Signals & Systems

1. Signals & Systems Lecture 17: Chapter 3 Spectrum Representation

2. Today's lecture • Error signal • Convergence of Fourier synthesis • Frequency modulation • Chirp formula

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

4. Convergence of Fourier Synthesis

5. 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

6. Sinusoid with Changing Frequency

7. 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 x(t) =Acos(ω0t + ) =Acosψ(t) ψ(t) = ω0t + and dψ(t)/dt = ω0 whereψ(t) denotes the time varying angle function

8. 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

9. 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

10. Frequency Modulation: Chirp Signals