2. (32 points) Symmetry properties of Fourier transform (a) (16 points) Determine which of the signals, whose Fourier transforms are depicted inFig 1 satisfy each of the following: i. x(t)

is even ii. x(t) is odd iii. x(t) is real iv. x(t) is complex (neither real, nor pure imaginary) v. x(t) is real and even vi. x(t) is imaginary and odd vii. x(t) is imaginary and even viii. There exists a non-zero wo such that e1aox(t) is real and even (b) (8 points) Using the properties of Fourier transform, determine whether the assertionsare true or false i. The convolution of a real and even signal and a real and odd signal is odd. i. The convolution of a signal and the same signal reversed is an even signal. i. If x(t) = x*(-t), then X(jw) is real. ii. If r(t) is a real signal with X (jw) its Fourier transform, then the Fourier transformsXe(jw) and X,(jw) of the even and odd components of x(t) satisfy the following: Xe(jw) = Re{X(jw)}