(Orthogonal and Unitary Matrices) You have learned in class that a real-valued orthogonal mairiz P has the property that PTP = PP\" = 1. In signal processing, oftentimes signals are represented in complex values. We can define for a complex-valued square matrix a similar concept called the unitary matriz. A unitary matrix Q satisfies the property that Q\"?Q = QQ* =1I, where \"/\" is the Hermitian operator that is a combination of both the transpose, \"7,\" and the complex conjugate, \"*.\" a) The discrete Fourier transform (DFT) matrix for a signal of length 3 is defined as Q3 = [an]k,ne{u,lg}? where g, = % exp(j %kn). Explicitly write out this 3-by-3 matrix. Simplify each entry but do not evaluate numerically the complex exponentials. b) Verify that Qg is a unitary matrix. ) Compute the forward discrete Fourier transform for signal x = [1.01,0.99,0.97]" to obtain a transformed signal z = Q3 x, where z = [z, 21, 2:2]1'. How large are the norms for 20, 21,4 and 2'2? d) Now, zero out z3 to obtain a new vector Zeompressed- Compute the inverse transform using x = Q" Zeompressed- 18 X similar to x?7 Can you guess why