Implementation of an Alternative High Performance Flip Flop to JK Flip Flop

Implementation of an Alternative High Performance Flip Flop to JK Flip Flop

1Ogunlere Samson, 2Ajaegbu Chigozirim, 3Kanu Richmond, 4Somefun Olawale

1234Computer Science Department, Babcock University, Ilishan-Remo, Nigeria

Abstract

– Computer performance is primarily affected by the processor and memory. If either one reaches its limits, the performance of the whole system degrades. As semiconductor technology advances, the performance gap between processor and memory has become one of the major issues in computer design. This situation causes a growing gap between processor and memory in the performance. Hence the design of an efficient and high performance memory element known as Flip-Flop Extension is of crucial importance in computer design. The analysis of the existing structures is necessary when the requirement of the Flip-Flop is for low-power and high- speed digital applications. This study is dedicated to the investigation of the existing conventional memory elements, the SET /RESET (SR) and JUMP-KEY (JK) Flip Flops

performance and the result is used to evaluate the design of an alternative Flip Flop known as Flip Flop Extension. It is evidence from the analysis, that the JK-FF Extension at 87.5% has one gate less than the conventional SR & JK-FFs. This is a great advantage in performance because fewer gates enhance performance in which numbers of gate/ transistors represent hardware cost.

Key Words: Conventional Flip Flop, Flip Flop Extension, Boolean Equation, K-Map, DeMorgans Theorem.

1. INTRODUCTION

   Flip-Flops are digital circuits with two stable, self- maintaining states that are used as storage/ memory elements such as Random Access Memory (RAM), Caches Memory and Read Only Memory (ROM). They are also very useful in the following electronic digital devices design; Sequence Detector, Data Synchronizer, Frequency Divider, Registers (data transfer), Counters and Registers in Central Processing Unit (CPU) for data transfer. They are derived from Sequential Logic Circuits which are the main electronics circuits that make the development of computers possible. The ability of computer systems to operate without the continuous human intervention is solely achieved through sequential logic circuits, the building blocks of Flip Flops [1].

2. MOTIVATION FOR THE STUDY

   It has been observed that computer performance is primarily affected by the processor and memory. If either one reaches its limits (which may initially be the memory), the performance of the whole system degrades. As semiconductor technology advances, the performance gap between processor – the Central Processing Unit (CPU) and main memory – the Random Access Memory (RAM) has become one of the major issues in computer design. In the

   past 45 years, an exponential rate of improvement has been witnessed in semiconductor technology. The processor performance increases at a rate of 60% per year while the memory performance increases just 10% per year [2; 3]. This situation causes a 50% growing gap between processor and memory in the performance as depicted in Figure 1.1. If memory fails to keep pace with the processors constant demands, the processor stalls in a wait state, and valuable processing time is lost. This imbalance has become one major bottleneck in further improving the computer performance.

   One reason memory system performance has consistently lagged processor performance is that memory systems typically consist of one or more chips that are designed and manufactured separately from the processor, and the performance of the interconnected multi-chip memory system is difficult to scale to achieve higher data-rate and lower access latency. Memory system data-rates are increasing with each new generation of memory devices at the rate of 100% every three years, and memory row cycle times are decreasing at a rate of approximately 7% per year [4]. The collective trends are increasing the ratio of row cycle times to the duration of data bursts on the data bus. This is why it is imperative to critically evaluate the existing conventional JK-FF and the need to bridge the speed gap between memory and processor by enhancing the memory speed through logical modification frameworks of the conventional JK-FF which utilizes 75% out of the 100% of its states.

   Figure 1.1: Processor and DRAM Memory Gap [2; 3]

3. DESIGNING SR AND JK-FLIP FLOPS

   We start by designing JK-FF from first principle Set and Reset Flip Flop (SR-FF) using NOR and NAND gate Configurations. When the forbidden states of an SR-Flip Flop (as shown in Table 1.1) are converted to toggling states,

   a JK-Flip Flop is so obtained. Hence, such a JK-Flip Flop retains the rest features of an SR-Flip Flop, such as its resting and active states; thus making a JK-Flip Flop to attain 75% active states utilization as against 50% active states utilization of an SR-Flip Flop on conversion of SR-FF to JK- FF. The remaining 25% rest state of SR-FF is being examined in this study.

   3.1. Design Logic Circuit of an SR-Flip Flop

   The Truth Table of Table 1.1 is converted into a K-Map in order to obtain the minimized logic equations of the SR-Flip Flop as depicted in Table 1.2.

   Logic equations (1a, 1b) & (1.1a, 1.2b) are derived from the K-Map and they can be used to construct the Flip Flop using the different gates as given by the equations. However, it is customary to use NOR and NAND gates to construct logic equations (1a, 1b) and (1.1a, 1.2b) respectively by converting the equations into these gates. The mathematical analysis of these equations as ascertained in [12] & [13] is as follows:

   Fig. 1.2(b): SR NAND gates Flip Flop Equation Omotosho & Ogunlere (2013)

4. PROPOSED FLIP FLOP EXTENSION

   Possibilities of alternative Flip Flops, to conventional JK- Flip Flop are proposed. These different JK-Flip Flops are tagged as highlighted in Table 1.2, and are stated according to our adopted convention as follows:

   Table 1.2: Proposed Flip Flop Extension Concept

   + = + ()

   = + (1)

   Put equation (1c) into (1a), we have

   +1 = + + (1)

   Complement equation (1d), we have

   Option 1: 87.5% utilization Flip Flops identified as JK- FF Extension 0; (for Rest 1 0); One state at rest.

   + +

   + +

   + =

   ()

   Option 2: 87.5% utilization Flip Flops identified as

   JK-

   + = + ()

   = + (1)

   FF Extension 1; (for Rest 0 1); One state at rest.

   Put equations (1f) into (1b), we have

   +1 = + + (1)

   Complement equation (1g), we have

   + = + + () Combining equations (1e) and (1h) results in the construction of SR-FF Logic Circuit Diagram using only NOR gates as shown in Figure 1.2(a). Similarly the same procedure is followed to obtain the logic circuit diagram of SR-FF using NAND Gates as shown in Figure 1.2(b).

   Fig. 1.2(a): SR NOR gates Flip Flop Equation Omotosho & Ogunlere (2013)

   Option 3: 100% utilization Flip Flops identified as XY- FF – No Rest States.

   The Truth Table of these possible options to conventional JK-Flip Flop tagged JK-FF Extensions is presented in Tables 1.3 and 1.4 with reference to Conventional JK-FF from where their simplified Boolean equations are derived.

   Q

   n +1

   Substitute equations (2.2b) & (2.2c) into equation (2.2), we have

5. MATHEMATICAL ANALYSIS OF JK-FF DESIGN

   The conversion of SR-FF to JK-FF as a Positive Logic Design is mathematically analyzed as follows using DeMorgans theorem, Boolean algebra rules and K-Map technique.

   5.1 JK-000, 001 Rest: Conventional JK Flip Flop at 75% active states utilization

   From K-map on JK Flip Flop at 75% active states utilization, the simplified equation of 2.1 is derived using NOR gate configuration as Positive Logic design.

   +1 = + (2.1)

   NAND GATES (1, 2, 3)

   +1 = + + + (2.2)

   + + +

   + + +

   + = (. )

   Combining equations (2.1a) and (2.2a) will produce NAND gate configuration of Figure 1.2a. Combining equations (2.1e) and (2.2e) will produce NOR gate configuration of Figure 1.2b.

   Figure 1.2 (a): Logic Circuit of Conventional JK-Flip Flop – 75% (NAND Gate Configuration.)

   Therefore, equations (2.1e) & (2.2e) can be combined as presented in Figure 1.2b

   + = + =

   (. )

   NOR GATES (1, 2, 3)

   .

   Figure 1.2 (b): Logic Circuit of Conventional JK-Flip Flop – 75% (NOR Gate Configuration)

   = + (2.1)

   Note that the positive logic design configuration is actually

   = + (2.1)

   the complement of the negative logic design configuration;

   Substitute equations (2.1b) & (2.1c) into equation (2.1), we have

   = + + + (2.1)

   which shows that either design will produce the same performance.

6. USING CHARACTERISTIC EQUATIONS TO SIMPLIFIED JK-FLIP FLOP DESIGN

   +1

   + + +

   + + +

   + = (. )

   +1 = + (2.2)

   NAND GATES (4, 5, 6)

   The complete Logic Circuit Diagrams based on the K-map analysis and the characteristic equations derived from the state diagram Excitation Table are constructed in Figures 1.2(c) and 1.2(d) respectively. These two circuit diagrams perform the same memory element functions.

   + = + =

   . . (. )

   NOR GATES (4, 5, 6)

   .

   = + (2.2)

   Equations 2.1 and 2.2 are used to produce the circuit diagram of Figure 1.2(d) which is the equivalent of circuit diagram in

   = + (2.2)

   Figure 1.2(c).

   Figure 1.2(c) Figure 1.2(d) Figure 1.2(c): NOR gates JK-FF based on K-map

   Figure 1.2(d): NOT, AND & OR gates JK-FF based on characteristic equations

7. OPTION 1: DESIGNING FLIP FLOP EXTENSION

   0 AT 87.5% (REST 1 0)

   +1 = + = ( + ) (3.1)

   NAND GATES (1, 2, 3)

   + = + = (. )

   Figure 2.1(a): Logic Circuit of JK-FF Extension 0 at 87.5% (NAND Gate Configuration)

   The complete Logic Circuit Diagrams based on the K-map analysis and the characteristic equations of 3.1 and 3.2 as derived from the state diagram excitation table are constructed in Figures 2.2(b) and 2.2(c) respectively. These two circuit diagrams perform the same memory element functions.

   NOR GATES

   .

   + = ( + ) =

   (

   ) (. )

   + +

   +1 = + (3.2)

   NAND GATES (4, 5)

   + = + = . (. )

   NOR GATES

   Figure 2.2(b) Figure 2.2(c)

   Figure 2.2(b): JK-FF Extension-0 at 87.5% using NAND gates based on K- map

   Figure 2.2(c): JK-FF Extension-0 at 87.5% using NOT, AND & OR gates based on characteristic equations

   = + (3.2)

   Substitute equation (3.2b) into equation (3.2), we have

   +1 = + + (3.2)

8. OPTION 2: DESIGNING FLIP FLOP EXTENSION 1 AT 87.5% (REST 0 1)

   +1

   = +

   = ( + ) (4.1)

   + =

   (. )

   NAND (1, 2, 3)

   + +

   Combining equations (3.1a) & (3.2a) to produce NAND gate configuration of Figure 2.1a; while the combination of equations (3.1b) & (3.2e) to produce NOR gate

   +

   = + = .

   (. )

   configuration is not feasible because equation (3.1b) = equation (3.2e).

   NOR

   + = ( + ) = + ( +

   ) (. )

   +1 = + (4.2)

   NAND (4, 5)

   + = + = (. )

9. OPTION 3: FLIP FLOP EXTENSION; NO REST

   NOR

   .

   STATES (100%) SUBSTITUTE J = X AND K = Y

   +1 = Y + + Y

   = + (4.2)

   = ( Y + ) + Y (5.1)

   Substitute equation (4.2b) into equation (4.2), we have

   +1

   = + + (4.2)

   = + + (. )

   NAND GATES (1, 2, 3, 4)

   +1 = Y + + Y

   +

   Combining equations (4.2a) & (4.1a) to produce NAND gate

   = Y . . Y

   (5.1)

   configuration of Figure 3.1a; while the combination of equations (4.1b) & (4.2e) to produce NOR gate configuration is not feasible because equation 4.1b = 4.2e.

   + == . . Y (. )

   NORGATES (4, 6, 7, 8)

   Replace

   = + (5.1)

   = X + (5.1)

   Figure 3.1(a): Logic Circuit of JK-FF Extension 1 at 87.5% (NAND Gate Configuration)

   = X + (5.1) Substitute equations (5.1b), (5.1c) & (5.1d) into equation (5.1a), we have

   The complete Logic Circuit Diagrams based on the K-map analysis and the characteristic equations of 4.2 and 4.1 as derived from the state diagram excitation Table are constructed in Figures 3.1(b) and 3.1(c) respectively. These two circuit diagrams perform the same memory element functions.

   + = X + + X + + + (. )

   + = X + + X + + + (. )

   +1 = + +

   = ( + ) + (5.2)

   NAND GATES (5, 6, 7, 8)

   + = + +

   = . . X . . (. )

   NOR GATES (1, 2, 3, 4)

   Replace

   = +

   (5.2)

   = + (5.2)

   = + (5.2)

   Figure 3.1(b) Figure 3.1(c)

   Figure 3.1(b): JK-FF Extension-1 at 87.5% using gates NAND based on

   Substitute equations (5.2b), (5.2c) & (5.2d) into equation (5.2), we have

   + = + + + + + (. )

   K-map

   + + + + +

   + + + + +

   Figure 3.1(c): JK-FF Extension-1 at 87.5% using NOT, AND & OR

   +

   = (. )

   gates based on characteristic equations

   Combining equations (5.1a) & (5.2a) will produce NAND gate configuration of Figure 4.1a; while combining

   equations (5.1f) &(5.2f) will produce NOR gate configuration of Figure 4.1b

   Figure 4.1(a): Logic Circuit of XY-FF No Rest at 100% (NAND Gate Configuration)

   Figure 4.1(b): Logic Circuit of XY-FF No Rest at 100% (NOR Gate Configuration)

   The complete Logic Circuit Diagrams based on the K-map analysis and the characteristic equations of 5.1 and 5.2 as derived from the state diagram excitation Table are constructed in Figures 4.1(c) and 4.1(d) respectively. These two circuit diagrams perform the same memory element functions.

   Figure 4.1 (c) Figure 4.1 (d)

   Figure 4.1(c): XY-FF Extension at 100% using NOR gates based on K- map

   Figure 4.1(d): XY-FF Extension 100% using NOT, AND & OR gates based on characteristic equations

10. SUMMARY OF THE VARIOUS FLIP FLOPS

    DESIGN

    Illustrated in Table 1.5 is the summary of the various Flip Flops design with the aim of verifying their gate structures and numbers.

11. CONCLUSIONS

Previous study revealed that very little research have been carried out on Flip Flops comparative analysis in the 100%, 87.5% and 75% active states utilization on digital device applications. In this paper, a new approach of designing memory element (Flip Flop) with its active states utilization of 87.5% and or 100% as against the conventional Flip Flops at 50% and 75% active states utilization which will enhance memory performance have been developed. This is evidence in section 7 and 8 where the Flip Flop Extensions at 87.5% active states utilization is designed with one gate less than the conventional JK-Flip Flop. The uniqueness of this study is that computer memory speed performance can be enhanced through conventional JK-FF modification just as it is currently being done with its processor counterpart. This is a great advantage in performance over the conventional Flip Flops because fewer gates enhance performance (i.e., gate delay represents performance). The 87.5% Flip Flop extension memory cell is also portable (less transistors) and cheaper because it requires fewer transistors as against the conventional Flip Flops. An important issue in digital device design is that numbers of transistors represent hardware cost because in essence, maximizing performance and minimising cost in digital devices are part of the factors in seeking alternative design on more efficient and effective Flip flops.

Efforts should be geared towards verifying the effectiveness and efficiency of these newly design Flip Flops Extension over the existing conventional Flip Flops.

REFERENCES

1. O. J. Omotosho, Fundamentals of Digital Systems, Franco-Ola publishers, 2012

2. Hennessy, J. L. & David A. P., (2007), Computer Architecture: A Quantitative Approach. 4th ed., p. 289. Elsevier Inc.

3. McCalpin, J. D., (2016), Memory Bandwidth and System Balance in HPC Systems, Invited Talk, International Conference for High Performance Computing, Networking, Storage, and Analysis. Available from: http://sites.utexas.edu/jdm4372/2016/11/22/sc16.

4. C. Kim, Memory World in the Next Decade, Memory Division, Device Solution Network Business, Samsung, Seminar

   presentation to Po-hang University of Science and Technology (POSTECH), Kyungpook, Korea,(2003

5. O. J. Omotosho and S. O. Ogunlere, Design Analysis and Circuit Enhancements of SR-Flip Flop, International Journal of Engineering Sciences and research Technology (IJESRT), ISSN: 2277-9655, 2013

6. O. J. Omotosho and S. O. Ogunlere, Analysis and Design of Different Flip Flops, Extension of Conventional JK-Flip Flops, International Journal of Engineering Sciences and research

Technology (IJESRT), ISSN: 2277-9655, 2013