
1 STANDARD COMBINATIONAL MODULES • DECODERS • ENCODERS • MULTIPLEXERS (Selectors) • DEMULTIPLEXERS (Distributors) • SHIFTERS Introduction to Digital Systems 9 { Standard Combinational Modules 2 BINARY DECODERS HIGH-LEVEL DESCRIPTION: Inputs: x = (xn−1; : : : ; x0); xj 2 f0; 1g Enable E 2 f0; 1g Outputs: y = (y2n−1; : : : ; y0); yi 2 f0; 1g 8 1 if (x = i) and (E = 1) > Function: yi = <> otherwise > 0 :> n−1 j x = X xj2 j=0 and i = 0; : : : ; 2n − 1 Introduction to Digital Systems 9 { Standard Combinational Modules 3 E y En 0 0 y 1 1 y x 2 0 0 2 x 1 1 Inputs Outputs x n-1 n-1 n-Input Binary Decoder y n n 2 -1 2 -1 Figure 9.1: n-INPUT BINARY DECODER. Introduction to Digital Systems 9 { Standard Combinational Modules 4 EXAMPLE 9.1: 3-INPUT BINARY DECODER E x2 x1 x0 x y7 y6 y5 y4 y3 y2 y1 y0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 2 0 0 0 0 0 1 0 0 1 0 1 1 3 0 0 0 0 1 0 0 0 1 1 0 0 4 0 0 0 1 0 0 0 0 1 1 0 1 5 0 0 1 0 0 0 0 0 1 1 1 0 6 0 1 0 0 0 0 0 0 1 1 1 1 7 1 0 0 0 0 0 0 0 0 - - - - 0 0 0 0 0 0 0 0 BINARY SPECIFICATION: Inputs: x = (xn−1; : : : ; x0); xj 2 f0; 1g E 2 f0; 1g Outputs: y = (y2n−1; : : : ; y0); yi 2 f0; 1g n Function: yi = E · mi(x) ; i = 0; : : : ; 2 − 1 Introduction to Digital Systems 9 { Standard Combinational Modules 5 EXAMPLE 9.2: IMPLEMENTATION OF 2-INPUT DECODER 0 0 0 0 y0 = x1x0E y1 = x1x0E y2 = x1x0E y3 = x1x0E E y 0 x y 0 1 y x 2 1 y 3 Figure 9.2: GATE NETWORK IMPLEMENTATION OF 2-INPUT BINARY DECODER. Introduction to Digital Systems 9 { Standard Combinational Modules 6 DECODER USES OPCODE field Instruction Other fields 4-Input Binary E=1 En Decoder 15 . 4 3 2 1 0 LOAD STORE ADD Decoded operations JUMP Figure 9.3: OPERATION DECODING. Introduction to Digital Systems 9 { Standard Combinational Modules 7 DECODER USES Cell referenced when address is Data input Data input 00000000000010 Binary cell E=1 0 1 Address 14 14 2 RAM Module Address 14 (2 x 1) 16383 Binary Decoder Read/write Read/write Data output (a) (b) Data output Figure 9.4: RANDOM ACCESS MEMORY (RAM): a) MODULE; b) ADDRESSING OF BINARY CELLS. Introduction to Digital Systems 9 { Standard Combinational Modules 8 BINARY DECODER AND or GATE • UNIVERSAL Example 9.5: x2x1x0 z2 z1 z0 000 0 1 0 001 1 0 0 010 0 0 1 011 0 1 0 100 0 0 1 101 1 0 1 110 0 0 0 111 1 0 0 Introduction to Digital Systems 9 { Standard Combinational Modules 9 (y7; : : : ; y0) = dec(x2; x1; x0; 1) z2(x2; x1; x0) = y1 y5 y7 z1(x2; x1; x0) = y0 y3 z0(x2; x1; x0) = y2 y4 y5 E=1 y 0 En 0 y 1 1 y z 2 2 2 x 0 y 0 3 3 x y z 1 1 4 1 4 x 2 y 2 5 Binary Decoder 5 y z 6 0 6 y 7 7 Figure 9.5: NETWORK IN EXAMPLE 9.5 Introduction to Digital Systems 9 { Standard Combinational Modules 10 DECODER NETWORKS: COINCIDENT DECODING x = (xleft; xright) xleft = (x7; x6; x5; x4) xright = (x3; x2; x1; x0) 4 x = 2 × xleft + xright y = DEC(xleft) w = DEC(xright) zi = AND(ys; wt) i = 24 × s + t Introduction to Digital Systems 9 { Standard Combinational Modules 11 x3 x2 x1 x0 0 1 0 0 4-Input Binary 1 En Decoder 15 . .4 3 2 1 0 w 4 1 z0 x4 0 1 y x5 2 0 1 x6 Decoder 0 z36 x7 4-Input Binary En 15 . 2 1 0 1 E z255 Figure 9.6: 8-INPUT COINCIDENT DECODER. Introduction to Digital Systems 9 { Standard Combinational Modules 12 n-INPUT COINCIDENT DECODER y = dec(xleft; E) w = dec(xright; 1) and and and z = ( (y2n=2−1; w2n=2−1); : : : ; (ys; wt); : : : ; (y0; w0)) Introduction to Digital Systems 9 { Standard Combinational Modules 13 x n/2-1 x0 1 En DECODER W w n/2 w 2 -1 0 wt z0 y0 xn/2 ys x n-1 DECODER Y En z n/2 y n/2 2 s+t 2 -1 E z n 2 - 1 Figure 9.7: n-INPUT COINCIDENT DECODER. Introduction to Digital Systems 9 { Standard Combinational Modules 14 TREE DECODING x = (xleft; xright) xleft = (x3; x2) xright = (x1; x0) Introduction to Digital Systems 9 { Standard Combinational Modules 15 4-INPUT TREE DECODER x x x3 x2 1 0 x=6: 0 1 1 0 1 0 Level 1 E En DEC 3 2 1 0 0 0 1 0 1 0 En 1 0 En 1 0 En 1 0 En Level 2 DEC 3 DEC 2 DEC 1 DEC 0 3 2 1 0 3 2 1 0 3 2 1 0 3 2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 z z z z 15 12 8 z6 =1 z4 0 Figure 9.8: 4-INPUT TREE DECODER. Introduction to Digital Systems 9 { Standard Combinational Modules 16 n-INPUT TREE DECODERr w = dec(xleft; E) dec dec dec z = ( (xright; w2n=2−1); : : : ; (xright; wt); : : : ; (xright; w0)) Introduction to Digital Systems 9 { Standard Combinational Modules 17 x x left right x n-1 xn/2 x n/2-1 x0 E En Level 1 DEC w n/2 2 - 1 w0 wt En En En n/2 Level 2 DEC 2 -1 DEC t DEC 0 z n z (n/2) z 2 - 1 2 t+s 0 Figure 9.9: n-INPUT TWO-LEVEL TREE DECODER. Introduction to Digital Systems 9 { Standard Combinational Modules 18 COMPARISON OF DECODER NETWORKS Coincident Tree Decoder modules 2 2k + 1 and gates 22k { Load per network input 1 decoder input 2k decoder inputs (max) Fanout per decoder output 2k and inputs 1 enable input Number of module inputs 2k + 2 + 22k+1 1 + k + 2k + k2k (related to number of connections) Delay tdecoder + tAND 2tdecoder Introduction to Digital Systems 9 { Standard Combinational Modules 19 EXAMPLE 9.6: 6-INPUT DECODER E x 0 0 z0 x 1 1 z x0 0 0 x z z 2 1 x 1 1 Decoder 7 1 E x 2 z Decoder 7 7 0 x3 1 1 x4 x5 Decoder 7 x3 0 1 0 z56 x4 1 z57 x5 z63 Decoder 7 Decoder 7 z63 (a) (b) Figure 9.10: IMPLEMENTATION OF 6-INPUT DECODER. a) COINCIDENT DECODER. b) TREE DECODER. Introduction to Digital Systems 9 { Standard Combinational Modules 20 EXAMPLE 9.6 (cont.) Coincident Tree Decoder modules 2 9 and gates 64 { Load per network input 1 decoder input 8 decoder inputs (max) Fanout per decoder output 8 and inputs 1 enable input Number of module inputs 136 36 Delay 3d 4d Introduction to Digital Systems 9 { Standard Combinational Modules 21 Decoder Cell array Data input Binary cell E=1 0 0 1 1 12 2 2 Address Tree Decoder 4095 4096 lines from decoder to cell array 4095 Read/write Data output (a) Figure 9.11: a) SYSTEM WITH TREE DECODER. Introduction to Digital Systems 9 { Standard Combinational Modules 22 Decoder E=1 Cell array Data input 0 6 0 Binary Decoder 63 1 12 Address E=1 0 6 Binary Decoder 63 4095 Read/write 128 lines from decoder Modified binary cell Data output to cell array equivalent to coincident decoder (b) Figure 9.11: b) SYSTEM WITH COINCIDENT DECODER. Introduction to Digital Systems 9 { Standard Combinational Modules 23 BINARY ENCODERS E En x 0 0 x y 1 1 0 0 y 1 1 Inputs y n-1 Outputs n-1 -Input Binary Encoder n 2 x n n 2 -1 2 -1 Ac A Figure 9.12: 2n-INPUT BINARY ENCODER. Introduction to Digital Systems 9 { Standard Combinational Modules 24 BINARY ENCODER: HIGH-LEVEL SPECIFICATION Inputs: x = (x2n−1; : : : ; x0); xi 2 f0; 1g, with at most one xi = 1 E 2 f0; 1g Outputs: y = (yn−1; : : : ; y0); yj 2 f0; 1g A 2 f0; 1g 8 i if (x = 1) and (E = 1) > i Function: y = <> otherwise > 0 :> 8 1 if (some x = 1) and (E = 1) > i A = <> otherwise > 0 :> n−1 j y = X yj2 j=0 and i = 0; : : : ; 2n − 1 Introduction to Digital Systems 9 { Standard Combinational Modules 25 EXAMPLE 9.7: FUNCTION OF AN 8-INPUT BINARY ENCODER E x7 x6 x5 x4 x3 x2 x1 x0 y y2 y1 y0 A 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 2 0 1 0 1 1 0 0 0 0 1 0 0 0 3 0 1 1 1 1 0 0 0 1 0 0 0 0 4 1 0 0 1 1 0 0 1 0 0 0 0 0 5 1 0 1 1 1 0 1 0 0 0 0 0 0 6 1 1 0 1 1 1 0 0 0 0 0 0 0 7 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - - - 0 0 0 0 0 Introduction to Digital Systems 9 { Standard Combinational Modules 26 BINARY SPECIFICATION OF ENCODER Inputs: x = (x2n−1; : : : ; x0); xi 2 f0; 1g, with at most one xi = 1 E 2 f0; 1g Outputs: y = (yn−1; : : : ; y0); yj 2 f0; 1g A 2 f0; 1g Function: yj = E · P(xk); j = 0; : : : ; n − 1 n A = E · P(xi); i = 0; : : : ; 2 − 1 Introduction to Digital Systems 9 { Standard Combinational Modules 27 EXAMPLE 9.8 y0 = E · (x1 x3 x5 x7) y1 = E · (x2 x3 x6 x7) y2 = E · (x4 x5 x6 x7) A = E · (x0 x1 x2 x3 x4 x5 x6 x7) Introduction to Digital Systems 9 { Standard Combinational Modules 28 x x x x x x x x 7 6 5 4 3 2 1 0 E y0 y1 y2 A Figure 9.13: IMPLEMENTATION OF AN 8-INPUT BINARY ENCODER.
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