1. Signals and Systems Lecture 10: Fourier Analysis of Periodic Signals

2. Today's lecture • Fourier Analysis of Triangular Wave • Convergence of Fourier Synthesis • Frequency Modulation

3. Triangular Wave

4. Triangular Wave ak = (e -jkπ – 1)/ k2π2

5. Triangular Wave

6. Convergence of Fourier Synthesis • Error Signal: • Worst-case error:

7. Convergence of Fourier Synthesis

8. General Waveforms • Waveforms can be synthesized by the equation x(t) = A0 + ∑Ak cos(2πfkt +k) • These waveforms maybe • constants • cosine signals ( periodic) • complicated-looking signals (not periodic) • So far we have dealt with signals whose amplitudes, phases and frequencies do not change with time

9. Frequency Modulation • Most real-world signals exhibit frequency change over time e.g. music. • Frequency of a signal may change linearly with time which sounds like a siren or chirp • Chirp signal: Signal whose frequency changes linearly with time from some low value to high value • Let ψ(t) = ω0t + and dψ(t)/dt = ω0 whereψ(t) denotes the time varying angle function

10. Stepped Frequency Sinusoids

11. Frequency Modulation • We can create a signal with quadratic angle function by defining • ψ(t) = 2πμt2 + 2πf0t +  instantaneous frequency = slope of the angle function ωi= dψ(t)/dt fi(t) = 1/2 π dψ(t)/dt fi(t) = 2μt + f0

12. Example 3.8: Synthesize a Chirp Formula Synthesize a frequency sweep from f1 = 220 Hz to f2 = 2320 Hz over a 3-second time interval. fi(t) = (f2 -f1)t / T2 + f1 ψi(t) = ∫ ωi(u) du t 0

13. Frequency Modulation: Chirp Signals

14. Assignment #2 • End Chapter Problems • P- 3.8 • P- 3.10 • P- 3.12 • P- 3.14 • P- 3.15 • Due on Tuesday 3rd March 2009