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Lecture 10: EE 221: Signals Analysis and Systems

Learn about the compact and combined trigonometric forms of Fourier series for periodic signals. Understand the magnitude and phase spectra, integration techniques, and Dirichlet conditions.

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Lecture 10: EE 221: Signals Analysis and Systems

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  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

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