c Let ψ j t 2 j 2 ψ 2 j t and ψ jn t ψ j t 2 j n 2 j 2 ψ 2 j t n Notice that

# C let ψ j t 2 j 2 ψ 2 j t and ψ jn t ψ j t 2 j n

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(c) Let ψ j ( t ) = 2 j/ 2 ψ 0 (2 j t ) and ψ j,n ( t ) = ψ j ( t - 2 - j n ) = 2 j/ 2 ψ 0 (2 j t - n ) Notice that the { ψ j,n ( t ) } are dyadic shifts of ψ j ( t ) with spacing 2 - j . What is span ( { ψ j,n ( t ) , n Z } )? (d) Let φ ( t ) be the standard sinc function φ ( t ) = sin( πt ) πt , φ n ( t ) = φ ( t - n ) . What is the span of the union of the { φ n ( t ) } and all of the { ψ j,k ( t ) } k for j = 0 , 1 , . . . , J ? (e) For a signal x ( t ), how can we obtain h x , ψ j,n i for all n using a filter-then-sample archi- tecture? 1 General hint for this question: Do everything in the Fourier domain. Start by sketching the continuous-time Fourier transform of ψ 0 ( t ) and ψ j ( t ) (when you get to part c). 2 Last updated 0:38, October 4, 2019
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