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Logic gate level Part 4: combinational devices

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K-maps & dont care conditions It isnt always necessary to process all possible input combinations, since some are never expected to be present Such input combinations are called dont care conditions, since we dont care about the outputs theyd produce should they ever be present

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K-maps & dont care conditions With dont care condition present, you can arbitarily choose either 1 or 0 for output Choice of output (1 or 0) for dont care conditions can aid in minimization Sigma notation for dont care conditions: (x,y,z) + d(a,b) where x,y,z,a and b all represent lines in the functions truth table We represent dont care conditions in a K-map with Xs

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Example K-map for X(a,b,c) = (2,4,6) + d(0,7) X1 1X1 With d.c. conditions, since we dont care, we can choose to include, or not include, boxes with X designations in K-maps X is wildcard condition – can be treated as either 1 or 0 In K-map above, if minterm 0 is treated as 1 and 7 as 0, get (0,2,4,6) = c

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Combinational Devices Many devices have input line called an enable, which acts like on/off switch – if enable is 0, all outputs are 0 regardless of other inputs – if enable is 1, output depends on input to function that specifies device AND gate can implement an enable

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AND gate as enable

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Selective inverter Has data line & invert line – If invert = 1, output is complement of data – If invert = 0, output is data unchanged Implement with XOR gate

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Multiplexer Device that selects one of several inputs to route to single output Consists of set of data lines & control lines – control lines determine which data input will be output – n control lines can control 2 n data lines

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8-input multiplexer Combination of control line (s0-s2) inputs determines which of 8 data lines (d0-d7) is expressed as output value

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Implementation of multiplexer Each data line ANDed with combination of control lines Result of ANDs is ORed together to get output Illustration on next slide shows 4-input version of this scheme; 8-input version (like previous example) would involve 8 4-input AND gates

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4-input mux implementation

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Binary decoder Takes input from control lines and sets one of several output lines to 1, rest to 0 Output value depends on input value(s)

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Binary decoder implementation

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Decoder with enable When enable line is 1, device operates normally When enable line is 0, all outputs are 0 Requires extra input to each AND gate

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Demultiplexer Routes single input value to one of several output lines Really just decoder with enable: input line connected to enable

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Building the CPU Control unit: portion of CPU that: – ensures synchronization of events – i.e. sending & receiving bits on the bus – selects next instruction – stores values in appropriate locations Made up of combinational devices

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Bus Internal bus: common path connecting all registers in a register machines CPU – each register composed of multiple bits, all of which can be transferred simultaneously to another register – bus composed of parallel wires – as many lines as there are bits in registers – may also include control lines indicating which registers should send & receive – action must be coordinated, as bits from only one register at a time can be broadcast over bus – coordination requires timing mechanism

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Putting it together Parallel AND gates used to connect registers to bus One input line to each gate is data waiting to be transmitted, second is select signal (CLOCK) Date transmitted only if select signal is 1

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Example: 2 8-bit registers tied to 8- bit bus with select (clock) signal

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Strobing When select signal is high, each gate allows signal to flow – clock (select) ANDed with each bit from registers – all bits transmitted simultaneously, and received simultaneously at destination register

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Clock Source of all select signals; generates pulses at fixed rate Normal state is low (0); transmits 1 at regular interval Speed measured in hertz: – 1 Hz = 1 cycle/second – 1 MHz = 1,000,000 cycles/second – 1 cycle ~ 1 step of fetch/execute cycle Time interval between pulses measured in fractions of seconds – for PC, typically nanoseconds (1 s ~ 1/1,000,000,000 second)

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Arithmetic Logic Unit Part of the computer that computes All operations performed using combinations of logic circuits – logical operations are performed by connecting operands bit-wise through ganged gates of appropriate type – arithmetic operations are also performed logically: operands connected bit-wise through ganged gates of appropriate type(s)

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Performing arithmetic operations Can break down any binary arithmetic operation into set of operations on pairs of bits Each unique pair of bits combines to produce – result (0 or 1) – carry (0 or 1) when result is larger than either of the two operands These 2 output bits can be viewed as single 2-bit number representing result of arithmetic operation on two 1-bit operands

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Performing arithmetic operations Example: addition of two operands produces the results shown in the truth table below: Notes on truth table – First output (Sum) same as XOR – Second output (Carry) same as AND

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Half adder A half adder is the logical construction that implements the truth table on the previous slide Half adders take single bits as input, produce 2 outputs

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Full adder Addition of two bits accomplishes only half the task of binary addition, unless were satisfied with working in single-digit numbers To complete an addition operation, we must add the carry to the next (left) bit Can construct full adder from half adders; carry from previous bit sum is ORed with carry from correct bit

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Implementation of full adder from two half-adders By chaining together several of these constructs, we can build adders for multiple- digit numbers

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Block diagram fo 4-digit adder

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Subtraction Result column (difference) can be expressed as: ab + ab Carry column: ab ABResult (A-B) carry

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Same device built with NAND gates

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Notes on adders If last bit in series of half adders produces an non-zero carry, result will overflow An adder constructed from a series of half adders is called a cascading adder – results cascade from right to left as previous carries must be determined before subsequent adds are performed

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Adder as black box Control unit delivers data to input registers in ALU Device select signal triggers ALU to perform addition & send result to result register Control unit causes result to be copied to its destination

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