ECE 513 - DIGITAL SIGNAL PROCESSING Cranos M. Williams Assigned: September 18, 2017 HOMEWORK 4 SOLUTION 1.DFT of Real Sequences with Special Symmetry: (20 Pts)Consider a N-point real se-quencex(n) with its N-point DFTX(k), where N is even. Assumex(n) satisfy thefollowing symmetryx(n+N2) =-x(n),n= 0,1,· · ·,N2-1.(1)(a) Calculate the even harmonics of the DFT. (Hint: That isX(k) whenkis even.)(b) Find aN/2-point complex sequencey(n), such that itsN/2-point DFTY(m)are identical to the odd harmonics ofX(k). (Hint:y(n) is a complex modulatedversion of the firstN/2 points ofx(n).) Solution: 1
ECE 513 - Homework 4 2 2.Circular and Linear Convolution: (20 Pts)Two sequences are given below.x(n)=3δ(n)-δ(n-2) + 2δ(n-3)y(n)=δ(n) + 5δ(n-1) + 4δ(n-2)-2δ(n-3)(a) Perform a circular convolution of the two sequencesx(n) andy(n) by hand.(b) Use the MATLAB routinesfftandifftto perform the circular convolution.Compare your results with the results you obtained in part (a).(c) Use the MATLAB functionconvto perform a linear convolution of the two se-quencesx(n) andy(n).
ECE 513 - Homework 4 3
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