Detailed Example of Single-Purpose Processor Design: The GCD Processor
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Understanding the GCD Algorithm
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Today, we are discussing the GCD or Greatest Common Divisor of two numbers. Can anyone tell me how we can calculate the GCD using Euclid's algorithm?
Isnβt it something about repeatedly subtracting the smaller number from the larger one until you find a zero?
Close! Itβs more efficient through a modulo operation where you compute A mod B and replace A with B and B with the remainder until B becomes zero. Remember this acronym: 'AMBR' for 'A, Mod, B, Replace.'
What happens when B becomes zero?
Great question! When B is zero, A holds the GCD. Thatβs the last non-zero B before halting.
So, is it correct that we are organizing this into an FSMD model?
Exactly! We will translate our algorithm into a Finite State Machine with Datapath, structuring our operations.
To summarize, the steps involve using the modulo operation and replacing the numbers in an iterative loop until one becomes zero.
FSMD Representation
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Let's refer back to our GCD algorithm. What variables do we need to set up in our FSMD?
We need registers for A, B, and the remainder, right?
Yes! Weβll have three registers: `A_reg`, `B_reg`, and `R_reg`. The modulo operation will happen in the datapath for A mod B. Can you remember what else we need?
We also need a comparator to check if B is zero.
Correct! The states involved in our FSMD will be IDLE, LOOP_CHECK, COMPUTE_MODULO, UPDATE_REGISTERS, and DONE. Letβs learn how these states will interact.
Do the states directly control the operations in the datapath?
Yes, they do! Each state corresponds to a particular operation in the datapath, coordinating the control signals for smooth processing. Remember this mnemonic: 'SOPUS' - States Operate to Process in the U.S. (Update & Show).
So, summarizing what we discussed today, we need to identify both registers and states that control the operations in our FSMD structure.
Datapath Construction and Controller Design
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Now that we've mapped out our FSMD, how do we build the datapath?
We should identify the interconnections between our registers and the functional units, right?
Exactly! The output of A_reg and B_reg will go into the modulo unit, and the result goes to R_reg. What about the control signals?
Weβll need signals to load registers and enable the modulo unit.
Spot on! Control signals will orchestrate our operations based on the states we have defined earlier. Letβs think about how all this is synchronized.
How do we generate those control signals?
Great question! We derive Boolean expressions based on our state transitions. Each state will produce specific control signals depending on current conditions. Remember the phrase: 'Control Signals Define Action.'
In summary, our datapath and controller must work in concert, ensuring that every operation executes smoothly according to our FSMD design.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, we dissect the design process of a single-purpose processor dedicated to computing the GCD of two 8-bit numbers. We outline the conversion of an algorithm into a Finite State Machine with Datapath (FSMD) structure, detailing the components required and the operational steps involved in the implementation.
Detailed
Detailed Summary
In this section, we delve into a practical implementation of a single-purpose processor (SPP) engineered specifically to compute the GCD (Greatest Common Divisor) of two 8-bit integers using Euclid's Algorithm. The design process is broken down into a systematic flow that includes:
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Algorithm Definition:
The GCD is calculated using the iterative method where while B is not equal to zero, the remainder of A divided by B is computed and assigned to B, while A adopts the previous B. This iterative algorithm is precise and efficient for the required calculations. -
Conversion to FSMD:
The algorithm is translated into a Finite State Machine with Datapath (FSMD) representation. This includes defining variables and the necessary operations: - Registers:
A_reg: Holds value AB_reg: Holds value BR_reg: Temporary storage for the remainder.
- Functional Units:
- An 8-bit modulo unit for the division operation.
- A comparator for checking if B is zero.
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States of FSM:
IDLE,LOOP_CHECK,COMPUTE_MODULO,UPDATE_REGISTERS,DONE. Each state has specific operations assigned to manage the data flow correctly.
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Datapath Component Identification:
The essential components of the datapath are identified, which include the registers and functional units like the modulo unit and the comparator. The necessary routing and connections between these elements are also established to create a coherent data flow. -
Controller State Diagram:
A state diagram is generated to depict transitions between states based on control signals (likestart_signal,B_is_zero), ensuring sequential control over the datapath operations. -
Control Signals Generation:
Boolean equations are derived for each control signal using the state transitions to facilitate smooth operation of the FSM.
By employing a methodical approach to convert the high-level algorithm into a hardware-specific implementation through FSMD, we exemplify how hardware design can be successfully executed for single-purpose processors in embedded systems.
Key Concepts
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GCD: The process of finding the highest integer that divides two integers without leaving a remainder.
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FSMD: A model that incorporates both the control sequence and the data processing elements within a digital system.
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Control Signals: Commands generated by the controller to manage operations in the datapath.
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Registers: Storage units that hold data temporarily during processing.
Examples & Applications
Using Euclid's algorithm, calculating the GCD of 48 and 18 involves swapping and calculating: 48 mod 18 = 12, then 18 mod 12 = 6, and finally 12 mod 6 = 0, yielding GCD = 6.
In an FSMD, the states involved in a GCD calculation might include IDLE for initialization, LOOP_CHECK to verify B's zero, and COMPUTE_MODULO to perform the actual calculation before updating registers.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
To find a GCD, just divide and see; the last non-zero's the key!
Stories
Once in a numberland, A and B went on a quest. They kept swapping numbers and calculating remainders until they found their common ground, the GCD!
Memory Tools
AMBR - A, Mod, B, Replace; use this for GCD pace.
Acronyms
SOPUS - States Operate to Process in the U.S. (Update & Show) to represent control flow.
Flash Cards
Glossary
- GCD
The Greatest Common Divisor, the largest number that divides two numbers without leaving a remainder.
- Euclid's Algorithm
An efficient method for computing the GCD of two integers by iterative replacement and remainder calculations.
- FSMD
Finite State Machine with Datapath; a model representing the control and data flow in digital systems.
- Modulo Operation
The operation that finds the remainder after division of one number by another.
- Controller
The component in an FSMD that generates control signals based on the current state and inputs.
- Datapath
The collection of registers and functional units that carry out the data processing tasks specified by the controller.
Reference links
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