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1 Lecture 15 Registers Counters Finite State Machine (FSM) design
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2 Registers Group of storage elements read/written as a unit Store related values (e.g. a binary word) Collection of flip-flops with common control Share clock, reset, set lines Example: Storage registers Shift registers Counters
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3 Storage registers Basic storage registers use flip-flops Example: 4 bit storage register RSRS DQDQDQ OUT1OUT2OUT3OUT4 CLK IN1IN2IN3IN4 RSRS DQ "0"
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4 Shift registers Hold successively sampled input values Delays values in time Example: 4-bit shift register Stores 4 input values in sequence DQDQDQDQ IN OUT1OUT2OUT3OUT4 CLK
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5 Shift register applications Parallel-to-serial conversion for signal transmission Pattern recognition (circuit recognizes 1001) parallel inputs parallel outputs serial transmission CLK DQDQDQDQ IN CLK OUT
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6 Counters: Ring counter Ring counter: Sequence is 1000, 0100, 0010, 0001 Assuming one of these patterns is the starting state DQDQDQDQ IN OUT1OUT2OUT3OUT4 CLK
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7 Counters: Johnson counter Johnson counter: Sequence is 1000, 1100, 1110, 1111, 0111, 0011, 0001, 0000 DQDQDQDQ IN OUT1OUT2OUT3OUT4 CLK
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8 Counters: Binary counter Has logic between flip-flops Flip low-order bit each clock cycle Flip next bit in cycle when low- order bit is 1 Flip next bit when both lower order bits are 1 Flip next bit when all lower order bits are 1
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9 Storing "states" for FSMs Combinational logic and storage elements Localized feedback loops Choice of storage elements alters the logic Combinational Logic Storage Elements Outputs State OutputsState Inputs Inputs
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10 Finite-state machines (FSMs) States: Possible storage-element values Transitions: Changes in state Clock synchronizes the state changes Sequential logic Sequences through a series of states Based on inputs and present state
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11 FSM design example: Counters 1. Draw a state diagram 2. Draw a state-transition table 3. Encode the next-state functions Minimize the logic using k-maps 4. Implement the design
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12 1. Draw a state diagram 010 100 110 011 001 000 101 111 3-bit up-counter
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13 2. Draw a state-transition table Like a truth-table State encoding is easy for counters Use count value current statenext state 00000011 10010102 20100113 30111004 41001015 51011106 61101117 71110000
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14 3. Encode next state functions Assume D flip-flops as state elements N1:= C1' N2:= C1C2' + C1'C2 := C1 xor C2 N3:= C1C2C3' + C1'C3 + C2'C3 := C1C2C3' + (C1' + C2')C3 := (C1C2) xor C3 C3C2C1N3N2N1 000001 001010 010011 011100 100101 101110 110111 111000 0011010100110101 C2 C3 C1 N3 1111000011110000 C2 C3 C1 N1 0110100101101001 C2 C3 C1 N2
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15 4. Implement the design 3 flip-flops hold state Counter is synchronously clocked Minimized logic computes next state
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16 What if we use T flip-flops? T flip-flops TQ C2C1C0N2N1N0 T2 T1 T0 000001 001010 010011 011100 100101 101110 110111 111000 0001000100010001 0101010101010101 1111111111111111 C1 C2 C0 T2 0 0 0 1 1 0 C1 C2 C0 T1 0 0 1 1 C1 C2 C0 T0 1 1 T0:= T1:= T2:= C0 C1 C0 1 Ti = 1 iff Ni ≠ Ci
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17 4. Implement the design TQTQTQ C0C1C2 CLK 1
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18 Example: 5-state counter Counter repeats 5 states in sequence Sequence is 000, 010, 011, 101, 110, 000 000 010 011 101 110 Step 1: State diagramStep 2: State transition table Assume D flip-flops CBAC+B+A+ 000010 001X X X 010011 011101 100X X X 101110 110000 111X X X Present State Next State
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19 3. Encode next state functions C B A 000XX1X1000XX1X1 C+ C B A 110XX0X1110XX0X1 B+ C B A 010XX1X0010XX1X0 A+ A+ = BC'C+ = AB+ = B' + A'C'
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20 4. Implement the design Recall that a D flip flop also produces Q’ so A’, B’, and C’ would all be available without any extra inverters
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21 Is our design robust? What if the counter starts in a 111 state? 001 100 111 Does our counter get stuck in invalid states??? 000 010 011 101 110
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22 5-state counter Back-annotate our design to check it 000 010 011 101 110 Draw state diagramFill in state transition table CBAC+B+A+ 000010 001110 010011 011101 100010 101110 110000 111100 Present State Next State 001 100 111 The proper methodology is to design your counter to be self-starting A+ = BC' C+ = A B+ = B' + A'C'
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23 Self-starting counters Invalid states should always transition to valid states Assures startup Assures bit-error tolerance Design your counters to be self-starting Draw all states in the state diagram Fill in the entire state-transition table May limit your ability to exploit don't cares Choose startup transitions that minimize the logic
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