By Hari Srivastava, H. L. Manocha
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Extra resources for A treatise on generating functions
DST-III: m, η = 1, 2 , . . , Ν DST-IV: m, η = 0, 1 , . . 11), the forward and inverse transforms for the D S T family can be explicitly expressed. v-i] = [S'/v-i], ! 10a), can be merged as (2/N) and moved to either the forward or the inverse transform. The basis functions of DST-I for Ν = 16 are shown in Fig. 2. 7 THE DIFFERENCE PROPERTY 1 2 3 4 5 6 7 8 9 1 10 1 11 1 12 1 13 14< 15' 1 Fig. 2 4 8 12 16 Basis functions for D S T - L Ν = 16. 5). We conclude this section by noting that although the shift property is derived for a unit time shift, transforms of more general time shifts can be obtained by the repeated application of the appropriate formulae.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 10). Are any of the DSTs interrelated? If so, how? 18. A simple convolution property for DCT-II has been developed by Chitprasert and Rao [F-7]. The only restriction is that one of the functions to be convolved must be real and even. This property has been utilized in human visual weighted progressive image transmission [PIT-12]. 6). 19. 10). 1 Introduction In Chapter 2, we have discussed some mathematical properties of the DCTs. We have also emphasized that they are not simply discretized versions of the Fourier cosine transform properties.
This limit of Ν -* o o , coupled with a finite support (assuming that the signal lasts for a finite duration), naturally evokes the transition from the discrete domain to the continuous domain. F o r example, the inner product of two N-dimensional vectors u a n d ν is defined as N = f ufa. 19) in fact approaches the inner product of the continuous function u(t) a n d v(t)9given by