Sectioned Convolution SCDWT

Sectioned ConvolutionandSCDWT N.C ShenAdvisor Prof : J.J Ding

1 2 3 4 5

i. Abstractii. Sectioned Convolutioniii. Sectioned Convolution in DWTiv. Efficiency Comparisonv. Future Work

Sectioned Convolution

how to perform the“ convolution ” !?

Frequency Response ofInput SignalFrequency Response ofFilterXFrequency Response ofConvolution 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.

Sectioned Convolution

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 Sectioned Convolution isOptimal Sectioned 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

SCDWT

* 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 SCDWT1/2 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 EISCDWT2*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 ofSCDWT 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 SCDWT

Efficiency Comparison of 1-D DWTInputLengthDWT SCDWT EISCDWT1024 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 SCDWT EISCDWT1024 - 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 SCDWT EISCDWT8 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 SCDWT EISCDWT8 - 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 EfficientConvolution with Short Input Output Delay”, AES 113THCONVENTION, LOS ANGELES, CA, USA, 2002 OCTOBER5–8

Thank you