ECE2030 Introduction to Computer Engineering Lecture 13: Building Blocks for Combinational Logic (4) Shifters, Multipliers Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering Georgia Tech

2 Basic Shifting Shift directions –Left (multiply by 2) –Right (divide by 2) Take floor value if the result is not an integer X X,Floor value of X (or  X  ) is the greatest integer number less than or equal to X, E.g. –  5/2  = 2 –  -3/2  = -2 Shift types –Logical (or unsigned) –Arithmetic (or signed)

3 Logical Shift Shift Left –MSB: Shifted out –LSB: Shifted in with a “0” –Examples: ( << 1) = ( << 3) = Shift right –MSB: Shifted in with a “0” –LSB: Shifted out –Examples: (Some ISA use triple “>” for logical right shift) ( >>> 1) = ( >>> 3) =

4 Arithmetic Shift Shift left –MSB: Shifted out, however, be aware of overflow/underflow –LSB: Shifted in with a “0” –Examples: (1100 << 1) = 1000 (1100 << 3) = 0000 (Incorrect!)  Underflow Shift right –MSB: Retain “sign bit” –LSB: Shifted out –Examples: (1100 >> 1) = 1110 (Retain sign bit) (1100 >> 3) = 1111 (  -4/8  = -1 )  Floor value of -0.5

5 Examples of Arithmetic Shift Arithmetic shift right by 1  Arithmetic shift left by 1  (= -65) Arithmetic shift left by 1 (i.e. x2)  (= +126  -130)  Underflow ! (= +66) Arithmetic shift left by 1 (i.e. x2)  (= -124  +132)  Overflow ! Overflow/Underflow

6 4-bit Logical Shifter S1S0D3D2D1D0 0XA3A2A1A0 100A3A2A1 11A2A1A00 A3 A2A1A0 D3 D2D1D0 S/NS S0 S1 L/R

7 4-bit Logical Shifter using 4-to-1 Mux 4-to-1 Mux s1 s0 S1S0D3D2D1D0 0XA3A2A1A0 100A3A2A1 11A2A1A00 D3 A2A3 4-to-1 Mux s1 s0 D2 A1 4-to-1 Mux s1 s0 D1 A0 4-to-1 Mux s1 s0 D0 S1 S0 Right Shift Left Shift

8 4-bit Arithmetic Shifter w/ 4-to-1 Mux 4-to-1 Mux s1 s0 S1S0D3D2D1D0 0XA3A2A1A0 10A3A3A2A1 11A2A1A00 D3 A2A3 4-to-1 Mux s1 s0 D2 A1 4-to-1 Mux s1 s0 D1 A0 4-to-1 Mux s1 s0 D0 S1 S0 Right Shift Left Shift

9 4-bit Arithmetic Shifter w/ 4-to-1 Mux 4-to-1 Mux s1 s0 S1S0D3D2D1D0 0XA3A2A1A0 10A3A3A2A1 11A2A1A00 D3 A2A3 4-to-1 Mux s1 s0 D2 A1 4-to-1 Mux s1 s0 D1 A0 4-to-1 Mux s1 s0 D0 S1 S0 Right Shift Left Shift Overflow/Underflow

10 4-bit Arithmetic Shifter w/ 4-to-1 Mux 4-to-1 Mux s1 s0 S1S0D3D2D1D0 0XA3A2A1A0 10A3A3A2A1 11A2A1A00 D3 A2A3 4-to-1 Mux s1 s0 D2 A1 4-to-1 Mux s1 s0 D1 A0 4-to-1 Mux s1 s0 D0 S1 S0 Right Shift Left Shift Overflow/Underflow Overflow Underflow Detection

11 Rotator S1S0D3D2D1D0 00A3A2A1A0 01 A3A2A1 10 A0A3A2 11 A1A0A3 4-to-1 Mux s1 s0 D3 A2A3 4-to-1 Mux s1 s0 D2 A1 4-to-1 Mux s1 s0 D1 A0 4-to-1 Mux s1 s0 D0 S1 S0

12 Barrel Shifter S2S1S0D3D2D1D0 000A3A2A1A0 001A3 A2A1 010A3 A2 011A3 100 A2A1A0 101A2A1A00 110A1A multiple bits Shift multiple bits at a time Left Shift Right Shift

13 Barrel Shifter Design w/ Mux (D3) S2S1S0D3D2D1D0 000A3A2A1A0 001A3 A2A1 010A3 A2 011A3 100 A2A1A0 101A2A1A00 110A1A to-1 Mux s1 s s1 s0 4-to-1 Mux 2-to-1 Mux 1 0 D3 A3 A2A1A0 S0 S1 S2 Replicate and change wiring of the two 4-to-1 Muxes for D2, D1 and D0

14 Barrel Shifter Design Alternative (16-bit) 2 3 Shifter 2 2 Shifter 2 1 Shifter 2 0 Shifter Left/Right S3 S2 S1 S0 16 (S3 S2 S1 S0) specifies the “shift amount” in binary 16 Output Number Input Number

15 Barrel Shifter Design w/ nMOSFET D3 D2 D1 D0 A3 S=0 (No Shift) S=1S=2S=3 A2 A1 A0 S=3 S=2 S=1

16 A3 Barrel Shifter Design w/ nMOSFET D3 D2 D1 D0 A3 S=0 (No Shift) S=1S=2S=3 A2 A1 A0 S=3 S=2 S=1 A3 A2

17 Barrel Shifter Design w/ nMOSFET D3 D2 D1 D0 A3 S=0 (No Shift)S=1S=2S=3 A2 A1 A0 S=3 S=2 S=1 = A3 = A2 = A1

18 Unsigned Binary Multiply 101 (5) X 111 (7) (35)

19 Unsigned Integer Multiplier (2-bit) s 2-bit by 2-bit carry carry out p0 a0b0 H.A. p1 c s a1b0a0b1 H.A. c s p2p3 a1b1

20 Unsigned Integer Multiplier (3-bit) 3-bit by 3-bit p0 a0b0 s F.A. p1 a1b0a0b1 0 co s ci c F.A. p2 s a2b0a1b1 co s ci c F.A. s co s ci a0b2 00 c F.A. p3 a2b1 co s ci c F.A. co s ci a1b2 0 s s s c p4 c F.A. co s ci a2b2 p5

21 4-bit Unsigned Integer Multiplier a0 b0 P0 a1 b0 a0 b1 + 0 P1 a2 b0 a1 b1 + a0 b2 + 0 P2 a3 b0 a2 b1 + a1 b2 + a0 b3 + 0 P3 a3 b1 + a2 b2 + a1 b3 + 0 P4 a3 b2 + a2 b3 + P5 a3 b3 + P6 P7 a0 b0 + Cin Cout Sum AB Full Adder a0b0 

22 Propagation Delay a0 b0 P0 a1 b0 a0 b1 + 0 P1 a2 b0 a1 b1 + a0 b2 + 0 P2 a3 b0 a2 b1 + a1 b2 + a0 b3 + 0 P3 a3 b1 + a2 b2 + a1 b3 + 0 P4 a3 b2 + a2 b3 + P5 a3 b3 + P6 P x4 Delay = 8 adders 8x8 Delay = 20 adders

23 a3 b3 + P7P6 Carry-Save Multiplier a0 b0 P0 a1 b0 a0 b1 + 0 P1 a2 b0 a1 b1 + a0 b2 + 0 P2 a3 b0 a2 b1 + a1 b2 + a0 b3 + 0 P3 a3 b2 + a2 b3 + P5 a3 b1 + a2 b2 a1 b P4

24 Propagation Delay of Carry-Save Multiplier a0 b0 P0 a3 b1 + a2 b2 a1 b3 + a3 b2 + a2 b3 + P5 a3 b3 + P7 a1 b0 a0 b1 + 0 P x4 Delay = 6 adders 8x8 Delay = 14 adders a2 b0 a1 b1 + a0 b2 + 0 P2 1 2 a3 b0 a2 b1 + a1 b2 + a0 b3 + 0 P P6P4

25 Signed Binary Multiply When the Multiplicand is negative (-3) (+9) Maintain the sign bits of the partial product

26 Signed Binary Multiply When the Multiplier is negative (+9) (-3) (-27) At the last step, 2’s complement the multiplicand before adding

27 Signed Binary Multiply When both the Multiplicand and Multiplier are negative (-9) (-3) (+27) At the last step, 2’s complement the multiplicand before adding Maintain the sign bits of the partial product

28 More Examples (1) (-6) (+5) (-30) Assume 8-bit numbers

29 More Examples (2) 0011 (+3) 1110 (-2) (-6) Assume 4-bit numbers

30 More Examples (3) (-4) (-32) (+128) Assume 8-bit numbers