1. by Ya Bao http://eent3.sbu.ac.uk/staff/baoyb/f oct 1 Fourier Series Fourier series: how to get the spectrum of a periodic signal. Fourier transform: how to get the spectrum of an aperiodic signal and the reverse. Any periodic signal can be expressed as an infinite series of sine and cosine functions of harmonics of the fundamental frequency of the signal.

2. by Ya Bao http://eent3.sbu.ac.uk/staff/baoyb/foct2 General form of the Fourier series General – single side a 0 is dc value. a n and b n are the Fourier coefficients. Double-sided complex exponential form

3. by Ya Bao http://eent3.sbu.ac.uk/staff/baoyb/foct3 Fourier coefficients

4. by Ya Bao http://eent3.sbu.ac.uk/staff/baoyb/foct4 Simplification by symmetry Time average = 0, a 0 = 0 Even symmetry, b n = 0 Odd symmetry, a n = 0 Skew symmetry, a n = b n = 0 for n is even but nonzero Symmetry maybe hidden by a dc level.

5. by Ya Bao http://eent3.sbu.ac.uk/staff/baoyb/foct5

6. 6 Example1 dc=0, a 0 = 0 g(t) = +V, 0 < t <= T/4 = -V, T/4 < t <= 3T/4 = +V, 3T/4 < t <= T Even symmetry.

7. by Ya Bao http://eent3.sbu.ac.uk/staff/baoyb/foct7

8. 8 The complete single-sided Fourier series expression for this square wave (its spectrum)

9. by Ya Bao http://eent3.sbu.ac.uk/staff/baoyb/foct9 Double-sided result.

10. by Ya Bao http://eent3.sbu.ac.uk/staff/baoyb/foct10 Example 2 A pulse train of mark/space ratio 1:2 a 0 = V/3

11. by Ya Bao http://eent3.sbu.ac.uk/staff/baoyb/foct11