Research Article FPGA Implementation of Optimal 3D-Integer

Research ArticleFPGA Implementation of Optimal 3D-Integer DCT Structure forVideo Compression

J Augustin Jacob1 and N Senthil Kumar2

1Kamaraj College of Engineering and Technology Virudhunagar 626001 India2Mepco Schlenk Engineering College Sivakasi 626001 India

Correspondence should be addressed to J Augustin Jacob mailtoaugusyahoocoin

Received 1 June 2015 Revised 15 September 2015 Accepted 17 September 2015

Academic Editor Marco Listanti

Copyright copy 2015 J A Jacob and N S Kumar This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A novel optimal structure for implementing 3D-integer discrete cosine transform (DCT) is presented by analyzing various integerapproximation methods The integer set with reduced mean squared error (MSE) and high coding efficiency are considered forimplementation in FPGA The proposed method proves that the least resources are utilized for the integer set that has shorter bitvalues Optimal 3D-integer DCT structure is determined by analyzing theMSE power dissipation coding efficiency and hardwarecomplexity of different integer sets The experimental results reveal that direct method of computing the 3D-integer DCT usingthe integer set [10 9 6 2 3 1 1] performs better when compared to other integer sets in terms of resource utilization and powerdissipation

1 Introduction

Nowadaysmost video compression algorithms rely on reduc-ing the spatial and temporal redundancy by motion compen-sation and prediction However these algorithms are complexand no symmetry exists between encoding and decodingblock This has made implementation of the algorithm morecomplex 3D-DCT based video coding [1] is consideredas an alternate to the existing standard video compressionalgorithms It eliminates some of the problems like blockingeffect caused by motion estimation algorithm which is lossyand time-consuming [2] For a video sequence that involvesfast motion object motion estimation may not yield correctmotion vector since full search cannot be done in a givenvideo stream

Few research efforts are made to enhance the 3D-DCTbased video codec [3ndash5] and made comparable to the stan-dard video compression algorithm If implementable struc-ture exists for 3D-integer DCT that will further acceleratethe encoding process A lossy compression scheme has beendeveloped by Zaharia et al [6] that apply 3D-DCT forcompressing 3D integral images and they showed that it

outperforms the JPEG standard Even though recent com-pression standards developed using discrete wavelet trans-form outperform the JPEG standard DCT is the preferredone because fast computation structures exist for DCT Itreflects the need for proposing new hardware for 3D-integerDCT However no attempt has been made to implement 3D-integer DCT algorithm It is essential to find the suitabilityof 3D-DCT based video coders in real time application byanalyzing the hardware complexity

Most standard video compression algorithms like MPEGand H26X adopt DCT as part of their standard This hadled to the development of many fast 1D- and 2D-DCTalgorithmsThe fundamental aim behind the development ofnew algorithm for DCT is to reduce the number of multipli-cations and additions In order to compute DCT for a giveninput sequence of length 119873 it requires 1198732 multiplicationsand 119873 (119873 minus 1) additions The fast DCT algorithm statedin [7] reduces the computational complexity to (1198732) log119873

2

multiplications and 119873 log2119873 additions A few algorithms

and implementation structure exist for computing real valued1D-DCT and 2D-DCT [8ndash23] Among them the algorithmpresented by Prado and Duhamel [16] is given significant

Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 204378 9 pageshttpdxdoiorg1011552015204378

2 The Scientific World Journal

importance because the study reveals that if an optimalalgorithm is obtained for 1D-DCT then the extension to thecorresponding 2D-DCT and 3D-DCT algorithm will alsobe optimal However implementing the real value transformbecomes more complex since the need of floating pointmultiplier is unavoidable even if it consumes more resourcesCham et al [24] have presented a simplified algorithm thatfirst converts the floating point to fixed point and thenperforms DCT However exact energy transformation willnot happen in this case because of the floating to fixed pointconversion The errors occurring during the computation of1D-DCT are propagated to the third dimension

Currently DCT with integer coefficients are of greatinterest because the design is simpler and implementedmoreefficiently An improvement over traditional real and fixedpoint implementation was proposed by Edirisuriya et al[25] In this paper DCT was computed using integer valuesSo there is no need to design floating point multiplier thatconsumes more resource and time The survey undoubtedlyshows the usage of integer DCT in 3D-DCT based video andimage compression algorithms However efforts to design thehardware for 3D-integer DCT are rare in the literature Afew approximation methods are available for deriving theequivalent integer DCT from real value DCT It is classifiedas indirect or C-matrix transformmethod proposed by Kwaket al [26] and direct method by Pei and Ding [27] Inthese papers the two approximation methods (direct andindirect) are considered for analysis and optimal integer setfor computing 3D-integer DCT is determined based on MSEand coding efficiency

Finally based on power dissipation and resource utiliza-tion optimal structure for 3D-integer DCT is determined

2 3D-Discrete Cosine Transforms

The discrete cosine transform (DCT) is a member of afamily of sinusoidal unitary transforms It found applicationsin digital signal processing and particularly in imagevideocompressionThe family of discrete trigonometric transformsconsists of 8 versions of DCT Each transform is identifiedas even or odd and of types I II III and IV All presentimage and video processing applications involve only eventypes of the DCT In particular DCT-II received muchattention in video compression applications because of itshigh energy packing ability and there exist fast computationstructures to compute DCT-II So throughout the text DCT-II was mentioned as DCT Equation (1) defines the one-dimensional-DCT and inverse DCT for a finite durationsignal 119891 of length119873

119903as

119865 (119877) = radic2

119873119903

119891119903

119873119903

sum

119903=0

119891 (119903)cos (2119903 + 1) 119877120587

2119873119903

0 le 119903 le 119873119903minus 1

119891 (119903) = radic2

119873119903

119891119903

119873119903

sum

119877=0

119865 (119877)cos (2119903 + 1) 119877120587

2119873119903

0 le 119877 le 119873119903minus 1

(1)

where

119891 [119903] =

1

radic2 119903 = 0

1 119903 = 1 2 119873119903minus 1

(2)

Usually image and video frames are two-dimensional innature Because of the orthogonality and separability prop-erty DCT can be extended to two dimensional forms The2D-DCT for a block of pixels of size 119873 times 119873 whose intensityvalues range between 0 and 255 is defined in

119865 (119877 119862) = radic4

119873119903sdot 119873119888

119891119903119891119888

119873119903

sum

119903=0

119873119888

sum

119888=0

119891 (119903 119888)cos (2119903 + 1) 119877120587

2119873119903

sdotcos (2119888 + 1) 119862120587

2119873119888

119891 (119903 119888) = radic4

119873119903sdot 119873119888

119891119903119891119888

119873119903

sum

119877=0

119873119888

sum

119862=0

119865 (119877 119862)

sdotcos (2119903 + 1) 119877120587

2119873119903

cos (2119888 + 1) 119862120587

2119873119888

(3)

where 119903 119888 119877 119862 = 0 1 119873119903119873119888minus 1 Consider

119891 [119903 119888] =

1

radic2 for 119903 119888 = 0

1 otherwise(4)

The equation for computing 2D-DCT is extended alongthe temporal domain to get the required expression forcomputing 3D-DCT It is defined in (5) and (7) Consider

119865 (119877 119862119863) = radic8

119873119903sdot 119873119888sdot 119873119889

119891119903119891119888119891119889

119873119903

sum

119903=0

119873119888

sum

119888=0

119873119889

sum

119889=0

119891 (119903 119888 119889)

sdotcos (2119903 + 1) 119877120587

2119873119903

cos (2119888 + 1) 119862120587

2119873119888

sdotcos (2119889 + 1)119863120587

2119873119889

(5)

where

119891119903 119891119888 119891119889=

1

radic2 for 119903 119888 119889 = 0

1 otherwise(6)

where 119865(119877 119862119863) and 119891(119903 119888 119889) represent the frequencydomain and time domain intensity values respectively Cor-respondingly the expression for finding inverse 3D-DCT isgiven as shown below

119891 (119903 119888 119889)

= radic8

119873119903sdot 119873119888sdot 119873119889

119891119903119891119888119891119889

119873119903

sum

119877=0

119873119888

sum

119862=0

119873119889

sum

119863=0

119865 (119877 119862119863)

sdotcos (2119903 + 1) 119877120587

2119873119903

cos (2119888 + 1) 119862120587

2119873119888

cos (2119889 + 1)119863120587

2119873119889

(7)

The Scientific World Journal 3

3 Integer Approximation of 3D-DCT UsingIndirect Method

In indirect method integer values are obtained using otherorthogonal transforms like the Walsh-Hadamard transformDCT can be implemented using WHT through a conversionmatrix shown in

119862 = 119879119873sdot 119879

119873

119879119873= 119862119873sdot 119879

119873

(8)

where 119862 represents discrete cosine transform and 119879119873is the

conversion matrix which converts the Walsh domain vector(119879) into DCT domain In indirect method there are totally11 different elements in the conversion matrix Substitutionof variable for each nonzero element in the matrix resultsin 11 variables denoted as 119860 119861 119862119863 119864 119865 119866119867 119868 119869 119870 Itis represented in (9) where

8is approximated conversion

matrix

8=

1

119860

(((((((((((

(

119860 0 0 0 0 0 0 0

0 119860 0 0 0 0 0 0

0 0 119861 119862 0 0 0 0

0 0 minus119862 119861 0 0 0 0

0 0 0 0 119863 minus119864 119867 119868

0 0 0 0 119865 119866 minus119869 119870

0 0 0 0 minus119870 119869 119866 119865

0 0 0 0 minus119868 minus119867 minus119864 119863

)))))))))))

)

(9)

Preserving the signs of the element of 8a searchwasmade to

find suitable integer values Also it has to satisfy the followingalgebraic equations

119863119865 minus 119864119866 minus 119869119867 + 119868119870 = 0 (10)

119863119870 + 119864119869 minus 119867119866 minus 119868119865 = 0 (11)

1198612+ 1198622= 1198632+ 1198642+ 1198672+ 1198682

= 1198652+ 1198662+ 1198692+ 1198702= 1198602

(12)

Equations (10) and (11) are conditions of orthogonality andthey ensure that rows of

8are orthogonal to each other

Equation (12) is for normality condition In order to make8resemble those of real valued transform constraints are set

on the variables119860 119861 119862119863 119864 119865 119866119867 119868 119869 119870Themagnitudesof the elements in 119879

8are compared and the following

inequalities are obtained

119863 gt 119866 gt 119869 gt 119867 gt 119870 gt 119865 gt 119868 gt 119864 gt 0 (13)

119869 gt 119862 ge 119867 minus 1 (14)

119861 ge 119863 minus 1 (15)

119860 gt 119861 (16)

All the integer solutions satisfying (10) to (12) under con-straints given by (13) to (16) will guarantee that the approx-imated conversion matrix

8is orthonormal and close to

the original conversion matrix 1198798 The generalized signal

flow graph of integer approximation using indirect methodis given in Figure 1 where

1198802= (

119861 119862

minus119862 119861)

1198804= (

119863 minus119864 119867 119868

119865 119866 minus119869 119870

minus119870 119869 119866 119865

minus119868 minus119867 minus119864 119863

)

(17)

In Figure 1 the lines indicated in blue color represent additionand dotted lines indicated in red color represent subtractionAdditional information regarding integer approximation canbe found in the work done by Britanak et al [28]

4 Integer Approximation UsingDirect Method

In direct method equivalent integer values are obtaineddirectly and it replaces the rational number in the DCTmatrix The approximated integer cosine transform matrix isgiven by

119862IDCT8

= 1198768sdot 1198818 (18)

where 1198768is a diagonal matrix with normalization factors

on its main diagonal and 1198818is an integer matrix It is seen

that totally there are 7 different elements in the DCT matrixThe same variables are used to represent the elements in theconversion matrix having the same magnitude Substitutinga variable for each nonzero element in the matrix results in7 variables denoted as 119860 119861 119862119863 119864 119865 119866 as it is shown in(19) Set of inequalities are formed so that orthogonality andnormality property of DCTmatrix is preserved in the integerdomain Consider

119862IDCT8

= 1198768

(((((((((((

(

119866 119866 119866 119866 119866 119866 119866 119866

119860 119861 119862 119863 minus119863 minus119862 minus119861 minus119860

119864 119865 minus119865 minus119864 minus119864 minus119865 119865 119864

119861 minus119863 minus119860 minus119862 119862 119860 119863 minus119861

119866 minus119866 minus119866 119866 119866 minus119866 minus119866 119866

119862 minus119860 119863 119861 minus119861 minus119863 119860 minus119862

119865 minus119864 119864 minus119865 minus119865 119864 minus119864 119865

119863 minus119862 119861 minus119860 119860 minus119861 119862 minus119863

)))))))))))

)

(19)

119860 (119861 minus 119862) minus 119863 (119861 + 119862) = 0 (20)

119860 gt 119861 gt 119862 gt 119863 gt 0 (21)

119860 gt 119864 gt 119865 gt 0 (22)

119860 gt 119866 gt 0 (23)

By solving (20) under set of constraints described in (21) to(23) different integer solutions set are obtained [22] Integer


w0

w4

w2

w6

w1

w5

w3

w7

x0

x1

x2

x3

x4

x5

x6

x7

(a)

w0

w4

w2

w6

w1

w5

w3

w7

C

C

B

B

U2

U4

A

A C0

C4

C2

C6

C1

C5

C3

C7

(b)

Figure 1 ((a) (b)) The signal flow graph of fast integer DCT using indirect method

sets with low mean squared error (MSE) and high trans-form coding efficiency (119899) are preferred to get the optimalsolution for 3D-integer DCT Fast computation structures areobtained by recursive sparse matrix factorization methodThe generalized signal flow graph of integer approximationusing direct integer DCT is given in Figure 2 where theparameters 119901 119903 119904 119906 V 119910 119911 are integers or dyadic rational

5 Criteria for Evaluation ofApproximated Integer DCT

In order to evaluate the approximation error between theinteger DCT and original transform matrix and to mea-sure the difference in performance in data compressionsome theoretical criteria are needed For this purpose theinput signal is frequently modeled as a first-order stationaryMarkov process (Markov-1) with zero-mean unit varianceand adjacent interelement correlation coefficient 120588 chosenbetween zero and one Then the input signal X is defined bya covariance matrix 119877

119909 whose elements are given by

[119877119909]119894119895= 120588|119894minus119895|

(24)

The matrix 119877119909is symmetric and Toeplitz The covariance

matrix 119877119910of the transformed vector y where y = 119860x is

obtained from (25)

119877119910= 119860119877119909119860119879 (25)

6 Mean Squared Error

For the evaluation of approximation error between theapproximated and original transform matrix the parametermean squared error (MSE) was used It is defined as followsLet us assume that 119880

119873is the original transform matrix and

119873is its approximation Then for a given input vector X of

length119873 the error vector is

119890 = 119880119873119909 minus

119873119909 = (119880

119873minus 119873) 119909 = 119863119909 (26)

G

G

F

F

E

E

p

p

p

p

sr

r

s

u

z

u

z

vy

y

v

C0

C4

C2

C6

C1

C5

C3

C7

x0

x1

x2

x3

x4

x5

x6

x7

Figure 2 The signal flow graph of fast integer DCT using directmethod

From (26) the MSE between the original and approximatedtransform can be defined by

1205981

119873119864 [119890119890119879] =

1

119873119864 [119909119879119863119879119863119909]

=1

119873119864 [Trace 119863119909119909

119879119863119879]

=1

119873Trace 119863119877

119909119863119879

(27)

where 119877119909is the covariance matrix of the input signal X

Thus to maintain the compatibility between the original andapproximated transform the MSE should be minimized

7 Transform Efficiency

Equation (28) defines the transform efficiency

120578 =sum119873minus1

119894=0

10038161003816100381610038161199031198941198941003816100381610038161003816

sum119873minus1

119894=0sum119873minus1

119895=0

10038161003816100381610038161003816119903119894119895

10038161003816100381610038161003816

100 (28)


where 119903119894119895are elements of 119877

119910 The transform efficiency mea-

sures the decorrelation ability of the transform The optimalKLT converts signal into completely uncorrelated coefficientsand it has transform efficiency 120578 = 100 for all values of 120588while the DCT has transform efficiency 120578 = 939911 for thecorrelation coefficient 120588 = 095

8 Structure for Computing 3D-IntegerDiscrete Cosine Transform

In order to reduce the hardware complexity optimal integersets from direct and indirect integer approximation arechosen based on number of multiplicationsadditions Thestructure that possesses minimum complexity is consideredfor computing 3D-integer DCT by taking 1D-integer DCTalong row column and temporal domainThe block diagramof the proposed 3D-integer DCT is shown in Figure 3 Tocompute 3D-integer DCT for the cube of dimension say8 times 8 times 8 the 1D-integer DCT is initially performed alongthe row wise and the computed values are stored in bufferldquo119860rdquo along the column wise The process is repeated for allthe rows of the cube starting from frame 1 to frame 8 Tohave clear visualization rows are marked with the same coloras shown in Figure 3 Here the buffer size and cube size areidentical The structure for computing DCT may be fromeither directmethod or indirectmethod Similarly 1D-integerDCT is computed for the values stored in buffer ldquo119860rdquo along therow wise and the results are stored along the column wise inbuffer ldquo119861rdquo this result in 2D-integer DCT Then perform onemore 1D-integerDCT for the values stored in buffer ldquo119861rdquo alongthe temporal direction that gives the 3D-integerDCT value asshown in Figure 3

9 Experimental Results

91 Determination of Optimal Integer Set for Computing 3D-Integer Discrete Cosine Transform In order to determinethe optimal integer set the performances of the proposed3D-IDCT are compared against the existing real valuedtransforms with respect to MSE and transform coding effi-ciency Different possible integer solutions exist for both thedirect and indirect method of computing 1D-IDCT and it issubjected to computing 3D-IDCT The MSE and transformcoding efficiency of the corresponding integer sets along withthe computational complexity are listed in Tables 1 and 2

The integer solutions whose MSE and coding efficiencyare very close to real value transform are considered forFPGA implementation Also it is observed that though theinteger set with higher bit solutions (5 6 7 and 8) yieldlow MSE and high coding efficiency it is not preferredfor implementation Because when computing 3D-integerDCT the size of registers (buffers ldquo119860rdquo and ldquo119861rdquo) holdingintermediate values becomes larger for higher bit solutionsthat directly increases the computational complexity (ie)higher bit length multiplier is required Further it is notedthat for integer set having zero and one as one of theelements variation in multiplicationadditions is observed

Here the number of multiplications and additions isestimated based on the structure shown in Figure 3 With

Table 1 MSE and coding efficiency of 3D-integer DCT computedvia WHT

3D-integer DCT coefficients(119860 119861 119862119863 119864 119865 119866119867 119868 119869 119870)

Meansquared error

Codingefficiency (120578)

Real valued transform 0 74883013 12 5 12 0 0 12 4 3 3 4Multiplications 54Additions 102

00236 722229

34 30 16 31 1 7 25 13 5 19 11Multiplications 60Additions 114

00031 730016


00011 741108

Table 2 MSE and coding efficiency of 3D-integer DCT computedusing direct method

S number3D-integer DCT

coefficients(119860119861119862119863119864119865119866)

Meansquared error

Codingefficiency

(120578)Real valued transform 0 748830

3-bit solution (multiplications 30 additions 78)1 5 3 2 1 3 1 1 00133 7018682 7 4 3 1 3 1 1 00155 691980

4-bit solution (multiplications 48 additions 78)3 10 9 6 2 3 1 1 00014 7481154 14 12 9 2 3 1 1 00024 7403395 12 10 6 3 3 1 1 00022 7332926 15 12 8 3 3 1 1 00014 739534





reference to the results obtained in Tables 1 and 2 the optimalinteger set is determined to be [10 9 6 2 3 1 1] because


1 2 3 4 5 6 7 8 12

8

3D-coefficient data

Temporal domain

2

8

12

8

1

8

12

PerformID-integer DCT(directindirect)



1 2 3 4 5 6 7 8C

R

R

R

R

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

Buffer register-ldquoBrdquo

Buffer register-ldquoArdquo

Figure 3 Block diagram of computing 3D-integer DCT

this integer set yields relatively low MSE and high codingefficiency when compared to real value transform

Further it was observed that if optimal integer set isused to encode the video sequence instead of real value 3D-DCT there is no much deviation in PSNR value Howeverit is noticed that the deviation is proportional to the MSEof the corresponding integer set For the optimal integer set

the maximum degradation in PSNR value was found to be001 db

10 FPGA Implementation of 3D-IDCT

The hardware design for computing 3D-integer DCT fora block of data 8 times 8 times 8 using the integer set [10 9 6


Power supplied to off-chip devices

Summary

Thermal margin

Junction temperature

Total on-chip power 0201W

254∘C

746∘C 393W

19∘CW

0

18

17

65

0000W

0036W

0034W

0131W

0000W

Transceiver

Core dynamic

Device static

IO

01

01

01

01

01

00

00

00Pow

er (W

)

Leak

age

Cloc

k

Logi

c

BRA

M

DSP PL

L

MM

CM

PHA

SER

PCIE IO

GTP

On-chip power by function (maximum 1V 27∘C)

Effective ΘJA

Figure 4 On-chip power utilization of 3D-integer DCT for the integer set [10 9 6 2 3 1 1]

2 3 1 and 1] was coded in Verilog Hardware DescriptionLanguage The functional behavior of the design was testedin Xilinx ISE simulator with sample data set Simulationsare also performed using MATLAB for the same data set forcorrectness The design was mapped on to Artix-7 FPGAboard The Artix-7 belongs to 28-nanometer (nm) processtechnology designed for low power products used in portablecommunication devicesThemaximumDCvalue of 3D-DCTwas found to be 4000 If normalization factors are neglectedin integer domain maximum of 17 bits are required to holdthe 3D-integer DCT value

As the value of elements in the integer set increases thenbit length of the processing elements also increases to showthat the least resources are utilized for the integer set that hasshorter bit values Synthesis was performed for the integerset [13 12 5 12 0 0 12 4 3 3 4] and comparison hasbeen made with the optimal integer set From the deviceutilization summary shown in Table 3 it was noticed thathigher resources are utilized for the integer set [13 12 5 120 0 12 4 3 3 4] It is due to the fact that for computing3D-integer set this integer set requires 25 bits however foroptimal integer set it requires only 17 bits So when bit lengthof the integer set increases then bit length of computationalunit (multiplicationaddition) also increases that leads tohigher resource utilization In order to estimate the powerconsumption of the design Xilinx Power Estimator (XPE)tool was used The distribution of on-chip power and totalpower of the design is shown in Figure 4

The total on-chip power reflects the heat dissipated fromthe chip If the device operates at 100MHz clock with the

Table 3 Device utilization summary

Device utilization

Optimalinteger set

[10 9 6 2 31 1]

Sample integerset

[13 12 5 12 0 012 4 3 3 4]

Number of slice registers(out of 437200) 975 1507

Number of slice LUT(out of 218600) 5396 7054

Number of fully used LUT-FFpairs (out of 6014) 357 146

Number of bonded IOBs (out of250) 213 269

Number of DSP slices(out of 900) 22 120

Clock 100MHz 10446MHzComputational complexityMultiplicationsadditions 4878 54102

total on-chip power of 0201W then the junction temperatureis 254∘C and it is well below the thermal margin of the targetFPGA device Also a comparison has been made betweenthe existing fixed point 2D-DCT algorithm based on Lofflermethod [22] and the proposed 3D-integer DCT algorithmin terms of device utilization It is identified that twelveinstances of fixed point 2D-DCTLoffler structures are neededto compute a fixed point 3D-DCT algorithm in accordance


Table 4 Comparison of device utilization summaries of the pro-posed method

Device utilization

Fixed point3D-DCT algorithm

based onLoffler method

[22]

Proposed3D-integer DCTalgorithm with theoptimal integer set[10 9 6 2 3 1 1]

Number of sliceregisters 5112 975

Number of slice LUT 27360 5396Number of fully usedLUT-FF pairs 15708 357

Bonded IOBs 274 213

with the fact that resource utilization is calculated and it isgiven in Table 4

It is clearly seen from Table 4 that the proposed 3D-integer DCT algorithm with optimal integer set [10 9 6 23 1 1] outperforms the fixed point 3D-DCT algorithm basedon Loffler method [22]

11 Conclusion

In this paper various integer sets from different approxima-tion methods for converting real to integer value transformsare analyzed in terms of MSE and coding efficiency Basedon that optimal integer set is chosen for computing 3D-integer DCT Further if optimal integer set was adopted toencode the video sequence then the deviation in PSNR withrespect to real value DCT was found to be 001 db Also anew hardware structure for computing the 3D-integer DCTis proposed and implemented the same in FPGA board Thesynthesis results reveal that the least resources are utilizedfor the integer set that has shorter bit values Also based onnumber of additions andmultiplications variation in resourceutilization is observed The experimental results reveal thatdirect method of computing the 3D-integer DCT using theinteger set [10 9 6 2 3 1 1] performs better when comparedto other integer sets in terms of resource utilization andpowerdissipation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Westwater and B Furht ldquoThree-dimensional DCT videocompression technique based on adaptive quantizersrdquo in Pro-ceedings of the 2nd IEEE International Conference onEngineeringof Complex Computer Systems pp 189ndash198 IEEE MontrealCanada October 1996

[2] D Le Gall ldquoMPEG a video compression standard for multime-dia applicationsrdquoCommunications of the ACM vol 34 no 4 pp46ndash58 1991

[3] N Bozinovic and J Konrad ldquoMotion analysis in 3D DCTdomain and its application to video codingrdquo Signal ProcessingImage Communication vol 20 no 6 pp 510ndash528 2005

[4] B Furht K Gustafson H Huang and O Marques ldquoAnadaptive three-dimensionalDCTcompression based onmotionanalysisrdquo in Proceedings of the ACM Symposium on AppliedComputing pp 765ndash768 ACM Melbourne Fla USA March2003

[5] J Augustin Jacob and N Senthil Kumar ldquoAn approach toadaptive 3D-DCT based motion level prediction algorithm forimproved 3D-DCT video codingrdquo Przegląd Elektrotechnicznyvol R 90 no 12 pp 95ndash99 2014

[6] R Zaharia A Aggoun andMMcCormick ldquoAdaptive 3D-DCTcompression algorithm for continuous parallax 3D integralimagingrdquo Signal Processing Image Communication vol 17 no3 pp 231ndash242 2002

[7] C W Kok ldquoFast algorithm for computing discrete cosinetransformrdquo IEEE Transactions on Signal Processing vol 45 no3 pp 757ndash760 1997

[8] G Plonka and M Tasche ldquoFast and numerically stable algo-rithms for discrete cosine transformsrdquo Linear Algebra and ItsApplications vol 394 pp 309ndash345 2005

[9] C Loeffer A Ligtenberg and G S Moschytz ldquoPractical fast 1-D DCT algorithms with 11 multiplicationsrdquo in Proceedings ofthe International Conference on Acoustics Speech and SignalProcessing pp 988ndash991 Glasgow Scotland May 1989

[10] D Hein and N Ahmed ldquoOn a real-time Walsh-Hadamardcosine transform image processorrdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 20 no 3 pp 453ndash457 1978

[11] S C Chan and K L Ho ldquoA new two-dimensional fast cosinetransform algorithmrdquo IEEE Transactions on Signal Processingvol 39 no 2 pp 481ndash485 1991

[12] H RWu andF J Paoloni ldquoA two-dimensional fast cosine trans-form algorithmbased onHoursquos approachrdquo IEEETransactions onSignal Processing vol 39 no 2 pp 544ndash546 1991

[13] V Britanak and K R Rao ldquoTwo-dimensional DCTDST uni-versal computational structure for 2119898 times 2

119899 block sizesrdquo IEEETransactions on Signal Processing vol 48 no 11 pp 3250ndash32552000

[14] E Feig and SWinograd ldquoFast algorithms for the discrete cosinetransformrdquo IEEE Transactions on Signal Processing vol 40 no9 pp 2174ndash2193 1992

[15] P Duhamel and C Guillemot ldquoPolynomial transform com-putation of the 2-D DCTrdquo in Proceedings of the InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo90) vol 3 pp 1515ndash1518 IEEE Albuquerque NM USA April1990

[16] J Prado and P Duhamel ldquoA polynomial-transform basedcomputation of the 2-D DCT with minimum multiplicativecomplexityrdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo96) vol 3pp 1347ndash1350 IEEE Atlanta Ga USA May 1996

[17] N I Cho I D Yun and S U Lee ldquoA fast algorithm for2-D DCTrdquo in Proceedings of the International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo91) pp 2197ndash2200 Toronto Canada May 1991

[18] N I Cho and S U Lee ldquoFast algorithm and implementation of2-D discrete cosine transformrdquo IEEE Transactions on Circuitsand Systems vol 38 no 3 pp 297ndash305 1991

[19] N I Cho and S U Lee ldquoA fast 4times4 DCT algorithm for therecursive 2-DDCTrdquo IEEE Transactions on Signal Processing vol40 no 9 pp 2166ndash2173 1992


[20] M I Cho I D Yun and S U Lee ldquoOn the regular structure forthe fast 2-D DCT algorithmrdquo IEEE Transactions on Circuits andSystems II Analog and Digital Signal Processing vol 40 no 4pp 259ndash266 1993

[21] A Mohsen O Sharifi-Tehrani and M Peyman ldquoOptimizinghardware simulation and realization of discrete cosine trans-form using VHDL hardware description languagerdquo AustralianJournal of Basic andApplied Sciences vol 5 pp 2040ndash2045 2011

[22] IMartisius D Birvinskas V Jusas andZ Tamosevicius ldquoA 2-DDCT hardware codec based on loeffler algorithmrdquo Elektronikair Elektrotechnika vol 113 pp 47ndash50 2011

[23] C-T Lin Y-C Yu and L-D Van ldquoCost-effective triple-modereconfigurable pipeline FFTIFFT2-D DCT processorrdquo IEEETransactions on Very Large Scale Integration Systems vol 16 no8 pp 1058ndash1071 2008

[24] W-K Cham C-S Choy andW-K Lam ldquoA 2-D integer cosinetransform chip set and its applicationrdquo IEEE Transactions onConsumer Electronics vol 38 no 2 pp 43ndash47 1992

[25] A Edirisuriya A Madanayake R J Cintra V S Dimitrovand N Rajapaksha ldquoA single-channel architecture for algebraicinteger-based 8times8 2-D DCT computationrdquo IEEE Transactionson Circuits and Systems for Video Technology vol 23 no 12 pp2083ndash2089 2013

[26] H S Kwak R Srinivasan and K R Rao ldquo119862-matrix transformrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 31 no 5 pp 1304ndash1307 2003

[27] S-C Pei and J-J Ding ldquoThe integer transforms analogous todiscrete trigonometric transformsrdquo IEEE Transactions on SignalProcessing vol 48 no 12 pp 3345ndash3364 2000

[28] V Britanak P C Yip and K R Rao Discrete Cosine andSine Transforms General Properties Fast Algorithms and IntegerApproximations chapter 3ndash5 Academic Press Oxford UK2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



importance because the study reveals that if an optimalalgorithm is obtained for 1D-DCT then the extension to thecorresponding 2D-DCT and 3D-DCT algorithm will alsobe optimal However implementing the real value transformbecomes more complex since the need of floating pointmultiplier is unavoidable even if it consumes more resourcesCham et al [24] have presented a simplified algorithm thatfirst converts the floating point to fixed point and thenperforms DCT However exact energy transformation willnot happen in this case because of the floating to fixed pointconversion The errors occurring during the computation of1D-DCT are propagated to the third dimension

Currently DCT with integer coefficients are of greatinterest because the design is simpler and implementedmoreefficiently An improvement over traditional real and fixedpoint implementation was proposed by Edirisuriya et al[25] In this paper DCT was computed using integer valuesSo there is no need to design floating point multiplier thatconsumes more resource and time The survey undoubtedlyshows the usage of integer DCT in 3D-DCT based video andimage compression algorithms However efforts to design thehardware for 3D-integer DCT are rare in the literature Afew approximation methods are available for deriving theequivalent integer DCT from real value DCT It is classifiedas indirect or C-matrix transformmethod proposed by Kwaket al [26] and direct method by Pei and Ding [27] Inthese papers the two approximation methods (direct andindirect) are considered for analysis and optimal integer setfor computing 3D-integer DCT is determined based on MSEand coding efficiency

Finally based on power dissipation and resource utiliza-tion optimal structure for 3D-integer DCT is determined

2 3D-Discrete Cosine Transforms

The discrete cosine transform (DCT) is a member of afamily of sinusoidal unitary transforms It found applicationsin digital signal processing and particularly in imagevideocompressionThe family of discrete trigonometric transformsconsists of 8 versions of DCT Each transform is identifiedas even or odd and of types I II III and IV All presentimage and video processing applications involve only eventypes of the DCT In particular DCT-II received muchattention in video compression applications because of itshigh energy packing ability and there exist fast computationstructures to compute DCT-II So throughout the text DCT-II was mentioned as DCT Equation (1) defines the one-dimensional-DCT and inverse DCT for a finite durationsignal 119891 of length119873

119903as

119865 (119877) = radic2

119873119903

119891119903

119873119903

sum

119903=0

119891 (119903)cos (2119903 + 1) 119877120587

2119873119903

0 le 119903 le 119873119903minus 1

119891 (119903) = radic2

119873119903

119891119903

119873119903

sum

119877=0

119865 (119877)cos (2119903 + 1) 119877120587

2119873119903

0 le 119877 le 119873119903minus 1

(1)

where

119891 [119903] =

1

radic2 119903 = 0

1 119903 = 1 2 119873119903minus 1

(2)

Usually image and video frames are two-dimensional innature Because of the orthogonality and separability prop-erty DCT can be extended to two dimensional forms The2D-DCT for a block of pixels of size 119873 times 119873 whose intensityvalues range between 0 and 255 is defined in

119865 (119877 119862) = radic4

119873119903sdot 119873119888

119891119903119891119888

119873119903

sum

119903=0

119873119888

sum

119888=0

119891 (119903 119888)cos (2119903 + 1) 119877120587

2119873119903

sdotcos (2119888 + 1) 119862120587

2119873119888

119891 (119903 119888) = radic4

119873119903sdot 119873119888

119891119903119891119888

119873119903

sum

119877=0

119873119888

sum

119862=0

119865 (119877 119862)

sdotcos (2119903 + 1) 119877120587

2119873119903

cos (2119888 + 1) 119862120587

2119873119888

(3)

where 119903 119888 119877 119862 = 0 1 119873119903119873119888minus 1 Consider

119891 [119903 119888] =

1

radic2 for 119903 119888 = 0

1 otherwise(4)

The equation for computing 2D-DCT is extended alongthe temporal domain to get the required expression forcomputing 3D-DCT It is defined in (5) and (7) Consider

119865 (119877 119862119863) = radic8

119873119903sdot 119873119888sdot 119873119889

119891119903119891119888119891119889

119873119903

sum

119903=0

119873119888

sum

119888=0

119873119889

sum

119889=0

119891 (119903 119888 119889)

sdotcos (2119903 + 1) 119877120587

2119873119903

cos (2119888 + 1) 119862120587

2119873119888

sdotcos (2119889 + 1)119863120587

2119873119889

(5)

where

119891119903 119891119888 119891119889=

1

radic2 for 119903 119888 119889 = 0

1 otherwise(6)

where 119865(119877 119862119863) and 119891(119903 119888 119889) represent the frequencydomain and time domain intensity values respectively Cor-respondingly the expression for finding inverse 3D-DCT isgiven as shown below

119891 (119903 119888 119889)

= radic8

119873119903sdot 119873119888sdot 119873119889

119891119903119891119888119891119889

119873119903

sum

119877=0

119873119888

sum

119862=0

119873119889

sum

119863=0

119865 (119877 119862119863)

sdotcos (2119903 + 1) 119877120587

2119873119903

cos (2119888 + 1) 119862120587

2119873119888

cos (2119889 + 1)119863120587

2119873119889

(7)




119862 = 119879119873sdot 119879

119873

119879119873= 119862119873sdot 119879

119873

(8)




matrix

8=

1

119860

(((((((((((

(

119860 0 0 0 0 0 0 0

0 119860 0 0 0 0 0 0

0 0 119861 119862 0 0 0 0

0 0 minus119862 119861 0 0 0 0

0 0 0 0 119863 minus119864 119867 119868

0 0 0 0 119865 119866 minus119869 119870

0 0 0 0 minus119870 119869 119866 119865


)))))))))))

)

(9)



119863119865 minus 119864119866 minus 119869119867 + 119868119870 = 0 (10)

119863119870 + 119864119869 minus 119867119866 minus 119868119865 = 0 (11)

1198612+ 1198622= 1198632+ 1198642+ 1198672+ 1198682

= 1198652+ 1198662+ 1198692+ 1198702= 1198602

(12)








119869 gt 119862 ge 119867 minus 1 (14)

119861 ge 119863 minus 1 (15)

119860 gt 119861 (16)





1198802= (

119861 119862

minus119862 119861)

1198804= (

119863 minus119864 119867 119868

119865 119866 minus119869 119870

minus119870 119869 119866 119865


)

(17)




119862IDCT8

= 1198768sdot 1198818 (18)




119862IDCT8

= 1198768

(((((((((((

(

119866 119866 119866 119866 119866 119866 119866 119866








)))))))))))

)

(19)

119860 (119861 minus 119862) minus 119863 (119861 + 119862) = 0 (20)

119860 gt 119861 gt 119862 gt 119863 gt 0 (21)

119860 gt 119864 gt 119865 gt 0 (22)

119860 gt 119866 gt 0 (23)



w0

w4

w2

w6

w1

w5

w3

w7

x0

x1

x2

x3

x4

x5

x6

x7

(a)

w0

w4

w2

w6

w1

w5

w3

w7

C

C

B

B

U2

U4

A

A C0

C4

C2

C6

C1

C5

C3

C7

(b)






[119877119909]119894119895= 120588|119894minus119895|

(24)



obtained from (25)

119877119910= 119860119877119909119860119879 (25)






119890 = 119880119873119909 minus

119873119909 = (119880

119873minus 119873) 119909 = 119863119909 (26)

G

G

F

F

E

E

p

p

p

p

sr

r

s

u

z

u

z

vy

y

v

C0

C4

C2

C6

C1

C5

C3

C7

x0

x1

x2

x3

x4

x5

x6

x7



1205981

119873119864 [119890119890119879] =

1

119873119864 [119909119879119863119879119863119909]

=1

119873119864 [Trace 119863119909119909

119879119863119879]

=1

119873Trace 119863119877

119909119863119879

(27)





120578 =sum119873minus1

119894=0

10038161003816100381610038161199031198941198941003816100381610038161003816

sum119873minus1

119894=0sum119873minus1

119895=0

10038161003816100381610038161003816119903119894119895

10038161003816100381610038161003816

100 (28)













Meansquared error



00236 722229


00031 730016


00011 741108



coefficients(119860119861119862119863119864119865119866)

Meansquared error

Codingefficiency










1 2 3 4 5 6 7 8 12

8

3D-coefficient data

Temporal domain

2

8

12

8

1

8

12




1 2 3 4 5 6 7 8C

R

R

R

R

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8











Summary

Thermal margin



254∘C

746∘C 393W

19∘CW

0

18

17

65

0000W

0036W

0034W

0131W

0000W

Transceiver

Core dynamic

Device static

IO

01

01

01

01

01

00

00

00Pow

er (W

)

Leak

age

Cloc

k

Logi

c

BRA

M

DSP PL

L

MM

CM

PHA

SER

PCIE IO

GTP


Effective ΘJA






Device utilization

Optimalinteger set

[10 9 6 2 31 1]

Sample integerset

[13 12 5 12 0 012 4 3 3 4]










Device utilization



[22]




Bonded IOBs 274 213



11 Conclusion




References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119862 = 119879119873sdot 119879

119873

119879119873= 119862119873sdot 119879

119873

(8)




matrix

8=

1

119860

(((((((((((

(

119860 0 0 0 0 0 0 0

0 119860 0 0 0 0 0 0

0 0 119861 119862 0 0 0 0

0 0 minus119862 119861 0 0 0 0

0 0 0 0 119863 minus119864 119867 119868

0 0 0 0 119865 119866 minus119869 119870

0 0 0 0 minus119870 119869 119866 119865


)))))))))))

)

(9)



119863119865 minus 119864119866 minus 119869119867 + 119868119870 = 0 (10)

119863119870 + 119864119869 minus 119867119866 minus 119868119865 = 0 (11)

1198612+ 1198622= 1198632+ 1198642+ 1198672+ 1198682

= 1198652+ 1198662+ 1198692+ 1198702= 1198602

(12)








119869 gt 119862 ge 119867 minus 1 (14)

119861 ge 119863 minus 1 (15)

119860 gt 119861 (16)





1198802= (

119861 119862

minus119862 119861)

1198804= (

119863 minus119864 119867 119868

119865 119866 minus119869 119870

minus119870 119869 119866 119865


)

(17)




119862IDCT8

= 1198768sdot 1198818 (18)




119862IDCT8

= 1198768

(((((((((((

(

119866 119866 119866 119866 119866 119866 119866 119866








)))))))))))

)

(19)

119860 (119861 minus 119862) minus 119863 (119861 + 119862) = 0 (20)

119860 gt 119861 gt 119862 gt 119863 gt 0 (21)

119860 gt 119864 gt 119865 gt 0 (22)

119860 gt 119866 gt 0 (23)



w0

w4

w2

w6

w1

w5

w3

w7

x0

x1

x2

x3

x4

x5

x6

x7

(a)

w0

w4

w2

w6

w1

w5

w3

w7

C

C

B

B

U2

U4

A

A C0

C4

C2

C6

C1

C5

C3

C7

(b)






[119877119909]119894119895= 120588|119894minus119895|

(24)



obtained from (25)

119877119910= 119860119877119909119860119879 (25)






119890 = 119880119873119909 minus

119873119909 = (119880

119873minus 119873) 119909 = 119863119909 (26)

G

G

F

F

E

E

p

p

p

p

sr

r

s

u

z

u

z

vy

y

v

C0

C4

C2

C6

C1

C5

C3

C7

x0

x1

x2

x3

x4

x5

x6

x7



1205981

119873119864 [119890119890119879] =

1

119873119864 [119909119879119863119879119863119909]

=1

119873119864 [Trace 119863119909119909

119879119863119879]

=1

119873Trace 119863119877

119909119863119879

(27)





120578 =sum119873minus1

119894=0

10038161003816100381610038161199031198941198941003816100381610038161003816

sum119873minus1

119894=0sum119873minus1

119895=0

10038161003816100381610038161003816119903119894119895

10038161003816100381610038161003816

100 (28)













Meansquared error



00236 722229


00031 730016


00011 741108



coefficients(119860119861119862119863119864119865119866)

Meansquared error

Codingefficiency










1 2 3 4 5 6 7 8 12

8

3D-coefficient data

Temporal domain

2

8

12

8

1

8

12




1 2 3 4 5 6 7 8C

R

R

R

R

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8











Summary

Thermal margin



254∘C

746∘C 393W

19∘CW

0

18

17

65

0000W

0036W

0034W

0131W

0000W

Transceiver

Core dynamic

Device static

IO

01

01

01

01

01

00

00

00Pow

er (W

)

Leak

age

Cloc

k

Logi

c

BRA

M

DSP PL

L

MM

CM

PHA

SER

PCIE IO

GTP


Effective ΘJA






Device utilization

Optimalinteger set

[10 9 6 2 31 1]

Sample integerset

[13 12 5 12 0 012 4 3 3 4]










Device utilization



[22]




Bonded IOBs 274 213



11 Conclusion




References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










w0

w4

w2

w6

w1

w5

w3

w7

x0

x1

x2

x3

x4

x5

x6

x7

(a)

w0

w4

w2

w6

w1

w5

w3

w7

C

C

B

B

U2

U4

A

A C0

C4

C2

C6

C1

C5

C3

C7

(b)






[119877119909]119894119895= 120588|119894minus119895|

(24)



obtained from (25)

119877119910= 119860119877119909119860119879 (25)






119890 = 119880119873119909 minus

119873119909 = (119880

119873minus 119873) 119909 = 119863119909 (26)

G

G

F

F

E

E

p

p

p

p

sr

r

s

u

z

u

z

vy

y

v

C0

C4

C2

C6

C1

C5

C3

C7

x0

x1

x2

x3

x4

x5

x6

x7



1205981

119873119864 [119890119890119879] =

1

119873119864 [119909119879119863119879119863119909]

=1

119873119864 [Trace 119863119909119909

119879119863119879]

=1

119873Trace 119863119877

119909119863119879

(27)





120578 =sum119873minus1

119894=0

10038161003816100381610038161199031198941198941003816100381610038161003816

sum119873minus1

119894=0sum119873minus1

119895=0

10038161003816100381610038161003816119903119894119895

10038161003816100381610038161003816

100 (28)













Meansquared error



00236 722229


00031 730016


00011 741108



coefficients(119860119861119862119863119864119865119866)

Meansquared error

Codingefficiency










1 2 3 4 5 6 7 8 12

8

3D-coefficient data

Temporal domain

2

8

12

8

1

8

12




1 2 3 4 5 6 7 8C

R

R

R

R

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8











Summary

Thermal margin



254∘C

746∘C 393W

19∘CW

0

18

17

65

0000W

0036W

0034W

0131W

0000W

Transceiver

Core dynamic

Device static

IO

01

01

01

01

01

00

00

00Pow

er (W

)

Leak

age

Cloc

k

Logi

c

BRA

M

DSP PL

L

MM

CM

PHA

SER

PCIE IO

GTP


Effective ΘJA






Device utilization

Optimalinteger set

[10 9 6 2 31 1]

Sample integerset

[13 12 5 12 0 012 4 3 3 4]










Device utilization



[22]




Bonded IOBs 274 213



11 Conclusion




References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















Meansquared error



00236 722229


00031 730016


00011 741108



coefficients(119860119861119862119863119864119865119866)

Meansquared error

Codingefficiency










1 2 3 4 5 6 7 8 12

8

3D-coefficient data

Temporal domain

2

8

12

8

1

8

12




1 2 3 4 5 6 7 8C

R

R

R

R

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8











Summary

Thermal margin



254∘C

746∘C 393W

19∘CW

0

18

17

65

0000W

0036W

0034W

0131W

0000W

Transceiver

Core dynamic

Device static

IO

01

01

01

01

01

00

00

00Pow

er (W

)

Leak

age

Cloc

k

Logi

c

BRA

M

DSP PL

L

MM

CM

PHA

SER

PCIE IO

GTP


Effective ΘJA






Device utilization

Optimalinteger set

[10 9 6 2 31 1]

Sample integerset

[13 12 5 12 0 012 4 3 3 4]










Device utilization



[22]




Bonded IOBs 274 213



11 Conclusion




References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 4 5 6 7 8 12

8

3D-coefficient data

Temporal domain

2

8

12

8

1

8

12




1 2 3 4 5 6 7 8C

R

R

R

R

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8











Summary

Thermal margin



254∘C

746∘C 393W

19∘CW

0

18

17

65

0000W

0036W

0034W

0131W

0000W

Transceiver

Core dynamic

Device static

IO

01

01

01

01

01

00

00

00Pow

er (W

)

Leak

age

Cloc

k

Logi

c

BRA

M

DSP PL

L

MM

CM

PHA

SER

PCIE IO

GTP


Effective ΘJA






Device utilization

Optimalinteger set

[10 9 6 2 31 1]

Sample integerset

[13 12 5 12 0 012 4 3 3 4]










Device utilization



[22]




Bonded IOBs 274 213



11 Conclusion




References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Summary

Thermal margin



254∘C

746∘C 393W

19∘CW

0

18

17

65

0000W

0036W

0034W

0131W

0000W

Transceiver

Core dynamic

Device static

IO

01

01

01

01

01

00

00

00Pow

er (W

)

Leak

age

Cloc

k

Logi

c

BRA

M

DSP PL

L

MM

CM

PHA

SER

PCIE IO

GTP


Effective ΘJA






Device utilization

Optimalinteger set

[10 9 6 2 31 1]

Sample integerset

[13 12 5 12 0 012 4 3 3 4]










Device utilization



[22]




Bonded IOBs 274 213



11 Conclusion




References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Device utilization



[22]




Bonded IOBs 274 213



11 Conclusion




References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article FPGA Implementation of Optimal 3D-Integer