DEVELOPMENT OF NOVEL ALGORITHM FOR HAAR TRANSFORM USING PAIRED TRANSFORM: IMPLEMENTATION ON TMS DSP PROCESSORS.

Paired transform splits the mathematical structure of many discrete unitary transforms, including the Fourier, Hadamard, Cosine and Hartley transforms, into the minimum number of short transforms. The discrete Haar transform that is very useful in many signal and image processing applications can be also calculated by the fast paired transform. The Haar transform can be considered as the particular case of the paired transforms, namely the 2-paired transform. In this research a Novel Fast algorithm for Haar transform is developed using paired transform: Paired Fast Haar Transform (Paired FHT). For illustration purposes relation between the Haar and paired transforms are described using the examples for 16, 8, 4 – point transforms are analyzed in detail. Finally the novel Haar transform: Paired Fast Haar transform is implemented using Code Compose Studio for TMS DSP processor Starter Kit (TMS DSK) TMS 320 C 6713 DSP processor for understanding the possible sampling rates.

FAST TRANSFORMS OF FOURIER CLASS IN OFDM TECHNOLOGY OF WIRELESS TRANSMISSION SYSTEMS

The ma­in mo­di­fi­ca­ti­ons and stan­dards of OFDM techno­logy that pro­vi­de high qua­lity com­mu­ni­ca­ti­on in mul­ti­path transmis­si­on of the transmit­ted sig­nal are highlighted. It is analyzed in the struc­tu­re of the transmit­ter of the com­mu­ni­ca­ti­on system ba­sed on OFDM techno­logy of exe­cu­ti­on of fast transforms of Fou­ri­er class. The ortho­go­nal freq­uency di­vi­si­on mul­tip­le­xing / de­mul­tip­le­xing functi­on is as­sig­ned to the fast com­pu­ter of transform, and the pre­co­der is used to re­du­ce the high pe­ak fac­tor in­he­rent in OFDM techno­logy. The ba­sic ele­ments and req­ui­re­ments for the com­pu­ters that per­form fast transforms in the struc­tu­ral sche­me of imple­men­ta­ti­on of OFDM techno­logy are de­ter­mi­ned. The re­la­ti­on bet­we­en the num­ber of sub­car­ri­ers and the si­ze of ba­sic transform of OFDM techno­logy is con­si­de­red. The pos­si­bi­lity of using Fou­ri­er, Hartley transforms and co­si­ne transforms in the pre­co­der has be­en fo­und out. The ba­sic sta­ges of the met­hod of construc­ting the struc­tu­ral sche­me of fast Fou­ri­er transforms ba­sed on cyclic con­vo­lu­ti­ons are for­mu­la­ted. The iden­ti­fi­ed steps inclu­de: bu­il­ding a has­hing ar­ray, de­ter­mi­ning a simpli­fi­ed has­hing ar­ray supple­men­ted by an ar­ray of signs, construc­ting and analyzing a ge­ne­ra­li­zed struc­tu­re of the ba­sis mat­rix, bu­il­ding blocks of in­put da­ta in­teg­ra­ti­on, bu­il­ding blocks of cyclic con­vo­lu­ti­ons, bu­il­ding blocks of com­bi­ning re­sults of cyclic con­vo­lu­ti­ons who­se out­puts are re­sults of di­rect/in­di­rect transforms of Fou­ri­er class ba­sed on cyclic con­vo­lu­ti­ons. The sta­ges of construc­ting and analyzing the ge­ne­ra­li­zed struc­tu­re of the ba­sic mat­rix are per­for­med on the ba­sis of a has­hing ar­ray, a simpli­fi­ed has­hing ar­ray, and an ar­ray of signs. The ta­bu­lar as­signment of the block-cyclic struc­tu­re of the ba­sic mat­rix spe­ci­fi­es the co­or­di­na­tes of the pla­ce­ment of the sign and the simpli­fi­ed val­ue of the first ele­ments of the cyclic sub­mat­ri­ces. An example for the ішяу N=16 of de­ter­mi­ning the has­hing ar­ray, the simpli­fi­ed has­hing ar­ray and the sign ar­ray, the block-cyclic struc­tu­re of the ba­sis mat­rix used in construc­ting the struc­tu­ral sche­me of the com­pu­ter is con­si­de­red. An example of a struc­tu­ral sche­me of a DHT-I of si­ze N=20, con­ta­ining fo­ur blocks of exe­cu­ti­on of a 4-po­int cyclic con­vo­lu­ti­on. The techniq­ue of construc­ting the struc­tu­re sche­me of com­pu­ters using cyclic con­vo­lu­ti­on blocks can be used to ef­fi­ci­ent per­form discre­te transforms of Fou­ri­er class in OFDM-ba­sed com­mu­ni­ca­ti­on systems. The pos­si­bi­lity of using struc­tu­ral construc­ti­on techniq­ue to au­to­ma­te the pro­cess of construc­ting struc­tu­ral sche­mes the transforms of Fou­ri­er class ba­sed on cyclic con­vo­lu­ti­ons has be­en es­tab­lis­hed.

Two Dimensional Short Time Hartley Transforms

The Hartley transform, as in the case of the Fourier transform, is not suitably applicable to non-stationary representations of signals whose statistical properties change as a function of time. Hence, different versions of 2-D short time Hartley transforms (STHT) are given in comparison with the short time Fourier transform (STFT). Although the two different versions of STHT defined here with their inverses are equally applicable, one of them is mathematically incorrect/incompatible due to the incorrect definition of the 2-D Hartley transform in literature. These definitions of STHTs can easily be extended to multi-dimensions. Computations of the STFT and the two versions of STHTs are illustrated based on 32 channels (traces) of synthetic seismic data consisting of 256 samples in each trace. Salient features of STHTs are incorporated.

Performance Comparison of Hartley Transform with Hartley Wavelet and Hybrid Hartley Wavelet Transforms for Image Data Compression

This paper proposes image compression using Hybrid Hartley wavelet transform. The paper compares the results of Hybrid Hartley wavelet transform with that of orthogonal Hartley transform and Hartley Wavelet Transform. Hartley wavelet is generated from Hartley transform and Hybrid Hartley wavelet is generated from Hartley transform combined with other orthogonal transform which contributes to local features of an image. RMSE values are calculated by varying local component transform in hybrid Hartley wavelet transform and changing the size of it. Sizes of local component transform is varied as N=8, 16, 32, 64. Experiments are performed on twenty sample color images of size 256x256x3. Performance of Hartley Transform, Hartley Wavelet transform and Hybrid Hartley wavelet Transform is compared in terms of compression ratio and bit rate. Performance of Hartley wavelet is 35 to 37% better than that of Hartley transform whereas performance of hybrid Hartley wavelet is still improved than Hartley wavelet transform by 15 to 20%. Hartley-DCT pair gives best results among all Hybrid Hartley Transforms. Using hybrid wavelet maximum compression ratio up to 32 is obtained with acceptable quality of reconstructed image.