a Compute h L n for 20 n 20 for L 1 2 3 4 Turn in a stem plot of your answers

A compute h l n for 20 n 20 for l 1 2 3 4 turn in a

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(a) Compute h L [ n ] for - 20 n 20 for L = 1 , 2 , 3 , 4. Turn in a stem plot of your answers and your code. To do this, you will need to approximate the integral h L [ n ] = 1 2 π Z π - π H L ( e ) e jωn d ω, for those values of n . You can do this using any numerical integration technique with which you are familiar, or by simply taking a sum on a very fine grid and normalizing appropriately to account for the width of the intervals. h [ - 20] and h [20] should be very close to zero. Also, since H L is real and even, you can replace e jωn in the expression above with cos( ωn ). (b) Using your answer above and the provided bspline functions, plot (a very good approx- imation of) the dual B-spline functions e b L ( t ) for L = 1 , 2 , 3 , 4. 1 Last updated 23:32, October 15, 2019
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4. Let x ( t ) be the signal x ( t ) = ( 1 / 2 , 0 t 10 , - sin( πt/ 10) , 10 t 20 Compute and use MATLAB to plot the solutions to min y ∈V L Z -∞ | x ( t ) - y ( t ) | 2 d t, for L = 1 , 2 , 3 , 4, where V L = Span( { b L ( t - n ) } n Z ) . To do this, you will need to integrate x ( t ) against the dual B-splines. You can compute this integral using Z -∞ x ( t ) e b L ( t - n ) d t = X = -∞ h L [ ] Z -∞ x ( t ) b L ( t - n - ) d t. So the first step is to write a function that numerically integrates x ( t ) with shifts of b L ( t ) over an appropriate range. 2 Last updated 23:32, October 15, 2019
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