1. 1 Roadmap SignalSystem Input Signal Output Signal characteristics Given input and system information, solve for the response Solving differential equation Calculate convolution Linear Time-invariant Invertible Causal Memory Bound classification transformation Special properties (even/odd, periodic) Math. description Energy & power

2. 2 ECE310 – Lecture 13 Fourier Series - CTFS 02/26/01

3. 3 Why a New Domain? It is often much easier to analyze signals and systems when they are represented in the frequency domain The entire subject of signals & systems consists primarily the following concepts: Writing signals as functions of frequency Looking at how systems respond to inputs of different frequencies Developing tools for switching between time- domain and frequency-domain representations Learning how to determine which domain is best suited for a particular problem

4. 4 Scenario for Selected Legs

5. 5 two legs selected

6. 6

7. 7 Leg1 Analysis (AAV from N to W)

8. 8 Leg1 Analysis (DW from W to E)

9. 9 Fourier Series & Fourier Transform They both represent signal in the form of a linear combination of complex sinusoids FS can only represent periodic signals for all time FT can represent both periodic and aperiodic signals for all time

10. 10 Limitations of FS Dirichlet conditions The signal must be absolutely integrable over the time, t 0 < t < t 0 + T F The signal must have a finite number of maxima and minima in the time, t 0 < t < t 0 + T F The signal must have a finite number of discontinuities, all of finite size in the time, t 0 < t < t 0 + T F

11. 11 The Fourier Series of x(t) over T F Fourier series (x F ) represents any function over a finite interval T F Outside T F, x F repeats itself periodically with period T F. x F is one period of a periodic function which matches the function x(t) over the interval T F. If x(t) is periodic with period = T 0 if T F =nT 0, then the Fourier series representation (TSR) equals to x(t) everywhere; if T F != nT 0, then FSR equals to s(t) only in the time period T F, not anywhere else.

12. 12 Examples

13. 13 The Fourier Series

14. 14 Some Parameters T F is the interval of signal x(t) over which the Fourier series represents f F = 1/T F is the fundamental frequency of the Fourier series representation n is called the “harmonic number” 2f F is the second harmonic of the fundamental frequency f F. The Fourier series representation is always periodic and is linear combinations of sinusoids at f F and its harmonics.

15. 15 Interpretation The FS coefficient tells us how much of a sinusoid at the nth harmonic of f F are in the signal x(t) In another word, how much of one signal is contained within another signal

16. 16 Calculation of FS Sinusoidal signal (ex 1,2) Non-sinusoidal signal (ex 3) Periodic signal over a non-integer number of periods (ex 4) Periodic signal over an integer number of periods (ex 4) Even and odd periodic signals (ex 5) Random signal (no known mathematical description) (ex 6)

17. 17 Example 1 – Finite Nonzero Coef x(t) = 2cos(400  t) over 0<t<10ms Band-limited signals Analytically Trignometric form and Complex form Graphically (p6-10~6-12)

18. 18 Example 2 – Finite Nonzero Coef x(t) = 0.5 - 0.75cos(20  t) + 0.5sin(30  t) over -100ms < t < 100ms Band-limited signals (p6-14,6-15)

19. 19 Example 3 – Infinite Nonzero Coef x(t) = rect(2t)*comb(t) over –0.5<t<0.5 When we have infinite nonzero coefficients, we tend to use magnitude and phase of the CTFS versus harmonic to present the CTFS (p6-19)

20. 20 Example 4 – Periodic Signal x(t) = 2cos(400pt) over 0<t<7.5ms Over a non-integer number of period p6-20

21. 21 Example 5 – Periodic Even/Odd Signals For a periodic even function, X[k] must be real and X s [k] must be zero for all k For a periodic odd function, X[k] must be imaginary and X c [k] must be zero for all k

22. 22 Example 6 – Random Signal Is it necessary to know the mathematical description of the signal in order to derive its CTFS? No Graphically (p6-24, 6-25)

23. 23 Convergence of the CTFS For continuous signal As N increases, CTFS approaches x(t) in that interval For signals with discontinuities As N increases, there is an overshoot or ripple near the discontinuities which does not decrease – Gibbs phenomenon When N goes to infinity, the height of the overshoot is constant but its width approaches zero, which does not contribute to the average power

24. 24 Example P6-35, 6-36

25. 25 Exercises 6.1.2

26. 26 Response of LTI System with Periodic Excitation Represent the periodic excitation using complex CTFS Since it’s an LTI system, the response can be found by finding the response to each complex sinusoid Example: RC lowpass circuit Magnitude and phase of V out [k]/V in [k] (p6-53)

27. 27 Properties of CTFS Linearity Time shifting Time reversal Time scaling Time differentiation Time integration Time multiplication Frequency shifting Conjugation Parseval’s theorem

28. 28 Parseval’s Theorem Only if the signal is periodic The average power of a periodic signal is the sum of the average powers in its harmonic components

29. 29 Summary - CTFS The essence of CTFS The limitation of CTFS The calculation of CTFS The convergence of CTFS Continuous signals Signals with discontinuities – Gibbs phenomena Properties of CTFS Especially Parseval’s theorem Application in LTI system

30. 30 Test 2