Design and Minimization of Reversible Circuits for a Data Acquisition and Storage System

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1 International Journal of Engineering and Technology Volume 2 No. 1, January, 2012 Design and Minimization of Reversible ircuits for a Data Acquisition and Storage System 1 Lafifa Jamal, 2 Farah Sharmin, 3 Md. Abdul Mottalib, 4 Hafiz Md. Hasan Babu 1,2,4 Department. of omputer Science & Engineering, University of Dhaka, Dhaka-1000, Bangladesh 3 Department of omputer Science and Information Technology Islamic University of Technology, Gazipur-1704, Dhaka, Bangladesh ABSTRAT Reducing power dissipation is the ultimate objective in the world of VLSI circuit design. onventional logic dissipates more power by losing bits of information whereas reversibility recovers bit loss from the unique input-output mapping. Thus reversible logic has become immensely popular research area and its applications have spread in various technologies. In this paper we have proposed the compact design of reversible circuits for a data acquisition and storage system. The design comprises with a compact reversible analog-to- digital converter and a reversible address register. In the way of designing this data acquisition and storage system we have proposed a reversible J-K flip-flop with asynchronous inputs, a reversible D flipflop and a reversible three state buffer register. All the reversible designs individually have less number of gates, garbage outputs and quantum cost compared with the existing ones and have outperformed those described in the literature. Moreover we have proposed some lower bounds for designing these reversible components of the compact data acquisition and storage system. Key words: Reversible Logic, Garbage Output, Flip-Flop, Quantum ost 1. INTRODUTION Reversible logic has spread its popularity in numerous technologies such as low power MOS design, optical information processing, DNA computing, bioinformatics, quantum computing[1],thermodynamics and nanotechnology over the last couple of years[2, 3, 4]. In the early 1960s R. Landauer demonstrated that losing bits of information causes loss of energy [5]. It is proved that the loss of each bit of information loses at least KT ln2 joules of energy where K is the Boltzmann s constant and T is the temperature at which the system is operating [5]. Information is lost when an input cannot be recovered from its output or vice-versa. Reversible logic has the feature to generate one to one correspondence between its input and output [6,7,8]. As a result no information is lost in reversible logic and zero power dissipation would be achieved only if the network consists of reversible gates [9]. Synthesis of reversible logic is more complicated than irreversible one as it imposes many design constraints [6]. A reversible circuit therefore should have the following attributes [7]: Since garbage outputs are not used as primary outputs so any realization technique should keep garbage as minimum as possible. Each reversible gate has a particular quantum cost so any realization technique should keep the number of reversible gates as minimum as possible. Input lines that are either 0 or 1 known as constant inputs should be as minimum as possible. A reversible data acquisition and storage system consists of a reversible analog-to digital converter which will convert analog signal into digital form, a reversible Random Access Memory (RAM) which is a two dimensional array of memory cells and a reversible synchronous counter which will point to the consecutive locations of memory. With the help of theorems and comparisons the efficiency of reversible logic synthesis of data acquisition and storage system has also been proved in this paper. The paper consists of the following sections: Section 2 describes about the reversible gate, some basic definition and quantum realization of some reversible circuits. Section 3 introduces the logic synthesis of reversible data acquisition and storage system and comparison with other existing researches. Finally the paper is concluded with the section 4. ISSN: IJET Publications UK. All rights reserved. 9

2 2. BAKGROUND In this section we have presented some definitions and basic idea on reversible logic for our future reference. Quantum realization of some popular reversible circuits has also been illustrated in this section. 2.1 Reversible Gate A reversible gate is an n n data stripe block which uniquely maps between input vector I v = ( I 0, I 1,..., I n ) and output vector O v = ( O 0, O 1,..., O n ) denoted as I v O v. means that every gate in the given circuit will take same amount of time for internal logic operations. All inputs to the circuit are known before the computation begins. Which means that the internal structure and each operation of the gate is known before the calculation. From the above definition, the delay of the logic circuit of Fig. 2 which consists of only one gate is obviously 1 as this is the only gate from its input to output line. 2.5 Fault Tolerant Gate A fault tolerant Gate is a reversible gate which constantly preserves same parity between input and output, more specifically an n n fault tolerant gate holds the following property: I 0 I 1 I 2... I n-1 = O 0 O 1 O 2... O n-1 (1) 2.2 Garbage Output Fig. 1: A k k Reversible Gate Every gate output that is not used as input to other gates or as a primary output is known as garbage. where I 0, I 1,..., I n-1 are input vectors and O 0, O 1,..., O n-1 are output vectors. The circuit which consists of all fault tolerant gates must preserve the parity pattern in its input and output vectors. 2.6 Feynman Double Gate Feynman Double Gate, (F2G) [12] respectively shown in Fig. 3, where I v = (A, B, ) and O v = (P=A, Q= A B, R= A ). The quantum cost of Feynman Double gate is 2 [11, 13] which has been shown is Fig. 4. Fig. 2: A Reversible Gate with One Garbage Output 2.3 Quantum ost Every quantum circuit is built from 1 1 and 2 2 quantum primitives and its cost is calculated as a total sum of 2 2 gates used since 1 1 gate costs nothing i.e. zero. Basically the quantum primitives are matrix operation which is applied on qubits state. All the gates of the form 2 2 has equal quantum cost and the cost is unity i.e. 1 [11]. Since every reversible gate is a combination of 1 1 or 2 2 quantum gate, therefore the quantum cost of a reversible circuit calculates the total number of 2 2 gates used. The quantum cost of Feynman gate in Fig. 2 is 1 and the quantum cost of Feynman Double gate in Fig. 3 is Delay The delay of a logic circuit is the maximum number of gates in a path from any input line to any output line. This definition is based on the following two assumptions [11]: Each gate performs the computation in one unit time. This Fig. 3: A 3 3 Reversible Feynman Double Gate Fig. 4: Quantum Realization of 3 3 Reversible Feynman Double Gate 2.7 Toffoli Gate Toffoli Gate, (TG) [14] respectively shown in Fig. 5, where I v = (A, B, ) and O v = (P = A, Q = B, R = AB ). In Fig. 6, V is the square root of NOT gate and V + is its hermitian. Thus VV + =I (an identity matrix, describing just a quantum wire). The quantum cost of Toffoli gate is therefore 5 [11,13]. ISSN: IJET Publications UK. All rights reserved. 10

3 Fig. 5: A 3 3 Reversible Toffoli Gate Fig. 9: A 3 3 Reversible Peres Gate Fig. 6: Quantum Realization of 3 3 Reversible Toffoli Gate 2.8 Fredkin Gate Fredkin Gate (FRG) [15] respectively shown in Fig. 7, where I v = (A, B, ) and O v = (P = A, Q = A'B A, R = A' AB). Quantum equivalent circuit of FRG has been shown in Fig. 8. Each dotted rectangle in this circuit is equivalent to one 2 2 FG and so the cost is 1 for that particular case. Thus the quantum cost of Fredkin gate turns out to be 5 like Toffoli gate [11]. Fig. 7: A 3 3 Reversible Fredkin Gate Fig. 8: Quantum Realization of 3 3 Reversible Fredkin Gate 2.9 Peres Gate Peres Gate (PG) [16] respectively shown in Fig. 9, where I v = (A, B, ) and O v = (P = A, Q = A B, R = AB ). The quantum cost of Peres gate is 4 [16] which has been shown is Fig. 10. Fig. 10: Quantum Realization of 3 3 Reversible Peres Gate 3. PROPOSED DESIGN OF REVERSIBLE DATA AQUISITION AND STORAGE SYSTEM In this section we have proposed a compact design of a data acquisition and storage system. In the process of proposing this reversible architecture we have proposed a reversible priority encoder for the Analog-to-Digital onverter in section 3.1, a reversible J-K flip-flop with two asynchronous inputs in section 3.2, a reversible D flipflop in section 3.3, a reversible synchronous address counter and finally a tri-state buffer register in section 3.4 and 3.5 respectively. 3.1 Proposed Design of Reversible Priority Encoder A 3-bit flash AD is known as parallel A/D converter. It is formed of a series of comparators, each one comparing the input signal to a unique reference voltage. The comparator outputs connect to the inputs of a priority encoder circuit, which then produces a binary output. A stable reference voltage V ref is provided by a precision voltage regulator as part of the converter circuit. As the analog input voltage exceeds the reference voltage at each comparator, the comparator outputs will sequentially saturate to a high state. The priority encoder generates a binary number based on the highest-order active input; ignoring all other active inputs. An n bit flash converter would require 2 n -1 comparators, 2 n resistors and the necessary logic. In this section we have proposed the reversible design of 8-to-3 priority encoder which has been shown in Fig. 11. This reversible design consists of 21 reversible gates having quantum cost of 50 and 20 garbage outputs. ISSN: IJET Publications UK. All rights reserved. 11

4 Table 1: Truth Table for a 8-to-3 Priority Encoder V A Q 3 Q 2 Q V X V X V X V X V X V X V X > 7 V X Output: Q 1 = ' 1 2 ' 3 4 ' 5 6 ' 7 Q 2 = ' 2 ' 3 3 ' 4 6 ' 7 7 Q 3 = ' 1 ' 2 ' 3 ' 4 Fig. 11: Proposed Design of Reversible 8 3 Priority Encoder A 2 n n bit reversible priority encoder can be constructed from cascading 8 3 reversible priority encoder, a precharging block for carrying 2 n carry signals on 2 n carry lines which is generated in each n bit input priority encoder unit and a zero detecting block coupled with 2 n n bit reversible priority encoder for detecting low signal levels at the input lines. 3.2 Proposed Design of Reversible J-K Flip-Flop with Two Asynchronous Inputs comparative result of different J-K flip-flops has been shown in Table 2. We have also proposed a reversible positive level triggered J-K flip-flop with two asynchronous inputs which has been shown in Fig. 13. The characteristics equation of reversible J-K flip-flop with two asynchronous inputs is PRE Q +.LR'. This proposed design is a compact one as it comprises with 5 reversible gates and the total quantum cost of this circuit is 17 and it produces 4 garbage outputs. In this section we have designed a reversible positive level triggered J-K flip-flop which is shown in Fig. 12 and a ISSN: IJET Publications UK. All rights reserved. 12

5 3.3 Proposed Design of Reversible D Flip- Flop Fig. 12: Proposed Design of Reversible J-K Flip-Flop Flip-Flops are the main building block from which many circuits can be constructed. Reversible D flip-flop has been used as the core component to store bits of data in reversible data register. We have designed an optimized D flip-flop in terms of quantum cost and garbage bits with respect to existing designs such as [20], [21] and [18]. One FRG and one F2G have been used to design the D flip-flop which is shown in Fig. 14. The proposed design is also a fault tolerant one. A comparative result of different D flipflops has been shown in Table 3. Fig. 13: Proposed Design of Reversible J-K Flip-Flop with Two Asynchronous Inputs Table 2: omparison of different reversible J-K Flip-Flop No. of Gates No. of Garbage Quantum ost Existing ircuit [17] Existing ircuit [18] Existing ircuit [19] Proposed ircuit Fig. 14: Proposed Design of Reversible D Flip-Flop Table 3: omparison of Different Reversible D Flip-Flop No. of Gates No. of Garbage Quantum ost Existing ircuit [20] Existing ircuit [21] Existing ircuit [18] Proposed ircuit Proposed Design of Reversible Synchronous ounter In this section, we have proposed a reversible design of a synchronous counter. In a synchronous counter all the flipflops are triggered simultaneously by the clock input pulses. Since the input pulses are applied to all the flipflops some means must be used to control when a flip-flop is to toggle and when it is to remain unaffected by a clock pulse. This is accomplished by using the J and K inputs. This synchronous counter is used as the address register to access the consecutive locations of a reversible memory for data storage. In this section we have proposed a reversible 4-bit synchronous counter using Algorithm 1 which has been shown in Fig. 15. Algorithm 1: n- bit Reversible Synchronous ounter 1. Provide HIGH input to both the J and K input of the rightmost 0 th J-K flip-flop. 2. Generate PG 1 = Q 0.OUNT and feed the signal to both the J and K input of the J-K flip-flop. 3. For i= 2 to n-1 (from right to left) Loop ISSN: IJET Publications UK. All rights reserved. 13

6 i. Generate PG i = Q i-1.pg i-1.ount ii. Feed P i to both the J and K input of the proposed i th J-K flip-flop End Loop 4. Return Q= (Q n-1, Q n-2,..., Q 1, Q 2 ) 5. End Fig. 16: Proposed Design of Reversible Tri-State Buffer Theorem 2: The n-bit proposed three state buffer register requires at least 2n reversible gates producing 2n garbage bits having quantum cost at least 7n. Fig. 15: Proposed Design of Reversible Synchronous ounter Theorem 1: The n-bit proposed synchronous counter requires 8n-4 gates producing 8n - 6 garbage bits having quantum cost of at least 26n-13 provided that n 2. Proof: The proposed reversible J-K flip-flop with two asynchronous inputs has shown in Fig. 13 produces 4 garbage outputs having quantum cost of 17. Here in the reversible design of synchronous counter, for each bit n (n 2) additional two Peres gates are used to design a three input AND gate producing 4 garbage bits. A Feynman gate was required for each bit except the most significant bit position. So the total number of gates required for the n-bit synchronous counter is = number of gates in n J-K flipflops + number of Feynman gates + number of Peres gates = 5n + (n-1) + (2n -3) = 8n - 4. Overall quantum cost of the circuit is = cost of n J-K flip-flops + cost of n-1 Feynman gates + cost of 2n - 3 Peres gates = 17n + (n-1) + 4.(2n - 3) = 26n -13. Total garbage bits from the proposed design = garbage bits generated from n J-K flip-flops + garbage bits generated by the Peres gates = 4n + (4n - 6) = 8n Proposed Design of Reversible Tri-State Buffer In this section we have designed a reversible three state buffer register with four positive edge triggered clocked D flip-flops which has been shown in Fig. 16. A buffer register is a data register which is used for the temporary storage of a word. It consists of n clocked D flip-flops all sharing a common clock pulse. All of the digits in the n bit data word are connected to the data register by an n bit data bus. Proof: The proposed reversible three state buffer register comprises with the proposed reversible D flip-flop which requires two reversible gates producing two garbage output. For each bit of information storage of the proposed buffer register a total quantum cost of 7 is required. Therefore the n-bit proposed three state buffer register would have the total 2n reversible gates producing total quantum cost of 7n and 2n garbage outputs. 4. ONLUSION In this paper we have presented a compact design of a reversible data acquisition and storage system. In the way of developing the reversible design we have proposed some sequential circuits as well. It has been established with the help of comparisons and theorems that the proposed designs are compact in terms of quantum cost, number of gates and number of garbage outputs. Since data acquisition and storage system can be used in many advanced applications such as reversible central processing unit design and quantum computers etc so we believe that our effort may find its future use in numerous technologies [1, 2, 3, 4]. REFERENES [1] M. Nielsen and I. huang, Quantum computation and quantum information, ambridge University Press, [2] R..Merkle, Reversible electronic logic using switches, Nanotechnology, 4: pp , [3] R..Merkle, Two types of mechanical reversible logic, Nanotechnology, 4: pp ,1993. [4] R.. Merkle and K. E. Drexler, Helical logic, Nanotechnology, 7: pp , ISSN: IJET Publications UK. All rights reserved. 14

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