P20) Find **all** the solutions of the following equations: (a) , (b) , (c) .

P21) Suppose is the discrete Fourier transform of a real signal . Find , , and .

P22) Suppose be a real signal with Fourier transform . Find and , where and . (Hint: Relate and to using shifts and time-reversals.)

P23) Without explicitly calculating any Fourier transforms, find , where .

P24) Let and let . Let and let . Compute the imaginary part of . Bonus: Under what condition is this imaginary part equal to zero?

P25) Let be even and suppose . Find , , and .

P26) Find the complex and real Fourier series where is the function of period satisfying , .

P27) Suppose . Find the complex and real Fourier series of the function where is the function of period satisfying if and if . (Start by drawing a picture of .)