previous midterm need explanation and answers for 6Qs

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THE UNIVERSITY OF CALGARY FACULTY OF SCIENCE / Math & Stat March 18, 2010 MIDTERM REDO Name: 10:00 – 10:30 AMAT 415 ID: A non-graphing calculator is permitted. No notes, no books, not other aids allowed. 1. Write down all the solutions to z8 = 1 in the form x+iy, AND plot them on the complex plane. (Make sure you have all of them!) 2. Compute the following integral, where curve is the counterclockwise circle jzj = 2: Ze3iz (z + 1)2 dz 3. One hundred data samples (x(0); x(1); : : : ; x(99)) are recorded at 5000 samples per second, and the 100-point DFT is computed to get transform values (y(0); y(1); : : : ; y(99)). The value at output y(40) corresponds to what frequency, in Hertz?4. The Fourier transform of sequence x = (: : : ; x(??1); x(0); x(1); x(2); : :
is the 1-periodic function: F() =Xn x(n)e2in: Find a sequence x with Fourier transform F() = e4i+ 3 cos(4). 5. Find a vector x that satises the convolution equation (3; 4; 5) (x) = (6; 17; 22; 15): You can assume all the vectors start at index n = 0.