i. motivation & examples
DESCRIPTION
I. Motivation & Examples. Output depends on current input and past history of inputs. “State” embodies all the information about the past needed to predict current output based on current input. State variables , one or more bits of information. - PowerPoint PPT PresentationTRANSCRIPT
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Sequential logic networks
I. Motivation & Examples
Output depends on current input and past history of inputs. “State” embodies all the information about the past needed to predict
current output based on current input.
– State variables, one or more bits of information. If the current State of the circuit is known at time t, what is the state of
the circuit at time (t+1)
Answer: the next state depends on current state and input
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Sequential logic networks
State table
– For each current-state, specify next-states as function of inputs
– For each current-state, specify outputs as function of inputs State diagram
– Graphical version of state table
I. Motivation & Examples
Describing sequential circuit
Example 1: TV channel control •Let the channel # represent the state of the circuit•Input are up/down on the channel control
1 2 3 4 99…u u u u u
d d d d d
on
u: up d: down
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Sequential logic networks
I. Motivation & Examples
Example 2: A sequential process that inputs an n-bit binary string and outputs 1 if the string contains an even number of 1’s
1 (final output) 01111 SLN
0 (final output) 0111 SLN
What represents the state of the circuit?• Case1:
State as the number of 1’s read so far (possibly infinite # of states)
•Case 2: Two states E and O•E (even): if the # of 1’s read so far is even•O (odd) if the # of 1’s read so far is odd
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Sequential logic networks
I. Motivation & Examples
Example 2: State Diagram for Case 1
1 2 3 2n…01/0 1/1 1/0 1/1 1/0
41/1
0/0 0/00/1 0/1 0/1 0/1
Example 2: State Diagram for Case 2
Input Output
OE
1/0
0/11/1 0/0
Input Output
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Sequential logic networks
I. Motivation & Examples
Example 2: State Diagram for Case 2
OE
1/0
0/11/1 0/0
Input Output
•Better design•Has less states
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Sequential logic networks
I. Motivation & Examples
Example 3: Discuss sequential n-bits comparator•Compare two n-bits numbers X=[Xn-1, …, X0], Y=[Yn-1, …, Y0]•Output 1 if X>Y•Use the basic 1-bit comparator designed in class
Xn-1 Xn-2 Xn-3 X2 X1 X0
Shift right
1-bitComparator
Ci
Yn-1 Yn-2 Yn-3 Y2 Y1 Y0
Shift rightFi
Xi Fi-1
Yi
Operation controlled by a clock to decide:
.when to shift input data
.when output Fi is stable
...
...
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Sequential logic networks
I. Motivation & Examples
Example 4: Discuss sequential n-bits adder•Add two n-bits numbers X=[Xn-1, …, X0], Y=[Yn-1, …, Y0]•Output S=X+Y where [Sn,Sn-1,…,S0]•Use the basic 1-bit adder with carry in and carry out
Xn-1 Xn-2 Xn-3 X2 X1 X0
Shift right
1-bitFull adder Ci
Yn-1 Yn-2 Yn-3 Y2 Y1 Y0
Shift rightCi
Xi Ci-1
Yi
Operation controlled by a clock to decide:
.when to shift input data
.when output are ready
Shift rightSn-1 Sn-2 S2 S1 S0...
...
...
Sn
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Sequential logic networks
Clock signals Sequential circuit are controlled by a clock signal Very important with most sequential circuits
– State variables change state at clock edge.
II. General Representation
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Sequential logic networks
General diagram of sequential circuit Sequential circuit are controlled by a clock signal Very important with most sequential circuits
– State variables change state at clock edge.
II. General Representation
SLN
Memory components
Inputi0i1…in
Outputo0o1…omCurrent states Next states
State variables: s0,s1, …sk
Feedback
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Sequential logic networks
Some important questions How to represent the states of a sequential circuit? How to memorize the (current and next) states? How to determine the next of the circuit? How to determine the outputs
– as a function F(state) of current state only?
– as a function F(input,state) of both input and current state?
The concept of STATE is very important
II. General Representation
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Sequential logic networks
Memory component
How do we represent the states? Memory component are used as state variables
– Goal: Memorize the current state of the circuit
– How are memory components implemented? Latch, Flip-flop are 1-bit memory component
II. General Representation
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Sequential logic networks
Bistable element The simplest sequential circuit Two states
– One state variable, say, Q (QN or Q_L the complement of Q)
HIGH LOW
LOW HIGH
III. Basic memory component
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Sequential logic networks
Bistable element
The simplest sequential circuit Two states
– One state variable, say, Q
LOW HIGH
HIGH LOW
III. Basic memory component
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Sequential logic networks
Bistable element: Analog analysis
Assume pure CMOS thresholds, 5V rail Theoretical threshold center is 2.5 V
III. Basic memory component
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Sequential logic networks
Bistable element: Analog analysis
Assume pure CMOS thresholds, 5V rail Theoretical threshold center is 2.5 V
2.5 V 2.5 V
2.5 V 2.5 V
III. Basic memory component
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Sequential logic networks
Bistable element: Analog analysis
Assume pure CMOS thresholds, 5V rail Theoretical threshold center is 2.5 V
2.5 V
2.5 V 2.5 V
2.0 V
2.0 V 4.8 V
2.5 V2.51 V4.8 V 0.0 V
0.0 V 5.0 V
III. Basic memory component
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Sequential logic networks
Bistable element: summary
If (Q=0), then input to Not gate 2 is 0
==> Output of Not gate 2 is 1 (Q_L =1)
==> The input of Not gate 1 is 1, so output of Not gate 1 is 0
==> Stable output (Q=0) and (Q_L = 1) If (Q=1), then input to Not gate 2 is 1
==> Output of Not gate 2 is 0 (Q_L =0)
==> The input of Not gate 1 is 0, so output Not gate 1 is
==> Stable output (Q=1) and (Q_L = 0)
II. General Representation
1
2
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Sequential logic networks
S-R Latch….
How to control it?
– Screwdriver
– Control inputs S-R latch
III. Basic memory component
Contradiction!!!!
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Sequential logic networks
S-R Latch….
III. Basic memory component
Set operation: SR 00 ----> 10, set the device output to Q=1 regardless of current value of Q Reset operation: SR 00 ----> 01, set the device output to Q=0 regardless of current value of Q
Hold operation: SR 10 ----> 00 or 01 ----> 00, Device output are the same as last output values
•Only one input value changes •Possible input changes:
•SR: 00 ---> 01 ---> 00 ---> 10 ---> 00 ….•Input SR = 11 is not allowed ( Both NOR gates output 0, i.e Q=Q’=0 )
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Sequential logic networks
S-R latch operation
III. Basic memory component
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Sequential logic networks
S-R latch timing parameters
Propagation delay Minimum pulse width
III. Basic memory component
Progation delay Minimum time to maintain signal at 1
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Sequential logic networks
S-R latch symbols
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Sequential logic networks
S-R latch with enable
III. Basic memory component
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Sequential logic networks
Sequential network architecture (revisited)
III. Basic memory component
Outputo1
omSLN
......
Inputi1
in
M1...
Mk
Components Mi are latches/Flip flops
Operation rules:•Memory components Mi must be in stable state before input changes•Only one input of the component Mi can change at a time
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Sequential logic networks
Charcteristics equation of S-R latch
III. Basic memory component
Definition: The characteristic equation specifies a flip-flop next state as a function of its current state and inputs
Notation: Let q represent the current state of the flip-flop and Q its next
S R q Q
0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1
0 10011 X
Q=q
Q=0
Q= 1
XNot allowed
Hold
ResetSet
Characteristics tableCharacteristics table
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Sequential logic networks
Charcteristics table (other representation)
III. Basic memory component
S R Q
0 00 11 01 1
q 01X
Q=qQ=0Q= 1Not allowed
HoldResetSet
Characteristics tableCharacteristics table
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Sequential logic networks
Charcteristics equation of S-R latch
III. Basic memory component
Use the characteristics table to get an excitation map of the flip flop
S R q Q
0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1
0 10011 X
Q=q
Q=0
Q= 1
X
Characteristics tableCharacteristics table
SRq
0
1
00 01 11 10X
X
1
11
Q
Use K-map method to derive the characteristics equation:
Q = S + R’q
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Sequential logic networks
Excitation table of SR flip flop
III. Basic memory component
The excitation table describes the input values of S and R that cause the corresponding transitions (q ---> Q) from current to next state
Types of transitions: q --->Q 0 ---> 00 ---> 11 ---> 01 ---> 1
Excitation table of S R latch
q ---> Q S R0 ---> 00 ---> 11 ---> 01 ---> 1
0 X1 00 1X 0
0 0 to holdcurrent value
0 1 to resetQ=0
OR
0 0 to holdcurrent value
1 0 to set Q=1
OR
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Sequential logic networks
JK Flip- Flop
III. Basic memory component
Recall: In SR flip flop, both input S, R cannot be 1 (SR=11)
This restriction is removed in a JK flip flop. The behavior of the JK flip flop is as follows:
J K q Q
0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1
0 10011 1
Q=q
Q=0
Q= 1
0Q = Q’ (Toggle)
Hold
ResetSet
Characteristics tableCharacteristics table
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Sequential logic networks
Charateristics of JK flip flop (other representation)
III. Basic memory component
C J K Q
1 0 01 0 11 1 01 1 1
q 01q'
Q =qQ =0Q = 1Q = q’ (Toggle)
HoldResetSet
Characteristics tableCharacteristics table
Characteristics table Characteristics table ( Clocked JK flip flop )
0 x x Disabled
J K Q
0 00 11 01 1
q 01q'
Q =qQ =0Q = 1Q = q’ (Toggle)
HoldResetSet
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Sequential logic networks
Charcteristics equation of Jk Flip flop
III. Basic memory component
Use the characteristics table to get an excitation map of the flip flop
Characteristics tableCharacteristics table
JKq
0
1
00 01 11 101
0
1
11
Q
Use K-map method to derive the characteristics equation:
Q = Jq’ + Kq
J K q Q
0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1
0 10011 1
Q=q
Q=0
Q= 1
0Q = Q’
0
0 0
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Sequential logic networks
Excitation table of JK flip flop
III. Basic memory component
The excitation table describes the input values of S and R that cause the corresponding transitions (q ---> Q) from current to next state
Types of transitions: q --->Q 0 ---> 00 ---> 11 ---> 01 ---> 1
Excitation table of JK flip flop
q ---> Q J K0 ---> 00 ---> 11 ---> 01 ---> 1
0 X1 XX 1X 0
0 0 to holdcurrent value
0 1 to resetQ=0
OR
0 0 to holdcurrent value
1 0 to set Q=1
OR
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Sequential logic networks
Excitation table of JK flip flop
III. Basic memory component
The excitation table describes the input values of S and R that cause the corresponding transitions (q ---> Q) from current to next state
Types of transitions: q --->Q 0 ---> 00 ---> 11 ---> 01 ---> 1
Excitation table of JK flip flop
q ---> Q J K0 ---> 00 ---> 11 ---> 01 ---> 1
0 X1 XX 1X 0
0 0 to holdcurrent value
0 1 to resetQ=0
OR
0 0 to holdcurrent value
1 0 to set Q=1
OR
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Sequential logic networks
JK Flip flop Symbols
III. Basic memory component
J
K
Q
QN
J
K
Q
Q
J
K
Q
QN
CK
J
K
Q
QN
CK
Clocked JK Flip flop
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Sequential logic networks
D Flip- flop ( Delay flip flop)
III. Basic memory component
This flip flop has only one control input. The D flip flop simply retains its input between clock pulses
D q Q
0 00 11 01 1
0 011
Q=d
Characteristics tableCharacteristics table
C D Q
1 0 1 1
Characteristics table Characteristics table ( Clocked D flip flop )
0 x
0 1 Q=dDisabled
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Sequential logic networks
Charcteristics equation of D Flip flop
III. Basic memory component
Use the characteristics table to get an excitation map of the flip flop
Characteristics tableCharacteristics table
Characteristics equation:
Q = D
D q Q
0 00 11 01 1
0 101
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Sequential logic networks
D Flip flop Symbols
III. Basic memory component
D Q
QN
D Q
Q
D Q
QN
CK
D Q
QN
CK
Clocked JK Flip flop
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Sequential logic networks
D latch
III. Basic memory component
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Sequential logic networks
D-latch operation
III. Basic memory component
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Sequential logic networks
D-latch timing parameters Propagation delay (from C or D) Setup time (D before C edge) Hold time (D after C edge)
III. Basic memory component
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Sequential logic networks
Edge-triggered D flip-flop behavior
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Sequential logic networks
Edge-triggered D flip-flop behavior
III. Basic memory component
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Sequential logic networks
D flip-flop timing parameters Propagation delay (from CLK) Setup time (D before CLK) Hold time (D after CLK)
III. Basic memory component
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Sequential logic networks
IV. Counters
A counter is a sequential-circuit that generates a predetermined number sequence over and over again
A counter can be used as
– a digital clock
– special sequence generator
– program counter
– pulse counter
Definitions
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Sequential logic networks
Examples
IV. Counters
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6
00 01 11 10
0000 0001 0011 0010 0110 0100
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Sequential logic networks
IV. Counters
Counters are often implemented by Flip flops. They are – synchronous if all flip flops are clocked by the same signal– ripple (asynchronous) individual flip flop are clocked at different
times Counters may be classified by other characteristics:
– mod N counter or divide-by- N counter, if counter has N distinct states (State = a number of the counted sequence)
– by the number of fli flops in the counter: n bit counter– Other types of counter:
binary up (or down) counter : successive states represent an increasing binary count
00 --> 01 --> 10 --> 11 --> 00 …….. gray code binary counter
00 --> 01 ---> 11 ---> 10 ---> 00
Types of counters
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Sequential logic networks
IV. Counters
Problem Statement
Design a sequential device to generate the sequence 0, 1, 2, 3 over and over again
There are 4 distinct states (divide-by-4) counter Encode the four states as follows;
0 encoded by 00
1 encoded by 01
2 encoded by 10
3 encoded by 11 Represent each binary bit of a code by a flip flop (in this example, let
us use JK flip flops to design the counter)
Intuitive Design of a counter
00 01 10 11
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Sequential logic networks
IV. Counters
Flip flop 0 changes state at every clock pulse
Flip flop 1 changes states every two clock pulses
Intuitive Design of a counter00
01
10
11
00
00
01
10
11
Flip flop 1 Flip flop 0
State transition flip flop 0 State transition flip flop 0
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Sequential logic networks
IV. Counters
Design using JK flip flops for states 0 and 1 of the counterIntuitive Design of a counter
J
K
Q
QN
CK
J
K
Q
QN
CK
1
1
EN
S0S1
S1S0 : 00 --> 01 --> 10 --> 11 --> 00 …….
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Sequential logic networks
IV. Counters
Design using JK flip flopsIntuitive Design of a 4 bit binary counter
S3S2S1S0 : 0000 --> 0001 --> 0010 --> 0011 --> 0100 --> 0101 … --> 1111 --> 0000
There are 16 states design requires four flip flops Synchronous design, all flip flops clocked by the same signal
S0 Changes state (toggles) every clock pulse S1 Changes state (toggles) when S0 = 1 S2 Changes state (toggles) when S1=1 and S0 = 1 S3 Changes state (toggles) when S2=1, S1=1 and S0=1
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Sequential logic networks
IV. Counters
Design using JK flip flopsIntuitive Design of a 4 bit binary counter
J
K
Q
QN
CK
J
K
Q
QN
CK
J
K
Q
QN
CK
J
K
Q
QN
CK
EN
S3 S2 S1 S0
1
1
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Sequential logic networks
State-machine structure (Mealy)
typically edge-triggered D flip-flops
output depends onstate and input
V. Sequential network design
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Sequential logic networks
State-machine structure (Moore)
output dependson state only
typically edge-triggered D flip-flops
V. Sequential network design
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Sequential logic networks
Q
Q’C
D Q
Q’C
S
R
Q
Q’C
J
K
Characteristic Table
S R q Q0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1
0 1 0 0 1 1 -- --
J K q Q0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1
0 1 0 0 1 1 1 0
D q Q0 0 0 1 1 0 1 1
0 0 1 1
Characteristic Equation Q = D
SRq 00 01 11 10
0
1 1
d
d 1
1
Q = S + R’q
JKq 00 01 11 10
0
1 1
1
1
1
Q = Jq’ + K’q
Transition Table(Excitation Table)
q Q D0 0 0 1 1 0 1 1
0 1 0 1
q Q S R0 0 0 1 1 0 1 1
0 d 1 0 0 1d 0
q Q J K0 0 0 1 1 0 1 1
0 d 1 d d 1d 0
D Flip flop S-R Flip flop J-K Flip flop
q : Current state
Q : Next state
V. Sequential network design
Flip Flop : summary
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Sequential logic networks
Characteristic table : For each input and state combination, define the next state of the flip flopCharacteristic equation: Define the next state (Q) as a function of current state and input to the flip flopTransition table (excitation table): For each transition type, define the inputs that cause the transition
V. Sequential network designFlip Flop : summary
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Sequential logic networks
Synchronous state machine : Major design steps
Step 1: Start from state diagram or word description
Step 2: Construct a State/Output table
Moore machine: one output per state (one output column)
Mealy machine: One output per state and for each input combination (one output column per input combination)
Step 3: Reduce the number of states in State/output table by removing redundant states (a state is redundant if for the same input combinations) it has the same next state and output as another state.
Step4: Encode the states in binary (for n states, log2n bits are required). Each bit in the code represents a flip flop.
Step5: Substitute corresponding binary codes to states in the State/Output table
Step6: Separate the state table into flip flop next state maps (one map for each bit or flip flop)
Step7: Use the flip flop next state map to derive flip flop excitation maps (this step depends on the type of flip flop used in the design)
Step8: Use the flip flop excitation maps to determine excitation equations for the flip flop (these equations define the input logic of the flip flop)
Step 9: Use the State/Output table to define the output logic circuit
Step10: Draw the circuit, including flip flop, flip flop input circuits and output circuit.
V. Sequential network designMajor design steps