Design and Analysis of 4-2 Compressor for Arithmetic Application
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The document discusses the design and analysis of two new inaccurate 4-2 compressors for use in multipliers, focusing on the reduction of power consumption, delay, and transistor count compared to exact designs. The proposed designs show significant improvements in performance metrics with accompanying simulation results, revealing a lower error rate and efficient computations suitable for applications in multimedia processing and digital signal operations. Overall, the work highlights the feasibility of inexact computing methodologies in creating low-power digital arithmetic circuits.
![Asian Journal of Applied Science and Technology (AJAST)
Volume 1, Issue 1, Pages 106-109, February 2017
© 2017 AJAST All rights reserved. www.ajast.net
Page | 106
Design and Analysis of 4-2 Compressor for Arithmetic Application
S.Sowmiya#
, K.Stella* and V.M.Senthilkumar+
#
PG Student, Department of Electronics and Communication Engineering, Vivekanandha College of engineering for Women, India.
*PG Student, Department of Electronics and Communication Engineering, Vivekanandha College of engineering for Women, India.
+
Associate Professor, Department of Electronics and Communication Engineering, Vivekanandha College of engineering for Women, India.
Article Received: 15 February 2017 Article Accepted: 24 February 2017 Article Published: 26 February 2017
1. INTRODUCTION
Commonly used multimedia applications have Digital Signal
Processing (DSP) blocks as their backbone. Most of these
DSP blocks implement the image and video processing
algorithms, where the ultimate output is either an image or a
video for human consumption [1].
The limited perception of human vision allows the outputs of
these algorithms to be numerically approximate rather than
accurate. This relaxation on numerical exactness provides
some freedom to carry out imprecise or approximate
computation. The freedom can be taken advantage of to come
up with low-power designs at different levels of design
abstraction, viz. logic, architecture, and algorithm.
Methodologies for inexact computing rely on the feature that
many applications can tolerate some loss of precision and
therefore, the solution can tolerate some degree of
uncertainty. However, inexact computing applications are
mostly implemented using digital binary logic circuits, thus
operating with a high degree of predictability and precision. A
framework based on a precise and specific implementation
can still be used with a methodology that intrinsically has a
lower degree of precision and an increasing uncertainty in
operation [4].
Addition and multiplication are widely used operations in
computer arithmetic for addition full-adder cells have been
extensively analyzed for approximate computing. The
paradigm of inexact computation relies on relaxing fully
precise and completely deterministic building blocks such as
a full adder when for example, implementing the bio-inspired
systems. This allows nature inspired computation to redirect
the existing design process of digital circuits and systems by
taking advantage of a decrease in complexity and cost with
possibly a potential increase in performance and power
efficiency.
2. EXACT COMPRESSOR
Fig.1. General Structure of 4-2 compressor
The following equation gives the outputs of 4-2 compressor,
which table 1 show its truth table. The common
implementation of a 4-2 compressor is accomplished by
utilizing two full adders are shown in Fig. 3.3.
Sum = x1 x2 x3 x4 c in…… (1)
C out = (x1 x2) x3+ (x1 x2) ` x1… (2)
Carry = (x1 x2 x3 x4) c in+(x1 x2 x3 x4) `
x4…… (3)
Fig.2. Implementation of 4-2 compressor
ABSTRACT
Imprecise computing is an attractive model for digital processing at nano metric scales. Inexact computing is particularly interesting for computer
arithmetic designs. This work deals about the design and analysis of two new inaccurate 4-2 compressors for utilization in a multiplier. These designs
rely on different features of compression, such that imprecision in computation is measured by the error rate and the so-called normalized error
distance can meet with respect to circuit-based figures of merit of a design in terms of number of transistors, delay and power consumption. The
proposed approximate compressors are proposed and analyzed in Dadda multiplier. Extensive simulation results are provided and an application of
the approximate multipliers to image processing is presented. The results proposed designs shows that reduced power dissipation, delay and transistor
count.
Keywords: Compressor, Dadda multiplier and Inexact computing.](https://image.slidesharecdn.com/ajast24-170318194909/85/Design-and-Analysis-of-4-2-Compressor-for-Arithmetic-Application-1-320.jpg)
![Asian Journal of Applied Science and Technology (AJAST)
Volume 1, Issue 1, Pages 106-109, February 2017
© 2017 AJAST All rights reserved. www.ajast.net
Page | 108
As shown in Table 1, the carry output in an exact compressor
has the same value of the input cin in 24 out of 32 states.
Therefore, an approximate design must consider this feature.
Table 2 shows the truth table of the first proposed
approximate compressor. The proposed design 1 has 12
incorrect outputs out of 32 outputs thus yielding an error rate
of 37.5%. This is less than the error rate using the best [2]
approximate full-adder cell.
3.2 Design 2
A second design of an approximate compressor is proposed to
further increase performance as well as reducing the error rate
can be ignored in the hardware design.
Fig.4. Gate level implementation of design-2
In this new design, carry uses the right hand side of (3) and cout
is always equal to cin; since c in is zero in the first stage, cout
and c in will be zero in all stages. So, cin and c out can be
ignored in the hardware design.
Sum` = ((x1 x2) `+ (x3 x4) `) ……............ (7)
Carry` = ((x1x2) ` + (x3x4) `) ` ……...........… (8)
Table 3: Truth Table of Design-2
X4 X3 X2 X1 Carry’ Sum'
0 0 0 0 0 1
0 0 0 1 0 1
0 0 1 0 0 1
0 0 1 1 0 1
0 1 0 0 0 1
0 1 0 1 1 0
0 1 1 0 1 0
0 1 1 1 1 1
1 0 0 0 0 1
1 0 0 1 1 0
1 0 1 0 1 0
1 0 1 1 1 1
1 1 0 0 0 1
1 1 0 1 1 1
1 1 1 0 1 1
1 1 1 1 1 1
Fig.4. Shows the gate level implementation of design 2
compressor and the expressions below describe its outputs.
The delay of the critical path of this approximate design is 2Δ,
so it is 1Δ less than the previous designs; moreover, a further
reduction in the number of gates is accomplished
Fig.5. Output waveform of design-2
4. DADDA MULTIPLIER
A multiplier using such a compression scheme is normally
referred to as a Dadda multiplier.
In this multipliers least reduction at each stage [4]. The
maximum height of consist of three stages.
In the first stage, partial product each stage is determined by
working back from final stage matrix is formed.
In the second stage, partial product matrix is reduced to a
height of two products.
In the final stage, this consists of two rows of partial products
are combined using of each stage should be in the order 2, 3,
carry propagation adder. Dadda multiplier less number of half
adders are required than the Wallace multiplier.
Fig.6. Output waveform of Dadda multiplier
5. RESULT ANALYSIS
The below Table 4 shows that design 2 4-2 compressor is
better than other compressor because of its reduced transistor
count and power dissipation.](https://image.slidesharecdn.com/ajast24-170318194909/85/Design-and-Analysis-of-4-2-Compressor-for-Arithmetic-Application-3-320.jpg)
![Asian Journal of Applied Science and Technology (AJAST)
Volume 1, Issue 1, Pages 106-109, February 2017
© 2017 AJAST All rights reserved. www.ajast.net
Page | 109
Table 4: Analysis of compressor
COMPRESSOR POWER
DISSIPATION
(µW)
TRANSISTOR
COUNT
Exact design 0.016140 82
Design1 0.010332 44
Design 2 0.010096 42
Table 5: Analysis of multiplier
MULTIPLIER
POWER
DISSIPATION
(mW)
DELAY
Exact design 9.6566 1.25ns
Design1 5.8186 1.42ns
Design 2 5.5080 84.3ps
The above Table 5 shows that design 2 multiplier is better
than other multiplier because of its reduced delay and power
dissipation.
6. CONCLUSION
The compressors are utilized in the reduction module of four
approximate multipliers. The approximate compressors show
a significant reduction in transistor count, power consumption
and delay compared with an exact design. In terms of
transistor count, the first design has a 46% improvement,
while the second design has a 49% improvement. In terms of
power dissipation, the first design has a 57% improvement
and the second design has a 60% improvement over CMOS
implementation at feature sizes of 32 nm. In terms of delay,
the second design has a 44% improvement compared to the
exact compressor and 35% improvement compared to the first
design on average at CMOS feature sizes of 32 nm. The
proposed multipliers show a significant improvement in terms
of power consumption and transistor count compared to an
exact multiplier.
REFERENCES
[1] Chang C., Gu J. and Zhang M. (2004), ‘Ultra
Low-Voltage Low- Power CMOS 4-2 and 5-2 Compressors
for Fast Arithmetic Circuits’, IEEE Transactions on Circuits
& Systems, Vol. 51, No. 10, pp. 1985-1997.
[2] Cheemalavagu S., Kormas P., Palme K.V., Akgul B.E.S.
and Chakrapani L.N. (2005), ‘A probabilistic CMOS switch
and its realization by exploiting noise’, in Proc. SOC , Perth,
Western Australia.
[3] Gu J., and Chang C.H., (2003), ‘Ultra Low-voltage,
low-power 4-2 compressor for high speed multiplications’, in
Proc. 36th IEEE Int. Symp. Circuits Systems, Bangkok,
Thailand.
[4] Gupta V., Mohapatra D., Park S.P., Raghu Nathan A. and
Roy K. (Aug 2011), ‘IMPACT: Imprecise adders for
low-power Approximate Computing’, Low Power
Electronics and Design (ISLPED) International Symposium.
[5] King J.E. and Swartz lander E. (1998), ‘Data dependent
truncated scheme for parallel multiplication,’ in Proceedings
of the Thirty First Alomar Conference on Signals and
Systems, pp. 1178–1182.
[6] Kulkarni P., Gupta P. and Ercegovac S.( 2011) , ‘Trading
accuracy for power in a multiplier architecture’, Journal of
Low Power Electronics, Vol. 7, No. 4, pp. 490--501.
[7] Liang J., Han J. and Lombardi F.(2013) ,‘New Metrics for
the Reliability of Approximate and Probabilistic Adders’,
IEEE Transactions on Computers ,Vol. 63, No. 9, pp. 1760 –
1771.
[8] Mahanadi H.R., Hamada A., Fakhraie S.M. and Lucas C.
(2010), ‘Bio-Inspired Imprecise Computational Blocks for
Efficient VLSI Implementation of Soft-Computing
Applications’, IEEE Transactions on Circuits and Systems
I:Regular Papers, Vol. 57, No. 4, pp. 850-862.
[9] Margala M. and Durdle N.G. (1999), ‘Low-power
low-voltage 4-2 compressors for VLSI applications,’ in Proc.
IEEE Alessandro Volta Memorial Workshop Low-Power
Design, pp. 84–90.
[10] Mustafa K., Vojin G. and Oklobdzij S. (2010), ‘Energy
efficient implementation of parallel CMOS multipliers with
improved compressors.’ Proc. of the 16th ACM/IEEE
international symposium on Low power electronics and
design- ACM.](https://image.slidesharecdn.com/ajast24-170318194909/85/Design-and-Analysis-of-4-2-Compressor-for-Arithmetic-Application-4-320.jpg)
Design and Analysis of 4-2 Compressor for Arithmetic Application
- 1. Asian Journal of Applied Science and Technology (AJAST) Volume 1, Issue 1, Pages 106-109, February 2017 © 2017 AJAST All rights reserved. www.ajast.net Page | 106 Design and Analysis of 4-2 Compressor for Arithmetic Application S.Sowmiya# , K.Stella* and V.M.Senthilkumar+ # PG Student, Department of Electronics and Communication Engineering, Vivekanandha College of engineering for Women, India. *PG Student, Department of Electronics and Communication Engineering, Vivekanandha College of engineering for Women, India. + Associate Professor, Department of Electronics and Communication Engineering, Vivekanandha College of engineering for Women, India. Article Received: 15 February 2017 Article Accepted: 24 February 2017 Article Published: 26 February 2017 1. INTRODUCTION Commonly used multimedia applications have Digital Signal Processing (DSP) blocks as their backbone. Most of these DSP blocks implement the image and video processing algorithms, where the ultimate output is either an image or a video for human consumption [1]. The limited perception of human vision allows the outputs of these algorithms to be numerically approximate rather than accurate. This relaxation on numerical exactness provides some freedom to carry out imprecise or approximate computation. The freedom can be taken advantage of to come up with low-power designs at different levels of design abstraction, viz. logic, architecture, and algorithm. Methodologies for inexact computing rely on the feature that many applications can tolerate some loss of precision and therefore, the solution can tolerate some degree of uncertainty. However, inexact computing applications are mostly implemented using digital binary logic circuits, thus operating with a high degree of predictability and precision. A framework based on a precise and specific implementation can still be used with a methodology that intrinsically has a lower degree of precision and an increasing uncertainty in operation [4]. Addition and multiplication are widely used operations in computer arithmetic for addition full-adder cells have been extensively analyzed for approximate computing. The paradigm of inexact computation relies on relaxing fully precise and completely deterministic building blocks such as a full adder when for example, implementing the bio-inspired systems. This allows nature inspired computation to redirect the existing design process of digital circuits and systems by taking advantage of a decrease in complexity and cost with possibly a potential increase in performance and power efficiency. 2. EXACT COMPRESSOR Fig.1. General Structure of 4-2 compressor The following equation gives the outputs of 4-2 compressor, which table 1 show its truth table. The common implementation of a 4-2 compressor is accomplished by utilizing two full adders are shown in Fig. 3.3. Sum = x1 x2 x3 x4 c in…… (1) C out = (x1 x2) x3+ (x1 x2) ` x1… (2) Carry = (x1 x2 x3 x4) c in+(x1 x2 x3 x4) ` x4…… (3) Fig.2. Implementation of 4-2 compressor ABSTRACT Imprecise computing is an attractive model for digital processing at nano metric scales. Inexact computing is particularly interesting for computer arithmetic designs. This work deals about the design and analysis of two new inaccurate 4-2 compressors for utilization in a multiplier. These designs rely on different features of compression, such that imprecision in computation is measured by the error rate and the so-called normalized error distance can meet with respect to circuit-based figures of merit of a design in terms of number of transistors, delay and power consumption. The proposed approximate compressors are proposed and analyzed in Dadda multiplier. Extensive simulation results are provided and an application of the approximate multipliers to image processing is presented. The results proposed designs shows that reduced power dissipation, delay and transistor count. Keywords: Compressor, Dadda multiplier and Inexact computing.
- 2. Asian Journal of Applied Science and Technology (AJAST) Volume 1, Issue 1, Pages 106-109, February 2017 © 2017 AJAST All rights reserved. www.ajast.net Page | 107 This design is not efficient because it produces at least 17 incorrect output of 32 possible combination that is error rate of this exact compressor design is above 53%. Table 1: Truth table of exact 4-2 compressor Cin X4 X3 X2 X1 Cout Carry Sum 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 3. PROPOSED COMPRESSOR 3.1. Design 1 In Design 1, the carry is simplified to cin by changing the value of the other 8 outputs. Carry`= C in……………………………......... (4) Sum` = c in` ((x1 x2) `+ (x3 x4) `)….... (5) C out` = ((x1 x2) ` +(x3 x4) `)…………......... (6) Eqs. (4) - (6) are the logic expressions for the outputs of the first design of the approximate 4-2 compressor proposed in this manuscript. The gate level structure of the first proposed design (Fig.3) shows that the critical path of this compressor has still a delay of 3Δ, so it is the same as for the exact compressor of Fig.2. Fig.3. Gate level implementation of design 1 Table 2: Truth table of design 1 compressor Cin X4 X3 X2 X1 Cout Carry Sum 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 However; the propagation delay through the gates of this design is lower than the one for the exact compressor. Therefore, the critical path delay in the proposed design is lower than in the exact design and moreover, the total number of gates in the proposed design is significantly less than exact compressor.
- 3. Asian Journal of Applied Science and Technology (AJAST) Volume 1, Issue 1, Pages 106-109, February 2017 © 2017 AJAST All rights reserved. www.ajast.net Page | 108 As shown in Table 1, the carry output in an exact compressor has the same value of the input cin in 24 out of 32 states. Therefore, an approximate design must consider this feature. Table 2 shows the truth table of the first proposed approximate compressor. The proposed design 1 has 12 incorrect outputs out of 32 outputs thus yielding an error rate of 37.5%. This is less than the error rate using the best [2] approximate full-adder cell. 3.2 Design 2 A second design of an approximate compressor is proposed to further increase performance as well as reducing the error rate can be ignored in the hardware design. Fig.4. Gate level implementation of design-2 In this new design, carry uses the right hand side of (3) and cout is always equal to cin; since c in is zero in the first stage, cout and c in will be zero in all stages. So, cin and c out can be ignored in the hardware design. Sum` = ((x1 x2) `+ (x3 x4) `) ……............ (7) Carry` = ((x1x2) ` + (x3x4) `) ` ……...........… (8) Table 3: Truth Table of Design-2 X4 X3 X2 X1 Carry’ Sum' 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 Fig.4. Shows the gate level implementation of design 2 compressor and the expressions below describe its outputs. The delay of the critical path of this approximate design is 2Δ, so it is 1Δ less than the previous designs; moreover, a further reduction in the number of gates is accomplished Fig.5. Output waveform of design-2 4. DADDA MULTIPLIER A multiplier using such a compression scheme is normally referred to as a Dadda multiplier. In this multipliers least reduction at each stage [4]. The maximum height of consist of three stages. In the first stage, partial product each stage is determined by working back from final stage matrix is formed. In the second stage, partial product matrix is reduced to a height of two products. In the final stage, this consists of two rows of partial products are combined using of each stage should be in the order 2, 3, carry propagation adder. Dadda multiplier less number of half adders are required than the Wallace multiplier. Fig.6. Output waveform of Dadda multiplier 5. RESULT ANALYSIS The below Table 4 shows that design 2 4-2 compressor is better than other compressor because of its reduced transistor count and power dissipation.
- 4. Asian Journal of Applied Science and Technology (AJAST) Volume 1, Issue 1, Pages 106-109, February 2017 © 2017 AJAST All rights reserved. www.ajast.net Page | 109 Table 4: Analysis of compressor COMPRESSOR POWER DISSIPATION (µW) TRANSISTOR COUNT Exact design 0.016140 82 Design1 0.010332 44 Design 2 0.010096 42 Table 5: Analysis of multiplier MULTIPLIER POWER DISSIPATION (mW) DELAY Exact design 9.6566 1.25ns Design1 5.8186 1.42ns Design 2 5.5080 84.3ps The above Table 5 shows that design 2 multiplier is better than other multiplier because of its reduced delay and power dissipation. 6. CONCLUSION The compressors are utilized in the reduction module of four approximate multipliers. The approximate compressors show a significant reduction in transistor count, power consumption and delay compared with an exact design. In terms of transistor count, the first design has a 46% improvement, while the second design has a 49% improvement. In terms of power dissipation, the first design has a 57% improvement and the second design has a 60% improvement over CMOS implementation at feature sizes of 32 nm. In terms of delay, the second design has a 44% improvement compared to the exact compressor and 35% improvement compared to the first design on average at CMOS feature sizes of 32 nm. The proposed multipliers show a significant improvement in terms of power consumption and transistor count compared to an exact multiplier. REFERENCES [1] Chang C., Gu J. and Zhang M. (2004), ‘Ultra Low-Voltage Low- Power CMOS 4-2 and 5-2 Compressors for Fast Arithmetic Circuits’, IEEE Transactions on Circuits & Systems, Vol. 51, No. 10, pp. 1985-1997. [2] Cheemalavagu S., Kormas P., Palme K.V., Akgul B.E.S. and Chakrapani L.N. (2005), ‘A probabilistic CMOS switch and its realization by exploiting noise’, in Proc. SOC , Perth, Western Australia. [3] Gu J., and Chang C.H., (2003), ‘Ultra Low-voltage, low-power 4-2 compressor for high speed multiplications’, in Proc. 36th IEEE Int. Symp. Circuits Systems, Bangkok, Thailand. [4] Gupta V., Mohapatra D., Park S.P., Raghu Nathan A. and Roy K. (Aug 2011), ‘IMPACT: Imprecise adders for low-power Approximate Computing’, Low Power Electronics and Design (ISLPED) International Symposium. [5] King J.E. and Swartz lander E. (1998), ‘Data dependent truncated scheme for parallel multiplication,’ in Proceedings of the Thirty First Alomar Conference on Signals and Systems, pp. 1178–1182. [6] Kulkarni P., Gupta P. and Ercegovac S.( 2011) , ‘Trading accuracy for power in a multiplier architecture’, Journal of Low Power Electronics, Vol. 7, No. 4, pp. 490--501. [7] Liang J., Han J. and Lombardi F.(2013) ,‘New Metrics for the Reliability of Approximate and Probabilistic Adders’, IEEE Transactions on Computers ,Vol. 63, No. 9, pp. 1760 – 1771. [8] Mahanadi H.R., Hamada A., Fakhraie S.M. and Lucas C. (2010), ‘Bio-Inspired Imprecise Computational Blocks for Efficient VLSI Implementation of Soft-Computing Applications’, IEEE Transactions on Circuits and Systems I:Regular Papers, Vol. 57, No. 4, pp. 850-862. [9] Margala M. and Durdle N.G. (1999), ‘Low-power low-voltage 4-2 compressors for VLSI applications,’ in Proc. IEEE Alessandro Volta Memorial Workshop Low-Power Design, pp. 84–90. [10] Mustafa K., Vojin G. and Oklobdzij S. (2010), ‘Energy efficient implementation of parallel CMOS multipliers with improved compressors.’ Proc. of the 16th ACM/IEEE international symposium on Low power electronics and design- ACM.