 2 . . I 0 I 1 y 7 y 7 =1 if E=1 & (I 2, I 1, I 0 )=(1,1,1) I 2 9
Decoder Application: universal set {Decoder, OR} Example: Implement the following functions with a 3-input decoder and OR gates. i) f 1 (a,b,c) = Σ m(1,2,4) ii) f 2 (a,b,c) = Σ m(2,3), iii) f 3 (a,b,c) = Σ m(0,5,6) 10
Decoder Application: universal set {Decoder, OR} Example: Implement functions i) f 1 (a,b,c) = Σ m(1,2,4) ii) f 2 (a,b,c) = Σ m(2,3), iii) f 3 (a,b,c) = Σ m(0,5,6) y 1 with a 3-input decoder and OR gates. y 2 E=1 y 4 f 1 y 2 y 0 0 c I 0 y 1 1 2 y 3 I 1 b f 2 3 . 4 . 5 a I 2 6 y 7 7 y 0 y 5 y 6 f 3 11
Decoders • OR minterms E=1 2:4 Decoder Minterm 11 AB A 10 AB B 01 AB 00 AB Y = AB + AB = A ⊕ B Y 12
Tree of Decoders Implement a 4-2 4 decoder with 3-2 3 decoders. y 0 0 d I 0 y 1 1 2 c I 1 3 4 5 b I 2 6 y 7 7 y 8 0 I 0 y 9 1 2 I 1 3 4 5 I 2 6 y 15 7 a 13
Tree of Decoders Implement a 6-2 6 decoder with 3-2 3 decoders. E y 0 E I 2, I 1, I 0 D 0 y 7 y 8 I 5, I 4, I 3 I 2, I 1, I 0 D 1 y 15 … … y 56 I 2, I 1, I 0 D 7 y 63 14
PI Q: A four variable switching function f(a,b,c,d) can be implemented using which of the following? A. 1:2 decoders and OR gates B. 2:4 decoders and OR gates C. 3:8 decoders and OR gates D. None of the above E. All of the above 15
2. Encoder • Definition (What is it?) • Logic Diagram (How is it realized?) • Priority Encoder (Special type of encoder) 16
iClicker: Definition of Encoder A. Any program, circuit or algorithm which encodes B. In digital audio technology, an encoder is a program that converts an audio WAV file into an MP3 file C. A device that convert a message from plain text into code D. A circuit that is used to convert between digital video and analog video E. All of the above 17
2. Encoder: Definition En 8 inputs 3 outputs I 0 0 y 0 0 1 2 y 1 1 3 4 2 y 2 5 6 I 7 7 A At most one I i = 1. (y n-1 ,.., y 0 ) = i if I i = 1 & E n = 1 (y n-1 ,.., y 0 ) = 0 otherwise. A = 1 if En = 1 and one i s.t. I i = 1 A = 0 otherwise. 18
Encoder Definition: A digital module that converts the assertion of a device to the binary address of the device. E I 2n-1… I 0 y n-1 … y 0 Encoder Description: A E At most one I i = 1. I 0 (y n-1 ,.., y 0 ) = i if I i = 1 & E = 1 0 y 0 0 1 2 y 1 (y n-1 ,.., y 0 ) = 0 otherwise. 1 3 4 2 y 2 A = 1 if E = 1 and one i s.t. I i = 1 5 6 I 7 7 A = 0 otherwise. 3 outputs A 8 inputs 19
Encoder: Logic Diagram En En y 0 y 1 I 2 I 1 I 3 I 3 I 5 I 6 I 7 I 7 En En y 2 I 4 A I 0 I 5 I 1 . I 6 . I 7 I 6 I 7 20
Priority Encoder: E I 0 0 y 0 0 1 2 y 1 I 3 3 1 Eo Gs 21
Priority Encoder: Definition Description: Input (I 2n-1 ,…, I 0 ), Output (y n-1 ,…, , y 0 ) (y n-1 ,…, , y 0 ) = i if I i = 1 & E = 1 & I k = 0 for all k > i (high bit priority) or E for all k< i (low bit priority) . E o = 1 if E = 1 & I i = 0 for all i, I 0 0 G s = 1 if E = 1 & i s.t. I i = 1 . y 0 E 0 1 2 1 y 1 3 (G s is like A, and E o tells us if 4 2 5 y 2 enable is true or not). 6 7 I 7 Eo Gs 22
Priority Encoder: Implement a 32-input priority encoder w/ 8 input priority encoders (high bit priority). E I 31-24 y 32, y 31, y 30 Gs Eo I 25-16 y 22, y 21, y 20 Gs Eo I 15-8 y 12, y 11, y 10 Gs Eo I 7-0 y 02, y 01, y 00 Gs Eo 23
Multiplexer • Definition • Logic Diagram • Application 24
3. Mux (Multiplexer) Definition: A digital module that selects one of data inputs according to the binary address of the selector. E Description If E = 1 y = D i where i = (S n-1 , .. , S 0 ) D 2n-1 -D 0 Else y y = 0 (Data input) S n-1,0 (Selector) 25
iClicker: Multiplexer Definition A. A device that interleaves two or more activities B. A communications device that combines several signals for transmission over a single medium C. A logic circuit that sends one of several inputs out over a single output channel. D. The circuit that uses a common communications channel for sending two or more messages or signals. E. All of the above 26
Multiplexer (Mux): Definition • Selects between one of N inputs to connect to the output. • log 2 N -bit select input – control input • Example: 2:1 Mux S D 0 0 Y D 1 1 S D 1 D 0 Y S Y D 0 0 0 0 0 0 D 1 0 0 1 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 27
PI Q: What is the output of the following MUX? A. 0 B. 1 C. Can’t say E =1 y 1 0 0 1 S=1 28
Multiplexer Definition: Example En D 0 0 S 1 S 0 y D 1 1 y D 2 2 D 3 3 S 1 S 0 29
Multiplexer: Logic Diagram • Tristates • Logic gates – For an N-input mux, – Sum-of-products form use N tristates – Turn on exactly one to Y D 0 D 1 00 01 11 10 S select the appropriate 0 0 0 1 1 input 1 0 1 1 0 Y = D 0 S + D 1 S S D 0 D 0 Y D 1 S D 1 Y 30
Multiplexer Application • Mux for a Boolean function with truth table as input A B Y 0 0 0 0 1 0 1 0 0 1 1 1 Y = AB A B 00 01 Y 10 11 31
Multiplexer: Application A A Y A B Y 0 0 0 0 0 0 0 1 0 Y Y = AB 1 0 0 1 1 B B 1 1 1 32
Multiplexer Application: universal set {Mux} Example 1: Given f (a,b,c) = Σ m (0,1,7) + Σ d(2), implement with an 8-input Mux. Id a b c f 0 0 0 0 1 1 0 0 1 1 2 0 1 0 - 3 0 1 1 0 4 1 0 0 0 5 1 0 1 0 6 1 1 0 0 7 1 1 1 1 33
Multiplexer Application: universal set {Mux} Example 1: Given f (a,b,c) = Σ m (0,1,7) + Σ d(2), implement with an 8-input Mux. En Id a b c f 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 2 2 0 1 0 - 0 3 y 3 0 1 1 0 0 4 0 5 4 1 0 0 0 6 0 5 1 0 1 0 7 1 6 1 1 0 0 S 2 S 1 S 0 7 1 1 1 1 a b c 34
Example 2: Given f (a,b,c) = Σ m (0,1,7) + Σ d(2), implement with 4-input Muxes. E D (c) a b c = 0 c = 1 0 D 0 (c) = 0 0 0 1 1 D 1 (c) = y 1 0 D 2 (c) = 2 1 1 D 3 (c) = 3 S 1 S 0 a b 35
Example 2: Given f (a,b,c) = Σ m (0,1,7) + Σ d(2), implement with 4-input Muxes. E D (c) a b c = 0 c = 1 1 0 D 0 (c) =1 0 0 1 1 0 1 0 1 - 0 D 1 (c) =0 y 1 0 0 0 D 2 (c) =0 0 2 1 1 0 1 D 3 (c) =c c 3 S 1 S 0 a b 36
Example 3: Given f (a,b,c) = Σ m (0,1,7) + Σ d(2), implement with 2- input Muxes. a 00 01 10 11 D (b,c) E 0 1 1 - 0 D 0 (b,c) 1 0 0 0 1 D 1 (b,c) 0 y 1 a 37
Example 3: Given f (a,b,c) = Σ m (0,1,7) + Σ d(2), implement with 2- input Muxes. a 00 01 10 11 D (b,c) E 0 1 1 - 0 D 0 (b,c) 1 0 0 0 1 D 1 (b,c) b ’ 0 y D 0 (b,c) = b ’ D 1 (b,c) = bc D 1 (b,c) 1 1 - 0 0 c c 1 0 0 1 a b b 38
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