column is f k above F H is the conjugate transpose of F and D is a diagonal

# Column is f k above f h is the conjugate transpose of

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column is f k above ( F H is the conjugate transpose of F ), and D is a diagonal matrix. Relate this to the fact that circular convolution in the time domain is multiplication in the DFT domain. 3. Suppose we have a machine that measures signals f ( t ) in L 2 ([0 , 1]) by taking integrals over different regions: y [ m ] = Z b m a m f ( t ) d t, 0 a m b m 1 , (3) for m = 1 , . . . , M . To recover f ( t ), we model it as being a polynomial of order N - 1, f ( t ) = x N - 1 t N - 1 + x N - 2 t N - 2 + · · · + x 1 t + x 0 . (4) 1 Last updated 11:21, October 31, 2019

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(a) Write a MATLAB function intmat.m that takes an M vector of lower limits a , a M vector of upper limits b , and a degree N and returns the M × N matrix A such that if f ( t ) has the form ( 4 ), applying A to 1 x = x N - 1 x N - 2 . . . x 0 results in evaluations of the measurements in ( 3 ), y = y [1] y [2] . . . y [ m ] = Ax . (b) The file hw8problem3.mat contains vectors a , b , and y of length 20 representing a particular set of lower limits, upper limits, and associated measurements. Find the polynomial of order 9 ( N = 10) that best describes these measurements in the least- squares sense. Plot your synthesized estimate ˆ f ( t ) as a function of time — that is, after estimating the coefficients ˆ x N - 1 , . . . , ˆ x 0 , form ˆ f ( t ) = ˆ x N - 1 t N - 1 + · · · + ˆ x 1 t + ˆ x 0 and plot it as a function of time on [0 , 1].
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