rec8.pdfrec8sol1Recitation 8 - EECS 451, Winter 2010Mar. 17, 2010OUTLINE• Using data windows• Practice problemsConcepts: Frequency analysis using data windows1) Compute unknown ωifromy[n] = A1cos(ω1n + θ1) + A2cos(ω2n + θ2).by using only {y[n], 0 ≤ n ≤ L − 1}• ComputeYk=N−1Xn=0y[n]ej2πnk/N.• Find peaks in |Yk| at k = k1, k2< N/2. Then, the discrete frequencies are: ωi=2πNki.2) We can think ofy[n] = w[n]³A1cos(ω1n + θ1) + A2cos(ω2n + θ2)´,where w[n] = 1, 0 ≤ n ≤ L − 1.•W (ejω) =sin(ωL/2)sin(ω/2)e−jω(L−1)/23) Resolving peaks.|ω1− ω2| >2πLto resolve peaks at ω1and ω2, because the first zero-crossing of W(ejω) occurs at ω =2πL.2Problems1) Consider a problem of analyzing frequency spectrum of y[n] = A1cos(ω1n+θ1)+A2cos(ω2n+θ2) using N-point DFT where the data is available for 0 ≤ n ≤ L − 1.a) Suppose we increased L. Explain the effect of a larger L.b) Explain the effect of a larger N.c) Explain the effect of using Hamming window.2) a) A filter is given by the following difference equation.y(n) + y(n − 1) + y(n − 1) = x(n) − x(n − 1) + x(n − 2).Find the frequency component which this filter rejects.b) A filter is given by the following difference equation.2y(n) + 3y(n − 1) + ay(n − 1) = x(n) + x(n − 1) + x(n − 2).Find a which makes the phase response be zero for all frequencies.3) Let x(n) be the real and even signal with period 6. Find the frequency component of theoutput of the system y(n) = x(n) − x(n − 1) + x(n − 3) − x(n − 4) using DTFS of