Sectioned Convolution SCDWT
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<strong>Sectioned</strong> <strong>Convolution</strong>and<strong>SCDWT</strong> N.C ShenAdvisor Prof : J.J Ding
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i. Abstractii. <strong>Sectioned</strong> <strong>Convolution</strong>iii. <strong>Sectioned</strong> <strong>Convolution</strong> in DWTiv. Efficiency Comparisonv. Future Work
<strong>Sectioned</strong> <strong>Convolution</strong>
how to perform the“ convolution ” !?
Frequency Response ofInput SignalFrequency Response ofFilterXFrequency Response of<strong>Convolution</strong> Result
Here is a problem.....
How many points of FFTshould be calculated !?
Signal LengthN + M - 1Filter Length
How to calculate the FFT !?
MultsAddsRadix - 2 N/2 log2N N log2NSplit - Radix N/4 (log2N+1) 3N/4 (log2N+1/3)
what is the mean of“ sectioned ” !?
we split the input signal intosection by section.
Overlap-Saved Method
x0[n]M-1Lnx1[n]M-1nL
x0[n]*h[n]M-1Lnx1[n]*h[n]M-1nL
what is the advantage ofoverlap-saved method !?
i. we do not have to wait until allof the signal has been received.
ii. it does not increase thesystem complexity.
<strong>Sectioned</strong> <strong>Convolution</strong>
The complexity of sectionedconvolution isL : sectioned length, M : filter length
The optimal sectioned lengthof sectioned convolution isL : sectioned length, M : filter length
what is the advantage ofsectioned convolution !?
i. Saving Energy
ii. Saving Time
iii. Fixed Hardware Architecture
The Complexity of <strong>Sectioned</strong> <strong>Convolution</strong> isOptimal <strong>Sectioned</strong> Length is
No matter how long the input signal is,the points of FFT depends on the filter length.
i. Saving Energyii.Saving Timeiii.Better Hardware Architecture
<strong>SCDWT</strong>
* kj is the input length in each level
1-Dimension DWTkj1/2 kj1/2 kj-1LP g(n)2↓approximatedpartx(n)HP h(n)2↓detail part
The Complexity of 1-Dimension DWT1time kj-point FFT for input signal+ 2 times kj-point FFT for filters+ 2 times kj-point FFT for DWT outputs
The Complexity of 1-Dimension <strong>SCDWT</strong>1/2 kj-1 times L-point FFT for input signal+ 2 times L-point FFT for filters+2*1/2 kj-1 times L-point FFT for DWT outputs
1-Dimension EIDWTLP g0(n)approximatedpartHP h0(n)+x(n)LP g1(n)detail part+HP h1(n)
The Complexity of 1-Dimension EI<strong>SCDWT</strong>2*1/4 kj-1 times L`-point FFT for input signal+ 4 times L`-point FFT for filters+4*1/4 kj-1 times L`-point FFT for DWT outputs
one of the advantages of<strong>SCDWT</strong> is........
Fixed Hardware Architecture
2-Dimension DWTLP g(n)2↓LP g(n)HP h(n)2↓2↓x(n)LP g(n)2↓HP h(n)2↓HP h(n)2↓
1/2 pj-1 pj 1/2 pj1/2 kj-1 1/2 kj-1 1/2 kj-11/2kjtimes pjpointFFT1/2 pjdownsampling1/2 pjkj1/2 kj1/2 pj timeskj-point FFTdownsampling
The Complexity of 2-Dimension DWT
1/2 kj-1 times1/2 pj-1L-point FFT withinput length 1/2 pj-1pj 1/2 pj1/2 kj-1 1/2 kj-1 1/2 kj-1downsampling1/2 pj timesL-point FFT withinput length 1/2 kj-11/2 pj1/2 pjkj1/2 kjdownsampling
The Complexity of 2-Dimension <strong>SCDWT</strong>
Efficiency Comparison of 1-D DWTInputLengthDWT <strong>SCDWT</strong> EI<strong>SCDWT</strong>1024 36.5 19.1 14.2512 73.0 28.4 21.2256 146.0 47.1 35.1128 292.1 84.5 63.0* DWT level is fixed, Filter length is fixed
Efficiency Comparison of 1-D DWTInputLenghtDWT <strong>SCDWT</strong> EI<strong>SCDWT</strong>1024 - 52.32% 38.98%512 - 38.96% 29.03%256 - 32.28% 24.05%128 - 28.94% 21.57%* DWT level is fixed, Filter length is fixed
Efficiency Comparison of 1-D DWTFilterLengthDWT <strong>SCDWT</strong> EI<strong>SCDWT</strong>8 73.0 28.4 21.216 75.3 35.0 28.932 80.1 41.9 36.164 90.0 50.4 44.6* DWT level is fixed, Input length is fixed
Efficiency Comparison of 1-D DWTFilterLengthDWT <strong>SCDWT</strong> EI<strong>SCDWT</strong>8 - 38.86% 29.03%16 - 46.51% 38.36%32 - 52.37% 45.14%64 - 56.06% 49.56%* DWT level is fixed, Input length is fixed
Conclusion and Future Work
No matter how long the input signal is,the points of FFT depends on the filter length.
Reference[1] Gerard P. M. Egelmeers and Piet C. W. Sommen, “A newmethod for efficient convolution in frequency domain bynonuniform partitioning for adaptive filtering”, IEEE Trans.Signal Process., Vol. 44, No.12, JUNE 1996[2] Jung Kap Kuk and Nam Ik Cho, “Block convolution witharbitrary delays using fast Fourier transform”, ISPACS 2005[3] Guillermo García, “Optimal Filter Partition for Efficient<strong>Convolution</strong> with Short Input Output Delay”, AES 113THCONVENTION, LOS ANGELES, CA, USA, 2002 OCTOBER5–8
Thank you