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FSM Design and Optimization

FSM Design and Optimization. Finite State Machine (FSM) A Digital Circuit, in general, can be subdivided into two parts: Combinational part – A circuit whose output is a function of its current inputs only

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FSM Design and Optimization

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  1. FSM Design and Optimization • Finite State Machine (FSM) • A Digital Circuit, in general, can be subdivided into two parts: • Combinational part – A circuit whose output is a function of its current inputs only • Sequential part – A circuit whose output is a function of its current inputs plus the past inputs [requires memory elements such as latches or flip-flops] • FSM is a mathematical abstraction of a Sequential Circuit • A System - comprising of inputs, outputs, and states while modeling time as discrete instants at which inputs or outputs can change • Synchronous FSM – when states and output transitions are synchronized with a clock (positive or negative edge) • Asynchronous FSM - when states and output transitions can occur at any time in response to input changes * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  2. FSM Design and Optimization • FSM Models • Mealy Model – Contains three components: • State Memory to store the current state S(t) • State Transition Function  to determine the next state S(t+1) depending upon the current state S(t) and the input X(t) • Output Function  which generates the output Y(t) as function of the current state S(t) and the input X(t) * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-01: Mealy Model of FSM Digital Systems-I

  3. FSM Design and Optimization • FSM Models – Cont’d • Moore Model –Similar to Mealy Model except thatOutput Function  which generates the output Y(t) as function of the current state S(t) only. • Both Mealy and Moore Models can be mapped into each other • Mealy Machines usually have fewer state variables (memory elements)- Widely used in Engineering Applications • Moore Machines are simpler to analyze mathematically * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-02: Moore Model of FSM Digital Systems-I

  4. FSM Design and Optimization • FSM Models – Cont’d • A Problem with Mealy Machine (as shown in Fig-01) – Output may have glitches. So, a slightly modified version of Mealy Machine is more commonly used. * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-03: Mealy FSM with Registered Output • All Digital Systems can be viewed as networks of FSMs ? Digital Systems-I

  5. FSM Design and Optimization • FSM Models – Cont’d • Autonomous FSM – Special FSM having no inputs, e.g. LFSR • Communicating FSMs – Two or more FSMs interacting with each other * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-04: Communicating FSMs Digital Systems-I

  6. FSM Design and Optimization • FSM Design Steps • Understand the Specifications • Problem Definition Using State Diagram and/or State Table • State Minimization – Removal of redundant internal states • State Assignment – Assigning binary codes to the states • Determination of State Transition Function and Output Function Equations • Logic Equation Minimization • Design Mapping to a given Technology or Device • Steps 3, 4 and 6 are Optimization Problems –valuable but not necessary steps * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  7. FSM Design and Optimization • FSM Design Steps – Cont’d • Step-1: Understanding the Specifications A Simple Vending Machine Design Example: [a] Accepts 1 or 2 Rupees Coins [b] Delivers a Pak-Cola bottle of drink costing Rupees 3 [c] Provides change where applicable * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-05: A Vending Machine Model Digital Systems-I

  8. FSM Design and Optimization • FSM Design Steps – Cont’d • Step-2: State Diagram Representation • Each State is represented as a circle with output arrows • Next to the arrow, input and outputs are given • For Vending Machine, FSM remains in state S0 until there is some coin, either of Rs. 1 or Rs. 2 inserted. • Upon such an event, depending upon the coin type, it switches to another state • FSM should not activate the Vend / Change driver unless the credit equals or exceed the Rs. 3 • A state transition diagram can be drawn as shown in Fig-07(Next Slide) * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-06: Notation used in State Diagram Representation Digital Systems-I

  9. FSM Design and Optimization • FSM Design Steps – Cont’d • Step-2: State Diagram Representation – Let us Complete it Inputs/Outputs = Rs.2:Rs.1/Vend:Change * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-07: State Diagram Representation of Vending Machine Digital Systems-I

  10. [a] [b] FSM Design and Optimization • FSM Design Steps – Cont’d • Step-3: State Minimization Equivalent States: Two states are said to be equivalent if they have identical next states and outputs. * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-08: State Minimization Step-03 [a] Cyclic State Diagram of VM [b] Reduced FSM for VM Digital Systems-I

  11. FSM Design and Optimization • FSM Design Steps – Cont’d • Step-3: State Minimization – Cont’d • Addition of Invalid State(s) due to State Assignment (Binary Codes) * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-09: Final Reduced FSM for VM Digital Systems-I

  12. FSM Design and Optimization • FSM Design Steps – Cont’d Step-4: State Assignment and State Transition Table * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  13. FSM Design and Optimization • FSM Design Steps – Cont’d Step-4: State Assignment and State Transition Table • Step-5: Determination of Logical Equations * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 ? Digital Systems-I

  14. FSM Design and Optimization • FSM Design Steps – Cont’d • Step-6-7: Simplification of Logic Equations and Hardware Implementation • Use of K Maps or Other Methods • Implementation is Technology Dependent * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-10: Implementation of VM FSM Digital Systems-I

  15. FSM Design and Optimization • FSM Design Example – Huffman Codec • Used for JPEG/MPEG Compression • Relies on known probability of a set of fixed symbols * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Table-04: Symbols with Their Binary Code and Frequency Fig-11: Huffman Tree Developed based on Symbol Frequency Digital Systems-I

  16. FSM Design and Optimization • FSM Design Example – Huffman Codec – Cont’d • Huffman Decoder Circuit Implementation as FSM * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-11: Huffman Decoder FSM State Diagram and FSM Implementation Digital Systems-I

  17. FSM Design and Optimization • FSM Optimization • Three Ways to Optimize the HW Complexity of FSM • State Minimization • State Assignment • Logic Equation Minimization • State Minimization Methods • State Merging by Observation • State Partitioning • Application of Implication Tables • State Merging by Observation • Vending Machine Example • Bit Sequence Detector * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  18. FSM Design and Optimization • FSM Optimization – Cont’d • State Merging by Observation • Bit Sequence Detector – A Circuit that generates an output Z = 1 when it detects a bit sequence from a serial data input D as 001, 010, 100, or 111. • S3 and S6 are Equivalent, and so are S4 and S5. Eliminate S5 and S6 * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-12: Bit Sequence Detector [a] State Diagram [b] State Table Digital Systems-I

  19. FSM Design and Optimization • FSM Optimization – Cont’d • State Merging by Observation – Cont’d • Bit Sequence Detector after State Minimization * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-13: Minimal State Bit Sequence Detector [a] Stat Table [b] State Diagram State Partitioning Digital Systems-I

  20. FSM Design and Optimization • FSM Optimization – Cont’d • State Partitioning • An FSM Example • Best Solution for this FSM takes only 5 States? Fig- 14: State Table for FSM of State Partitioning Example * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  21. FSM Design and Optimization • FSM Optimization – Cont’d • State Partitioning – Cont’d • An FSM Example • Step-1: State Partitioning by Outputs – Divide the states into sets with identical outputs • Step-2: State Partitioning with Next States – For states in each set, find their next states separately * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  22. FSM Design and Optimization • FSM Optimization – Cont’d • State Partitioning – Cont’d • An FSM Example • Step-3: Repartitioning based on Next States – After Step-2, two things have happened: next state group for C (input = 0) now belongs to B2, however, next state group for A (input = 1) now belongs to no single state group, so, A partition has become invalid * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 NOW all the next state groupings belong to some single state partition/group. WHAT is Final Partitioning? Digital Systems-I

  23. FSM Design and Optimization • FSM Optimization – Cont’d • State Partitioning – Cont’d • An FSM Example • Finally We got a State Partitioning Where Next outputs are the same for each state in the same state partition/group AND Next states are the same for each state in the same set/group * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Final Optimized FSM has got only Five States……………….! Digital Systems-I

  24. FSM Design and Optimization • FSM Optimization – Cont’d Self-study Exercise:Application of Implication Table: Easy to Computerize and Suitable for Larger FSM Optimization * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 State Assignment ASM Chart FSM Synthesis Next Week: Digital Systems-I

  25. FSM Design and Optimization • FSM Optimization – Cont’d • State Assignment • Assigning Unique Binary Codes to the States of a Minimized FSM • State Minimization has a Unique Technology-Independent Solution, however, State Assignment Depends on • Technology such as PLA, ROM, PAL, logic gates • Type of storage circuit, D-latches or FF • For a FSM of r Rows (States), with n-bit State Variables, All Possible Permutations are N = 2n ! / (2n-r)! • Many, among above Assignments, are just Rearrangements, according to McCluskey, Number of Distinct Assignments is Reduced to ND = (2n -1)! / (2n-r)!* n! * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  26. FSM Design and Optimization • FSM Optimization – Cont’d • State Assignment – Cont’d • Even the number given by McCluskey is still very large • Very Complex Problem, called Intractable or np-Complete. Such a problem usually have no optimal solution but some solution based on heuristics (thumb rules or simple rules) • Aim here would be to have Rules that provide maximum number of 1’s in adjacent cells of next-state truth table for better k-map reduction * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  27. FSM Design and Optimization • FSM Optimization – Cont’d • State Assignment – Cont’d • Rule-1: States with the same next state for a given input condition should be assigned codes differing in one (binary) bit position only. For Example, • Rule-2: Next States of a single state should be given logically adjacent state assignments. For Example, * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  28. FSM Design and Optimization • FSM Optimization – Cont’d • State Assignment – Cont’d Example-01: Consider the Bit Sequence Detector FSM • Applying Rule 2, S1 and S2 should be assigned logically adjacent codes, so, let S1 = 100 and S2 = 101 • Applying Rule 1, S3 and S4 both have the same next state with given input condition, so, S3 and S4 are assigned logically adjacent codes. S3 = 110 and S4 = 111 • S0 can be assigned 000 (arbitrary), and unassigned states would be 010, 011, and 001 * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-15: Bit Sequence Detector FSM Digital Systems-I

  29. FSM Design and Optimization • FSM Optimization – Cont’d • State Assignment – Cont’d • One-Hot State Assignment • Sometimes, instead of log2 r bi-stable latches, it is more efficient (and convenient as well ) to have r latches/flip-flops, i.e. one for each state. It is called One-Hot State Assignment. • At any time, only one FF will be set (FF corresponding to the state where FSM lies at that instant) • No State Assignment is required • One-hot state assignment is particularly suitable for FPGA (LUT and MUX based Architecture) implementation of FSM • Number of FF required is much higher than Min. Length State Encoding • Slower in Operation as compared to other option. * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  30. FSM Design and Optimization • FSM Implementation- HW Considerations • Implementation Alternatives • Standard ICs – Suitable for Simple Designs • PROM –Suitable for many Outputs/States, No Logic Minimization needed, Exhaustive Implementation for all Possible Input Combinations, Size grows Exponentially • CPLDs/FPGAs –More Suitable for most of FSM Implementations Fig-16: Generic Block Diagram of FSM * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-17: Implementation of FSM with PROM Digital Systems-I

  31. FSM Design and Optimization • FSM Implementation- HW Considerations – Cont’d • Asynchronous Inputs – A Possible Source of Race Condition • Asynchronous input “A” to FSM, while making transition from “0” to “1”, as shown above may give rise to a wrong state transition • SOLUTION: Synchronize all the Asynchronous Inputs to FSM using a Latch clocked by the FSM clock Fig-18: Asynchronous Inputs to FSM * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-19: Synchronizing Asynchronous Inputs Digital Systems-I

  32. FSM Design and Optimization • FSM Implementation- HW Considerations – Cont’d • Types of Flip-Flops at Output • Outputs of Programmable Macro-Cells or LEs of CPLDs/FPGAs are Configurable • Inverting/Non-Inverting • Register or Combinational • D Flip-Flop, S-R FF, J-K FF, or T-FF, any type is Possible • T-FF or J-K FF can Produce more Efficient Implementation (fewer product terms in Boolean Equations) • Better CAD toolsmake better choice automatically * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  33. [a] [b] FSM Design and Optimization • Algorithmic State Machine (ASM) Chart • An Alternative Method to Represent FSM based on Flow-Chart Notation – Popularized by Christopher Clare “ Designing Logic Systems Using State Machines” Key Features of ASM: • FSM is in one State Block per state time (Clock Cycle) • Single Entry Point for each State Block • For each combination of inputs, only one unambiguous exit path • Outputs asserted high, low, high-impedance until the next clock cycle * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-20: ASM Chart [a] ASM Elements [b] An ASM Block Digital Systems-I

  34. FSM Design and Optimization • Algorithmic State Machine (ASM) Chart – Cont’d • ASM Construction Rules Must Follow these Rules: • Each State can have one and only one State Box • Outputs depending on the Current State only (Moore Model) are represented by Square Box • Outputs depending on the Inputs (and of course the Current State), as in Mealy Model, are represented by Rounded Box • Decision Box contains the Conditions for the Input Variables * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-21: ASM Multi-Way Decision Block Simplification Digital Systems-I

  35. FSM Design and Optimization • Algorithmic State Machine (ASM) Chart – Cont’d • ASM Advantages over (Bubble) State Diagram • ASM Chart reflects HW Algorithm better than (Bubble) State Diagram Representation of FSM • Easier to Follow and Understand • ASM Chart avoids Transition Conflicts that could Occur in State Diagram Representation of FSM EXAMPLE: Inputs I3I2I1I0 = 1101, 1011, and 1111 all will make both transitions to be True. * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Fig-22: Possible Conflicts in State Diagram Representation of an FSM Digital Systems-I

  36. Rs.0 Rs.2 Rs.2 Rs.1 Rs.1 Rs.2 Rs.1 Rs.1 Rs.2 FSM Design and Optimization • Algorithmic State Machine (ASM) Chart – Cont’d • ASM Representation of Vending Machine * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 [a] [b] Fig-23: Mealy Model of Vending Machine [a] State Diagram [b] ASM Chart Digital Systems-I

  37. FSM Design and Optimization • FSM Design Using Verilog HDL • Mealy FSM and Its RTL Coding Fig-24: Mealy FSM to be Coded in Verilog HDL * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  38. FSM Design and Optimization • FSM Design Using Verilog HDL – Cont’d • Mealy FSM and Its RTL Coding – Cont’d * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 …Cont’d on Next Slide Digital Systems-I

  39. FSM Design and Optimization • FSM Design Using Verilog HDL – Cont’d • Mealy FSM and Its RTL Coding – Cont’d …Cont’d from Prev. Slide * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 …Cont’d on Next Slide Digital Systems-I

  40. FSM Design and Optimization • FSM Design Using Verilog HDL – Cont’d • Mealy FSM and Its RTL Coding – Cont’d …Cont’d from Prev. Slide * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 …Cont’d on Next Slide Digital Systems-I

  41. FSM Design and Optimization • FSM Design Using Verilog HDL – Cont’d • Mealy FSM and Its RTL Coding – Cont’d …Cont’d from Prev. Slide * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

  42. FSM Design and Optimization • FSM Design Using Verilog HDL – Cont’d • Moore FSM and Its RTL Coding * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 …Cont’d on Next Slide Digital Systems-I

  43. FSM Design and Optimization • FSM Design Using Verilog HDL – Cont’d • Moore FSM and Its RTL Coding …Cont’d from Prev. Slide * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 …Cont’d on Next Slide Digital Systems-I

  44. FSM Design and Optimization • FSM Design Using Verilog HDL – Cont’d • Moore FSM and Its RTL Coding …Cont’d from Prev. Slide * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 …Cont’d on Next Slide Digital Systems-I

  45. FSM Design and Optimization • FSM Design Using Verilog HDL – Cont’d • Moore FSM and Its RTL Coding …Cont’d from Prev. Slide * Chapter # 5 and Peter Cheung Lecture Notes-DSD-06 Digital Systems-I

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