Lecture 10: EE 221: Signals Analysis and Systems

1. Lecture 10:EE 221: Signals Analysis and Systems Instructor: Dr. Ghazi Al Sukkar Dept. of Electrical Engineering The University of Jordan Email: ghazi.alsukkar@ju.edu.jo

2. Compact (Combined) trigonometric Form • For a periodic signal with period the compact trigonometric form of Fourier series is: • How? For real : or But Hence, (The Magnitude spectrum is an even function of ) (The phase spectrum is an odd function of )

3. Cont.. • From the exponential form The compact form is: Hence: (The average value)

4. Cont.. represents the single-sided magnitude spectrum.

5. Trigonometric form of Fourier Series • The trigonometric form of a periodic signal with period is: • How? Return back to the compact trigonometric form: From the trigonometric identity:

6. Cont.. • But: Let and

7. Cont.. • and But:

8. Cont.. • Example: Find the trigonometric form of Fourier series for .

9. Cont.. Integration by parts or from tables: Again integration by parts of from tables:

10. Cont.. • To find the combined trigonometric form, we use the identity:

11. Existence of Fourier Series (Dirichlet Conditions) • A periodic signal can be expanded into a Fourier series if it satisfies Dirichlet conditions (sufficient but are not necessary): • has at most a finite number of discontinuities in one period. • has at most a finite number of maxima and minima in one period. • is bounded or (absolutely integrable) this will include the singularity functions. • If satisfies the Dirichlet conditions then the corresponding Fourier series is convergent, i.e., • At points of discontinuity, the series converges to the mean of the limits approached by from the right and from the left, i.e., Where: and

12. Cont.. • Example: • Using Fourier series But