(a) Table 8.1 lists a number of symmetry properties of the discrete Fourier series for periodic sequences, several of which we repeat here. Prove that each of these properties is true. In carrying out your proofs, you may use the definition of the discrete Fourier series and any previous property in the list. (For example, in proving property 3, you may use properties 1 and 2.)

Sequence                                 Discrete Fourier series

1. x*[n]                                                X*[– k]

2. x*[– n]                                             X*[k]

3. Re{x[n]}                                         X_e[k] 

4. jJ m{x [n]}                          X₀[k]

(b) From the properties proved in part (a), show that for a real periodic sequence x[n], the following symmetry properties of the discrete Fourier series hold:

1. Re{X[k]} = Re{X[– k]}

2. Jm{X[k]} = − Jm{X[– k]}

3. |X[k]| = |X[– k]|

4. < X[k] = − < X[– k]

Transcribed Image Text:

Periodic Sequence (Period N) DFS Cocfficients (Period N) *(k] 10. *|-n) X.\k] = }(X(k] + **(-k) X,\k] = }(X[k] – X*l-k]) Ret X|k}} 11. Rejšļn}} 12. jJm{i[n]} 13. i|n] = }(F[2] + * *|–n]) 14. in] = }(5[n] – i '(-n]) jIm[X\k]} Properties 15-17 apply only when x[1] is real. X\k] = X'[-k] Ref X[k]} = Re{ X[-k]} Im X\k]} = -Im[ X|-k]i X|k]| = |X|-k]| «X\k] = -4X|-k] Ret X|k]} %3D 15. Symmetry properties for k[x) real. 16. i,[n] = }(¥[»] + š[-n]) 17. šoln) = }(š[1] – i[-n]) jIm{X\k]}