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CSE140L: Components and Design Techniques for Digital Systems Lab

Timing, Mux, Demux, Adders

Tajana Simunic Rosing

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Outline• Pass gates (Appendix B)• Muxes & Demuxes (chap 4.2 pp. 171-183)• Adders (chap 5.6)• Non-ideal gate behavior (3.5)

– Rise/fall time– Delay– Pulse width

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CSE140: Components and Design Techniques for Digital Systems

Muxes and demuxes

Tajana Simunic Rosing

4

Pass transistor – Mux building block

• Connects X & Y when A=1, else X & Y disconnected– A_b = not(A)

Fig source: Prof. Subhashish Mitra

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Multiplexor (Mux)• Mux routes one of its N data inputs to its one output,

based on binary value of select inputs• 4 input mux needs 2 select inputs to indicate which input to route

through• 8 input mux 3 select inputs • N inputs log2(N) selects

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Mux Internal Design

• Selects input to connect to Y– selA == 1: connects A to Y– selB == 1: connects B to Y

Fig source: Prof. Subhashish Mitra

2×1

i1i0

s01

d

2×1

i1i0

s00

d

2×1

i1i0

s0

d

2x1 mux

AB Y YA

BAB

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2 -1I0I1I2I3I4I5I6I7

A B C

8:1mux

Z

I0I1I2I3

A B

4:1mux

ZI0I1

A

2:1mux Z

k=0

n

Multiplexers/selectors• 2:1 mux: Z = A'I0 + AI1• 4:1 mux: Z = A'B'I0 + A'BI1 + AB'I2 + ABI3• 8:1 mux: Z = A'B'C'I0 + A'B'CI1 + A'BC'I2 + A'BCI3 +

AB'C'I4 + AB'CI5 + ABC'I6 + ABCI7

• In general: Z = Σ (mkIk)

– in minterm shorthand form for a 2n:1 Mux

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N-bit Mux Example

• Four possible display items– Temperature (T), Average miles-per-gallon (A), Instantaneous mpg (I),

and Miles remaining (M) -- each is 8-bits wide– Choose which to display using two inputs x and y– Use 8-bit 4x1 mux

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Multiplexers as general-purpose logic• A 2n-1:1 multiplexer can implement any function of n variables

– with n-1 variables used as control inputs and– the data inputs tied to the last variable or its complement

• Example: F(A,B,C) = ABC + ABC’+A’BC+AB’C

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C0 C1 C2 Function Comments0 0 0 1 always 10 0 1 A + B logical OR0 1 0 (A • B)' logical NAND0 1 1 A xor B logical xor1 0 0 A xnor B logical xnor1 0 1 A • B logical AND1 1 0 (A + B)' logical NOR1 1 1 0 always 0

Mux example: Logical function unit

C2C0 C1

01234567S2

8:1 MUX

S1 S0

F

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1:2 Decoder:O0 = G • S’O1 = G • S

2:4 Decoder: O0 = G • S1’ • S0’O1 = G • S1’ • S0O2 = G • S1 • S0’O3 = G • S1 • S0

3:8 Decoder: O0 = G • S2’ • S1’ • S0’O1 = G • S2’ • S1’ • S0O2 = G • S2’ • S1 • S0’O3 = G • S2’ • S1 • S0O4 = G • S2 • S1’ • S0’O5 = G • S2 • S1’ • S0O6 = G • S2 • S1 • S0’O7 = G • S2 • S1 • S0

Demultiplexers/decoders• Decoders/demultiplexers: general concept

– single data input, n control inputs, 2n outputs– control inputs (called “selects” (S)) represent binary index of

output to which the input is connected– data input usually called “enable” (G)

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Gate level implementation of demultiplexers• 1:2 decoders

• 2:4 decoders

1:2 Decoder:O0 = G • S’O1 = G • S

2:4 Decoder: O0 = G • S1’ • S0’O1 = G • S1’ • S0O2 = G • S1 • S0’O3 = G • S1 • S0

Demultiplexers as general-purpose logic (cont’d)F1 = A'BC'D + A'B'CD + ABCDF2 = ABC'D' + ABCF3 = (A' + B' + C' + D')

A B

0 A'B'C'D'1 A'B'C'D2 A'B'CD'3 A'B'CD4 A'BC'D'5 A'BC'D6 A'BCD'7 A'BCD8 AB'C'D'9 AB'C'D10 AB'CD'11 AB'CD12 ABC'D'13 ABC'D14 ABCD'15 ABCD

4:16DECEnable

C D

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C0 C1 C2 Function Comments0 0 0 1 always 10 0 1 A + B logical OR0 1 0 (A • B)' logical NAND0 1 1 A xor B logical xor1 0 0 A xnor B logical xnor1 0 1 A • B logical AND1 1 0 (A + B)' logical NOR1 1 1 0 always 0

Mux example: Logical function unit• Multi-purpose function block

– 3 control inputs to specify operation to perform on operands– 2 data inputs for operands– 1 output of the same bit-width as operands

C2C0 C1

01234567S2

8:1 MUX

S1 S0

F

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CSE140: Components and Design Techniques for Digital Systems

Arithmetic circuits

Tajana Simunic Rosing

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Design example: 1-bit binary adder

• Inputs: A, B, Carry-in• Outputs: Sum, Carry-out

– Sum = A xor B xor Cin– Cout= A B + A Cin + B Cin

= A B + Cin (A xor B)

AB

CinCout

SA B Cin Cout S0 0 0 0 0 0 0 1 0 10 1 0 0 1 0 1 1 1 01 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1

A A A A AB B B B B

S S S S S

CinCout

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A

A

B

BCin Cout

@0@0

@0@0

@N

@1

@1

@N+1

@N+2

latearrivingsignal

two gate delaysto compute Cout

4 stageadder

A0B0

0

S0 @2

A1B1

C1 @2

S1 @3

A2B2

C2 @4

S2 @5

A3B3

C3 @6

S3 @7Cout @8

Ripple-carry adder critical delay path

T0 T2 T4 T6 T8

S0, C1 Valid S1, C2 Valid S2, C3 Valid S3, C4 Valid

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Carry-lookahead• Evaluate Sum and Ci+1

– Sum = Ai xor Bi xor Ci– Ci+1 = Ai Bi + Ai Ci + Bi Ci

= Ai Bi + Ci (Ai xor Bi)

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G3

C0

C0

C0

C0

P0

P0

P0

P0G0

G0

G0

G0C1 @ 3

P1

P1

P1

P1

P1

P1

G1

G1

G1

C2 @ 3

P2

P2

P2

P2

P2

P2

G2

G2

C3 @ 3

P3

P3

P3

P3

C4 @ 3

Pi @ 1 gate delay

Ci Si @ 2 gate delays

BiAi

Gi @ 1 gate delay

increasingly complexlogic for carries

Carry-lookahead implementation• Adder with propagate and generate outputs

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A0B0

0

S0 @2

A1B1

C1 @2

S1 @3

A2B2

C2 @4

S2 @5

A3B3

C3 @6

S3 @7Cout @8

A0B0

0

S0 @2

A1B1

C1 @3

S1 @4

A2B2

C2 @3

S2 @4

A3B3

C3 @3

S3 @4

C4 @3 C4 @3

Carry-lookahead implementation (cont’d)• Carry-lookahead logic generates individual carries

– sums computed much more quickly in parallel– however, cost of carry logic increases with more stages

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Lookahead Carry UnitC0

P0 G0P1 G1P2 G2P3 G3 C3 C2 C1

C0

P3-0 G3-0

C4

@3@2@4

@3@2@5

@3@2@5

@3@2

@4

@5@3

@0C16

A[15-12] B[15-12]C12

S[15-12]

A[11-8] B[11-8]C8

S[11-8]

A[7-4] B[7-4]C4

S[7-4]@7@8@8

A[3-0] B[3-0]C0

S[3-0]

@0

@4

4 4

4P G

4-bit Adder

4 4

4P G

4-bit Adder

4 4

4P G

4-bit Adder

4 4

4P G

4-bit Adder

Carry-lookahead adderwith cascaded carry-lookahead logic

• Carry-lookahead adder– 4 four-bit adders with internal carry lookahead– second level carry lookahead unit extends lookahead to 16 bits

G = G3 + P3 G2 + P3 P2 G1 + P3 P2 P1 G0

P = P3 P2 P1 P0

C1 = G0 + P0 C0C2 = G1 + P1 G0 + P1 P0 C0

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4-Bit Adder[3:0]

C0C4

4-bit adder[7:4]

1C8

0C8

five2:1 mux

01010101

adder low

adderhigh

01

4-bit adder[7:4]

C8 S7 S6 S5 S4 S3 S2 S1 S0

Carry-select adder• Redundant hardware to make carry calculation go faster

– compute two high-order sums in parallel while waiting for carry-in– one assuming carry-in is 0 and another assuming carry-in is 1– select correct result once carry-in is finally computed

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CSE140: Components and Design Techniques for Digital Systems

Timing and delay

Tajana Simunic Rosing

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F is not always 0pulse 3 gate-delays wide

D remains high forthree gate delays after

A changes from low to high

FA B C D

When is non-ideal gate behavior a good thing?

• Can be useful — pulse shaping circuits• Can be a problem — incorrect circuit operation• Example: pulse shaping circuit

– A’ • A = 0– delays matter

initially undefined

close switch

open switch

+

open switch

resistorA B

CD

Oscillatory behavior• Another pulse shaping circuit

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More complex gates

What we’ve covered thus far

• Xilinx Virtex II Pro board and tools• Transistor design• Delay estimates• Pass transistors• Muxes• Demuxes• Adders

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