1. EELE 367 – Logic Design Module 4 – Combinational Logic Design with VHDL Agenda 1.Decoders/Encoders 2.Multiplexers/Demultiplexers 3.Tri-State Buffers 4.Comparators 5.Adders (Ripple Carry, Carry-Look-Ahead) 6.Subtraction 7.Multiplication 8.Division (brief overview)

2. Module 4: Combinational Logic Design with VHDL 2 Integrated Circuit Scaling Integrated Circuit Scales Example# of Transistors SSI - Small Scale Integrated CircuitsIndividual Gates10's MSI- Medium Scale Integrated Circuits Mux, Decoder 100's LSI- Large Scale Integrated Circuits RAM, ALU's 1k - 10k VLSI- Very Large Scale Integrated CircuitsuP, uCNT100k - 1M ULSI- Ultra Large Scale Integrated CircuitsModern uP's> 1M SoC- System on ChipMicrocomputers SoP - System on Package Different technology blending - we use the terms SSI and MSI. Everything larger is typically just called "VLSI" - VLSI covers design that can't be done using schematics or by hand.

3. Module 4: Combinational Logic Design with VHDL 3 Decoders Decoders - a decoder has n inputs and 2 n outputs - one and only one output is asserted for a given input combination ex) truth table of decoder Input Output 000001 010010 100100 111000 - these are key circuits for a Address Decoders

4. Module 4: Combinational Logic Design with VHDL 4 Decoder Decoder Structure - The output stage of a decoder can be constructed using AND gates - Inverters are needed to give the appropriate code to each AND gate - Using AND/INV structure, we need: 2 n AND gates n Inverters Showing more inverters than necessary to illustrate concept

5. Module 4: Combinational Logic Design with VHDL 5 Decoders Decoders with ENABLES - An Enable line can be fed into the AND gate - The AND gate now needs (n+1) inputs - Using positive logic: EN = 0, Output = 0 EN =1, Output depends on input code

6. Module 4: Combinational Logic Design with VHDL 6 Decoders Decoder Example - Let's design a 2-to-4 Decoder using Structural VHDL - We know we need to describe the following structure: - We know what we'll need: 2 n AND gates = 4 AND gates n Inverters = 2 Inverters Showing more inverters than necessary to illustrate concept

7. Module 4: Combinational Logic Design with VHDL 7 Decoder Decoder Example - Let's design the inverter using concurrent signal assignments…. entity inv is port (In1: inSTD_LOGIC; Out1 : outSTD_LOGIC); end entity inv; architecture inv_arch of inv is begin Out1 <= not In1; end architecture inv_arch;

8. Module 4: Combinational Logic Design with VHDL 8 Decoders Decoder Example - Let's design the AND gate using concurrent signal assignments…. entity and2 is port (In1,In2 : inSTD_LOGIC; Out1 : outSTD_LOGIC); end entity and2; architecture and2_arch of and2 is begin Out1 <= In1 and In2; end architecture and2_arch;

9. Module 4: Combinational Logic Design with VHDL 9 Decoders Decoder Example - Now let's work on the top level design entity called "decoder_2to4" entity decoder_2to4 is port (A,B : inSTD_LOGIC; Y0,Y1,Y2,Y3: outSTD_LOGIC); end entity decoder_2to4;

10. Module 4: Combinational Logic Design with VHDL 10 Decoders Decoder Example - Now let's work on the top level design architecture called "decoder_2to4_arch" architecture decoder_2to4 _arch of decoder_2to4 is signal A_n, B_n : STD_LOGIC; component inv port (In1 : inSTD_LOGIC; Out1 : outSTD_LOGIC); end component; component and2 port (In1,In2 : inSTD_LOGIC; Out1 : outSTD_LOGIC); end component; begin ………

11. Module 4: Combinational Logic Design with VHDL 11 Decoders Decoder Example - cont…. begin U1 : invport map (A, A_n); U2 : invport map (B, B_n); U3 : and2port map (A_n, B_n, Y0); U4 : and2port map (A, B_n, Y1); U5 : and2port map (A_n, B, Y2); U6 : and2port map (A, B, Y3); end architecture decoder_2to4 _arch;

12. Module 4: Combinational Logic Design with VHDL 12 Encoders Encoder - an encoder has 2 n inputs and n outputs - it assumes that one and only one input will be asserted - depending on which input is asserted, an output code will be generated - this is the exact opposite of a decoder ex) truth table of binary encoder Input Output 0001 00 0010 01 0100 10 1000 11

13. Module 4: Combinational Logic Design with VHDL 13 Encoders Encoder - an encoder output is a simple OR structure that looks at the incoming signals ex) 4-to-2 encoder I3 I2 I1 I0 Y1 Y0 0 0 0 10 0 0 0 1 00 1 0 1 0 01 0 1 0 0 01 1 Y1 = I3 + I2 Y0 = I3 + I1

14. Module 4: Combinational Logic Design with VHDL 14 Encoders Encoders in VHDL - 8-to-3 binary encoder modeled with Structural VHDL entity encoder_8to3_binary is generic (t_delay : time := 1.0 ns); port (I : in STD_LOGIC_VECTOR (7 downto 0); Y : out STD_LOGIC_VECTOR (2 downto 0) ); end entity encoder_8to3_binary; architecture encoder_8to3_binary_arch of encoder_8to3_binary is component or4 port (In1,In2,In3,In4: in STD_LOGIC; Out1: out STD_LOGIC); end component; begin U1 : or4 port map (In1 => I(1), In2 => I(3), In3 => I(5), In4 => I(7), Out1 => Y(0) ); U2 : or4 port map (In1 => I(2), In2 => I(3), In3 => I(6), In4 => I(7), Out1 => Y(1) ); U3 : or4 port map (In1 => I(4), In2 => I(5), In3 => I(6), In4 => I(7), Out1 => Y(2) ); end architecture encoder_8to3_binary_arch;

15. Module 4: Combinational Logic Design with VHDL 15 Encoders Encoders in VHDL - 8-to-3 binary encoder modeled with Behavioral VHDL entity encoder_8to3_binary is generic (t_delay : time := 1.0 ns); port (I : in STD_LOGIC_VECTOR (7 downto 0); Y : out STD_LOGIC_VECTOR (2 downto 0) ); end entity encoder_8to3_binary; architecture encoder_8to3_binary_arch of encoder_8to3_binary is begin ENCODE : process (I) begin case (I) is when "00000001" => Y <= "000"; when "00000010" => Y <= "001"; when "00000100" => Y <= "010"; when "00001000" => Y <= "011"; when "00010000" => Y <= "100"; when "00100000" => Y <= "101"; when "01000000" => Y <= "110"; when "10000000" => Y <= "111"; when others => Y <= "ZZZ"; end case; end process ENCODE; end architecture encoder_8to3_binary_arch;

16. Module 4: Combinational Logic Design with VHDL 16 Priority Encoders Priority Encoder - a generic encoder does not know what to do when multiple input bits are asserted - to handle this case, we need to include prioritization - we decide the list of priority (usually MSB to LSB) where the truth table can be written as follows: ex) 4-to-2 encoder I3 I2 I1 I0 Y1 Y0 1 x x x1 1 0 1 x x1 0 0 0 1 x0 1 0 0 0 10 0 - we can then write expressions for an intermediate stage of priority bits “H” (i.e., Highest Priority): H3 = I3 H2 = I2∙I3’ H1 = I1∙I2’∙I3’ H0 = I0∙I1’∙I2’∙I3’ - the final output stage then becomes: Y1 = H3 + H2 Y0 = H3 + H1

17. Module 4: Combinational Logic Design with VHDL 17 Priority Encoders Priority Encoders in VHDL - 8-to-3 binary priority encoder modeled with Behavioral VHDL - If/Then/Else statements give priority - Concurrent Conditional Signal Assignments give priority entity encoder_8to3_priority is generic (t_delay : time := 1.0 ns); port (I : in STD_LOGIC_VECTOR (7 downto 0); Y : out STD_LOGIC_VECTOR (2 downto 0) ); end entity encoder_8to3_priority; architecture encoder_8to3_priority_arch of encoder_8to3_priority is begin Y <= "111" when I(7) = '1' else -- highest priority code "110" when I(6) = '1' else "101" when I(5) = '1' else "100" when I(4) = '1' else "011" when I(3) = '1' else "010" when I(2) = '1' else "001" when I(1) = '1' else "000" when I(0) = '1' else -- lowest priority code "ZZZ"; end architecture encoder_8to3_priority_arch;

18. Module 4: Combinational Logic Design with VHDL 18 Multiplexer Multiplexer - gates are combinational logic which generate an output depending on the current inputs - what if we wanted to create a “Digital Switch” to pass along the input signal? - this type of circuit is called a “Multiplexer” ex) truth table of Multiplexer SelOut 0 A 1 B

19. Module 4: Combinational Logic Design with VHDL 19 Multiplexer Multiplexer - we can use the behavior of an AND gate to build this circuit: X∙0 = 0 “Block Signal” X∙1 = X “Pass Signal” - we can then use the behavior of an OR gate at the output state (since a 0 input has no effect) to combine the signals into one output

20. Module 4: Combinational Logic Design with VHDL 20 Multiplexer Multiplexer - the outputs will track the selected input - this is in effect, a “Switch” ex) truth table of Multiplexer Sel A BOut 0 0 x 0 0 1 x 1 1 x 0 0 1 x 1 1 - an ENABLE line can also be fed into each AND gate

21. Module 4: Combinational Logic Design with VHDL 21 Multiplexer Multiplexers in VHDL - Structural Model entity mux_4to1 is port (D : in STD_LOGIC_VECTOR (3 downto 0); Sel : in STD_LOGIC_VECTOR (1 downto 0); Y : out STD_LOGIC); end entity mux_4to1; architecture mux_4to1_arch of mux_4to1 is signal Sel_n : STD_LOGIC_VECTOR (1 downto 0); signal U3_out, U4_out, U5_out, U6_out : STD_LOGIC; component inv1 port (In1: in STD_LOGIC; Out1: out STD_LOGIC); end component; component and3 port (In1,In2,In3 : in STD_LOGIC; Out1: out STD_LOGIC); end component; component or4 port (In1,In2,In3,In4: in STD_LOGIC; Out1: out STD_LOGIC); end component; begin U1 : inv1 port map (In1 => Sel(0), Out1 => Sel_n(0)); U2 : inv1 port map (In1 => Sel(1), Out1 => Sel_n(1)); U3 : and3 port map (In1 => D(0), In2 => Sel_n(1), In3 => Sel_n(0), Out1 => U3_out); U4 : and3 port map (In1 => D(1), In2 => Sel_n(1), In3 => Sel(0), Out1 => U4_out); U5 : and3 port map (In1 => D(2), In2 => Sel(1), In3 => Sel_n(0), Out1 => U5_out); U6 : and3 port map (In1 => D(3), In2 => Sel(1), In3 => Sel(0), Out1 => U6_out); U7 : or4 port map (In1 => U3_out, In2 => U4_out, In3 => U5_out, In4 => U6_out, Out1 => Y); end architecture mux_4to1_arch;

22. Module 4: Combinational Logic Design with VHDL 22 Multiplexer Multiplexers in VHDL - Structural Model w/ EN entity mux_4to1 is port (D : in STD_LOGIC_VECTOR (3 downto 0); Sel : in STD_LOGIC_VECTOR (1 downto 0); EN : in STD_LOGIC; Y : out STD_LOGIC); end entity mux_4to1; architecture mux_4to1_arch of mux_4to1 is signal Sel_n : STD_LOGIC_VECTOR (1 downto 0); signal U3_out, U4_out, U5_out, U6_out : STD_LOGIC; component inv1 port (In1: in STD_LOGIC; Out1: out STD_LOGIC); end component; component and4 port (In1,In2,In3,In4: in STD_LOGIC; Out1: out STD_LOGIC); end component; component or4 port (In1,In2,In3,In4: in STD_LOGIC; Out1: out STD_LOGIC); end component; begin U1 : inv1 port map (In1 => Sel(0), Out1 => Sel_n(0)); U2 : inv1 port map (In1 => Sel(1), Out1 => Sel_n(1)); U3 : and4 port map (In1 => D(0), In2 => Sel_n(1), In3 => Sel_n(0), In4 => EN, Out1 => U3_out); U4 : and4 port map (In1 => D(1), In2 => Sel_n(1), In3 => Sel(0), In4 => EN, Out1 => U4_out); U5 : and4 port map (In1 => D(2), In2 => Sel(1), In3 => Sel_n(0), In4 => EN, Out1 => U5_out); U6 : and4 port map (In1 => D(3), In2 => Sel(1), In3 => Sel(0), In4 => EN, Out1 => U6_out); U7 : or4 port map (In1 => U3_out, In2 => U4_out, In3 => U5_out, In4 => U6_out, Out1 => Y); end architecture mux_4to1_arch;

23. Module 4: Combinational Logic Design with VHDL 23 Multiplexer Multiplexers in VHDL - Behavioral Model w/ EN entity mux_4to1 is port (D : in STD_LOGIC_VECTOR (3 downto 0); Sel : in STD_LOGIC_VECTOR (1 downto 0); EN : in STD_LOGIC; Y : out STD_LOGIC); end entity mux_4to1; architecture mux_4to1_arch of mux_4to1 is begin MUX : process (D, Sel, EN) begin if (EN = '1') then case (Sel) is when "00" => Y <= D(0); when "01" => Y <= D(1); when "10" => Y <= D(2); when "11" => Y <= D(3); when others => Y <= 'Z'; end case; else Y <= 'Z'; end if; end process MUX; end architecture mux_4to1_arch;

24. Module 4: Combinational Logic Design with VHDL 24 Demultiplexer Demultiplexer - this is the exact opposite of a Mux - a single input will be routed to a particular output pin depending on the Select setting ex) truth table of Demultiplexer Sel Y0 Y1 0 In 0 1 0 In

25. Module 4: Combinational Logic Design with VHDL 25 Demultiplexer Demultiplexer - we can again use the behavior of an AND gate to “pass” or “block” the input signal - an AND gate is used for each Demux output

26. Module 4: Combinational Logic Design with VHDL 26 Demultiplexer Demultiplexers in VHDL - Structural Model entity demux_1to4 is port (D : in STD_LOGIC; Sel : in STD_LOGIC_VECTOR (1 downto 0); EN : in STD_LOGIC; Y : out STD_LOGIC_VECTOR (3 downto 0)); end entity demux_1to4; architecture demux_1to4_arch of demux_1to4 is signal Sel_n : STD_LOGIC_VECTOR (1 downto 0); component inv1 port (In1: in STD_LOGIC; Out1: out STD_LOGIC); end component; component and4 port (In1,In2,In3,In4: in STD_LOGIC; Out1: out STD_LOGIC); end component; begin U1 : inv1 port map (In1 => Sel(0), Out1 => Sel_n(0)); U2 : inv1 port map (In1 => Sel(1), Out1 => Sel_n(1)); U3 : and4 port map (In1 => D, In2 => Sel_n(1), In3 => Sel_n(0), In4 => EN, Out1 => Y(0)); U4 : and4 port map (In1 => D, In2 => Sel_n(1), In3 => Sel(0), In4 => EN, Out1 => Y(1)); U5 : and4 port map (In1 => D, In2 => Sel(1), In3 => Sel_n(0), In4 => EN, Out1 => Y(2)); U6 : and4 port map (In1 => D, In2 => Sel(1), In3 => Sel(0), In4 => EN, Out1 => Y(3)); end architecture demux_1to4_arch;

27. Module 4: Combinational Logic Design with VHDL 27 Demultiplexer Demultiplexers in VHDL - Behavioral Model with High Z Outputs entity demux_1to4 is port (D : in STD_LOGIC; Sel : in STD_LOGIC_VECTOR (1 downto 0); EN : in STD_LOGIC; Y : out STD_LOGIC_VECTOR (3 downto 0)); end entity demux_1to4; architecture demux_1to4_arch of demux_1to4 is begin DEMUX : process (D, Sel, EN) begin if (EN = '1') then case (Sel) is when "00" => Y <= 'Z' & 'Z' & 'Z' & D; when "01" => Y <= 'Z' & 'Z' & D & 'Z'; when "10" => Y <= 'Z' & D & 'Z' & 'Z'; when "11" => Y <= D & 'Z' & 'Z' & 'Z'; when others => Y <= "ZZZZ"; end case; else Y <= "ZZZZ"; end if; end process DEMUX; end architecture demux_1to4_arch;

28. Module 4: Combinational Logic Design with VHDL 28 Tri-State Buffers Tri-State Buffers - Provides either a Pass-Through or High Impedance Output depending on Enable Line - High Impedance (Z) allows the circuit to be connected to a line with multiple circuits driving/receiving - Using two Tri-State Buffers creates a "Bus Transceiver" - This is used for "Multi-Drop" Buses (i.e., many Drivers/Receivers on the same bus) ex) truth table of Tri-State Buffer ex) truth table of Bus Transceiver ENB Out Tx/RxMode 0 Z 0 Receive from Bus (Rx) 1 In 1 Drive Bus (Tx)

29. Module 4: Combinational Logic Design with VHDL 29 Tri-State Buffers Tri-State Buffers in VHDL - 'Z' is a resolved value in the STD_LOGIC data type defined in Package STD_LOGIC - Z & 0 = 0 - Z & 1 = 1 - Z & L = L - Z & H = H TRISTATE: process (In1, ENB) begin if (ENB = '1') then Out1 <= 'Z'; else Out1 <= In1; end if; end process TRISTATE;

30. Module 4: Combinational Logic Design with VHDL 30 Comparators Comparators - a circuit that compares digital values (i.e., Equal, Greater Than, Less Than) - we are considering Digital Comparators (Analog comparators also exist) - typically there will be 3-outputs, of which only one is asserted - whether a bit is EQ, GT, or LT is a Boolean expression - a 2-Bit Digital Comparator would look like: (A=B) (A>B) (A<B) A B EQGTLT 0 010 0 EQ = (A  B)' 0 10 0 1 GT = A·B' 1 0 0 1 0 LT = A'·B 1 1 1 0 0

31. Module 4: Combinational Logic Design with VHDL 31 Comparators Non-Iterative Comparators - "Iterative" refers to a circuit make up of identical blocks. The first block performs its operation which produces a result used in the 2nd block and so on. - this can be thought of as a "Ripple" effect - Iterative circuits tend to be slower due to the ripple, but take less area - Non-Iterative circuits consist of combinational logic executing at the same time "Equality" - since each bit in a vector must be equal, the outputs of each bit's compare can be AND'd - for a 4-bit comparator: EQ = (A3  B3)' · (A2  B2)' · (A1  B1)' · (A0  B0)'

32. Module 4: Combinational Logic Design with VHDL 32 Comparators Non-Iterative Comparators "Greater Than" - we can start at the MSB (n) and check whether A n >B n. - If it is, we are done and can ignore the rest of the LSB's. - If it is NOT, but they are equal, we need to check the next MSB bit (n-1) - to ensure the previous bit was equal, we include it in the next LSB's logic expression: Steps- GT = A n ·B n ' (this is ONLY true if A n >B n ) - if it is NOT GT, we go to the n-1 bit assuming that A n = B n (A n  B n )' - we consider A n-1 >B n-1 only when A n = B n [i.e., (A n  B n )' · (A n-1 ·B n-1 ') ] - we continue this process through all of the bits - 4-bit comparator GT =(A3·B3') + (A3  B3)' · (A2·B2') + (A3  B3)' · (A2  B2)' · (A1·B1') + (A3  B3)' · (A2  B2)' · (A1  B1)' · (A0·B0')

33. Module 4: Combinational Logic Design with VHDL 33 Comparators Non-Iterative Comparators "Less Than" - since we assume that if the vectors are either EQ, GT, or LT, we can create LT using: LT = EQ' · GT' Iterative Comparators - we can build an iterative comparator by passing signals between identical modules from MSB to LSB ex) module for 1-bit comparator EQ out = (A  B)' · EQ in - EQ out is fed into the EQ in port of the next LSB module - the first iterative module has EQ in set to '1'

34. Module 4: Combinational Logic Design with VHDL 34 Comparators Comparators in VHDL - Structural Model entity comparator_4bit is port (In1, In2 : in STD_LOGIC_VECTOR (3 downto 0); EQ, LT, GT : out STD_LOGIC); end entity comparator_4bit; architecture comparator_4bit_arch of comparator_4bit is signal Bit_Equal : STD_LOGIC_VECTOR (3 downto 0); signal Bit_GT : STD_LOGIC_VECTOR (3 downto 0); signal In2_n : STD_LOGIC_VECTOR (3 downto 0); signal In1_and_In2_n : STD_LOGIC_VECTOR (3 downto 0); signal EQ_temp, GT_temp : STD_LOGIC; component xnor2 port (In1,In2: in STD_LOGIC; Out1: out STD_LOGIC); end component; component or4 port (In1,In2,In3,In4: in STD_LOGIC; Out1: out STD_LOGIC); end component; component nor2 port (In1,In2: in STD_LOGIC; Out1: out STD_LOGIC); end component; component and2 port (In1,In2: in STD_LOGIC; Out1: out STD_LOGIC); end component; component and3 port (In1,In2,In3: in STD_LOGIC; Out1: out STD_LOGIC); end component; component and4 port (In1,In2,In3,In4: in STD_LOGIC; Out1: out STD_LOGIC); end component; component inv1 port (In1: in STD_LOGIC; Out1: out STD_LOGIC); end component;

35. Module 4: Combinational Logic Design with VHDL 35 Comparators Comparators in VHDL Cont… begin -- "Equal" Circuitry XN0 : xnor2 port map (In1(0), In2(0), Bit_Equal(0)); -- 1st level of XNOR tree XN1 : xnor2 port map (In1(1), In2(1), Bit_Equal(1)); XN2 : xnor2 port map (In1(2), In2(2), Bit_Equal(2)); XN3 : xnor2 port map (In1(3), In2(3), Bit_Equal(3)); AN0 : and4 port map (Bit_Equal(0), Bit_Equal(1), Bit_Equal(2), Bit_Equal(3), Eq); -- 2nd level of "Equal" Tree AN1 : and4 port map (Bit_Equal(0), Bit_Equal(1), Bit_Equal(2), Bit_Equal(3), Eq_temp); -- "Greater Than" Circuitry IV0 : inv1 port map (In2(0), In2_n(0)); -- creating In2' IV1 : inv1 port map (In2(1), In2_n(1)); IV2 : inv1 port map (In2(2), In2_n(2)); IV3 : inv1 port map (In2(3), In2_n(3)); AN2 : and2 port map (In1(3), In2_n(3), In1_and_In2_n(3)); -- creating In1 & In2' AN3 : and2 port map (In1(2), In2_n(2), In1_and_In2_n(2)); AN4 : and2 port map (In1(1), In2_n(1), In1_and_In2_n(1)); AN5 : and2 port map (In1(0), In2_n(0), In1_and_In2_n(0)); AN6 : and2 port map (Bit_Equal(3), In1_and_In2_n(2), Bit_GT(2)); AN7 : and3 port map (Bit_Equal(3), Bit_Equal(2), In1_and_In2_n(1), Bit_GT(1)); AN8 : and4 port map (Bit_Equal(3), Bit_Equal(2), Bit_Equal(1), In1_and_In2_n(0), Bit_GT(0)); OR0 : or4 port map (In1_and_In2_n(3), Bit_GT(2), Bit_GT(1), Bit_GT(0), GT); OR1 : or4 port map (In1_and_In2_n(3), Bit_GT(2), Bit_GT(1), Bit_GT(0), GT_temp); -- "Less Than" Circuitry ND0 : nor2 port map (EQ_temp, GT_temp, LT); end architecture comparator_4bit_arch;

36. Module 4: Combinational Logic Design with VHDL 36 Comparators Comparators in VHDL - Behavioral Model entity comparator_4bit is port (In1, In2 : in STD_LOGIC_VECTOR (3 downto 0); EQ, LT, GT : out STD_LOGIC); end entity comparator_4bit; architecture comparator_4bit_arch of comparator_4bit is begin COMPARE : process (In1, In2) begin EQ <= '0'; LT <= '0'; GT <= '0'; -- initialize outputs to '0' if (In1 = In2) then EQ <= '1'; end if; -- Equal if (In1 < In2) then LT <= '1'; end if; -- Less Than if (In1 > In2) then GT <= '1'; end if; -- Greater Than end process COMPARE; end architecture comparator_4bit_arch;

37. Module 4: Combinational Logic Design with VHDL 37 Ripple Carry Adder Addition – Half Adder - one bit addition can be accomplished with an XOR gate (modulo sum 2) 0 1 0 1 +0 +0+1+1 0 1 1 10 - notice that we need to also generate a “Carry Out” bit - the “Carry Out” bit can be generated using an AND gate - this type of circuit is called a “Half Adder” - it is only “Half” because it doesn’t consider a “Carry In” bit

38. Module 4: Combinational Logic Design with VHDL 38 Ripple Carry Adder Addition – Full Adder - to create a full adder, we need to include the “Carry In” in the Sum Cin A BCoutSum 0 0 0 0 0 0 0 1 0 1 Sum = A  B  Cin 0 1 0 0 1 Cout = Cin∙A + A∙B + Cin∙B 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 - you could also use two "Half Adders" to accomplish the same thing

39. Module 4: Combinational Logic Design with VHDL 39 Ripple Carry Adder Addition – Ripple Carry Adder - cascading Full Adders together will allow the Cout’s to propagate (or Ripple) through the circuit - this configuration is called a Ripple Carry Adder

40. Module 4: Combinational Logic Design with VHDL 40 Ripple Carry Adder Addition – Ripple Carry Adder - What is the delay through the Full Adder? - Each Full Adder has the following logic: Sum = A  B  Cin Cout = Cin∙A + A∙B + Cin∙B - t Full-Adder will be the longest combinational logic delay path in the adder

41. Module 4: Combinational Logic Design with VHDL 41 Ripple Carry Adder Addition – Ripple Carry Adder - What is the delay through the entire iterative circuit? - Each Full Adder has the following logic: t RCA = n·t Full-Adder - the delay increases linearly with the number of bits - different topologies within the full-adder to reduce delay (Δt) will have a n·Δt effect

42. Module 4: Combinational Logic Design with VHDL 42 Carry Look Ahead Adders Addition – Carry Look Ahead Adder - We've seen a Ripple Carry Adder topology (RCA) - this is good for simplicity and design-reuse - however, the delay increases linearly with the number of bits t RCA = n·t Full-Adder - different topologies within the full-adder to reduce delay (Δt) will have a n·Δt effect - the linear increase in delay comes from waiting for the Carry to Ripple through

43. Module 4: Combinational Logic Design with VHDL 43 Carry Look Ahead Adders Addition – Carry Look Ahead Adder - to avoid the ripple, we can build a Carry Look-Ahead Adder (CLA) - this circuit calculates the carry for all Full-Adders at the same time - we define the following intermediate stages of a CLA: Generate"g", an adder (i) generates a carry out (C i+1 )under input conditions A i and B i independent of A i-1, B i-1, or Carry In (C i ) A i B i C i+1 0 0 0 0 1 0 we can say that:g i = A i ·B i 1 0 0 1 1 1 remember, g does NOT consider carry in (C i )

44. Module 4: Combinational Logic Design with VHDL 44 Carry Look Ahead Adders Addition – Carry Look Ahead Adder Propagate"p", an adder (i) will propagate (or pass through) a carry in (C i ) depending on input conditions A i and B i, : C i A i B i C i+1 0 0 0 0 0 0 1 0 p i is defined when there is a carry in, 0 1 0 0 so we ignore the row entries where C i =0 0 1 1 1 1 0 0 0 if we only look at the C i =1 rows 1 0 1 1 we can say that: 1 1 0 1 p i = (A i +B i )·C i 1 1 1 1

45. Module 4: Combinational Logic Design with VHDL 45 Carry Look Ahead Adders Addition – Carry Look Ahead Adder - said another way, Adder(i) will "Generate" a Carry Out (C i+1 ) if: g i = A i ·B i and it will "Propagate" a Carry In (C i ) when p i = (A i +B i )·C i - a full expression for the Carry Out (C i+1 ) in terms of p and g is given by: C i+1 = g i +p i ·C i - this is good, but we still generate Carry's dependant on previous stages (i-1) of the iterative circuit

46. Module 4: Combinational Logic Design with VHDL 46 Carry Look Ahead Adders Addition – Carry Look Ahead Adder - We can eliminate this dependence by recursively expanding each Carry Equation ex) 4 bit Carry Look Ahead Logic C 1 = g 0 +p 0 ·C 0 (2-Level Product-of-Sums) C 2 = g 1 +p 1 ·C 1 C 2 = g 1 +p 1 ·(g 0 +p 0 ·C 0 ) C 2 = g 1 +p 1 ·g 0 +p 1 ·p 0 ·C 0 (2-Level Product-of-Sums) C 3 = g 2 +p 2 ·C 2 C 3 = g 2 +p 2 ·(g 1 +p 1 ·g 0 +p 1 ·p 0 ·C 0 ) C 3 = g 2 +p 2 ·g 1 +p 2 ·p 1 ·g 0 +p 2 ·p 1 ·p 0 ·C 0 (2-Level Product-of-Sums) C 4 = g 3 +p 3 ·C 3 C 4 = g 3 +p 3 ·(g 2 +p 2 ·g 1 +p 2 ·p 1 ·g 0 +p 2 ·p 1 ·p 0 ·C 0 ) C 4 = g 3 +p 3 ·g 2 +p 3 ·p 2 ·g 1 +p 3 ·p 2 ·p 1 ·g 0 +p 3 ·p 2 ·p 1 ·p 0 ·C 0 (2-Level Product-of-Sums) - this gives us logic expressions that can generate a next stage carry based upon ONLY the inputs to the adder and the original carry in (C 0 )

47. Module 4: Combinational Logic Design with VHDL 47 Carry Look Ahead Adders Addition – Carry Look Ahead Adder - the Carry Look Ahead logic has 3 levels 1) g and p logic 2) product terms in the C i equations 3) sum terms in the C i equations - the Sum bits require 2 levels of Logic 1) A i  B i  C i NOTE:A Full Adder made up of 2 Half Adders has 3 levels. But the 3rd level is used in the creation of the Carry Out bit. Since we do not use it in a CLA, we can ignore that level. - So a CLA will have a total of 5 levels of Logic

48. Module 4: Combinational Logic Design with VHDL 48 Carry Look Ahead Adders Addition – Carry Look Ahead Adder - the 5 levels of logic are fixed no matter how many bits the adder is (really?) - In reality, the most significant Carry equation will have i+1 inputs into its largest sum/product term - this means that Fan-In becomes a problem since real gates tend to not have more than 4-6 inputs - When the number of inputs gets larger than the Fan-In, the logic needs to be broken into another level ex) A+B+C+D+E = (A+B+C+D)+E - In the worst case, the logic Fan-In would be 2. Even in this case, the delay associated with the Carry Look Ahead logic would be proportional to log 2 (n) - Area and Power are also concerns with CLA's. Typically CLA's are used in computationally intense applications where performance outweighs Power and Area.

49. Module 4: Combinational Logic Design with VHDL 49 Carry Look Ahead Adders Adders in VHDL - (+) and (-) are not defined for STD_LOGIC_VECTOR - The Package STD_LOGIC_ARITH gives two data types: UNSIGNED (3 downto 0) := "1111"; -- +15 SIGNED (3 downto 0) := "1111"; -- -1 - these are still resolved types (STD_LOGIC), but the equality and arithmetic operations are slightly different depending on whether you are using Signed vs. Unsigned Considerations - when adding signed and unsigned numbers, the type of the result will dictate how the operands are handled/converted - if assigning to an n-bit, SIGNED result, an n-1 UNSIGNED operand will automatically be converted to signed by extending its vector length by 1 and filling it with a sign bit (0)

50. Module 4: Combinational Logic Design with VHDL 50 Carry Look Ahead Adders Adders in VHDL ex)A,B: inUNSIGNED (7 downto 0); C: inSIGNED (7 downto 0); D: inSTD_LOGIC_VECTOR (7 downto 0); S: outUNSIGNED (8 downto 0); T: outSIGNED (8 downto 0); U: outSIGNED (7 downto 0); S(7 downto 0) <= A + B;-- 8-bit UNSIGNED addition, not considering Carry S <= ('0' & A) + ('0' & B);-- manually increasing size of A and B to include Carry. Carry will be kept in S(9) T <= A + C;-- T is SIGNED, so A's UNSIGNED vector size is increased by 1 and filled with '0' as a sign bit U <= C + SIGNED(D);-- D is converted (considered) to SIGNED, not increased in size U <= C + UNSIGNED(D);-- D is converted (considered) to UNSIGNED, not increased in size

51. Module 4: Combinational Logic Design with VHDL 51 Subtraction Half Subtractor - one bit subtraction can be accomplished using combinational logic (A-B) A BBout D 0 0 0 0 0 1 1 1 D = A  B 1 0 0 1 B out = A'·B 1 1 0 0

52. Module 4: Combinational Logic Design with VHDL 52 Subtraction Full Subtractor - to create a full Subtractor, we need to include the “Borrow In” in the Difference (A-B-B in ) A B BinBout D 0 0 0 0 0 0 0 1 1 1 D = A  B  B in 0 1 0 1 1 B out = A'∙B + A'∙B in + B∙B in 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 - notice this is very similar to addition. - The Sum and Difference Logic are identical - The Carry and Borrow Logic are close

53. Module 4: Combinational Logic Design with VHDL 53 Subtraction Subtraction - Can we manipulate the subtraction logic so that Full Adders can be used as Full Subtractors? Addition Subtraction S = A  B  C in D = A  B  B in C out = A∙B + A∙C in + B∙C in B out = A'∙B + A'∙B in + B∙B in - Let's manipulate B out to try to get it into a form similar to C out B out = A'∙B + A'∙B in + B∙B in B out ' = (A+B') ∙ (A+B in ') ∙ (B'+B in ') Generalized DeMorgan's Theorem Now Multiply Out the Terms B out ' = (A∙A∙B')+(A∙B'∙B in ')+(A∙B'∙B')+(B'∙B'∙B in ')+(A∙A∙B in ')+(A∙B in '∙B in ')+(A∙B'∙B in ')+(B'∙B in '∙B in ') Now Remove Redundant Terms B out ' = (A∙B')+(A∙B'∙B in ')+(A∙B in ')+(B'∙B in ') B out ' = (A∙B')+(A∙B in ')+(B'∙B in ')

54. Module 4: Combinational Logic Design with VHDL 54 Subtraction Subtraction - Now we have similar expressions for Cout and Bout where Addition Subtraction C out = A∙B + A∙C in + B∙C in B out ' = A∙B' + A∙B in ' + B'∙B in ' - But this requires the Subtrahend and B in be inverted, how does this effect the Sum/Difference Logic? Addition Subtraction S = A  B  C in D = A  B  B in - remember that both inputs of a 2-input XOR can be inverted without changing the logic function which gives us: S = A  B  C in D = A  B'  B in '

55. Module 4: Combinational Logic Design with VHDL 55 Subtraction Subtraction - After all of this manipulation, we are left with Addition Subtraction S = A  B  C in D = A  B'  B in ' C out = A∙B + A∙C in + B∙C in B out ' = A∙B' + A∙B in ' + B'∙B in ' - This means we can use "Full Adders" for subtraction as long as: 1) The Subtrahend is inverted 2) B in is inverted 3) B out is inverted - In a ripple carry subtractor, intermediate B out 's are fed into B in 's, which is a double inversion - We can now invert by the first B in and the last B out by inserting a '1' into the first B in of the chain

56. Module 4: Combinational Logic Design with VHDL 56 Subtraction Subtraction - this gives us the minimal logic for a "Ripple Carry Subtractor" using "Full Adders" X-Y

57. Module 4: Combinational Logic Design with VHDL 57 Multipliers Multipliers - binary multiplication of an individual bit can be performed using combinational logic: A * B P 0 0 0 0 1 0 we can say that:P = A·B 1 0 0 1 1 1 - for multi-bit multiplication, we can mimic the algorithm that we use when doing multiplication by hand ex)1 2 this number is the "Multiplicand" x3 4 this number is the "Multiplier" 4 8 1) multiplicand for digit (0) + 3 6 2) multiplicand for digit (1) 4 0 8 3) Sum of all multiplicands - this is called the "Shift and Add" algorithm

58. Module 4: Combinational Logic Design with VHDL 58 Multipliers "Shift and Add" Multipliers - example of Binary Multiplication using our "by hand" method 11 1 0 1 1 - multiplicand x 13 x1 1 0 1 - multiplier 33 1 0 1 1 11 0 0 0 0 - these are the individual multiplicands 1 0 1 1 + + 1 0 1 1 1 4 3 1 0 0 0 1 1 1 1 - the final product is the sum of all multiplicands - this is simple and straight forward. BUT, the addition of the individual multiplicand products requires as many as n-inputs. - we would really like to re-use our Full Adder circuits, which only have 3 inputs.

59. Module 4: Combinational Logic Design with VHDL 59 Multipliers "Shift and Add" Multipliers - we can perform the additions of each multiplicand after it is created - this is called a "Partial Product" - to keep the algorithm consistent, we use "0000" as the first Partial Product 1 0 1 1 - Original multiplicand x1 1 0 1 - Original multiplier 0 0 0 0- Partial Product for 1st multiply 1 0 1 1 - Shifted Multiplicand for 1st multiply 1 0 1 1- Partial Product for 2nd multiply 0 0 0 0  - Shifted Multiplicand for 2nd multiply 0 1 0 1 1- Partial Product for 3rd multiply 1 0 1 1   - Shifted Multiplicand for 3rd multiply 1 1 0 1 1 1- Partial Product for 4th multiply 1 0 1 1    - Shifted Multiplicand for 4th multiply 1 0 0 0 1 1 1 1 - the final product is the sum of all multiplicands

60. Module 4: Combinational Logic Design with VHDL 60 Multipliers "Shift and Add" Multipliers - Graphical view of product terms and summation

61. Module 4: Combinational Logic Design with VHDL 61 Multipliers "Shift and Add" Multipliers - Graphical View of interconnect for an 8x8 multiplier. Note the Full Adders

62. Module 4: Combinational Logic Design with VHDL 62 Multipliers "Sequential" Multipliers - the main speed limitation of the Combinational "Shift and Add" multiplier is the delay through the adder chain. - in the worst case, the number of delay paths through the adders would be [n + 2(n-2)] ex)4-bit= 8 Full Adders 8-bit= 20 Full Adders - we can decrease this delay by using a register to accumulate the incremental additions as they take place. - this would reduce the number of operation states to [n-1] "Carry Save" Multipliers - another trick to speed up the multiplication is to break the carry chain - we can run the 0th carry from the first row of adders into adder for the 2nd row - a final stage of adders is needed to recombine the carrys. But this reduces the delay to [n+(n-2)]

63. Module 4: Combinational Logic Design with VHDL 63 Multipliers "Carry Save" Multipliers

64. Module 4: Combinational Logic Design with VHDL 64 Signed Multipliers Multipliers - we leaned the "Shift and Add" algorithm for constructing a combinational multiplier - but this only worked for unsigned numbers - we can create a signed multiplier using a similar algorithm Convert to Positive - one of the simplest ways is to first convert any negative numbers to positive, then use the unsigned multiplier - the sign bit is added after the multiplication following: pos x pos = pos Remember 0=pos and 1=neg is 2's comp so this is an XOR pos x neg = neg neg x pos = neg neg x neg = pos

65. Module 4: Combinational Logic Design with VHDL 65 Signed Multipliers 2's Comp Multiplier - remember that in a "Shift and Add', we created a shifted multiplicand - the shifted multiplicand corresponded to the weight of the multiplier bit - we can use this same technique for 2's comp remembering that - the MSB of a 2's comp # is -2 (n-1) - we also must remember that 2's comp addition must - be on same-sized vectors - the carry is ignored - we can make partial products the same size as shifted multiplicands by doing a "2's comp sign extend" ex) 1011 = 11011 = 1110111 - since the MSB has a negative weight, we NEGATE the shifted multiplicand for that bit prior to the last addition.

66. Module 4: Combinational Logic Design with VHDL 66 Signed Multipliers 2's Comp Shift and Add Multipliers - we can perform the additions of each multiplicand after it is created - this is called a "Partial Product" - to keep the algorithm consistent, we use "0000" as the first Partial Product 1 0 1 1 - Original multiplicand x1 1 0 1 - Original multiplier 0 0 0 0 0- Partial Product for 1st multiply w/ Sign Extension 1 1 0 1 1 - Shifted Multiplicand for 1st multiply w/ Sign Extension 1 1 1 0 1 1- Partial Product for 2nd multiply w/ Sign Extension 0 0 0 0 0  - Shifted Multiplicand for 2nd multiply w/ Sign Extension 1 1 1 1 0 1 1- Partial Product for 3rd multiply w/ Sign Extension 1 1 0 1 1   - Shifted Multiplicand for 3rd multiply w/ Sign Extension 1 1 1 0 0 1 1 1- Partial Product for 4th multiply w/ Sign Extension 0 0 1 0 1    - NEGATED Shifted Multiplicand for 4th multiply w/ Sign Extension 1 0 0 0 0 1 1 1 1 - the final product is the sum of all multiplicands ignore Carry_Out

67. Module 4: Combinational Logic Design with VHDL 67 Division Division - "Repeated Subtraction" - a simple algorithm to divide is to count the number of times you can subtract the divisor from the dividend - this is slow, but simple - the number of times it can be subtracted without going negative is the "Quotient" - if the subtracted value results in a zero/negative number, whatever was left prior to the subtraction is the "Remainder"

68. Module 4: Combinational Logic Design with VHDL 68 Division Division - "Shift and Subtract" - Division is similar to multiplication, but instead of "Shift and Add", we "Shift and Subtract"