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# HW7 - II(3pts For any integers m,n compute âˆ âˆž âˆž âˆ’...

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Homework #7 Due: Thursday, April 8, 2010 Note : Use of Matlab (or any other software) is not permitted. I. (3pts) Assume f:R R is a square-integrable function whose Fourier transform is supported in [0,2 σ ] . Assume we know the samples {f(nT) , n=…,-2,-1,0,1,2,… } for T=1/(2 ) . Show how to synthesize f(x) from this set of samples. Hint : Note that f is not σ -bandlimited. Instead consider the auxiliary function g(x)=e -2 π i σ x f(x) and try to reconstruct first g(x) and then f(x).
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Unformatted text preview: II. (3pts) For any integers m,n compute: âˆ« âˆž âˆž âˆ’ âˆ’ âˆ’ dt n t m t ) ( sinc ) ( sinc III. (4pts) Let Îµ >0, Ïƒ >0, B>0 be given. Construct a continuously differentiable function f:R â†’ R with a piecewise smooth absolutely integrable Fourier transform F such that . | ) ( | , , ) ( ) ( 2 B f and s for s F dt t f > < < âˆ’ = < âˆ« âˆž âˆž âˆ’ Îµ (Hint: See Exercise 8.28) Total : 10 pts...
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